Online.
Deterministic and Stochastic Modelling of
Ocean Surface Waves
Deterministisch en Stochastisch Modelleren van Oceanische
Oppervlaktegolven
Deterministic and Stochastic Modelling of
Ocean Surface Waves
Proefschrift
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. ir. K. C. A. M. Luyben,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen op vrijdag 6 juni 2014 om 12:30 uur
door
Pieter Bart SMIT
civiel ingenieur
geboren te Zoetermeer.
Dit proefschrift is goedgekeurd door de promotor:
Prof. dr. ir. G. S. Stelling
Samenstelling promotiecommissie:
Rector Magnificus,
Prof. dr. ir. G. S. Stelling,
Prof. dr. ir. A. J. H. M. Reniers,
Prof. dr. ir. R. H. M. Huijsmans,
Prof. dr. ir. D. Roelvink,
Prof. dr. A. E. P. Veldman,
Dr. ir. T. T. Janssen,
Dr. ir. M. Zijlema,
Prof. dr. ir. A. Mynett,
voorzitter
Technische Universiteit Delft, promotor
Technische Universiteit Delft
Technische Universiteit Delft
Unesco-IHE institute for water education
Rijksuniversiteit Groningen
Theiss Research
Technische Universiteit Delft
Technische Universiteit Delft, reservelid
Published by:
VSSD, Delft, the Netherlands
Front & Back:
Based on a photograph by E. Dupont.
c 2014 by P.B. Smit
Copyright ISBN 978-90-6562-354-6
An electronic version of this dissertation is available at
http://repository.tudelft.nl/.
Abstract
Deterministic and Stochastic Modelling of Ocean Surface Waves
Predicting the mean wave statistics in the nearshore, for instance the significant wave
height, has predominantly been the domain of operational stochastic wave models
based on the radiative transport (or energy balance) equation. Although reasonably
successful in the nearshore, these models were originally developed for oceanic scales,
and necessarily neglect or parametrise processes that are only significant in shallow
water, such as the linear processes of interference and diffraction, or the nonlinear
triad wave-wave interactions and dissipation due to wave breaking. In this dissertation we investigate the possibility of predicting the wave statistics on small scales in
strongly non-linear conditions, such as found in the surfzone, using the recently developed Surface WAves till SHore (SWASH) model, whereas on larger scales we pursue a
generalisation of existing stochastic models by incorporating coherent effects, hereby
extending these models to include interference and diffractive effects.
First we determine whether non-hydrostatic models, and the SWASH model in
particular, can be used to predict the wave statistics, such as the mean wave height en
period, in the surfzone. Specifically, we consider how to incorporate dissipation due
to wave breaking in non-hydrostatic models in an efficient and accurate way. Here
we strive not only to capture the bulk statistics as encompassed by the mean wave
height and period, but also the spectral evolution, and the development of the higher
order nonlinear statistics in a dissipative surfzone. Hereto the so-called Hydrostatic
Front Approximation is proposed, which enforces a hydrostatic pressure distribution
in the water column below the front of a breaking wave so that, based on the analogy
between a hydraulic jump and a turbulent bore, energy dissipation can be accounted
for by ensuring conservation of mass and momentum using shock capturing numerics.
The model is verified with observations of the mean wave heights and periods for
irregular, unidirectional waves in wave flumes, and with observations of such bulk
statistics for short-crested wave propagation in a wave basin. The results demonstrate
that the model can accurately predict the bulk parameters as well as wave-driven
horizontal circulations. Moreover, our results show that, without specific calibration,
the model accurately predicts not only the second-order bulk statistics, but also
the details of the spectral evolution, as well as the higher-order statistics (skewness
and asymmetry) of the waves. Monte Carlo simulations show that the model can
capture the principal features of the wave probability density function in the surfzone,
and that the spectral distribution of dissipation in SWASH is proportional to the
frequency squared, which is consistent with observations reported by earlier studies.
These results show that relatively efficient non-hydrostatic models such as SWASH
can be successfully used to parametrise surfzone wave processes.
Outside the surfzone the wave motion is often weakly influenced by nonlinear
processes and is therefore to a good approximation linear. This allows for a closed
stochastic description of the wave motion that can be applied on large scales. Howv
vi
ever, conventional third generation stochastic wave-models based on the Radiative
Transport Equation (RTE) are based on the premise that waves propagating at mutual angles are independent, and linear processes such as interference (e.g. standing
waves) and diffraction are therefore not accounted for. The second part of this thesis
therefore focuses on the derivation and verification of a new stochastic wave model
that, unlike traditional models based on the RTE, can account for fast-scale variations in the wave statistics that occur in these focal zones by including coherent
interference that occurs between crossing waves.
Hereto, on the premise of dispersive wave motion over slowly varying topography,
a deterministic equation that governs the linear wave motion is derived. Based on this
deterministic equation, we subsequently derive an evolution equation for the secondorder statistics in terms the Wigner – or Coupled Mode – spectrum that governs the
evolution of the complete second-order statistics, including coherent interference. The
resulting Quasi-Coherent (QC) approximation reduces to the RTE in case of quasihomogeneous statistics, and therefore embodies a natural generalisation of quasihomogeneous theory to include effects of coherent interference.
The model is verified through comparison to analytic solutions, and laboratory
and field observations. We discuss the differences with the radiative transfer equation and the limitations of our approximation, and illustrate the model’s ability to
resolve coherent interference structures in wave fields such as those typically found
in refractive focal zones and around obstacles. Moreover, we demonstrate that a
robust numerical implementation of the QC approximation takes the form of the
RTE including an additional scattering source term that accounts for the coherent
interference in the field. Consequently, the QC-approximation derived in the present
thesis can be incorporated into existing stochastic wave models based on the RTE.
In conclusion: in the present study we considered prediction of nearshore wave
statistics by further development of deterministic and stochastic wave models. The
resulting deterministic model SWASH, with proper treatment of wave breaking, is
suited for application in the surfzone, whereas the stochastic QC model bridges the
gap between traditional stochastic models, valid for oceanic scales, and deterministic
models. Both models are validated with empirical data, confirming that they form a
robust and complementary set of models for the prediction of wave statistics in the
nearshore.
Samenvatting
Deterministisch en Stochastisch Modelleren van Oceanische Oppervlaktegolven.
Tot op heden is het voorspellen van de golf statistiek nabij de kust, bijvoorbeeld de
significante golfhoogte, voornamelijk het domein van stochastische golfmodellen gebaseerd op de stralingstranport (ook wel energie balans) vergelijking. Alhoewel deze
modellen met succes zijn toegepast in het kustgebied, zijn deze modellen van origine
ontwikkeld voor grootschalige toepassingen op de open oceaan, en dientengevolge verwaarlozen of parametriseren ze processen die alleen significant zijn in ondiep water,
zoals de lineaire processen van interferentie en diffractie, of de niet-lineaire drie-golf
wisselwerkingen en dissipatie door golfbreken. In dit proefschrift onderzoeken we de
mogelijkheid om de golfstatistiek voor kleinschalige maar sterk niet lineaire situaties, zoals in de brandingszone, te voorspellen met behulp van het recent ontwikkelde
“Oppervlakte golven tot de kust” (SWASH1 ) model, terwijl we voor grootschaliger
toepassingen een generalisatie van de bestaande stochastische aanpak nastreven, zodat deze rekening houdt met diffractie en interferentie effecten.
Allereerst wordt onderzocht of een niet-hydrostatische model, en in het bijzonder
het SWASH model, toepasbaar is om de golfstatistiek, zoals de gemiddelde golfhoogte en golfperiode, te voorspellen in de brandingszone. In het bijzonder kijken
we hoe de dissipatie die naar aanleiding van golfbreken optreedt efficiënt en accuraat
kan worden meegenomen. Om dit te bereiken wordt de Hydrostatische Front Approximatie (HFA) voorgesteld waarin de drukverdeling in de water kolom onder een
brekende golf hydrostatisch wordt verondersteld en, gebaseerd op de analogie tussen
een brekende golf en een watersprong, golfenergie gedissipeerd wordt door te vereisen
dat massa en impuls in het numerieke model behouden blijven over de resulterende
schokgolf. Dit met als doel niet alleen een accurate weergave van de bulkstatistiek,
maar ook een natuurgetrouwe weergave van de evolutie van het golfspectrum en de
hogere orde statistiek in een dissipatieve brandingszone.
Het ontwikkelde model is geverifieerd met observaties van de gemiddelde golfhoogte en golfperiode voor onregelmatige, uni-directionele golven in golfgoten, en
met observaties van deze parameters voor kortkammige golven in een golfbasin. Uit
deze verificatie blijkt dat het resulterende model zowel de bulk parameters als de
golfgedreven stroming accuraat kan voorspellen. Bovendien laat deze studie zien
dat het model zonder verdere kalibratie naast de bulk tweede orde statistiek ook
de spectrale evolutie van het variantie spectrum nauwgezet beschrijft (inclusief niet
lineaire contributies), alsmede de hogere orde statistiek (asymmetrie en scheefheid)
van de golven. Monte-Carlo simulaties laten zien dat het model ook de karakteristieke eigenschappen van de kansdichtheid functie reproduceert behorende bij het
vrije oppervlak in de brandingszone. Daarnaast blijkt dat de resulterende spectrale
1 Afgeleid
van de Engelse vertaling: “Surface WAves till SHore”
vii
viii
verdeling van de dissipatie door golfbreken proportioneel is met de frequentie in het
kwadraat, consistent met eerdere observaties in de literatuur. Deze resultaten bevestigen dat een niet-hydrostatisch model zoals SWASH de effecten van de dominante
golf processen in de brandingszone met grote nauwkeurigheid kan reproduceren.
Buiten de brandingszone wordt de golfbeweging zwak beïnvloed door niet-lineaire
processen en is daarom bij benadering lineair. Dit maakt een gesloten stochastische
benadering die grootschalig toepasbaar is mogelijk. Echter, conventionele derde generatie stochastische modellen gebaseerd op de stralingstransport vergelijking (RTE2 )
veronderstellen daarnaast ook dat golven die in verschillende richtingen propageren
statistisch onafhankelijk zijn. Dientengevolge worden ook lineaire processen zoals
interferentie (e.g. staande golven) en diffractie – van belang in convergentiezones
– verwaarloosbaar geacht. Het tweede deel van dit proefschrift is daarom gewijd
aan de afleiding en verificatie van een nieuw stochastisch model dat, in tegenstelling
tot modellen gebaseerd op de RTE, rekening houdt met coherente interferentie, en
dus rekening houdt met de snelle variatie in de golfstatistiek die kan optreden in
convergentiezones.
Het model is afgeleid onder de aanname van dispersieve golfbeweging over langzaam variërende topografie, van waaruit een deterministische vergelijking die de lineaire golfbeweging beschrijft is opgesteld. Uit deze deterministische beschrijving is
een vergelijking voor de evolutie van het Wigner – of “coupled mode” – spectrum
afgeleid, die de evolutie van de complete tweede orde statistiek beschrijft, inclusief
coherente interferentie. De resulterende Quasi-Coherente (QC) benadering reduceert
tot de conventionele RTE wanneer de golfstatistiek quasi-homogeen is, en vormt dus
als zodanig een natuurlijke generalisatie van de bestaande quasi-homogene theorie
om deze uit te breidden met de effecten van coherente interferentie.
Verificatie van het model wordt bereikt door middel van een vergelijking met
analytische oplossingen, en met laboratorium en veld observaties. We bediscussiëren
de verschillen met de stralingstransport vergelijking, de limitaties van de huidige
aanpak en demonstreren dat deze coherente interferentie structuren kan beschrijven
zoals optreden in convergentiezones en rondom obstakels. Bovendien demonstreren
we dat een robuuste numerieke implementatie van de QC benadering de vorm heeft
van de RTE aangevuld met een verstrooiingsterm die rekening houdt met de coherente
interferentie in het golfveld. Dientengevolge kan de QC benadering afgeleid in dit
proefschrift toegevoegd worden aan bestaande derde generatie golfmodellen gebaseerd
op de RTE.
Concluderend: in de huidige studie beschouwden we de voorspelling van de golfstatistiek in de kustzone door een verdere ontwikkeling van deterministische en stochastische golf modellen. Het resulterende deterministische model SWASH, aangevuld met HFA, is toepasbaar in de brandingszone, terwijl het stochastische QC model
het gat tussen conventionele stochastische modellen – geschikt voor applicaties op de
open oceaan – en deterministische modellen overbrugt. De verificatie van beidde
modellen met empirische data bevestigd dat ze een robuuste en complementaire set
van modellen vormen om de golf statistiek in de kustzone te voorspellen.
2 Afgeleid
van de Engelse vertaling: “Radiative Transport Equation”
Contents
Abstract
v
Samenvatting
vii
1
Introduction
1.1 Objective and Outline . . . . . . . . . . . . . . . . . . . . . . . . . .
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Deterministic and stochastic wave models
2.1 Deterministic wave models in intermediate and shallow waters . . . .
2.2 Stochastic wave models in shallow waters . . . . . . . . . . . . . . . .
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Wave breaking in a non-hydrostatic wave model
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . .
3.2 Non-hydrostatic modelling . . . . . . . . . . . . .
3.3 Wave breaking approximations. . . . . . . . . . . .
3.4 Monochromatic wave breaking over a sloping beach
3.5 Random waves breaking over barred topography . .
3.6 Short-crested waves over 2D topography . . . . . .
3.7 Discussion . . . . . . . . . . . . . . . . . . . . . . .
3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . .
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3.A Turbulent stress approximations . . . . . . . . . . . . . . . . . . . . .
3.B Wave generating boundary conditions . . . . . . . . . . . . . . . . . .
3.C Wave breaking initiation criterion . . . . . . . . . . . . . . . . . . . .
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Non-hydrostatic modelling of surfzone
4.1 Introduction. . . . . . . . . . . . . . .
4.2 Model description. . . . . . . . . . . .
4.3 Experiment and model setup. . . . . .
4.4 Results . . . . . . . . . . . . . . . . .
4.5 Discussion . . . . . . . . . . . . . . . .
4.6 Conclusions . . . . . . . . . . . . . . .
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4.A Frequency dispersion in SWASH . . . . . . . . . . . . . . . . . . . . .
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evolution of inhomogeneous wave statistics
Introduction. . . . . . . . . . . . . . . . . . . . .
Evolution of correlators . . . . . . . . . . . . . .
Evolution of coherent wave structures . . . . . . .
Discussion . . . . . . . . . . . . . . . . . . . . .
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Contents
5.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Appendices
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5.A Operator definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.B Relation to geometric optics . . . . . . . . . . . . . . . . . . . . . . . 99
5.C Boundary condition for wide-angle diffraction . . . . . . . . . . . . . 100
6
Narrow-band wave statistics over nearshore topography
6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Evolution of inhomogeneous wave fields . . . . . . . . . . .
6.3 Wave deformation by an elliptical shoal . . . . . . . . . . .
6.4 Swell over submarine canyons . . . . . . . . . . . . . . . .
6.5 Discussion and conclusions . . . . . . . . . . . . . . . . . .
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6.A Fourier transform operators . . . . . . . . . . . . . . . . . . . . . . . 121
6.B Discrete model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
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Conclusions and outlook
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7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
References
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List of figures
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Acknowledgements
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List of publications
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Curriculum vitae
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1
Introduction
The sight – and sound – of ocean waves violently breaking on the shore during a severe
storm is impressive. The wild, chaotic and unpredictable appearance of the ocean is
built of a superposition of many waves, each with its own characteristic height, length
and period. These waves are generated by wind and they travel the ocean basins
propagating freely, nearly frictionless, under the influence of gravity. With periods of
approximately one until twenty seconds, heights in the order of centimetres to (tens
of) metres, and horizontal scales of several to hundreds of metres, these are known as
wind-generated surface gravity waves, often referred to more simply – as done here
– as ocean waves.
The necessity of nearshore wave prediction became urgent during World War II,
when knowledge of wave conditions at the landing beaches was a matter of life and
death (see Parker, 2010). Besides military applications, wave prediction in shallow
water was – and is – of economic interest (e.g. wave conditions at the port entrance), relevant for engineering (e.g. design wave conditions) and important for
scientific research (e.g. understanding nearshore processes). Sometimes detailed
(phase-resolved) information of the wave evolution is needed (e.g. impact forces on
a structure Gomez-Gesteira and Dalrymple, 2004), but for many applications statistical information on the wave field suffices, represented for instance by mean wave
height or period, or a wave spectrum (e.g. Wise Group, 2007).
Conceptually, there are two different – but related – techniques to predict the wave
statistics. The starting point is the same: a simplified description of the dynamics of a
fluid with a free-surface in terms of a set of partial differential equations, the solution
of which depends on boundary and initial conditions, and on the environmental
parameters such as e.g. wind, bathymetry, ambient currents, and variations therein.
Given the initial and boundary conditions, and medium variations, the prediction
of a wave variable – say the surface elevation η(x0 , t0 ) at a certain location x0 and
time t0 – then reduces to the (often formidable) task of solving the equations by
mathematical or numerical means.
This deterministic approach can also be used to calculate wave statistics. By
1
2
1. Introduction
considering N different realizations of the model input (drawn from their respective statistical distributions) and successively solving the equations for each input,
an ensemble of observations can be generated, from which the statistics of e.g. the
free-surface elevation (mean, variance etc.) can be discerned. This technique, often
referred to as Monte-Carlo analysis, is straightforward to apply, but requires either
large ensembles, or (assuming ergodicity) a single simulation with a long duration.
Its major advantage is that the physical processes which are included in the underlying model are also encapsulated in the statistical results, which in particular for
strongly nonlinear waves prevents the need for additional statistical closure approximations. The downside is that certain processes (most notably wave generation
by wind) are difficult to include, and accurate representation of small-scale motion
(e.g. turbulence), is not feasible in large domains due to computational constraints.
Hence, the more advanced nearshore models, such as the well established Boussinesq
models (e.g. Madsen et al., 1991; Nwogu, 1993; Wei et al., 1995; Madsen et al., 2006;
Klopman et al., 2010) or the more recently developed non-hydrostatic models (e.g.
Stelling and Zijlema, 2003; Yamazaki et al., 2009; Zijlema et al., 2011b; Ma et al.,
2012; Cui, 2013), necessarily parametrise some processes such as turbulence, bedfriction and most importantly depth-induced wave breaking. Generally, prediction
of bulk statistical parameters is reasonable in these models. However, describing the
evolution of higher-order statistics, as encompassed by e.g. the wave-spectrum or
third-order moments (e.g. skewness or asymmetry), is more elusive, as this critically
depends on the interplay between breaking-induced dissipation and nonlinear processes (e.g. Chen et al., 1997). In this regard, correctly accounting for the rate of
wave energy dissipation in breaking waves is critical.
Alternatively, the deterministic equations can be used to derive stochastic evolution equations for the second-order moments of the free surface, usually expressed
in terms of the variance density spectrum (e.g. Hasselmann, 1962; Willebrand, 1975;
Komen et al., 1994). In this way, there is no need to generate an ensemble, or predict
long time series to obtain wave statistics. Moreover, because the wave statistics often
change on slow scales, such models can be applied efficiently on spatial scales ranging from ocean basins to coastal regions. Prediction of bulk statistics, such as mean
wave heights and – to a lesser extent – periods, are often reasonable (Wise Group,
2007), and over extended regions including locally generated waves, stochastic models are the only feasible alternative. However, the downside of a stochastic approach
is that it is restricted to near-Gaussian and quasi-homogeneous statistics, which in
particular in shallow water might be overly restrictive. In such regions, not only do
waves develop non-Gaussian statistics, but - through interaction with the topography
- inhomogeneous effects can become important as well. For instance, the statistics
of coherent interference patterns found in wave focal zones resulting in locally enhanced sea states and caustics can only be accurately represented when accounting
for strongly inhomogeneous effects in wave statistics. However, such effects are completely ignored in state-of-the-art (operational) wave models, which are invariably
based on some form of an energy balance equation, implying quasi-homogeneous
statistics (Janssen et al., 2008).
In the present work, developments are considered in both the deterministic and
3
stochastic approach, with a particular focus on wave modelling in the nearshore, including the surfzone. We continue development and verification of the recently introduced non-hydrostatic models, which form a promising alternative to the Boussinesq
wave models for simulating nonlinear wave dynamics and statistics in the surfzone.
In particular, we consider depth-induced wave breaking in the non-hydrostatic wave
model Simulating Waves till SHore1 , or SWASH (Zijlema et al., 2011b), and consider
whether a relatively simple and efficient parameterisation of wave breaking, which
locally reduces the equations to the Nonlinear Shallow Water Equations (NSWE)
and represents the bore as a shock wave, can not only reproduce bulk parameters,
but also capture the details of the surfzone energy balance.
The other focus of this work is to develop a generalized stochastic modelling
framework that allows the evolution of cross-correlation information radiated into
the region or developed while propagating through a variable medium. This framework is applied to develop a more general form of the radiative transfer equation that
incorporates inhomogeneous and diffraction effects and which is referred to as the
Quasi Coherent (QC) approximation. Since the resulting QC model is a generalization of the widely used energy balance equation, the model equation is compatible
with existing operational models. However, with the inclusion of coherent interference effects (and diffraction) it is more accurate in predicting evolution of swell wave
statistics in coastal regions with strong medium variations. Moreover, and in contrast
to traditional models based on the radiative transfer equation, this approach can resolve the fine structure of alongshore standing waves in focal zones, which affects the
wave-induced flows, and thus transport processes.
1 In
which the author has been involved in its development.
4
1. Introduction
1.1. Objective and Outline
The overarching objective of the present study is to advance the predictive capability and understanding of wave dynamics - and in particular the evolution of wave
statistics – in shallow water, through (a) further development and verification of the
non-hydrostatic modelling framework, and (b) the development of a new stochastic
wave model that accounts for wave coherence and is applicable to wave propagation
over extended 2D topography.
The base of this thesis is formed by a set of journal papers2 that resulted from
these efforts. The articles are included almost verbatim (with minor updates to
correct for small errors) as four separate chapters, each of which can be read independently. This necessarily entails that information from the introductory chapters
(1 and 2), and between the main chapters, is repeated to retain individual legibility.
The outline of this work is therefore as follows. Following the introduction (this
chapter), and to sketch the context of the developments within the present work, we
shall discuss developments in both deterministic and stochastic wave models, with a
focus on nearshore wave modelling (Chapter 2).
The next two chapters (3 and 4) concern the further development and verification of the deterministic SWASH model. The introduction of a new wave breaking
parametrisation, which locally reduces the governing equations to the nonlinear shallow water equations, is the subject of Chapter 3. Through comparison with flume
experiments, we demonstrate that this approach reproduces observed wave heights
well, while allowing for a much reduced vertical resolution, thus improving efficiency.
Moreover observations of short-crested waves in a wave basin, including wave heights,
wave spectra, and wave-driven currents, are also reproduced well. Next, we consider
the model representation of the detailed nonlinear wave dynamics and statistics in a
dissipative surfzone (Chapter 4).
Subsequently, we shall develop and discuss a generalization of spectral wave models that includes the effects of coherent interferences on wave statistics (Chapter 5).
Using multiple scales, we approximate the transport equation for the (complete)
second-order wave correlation matrix. The resulting model, which accounts for the
generation and propagation of coherent interferences in a variable medium (e.g. in
wave focal zones), is validated through comparison with analytic solutions and laboratory observations. In addition, we discuss the differences with the radiative transfer
equation and the limitations of our approximation. Chapter 6 concerns further development of the model, with a particular focus on implementation and verification
by means of a field data set of swell waves incident on submarine canyons.
Lastly, discussions and conclusions on individual developments concerning the
stochastic and deterministic models are provided separately at the end of each chapter. Discussions and conclusions of the overall thesis, including an outlook to future
developments, are presented in Chapter 7.
2 Three
of which have been published, and one will be submitted later in 2014.
2
Deterministic and stochastic
wave models
In the early days of wave research, near the end of the 19th century, attention was
primarily focused on investigating analytical descriptions of waves of constant form
propagating through a uniform medium, with important contributions due to Stokes,
Airy, Cauchy, Korteweg and de Vries and others (see Craik, 2004, for a historical
overview). These studies remain invaluable today as their principal results, e.g. the
linear approximation introduced by Airy, or the seminal work by Stokes on nonlinear waves, advanced our understanding of wave motion greatly, and in one form or
another still influence our physical interpretation thereof. However, the extension of
their methods to nonlinear irregular wave motion, propagating through an inhomogeneous medium, invariably results in a set of coupled partial (or ordinary) differential
equations which (most likely) cannot be solved by analytical means. Therefore, with
the introduction of widespread computing facilities during the latter half of the 20th
century, focus shifted to solving (approximate) equations governing water wave dynamics using numerical approximations.
Nevertheless, computational limitations prevented – as they still do – using the basic primitive equations, the (Reynolds-averaged) Navier-Stokes equations, to resolve
all scales of motion (up to turbulence) on spatial and temporal scales of interest1 .
Instead, often a considerable simplification is pursued in describing the free-surface as
a single-valued function of the horizontal coordinates, and neglecting friction (leading to the Euler equations), or parametrising the effects of turbulence by a suitable
closure assumption. Wave models often reduce the problem further by considering
irrotational flow governed by the potential equations (i.e. the Laplace and Bernoulli
equations combined with the dynamic and kinematic boundary conditions, see e.g.
Mei et al., 2005; Dingemans, 1997).
1 Although
Volume Of Fluid (VOF) methods or Smoothed Particle Hydrodynamics (SPH) models
can now be feasibly applied on a 1D surfzone on scales of several wave lengths and periods (e.g.
Dalrymple and Rogers, 2006; Torres-Freyermuth et al., 2007).
5
6
2. Deterministic and stochastic wave models
If the vertical excursion of the free surface remains small (compared with local
depth and wave length), and the topography changes slowly (compared with the
wave length), this gives rise to linear wave theory, which has proven to be remarkably successful (even within the highly nonlinear surfzone). For instance, the linear
approximation to energy propagation (see e.g. Dingemans, 1997; Mei et al., 2005)
is the foundation of stochastic operational wave models. For complex situations,
e.g. with steep topography, or for strongly nonlinear waves, linear models have been
augmented with a myriad of more involved (non)linear models (deterministic and
stochastic), each with its own region of applicability (e.g. deep or shallow water) depending on underlying assumptions on the relative importance of the relative slope,
wave nonlinearity and wave dispersion.
In what follows we will discuss deterministic and stochastic models applicable
to (non)linear waves in intermediate to shallow waters. This results in a discussion
of the relation to and the advantages of non-hydrostatic models such as SWASH
when compared with other deterministic models in Section 2.1, with the purpose
of sketching the context of the work presented in Chapters 3 and 4 of this thesis.
Analogously, we discuss stochastic models, and the context of the improvements that
are pursued thereon in section 2.2.
2.1. Deterministic wave models in intermediate and
shallow waters
In shallow water, the depth over wavelength ratio (µ) reduces significantly (µ → 0)
and water wave motion becomes weakly dispersive, with small vertical variations in
the horizontal particle velocities, and a near linear variation in the vertical velocities.
In this regime, nonlinear effects, expressed in terms of the amplitude to depth ratio
(δ), become comparable with dispersive effects. The problem of wave evolution in
the nearshore is then usually approached under the assumption that nonlinear effects
dominate (resulting in the NSWE), or that they are of similar importance. The latter
limit, setting δ = O(µ2 ), gave rise to first models suitable for wave propagation over
uneven topography, the so-called Boussinesq models (Peregrine, 1967). In their classic
form these types of models consider a series solution to eliminate the vertical crossspace in lieu of the appearance of additional corrections to the nonlinear shallow
water equations to account for dispersion.
Since their inception, these models, and somewhat comparable models2 based on
Serre (1953) or Green and Naghdi (1976), have undergone rapid developments, with
research focusing on extending dispersive properties of such models (e.g. Witting,
1984; Madsen et al., 1991), deriving so called fully nonlinear models3 (e.g. Wei et al.,
1995), wave-current interaction (e.g. Chen et al., 1999), wave breaking (e.g. Kennedy
et al., 2000; Schäffer et al., 1993; Tonelli and Petti, 2010; Tissier et al., 2012) and
wave run-up (e.g. Fuhrman and Madsen, 2008). However, in intermediate to deep
2 In
the remainder of this section we will, for brevity, include these models under the umbrella of
Boussinesq models.
3 Such models make no a-priori assumption on the ordering of µ and δ, although µ is still assumed
to be small, see Kirby (1996).
2.1. Deterministic wave models in intermediate and shallow waters
7
water, the classic Boussinesq approach becomes less attractive, as one has to consider
increasingly complex equations containing higher-order (cross-) derivatives in time
and space4 . This not only affects model complexity, but can also introduce inherent
unstable behaviour into the equations (i.e. the wave energy grows without bounds,
Løvholt and Pedersen, 2009), although recasting the Boussinesq equations into different form can alleviate such problems (e.g. the Hamiltonian model by Klopman
et al., 2010, ensures conservation of energy). Good introductions into Boussinesq
models are found in Kirby (1996), Madsen and Schäffer (1999) and a recent overview
of the state of the art in Brocchini (2013).
Arguably, direct numerical evaluation of the Boussinesq, potential, or Euler equations, ignores that there are typically multiple scales of motion, one (or more) slow
scales related to e.g. changes in the medium (ambient currents and bathymetry)
or nonlinearity, and one fast scale related to the wave motion proper (resolving the
phases of the individual waves). The existence of such scales can be exploited through
a perturbation type solution of the wave variables (e.g. free-surface, velocity potential). For instance, the free surface elevation can be expressed a Fourier sum with
free-surface modes of the form an (x, t) exp [iωn − iφ(x)], where an (x) denotes a slowly
varying amplitude (incorporating the effects on slower scales), ωn is the modal frequency, φ the wave function and ∇x Φ = kn is the modal wavenumber. Assuming
that angles are restricted to small angles with respect to a reference direction (e.g the
coast-normal), directional waves can be included by utilizing an angular spectrum
expansion for each mode (e.g. Dalrymple and Kirby, 1988; Janssen et al., 2006), or
a parabolization of the governing equations (e.g. Kaihatu and Kirby, 1995). Substitution of the assumed form into the governing equations then results in a linear
problem for the fast scale motion (often formalized using multiple scale analysis) that
can often be solved explicitly, and a set of coupled (due to nonlinearity) ordinary differential equations for the modal amplitudes that – as long as medium variations and
nonlinearity remain weak – can be integrated over the slow scales of variation.
For instance, evolution equations based on the lowest-order Boussinesq theory
(Peregrine, 1967) were used by e.g. Freilich and Guza (1984), to model unidirectional
nonlinear wave shoaling on laterally uniform topography, and Herbers and Burton
(1997), for short-crested wave propagation. Advances have since then focussed on
inclusion of full linear dispersion (e.g. Kaihatu and Kirby, 1995; Bredmose et al.,
2005; Janssen et al., 2006), improved nonlinear behavior (e.g. inclusion of arbitrary
resonance mismatch Bredmose et al., 2005; Janssen et al., 2006), and inclusion of
(weakly) two-dimensional topography (e.g. Kaihatu and Kirby, 1995; Janssen et al.,
2006). In intermediate to deep water (if cubic interactions are included, Janssen
et al., 2006) these amplitude evolution models are more accurate, and more efficient
than time-domain models. Moreover, they form a suitable framework for deriving
stochastic models, to which we shall return later. However, in regions where such
a multiple scale approach breaks down, for instance where medium variations are
rapid or is nonlinearity strong, the validity of these multiple-scale (or WKB-like)
models is obviously limited. A model based on a direct numerical evolution of the
4 This
can be seen for instance by comparing the governing equations derived by Peregrine (1967)
to the model presented by Madsen et al. (2006).
8
2. Deterministic and stochastic wave models
primitive equations does not have such restrictions, and is thus more accurate in
regions where the WKB approximations break down (e.g. wave reflection, wave
runup, steep bathymetry).
For shallow-water waves, in which the vertical structure of the flow is fairly uniform, an obvious alternative to the Boussinesq approach would be to evaluate the
Euler equations with a coarse vertical grid resolution. Extension to deep water
(adding dispersion effects) would then merely require an increase in vertical grid resolution. However, common numerical solution techniques (central differences and a
staggered layout between pressure points and velocity components) would render this
approach very inefficient. For instance, resolving the phase celerity with a relative
error of 5% (compared with linear theory) for kd ≈ 1 requires O(10) points in the
vertical, which is an order of magnitude more demanding than Boussinesq models of
similar accuracy (at comparable horizontal resolution).
However, due to two recent developments this approach is not only feasible, but in
fact shows great potential. First, a pressure decomposition into a hydrostatic and a
dynamic (non-hydrostatic) contribution allows for an efficient fractional-step integration, where the velocity field is first predicted using the shallow water equations, and
subsequently, by solving a discrete Poisson equation for the dynamic pressure, corrected to ensure that the velocity field is divergence-free (Casulli and Stelling, 1998).
Second, discretization of the vertical pressure gradients by means of a Hermitian
method such as the Keller-box or Preissmann Scheme (e.g. Lam and Simpson, 1976)
allows to make use of the smooth vertical pressure profile by effectively using a finite
series of spline functions as approximations. Combining these two approximations,
Stelling and Zijlema (2003) achieves good dispersive properties for a non-hydrostatic
model with a small number of vertical pressure points (1–3), and a computational
effort comparable to Boussinesq models. Compared to the Boussinesq approach in
general5 , this makes the non-hydrostatic approach an attractive alternative because:
(i) it can handle rotational flows (important for wave-current interaction); (ii) the
governing equations are very similar to the basic primitive equations (facilitating
physical interpretation); (iii) the model contains at most second-order derivatives
thus simplifying the numerical implementation; (iv) extension to deeper water requires merely an increase of the vertical resolution (which provides flexibility); and
(v) the method is easily incorporated to an existing shallow-water solver.
Since its initial inception, the non-hydrostatic approach6 has rapidly gained traction, culminating in the release of codes ready for general use (e.g SWASH and
NHWave, Zijlema et al., 2011b; Ma et al., 2012), and inclusion into existing shallowwater models (e.g. H2Ocean, XBeach7 and NEOWAVE8 , Cui et al., 2012; Smit
et al., 2009; Yamazaki et al., 2009). Recent developments include implementation on
unstructured grids(Cui et al., 2012; Wei and Jia, 2014) and improvement of the dispersive properties (Cui, 2013). Yet, within the present context, the most important
development has been their extension to include dissipative surfzone waves, which
5 Noting
that this also depends on the particular formulation one is comparing with.
now on, unless stated otherwise, when referring to non-hydrostatic models, we specifically
refer to those based on the framework presented in Stelling and Zijlema (2003).
7 eXtreme Beach behavior.
8 Non-hydrostatic Evolution of Ocean WAVE.
6 From
2.1. Deterministic wave models in intermediate and shallow waters
9
requires the inclusion of the dominant and arguably most poorly understood process
in the surfzone: depth-induced breaking.
Extension of non-hydrostatic models to the surfzone
When solving the Navier-Stokes equations directly, wave breaking is an emerging
property of the equations and thus intrinsically included. However, adopting a
single-valued representation of the free-surface – as done by non-hydrostatic models
– implies that processes such as overturning, air entrainment, and wave-generated
turbulence are no longer resolved. Instead, the dissipation of organized wave energy
through turbulence into heat must be considered as an unresolved sub-grid process,
the integral properties of which (i.e. the dissipation rate) are to be captured by
model approximations.
By observing that both spilling and plunging breakers eventually evolve into a
quasi-steady bore, in which the entire front-face of the wave is turbulent (Peregrine,
1983), breaking wave dynamics becomes somewhat analogous to a hydraulic jump
(e.g. Lamb, 1932). Consequently, its integral properties (rate of energy dissipation,
jump height) are reasonably well estimated by regarding the breaking wave as a discontinuity in the flow variables (free surface, velocities). If the underlying model
equations permit the development of such bores (or shock waves), proper numerical treatment of such a discontinuity (conservation of mass and momentum) can be
used to determine the energy dissipation of waves in the surfzone. For example, Hibberd and Peregrine (e.g. 1978) used the (non-dispersive) NSWE to investigate the
runup on a beach. Until relatively recently, this approach has not been pursued in
nearshore wave models, because dispersive terms in the Boussinesq equations preclude the formation of shocks; nonlinear amplitude dispersion effects (responsible for
wave steepening) are balanced by linear dispersive effects thus stabilizing the wave
profile (Schäffer et al., 1993). In addition, the dispersive terms cannot be expressed
in conservative form, which prohibits the application of standard shock capturing
numerical methods. Consequently, Boussinesq models often parametrise wave breaking by including locally enhanced horizontal diffusion of momentum (thus dissipating
energy e.g. Kennedy et al., 2000), or by including the effect of a surface roller (e.g.
Schäffer et al., 1993).
The similarity with shallow-water models suggests that for non-hydrostatic models the bore analogy combined with a shock-capturing method can be used to simulate surfzone waves. Zijlema and Stelling (2008) used the shock-capturing method of
Stelling and Duinmeijer (2003) and demonstrated, by comparison to flume observations, that this is indeed feasible. Moreover, the resulting model reproduces the runup
of waves (also by virtue of Stelling and Duinmeijer, 2003), such as the maximum
runup of solitary waves on a plane beach (Synolakis, 1986), as demonstrated in Smit
(2008). Subsequently, numerous researchers have used non-hydrostatic formulations
– albeit with different underlying numerical implementations – to describe not only
surfzone waves (e.g. Ma et al., 2012; Yamazaki et al., 2009), but also tsunami propagation, generation, and the resulting inundation (Yamazaki et al., 2011; Tehranirad
et al., 2012; Cui, 2013).
The most attractive feature of the bore approach is that no model parameters are
10
2. Deterministic and stochastic wave models
required to predict the onset or dissipation rate of wave breaking. From a physical
viewpoint, disadvantages are that all breaking waves transition into a sawtooth shape,
resulting in exaggerated wave asymmetry compared to observations, and that the
turbulent energy generated in the breaking process is not accounted for. A further
practical disadvantage is that correctly predicting the location of incipient breaking
requires disproportionally high vertical resolution (10–20 points) when compared to
vertical mesh requirements outside the surfzone (2–3 points). The high resolution
is required because the characteristic vertical length scale of the wave motion is no
longer the wave length L, as in intermediate to deep water, but the wave height
H. Since H/L ∼ O(10−1 ), this generally poses a higher restriction on the vertical
resolution. When using lower resolutions, the waves do still break in the model, but
because particle velocities near the surface are underestimated (see Chapter 3), the
kinematic instability that initiates breaking (particle velocity exceeds wave celerity)
is shifted shoreward, and the onset of dissipation is delayed towards shallower water.
These stringent mesh requirements pose a severe constraint on the feasibility of the
bore concept in engineering practice, as a comparatively small region prohibits the
use of an otherwise acceptable coarse vertical resolution.
Hence, for efficiency reasons, dissipation based on an enhanced eddy viscosity were
reintroduced in non-hydrostatic models, either using the Smagorinsky subgrid model
(Smagorinsky, 1963; Jacobs, 2010; Ma et al., 2012), or a mixing-length hypothesis
(Zijlema et al., 2011b) to estimate the eddy viscosity. Although effective in the
surfzone, these formulations must be calibrated, and are too dissipative outside the
surfzone for short waves in the tail of the spectrum (which possess relatively high
curvature). Therefore – as in Boussinesq models – not only additional dissipation
is required, it must also be localized (or ’triggered’) in the surfzone; in other words,
inclusion or adaption of one of the existing breaker models. Yet, the fact that in
the approximation of the nonlinear shallow-water (NWS) equations, by lack of a
stabilizing dispersive mechanism, all waves develop into breakers, suggests that in
the vicinity of of the breaking wave front, we can simplify the governing equations to
the NSW equations. In this work we investigate the effect of an enforced hydrostatic
pressure distribution near the wave front and how it can be used to force the transition
into a bore. Similar developments have been tried in Boussinesq models, where - by
disabling the dispersive terms over the extent of the wave profile once the wave is
deemed breaking - the NSW equations are used to model the transition into a bore
(e.g. Tonelli and Petti, 2010). The primary advantage of this general approach is that
without any calibration the dissipation rate in the breaking wave is ensured to agree
with that of a hydraulic jump. However, the location of incipient wave breaking does
need to be parametrised.
It is not unreasonable to assume (and results within Chapter 3 corroborate this)
that for long-crested waves this approach will give a reasonable approximation of
mean dissipation rates and consequently bulk wave parameters (characteristic wave
height, mean period). After all, using the bore analogy to derive (stochastic) dissipation rates due to depth induced breaking – including a hydrostatic pressure assumption near the bore front – has been successfully used in phase-averaged models since
the work by Battjes and Janssen (1978) and later Thornton and Guza (1983). To
11
2.2. Stochastic wave models in shallow waters
achieve this, the only required step is to locally enforce hydrostatic pressure under a
breaking wave. Whether or not this approach is equally successful in predicting bulk
parameters for a short-crested wave field is not clear, primarily since the notion of a
’breaking wave’, and thus the determination of where to enforce hydrostatic pressure,
is not obvious. Naturally, doing this consistently is important, not only for estimating
wave dissipation, but subsequently also for driving of mean flows (through radiation
stress gradients) and transport processes. Additionally, accurately resolving breaking wave dynamics in the surfzone is important for understanding of the spectral
evolution (harmonic generation and driving of infragravity waves) and higher-order
statistical quantities such as skewness and asymmetry. These aspects critically depend on the interplay between breaking-induced dissipation, and nonlinear processes
(e.g. Chen et al., 1997). Since it is not clear whether such detailed information
is resolved in non-hydrostatic models, we investigate the detailed representation of
wave dissipation, nonlinearity, and the subsequent spectral evolution in a dissipative
surfzone, in this new class of models.
2.2. Stochastic wave models in shallow waters
Stochastic wave models stand at the heart of modern operational global wave prediction systems. Examples include WAM9 model (The WAMDI Group, 1988), or
WaveWatch (Tolman, 1991), which are routinely used for global and regional-scale
wave predictions. These models are fundamentally based on the premise that the
wave field (say the free-surface elevation ζ(x, t)) as a function of space x and time t,
can be represented as a zero-mean, quasi-homogeneous and quasi-stationary Gaussian process, which can be fully described by its slowly varying (compared with a
typical wave length and period) variance density spectrum E(f, θ, x, t) (Komen et al.,
1994). It essentially describes the distribution of variance hζ 2 i (where h. . . i denotes
the ensemble average), over its spectral components with frequency f and direction
θ.
The evolution of the wave spectrum is then obtained by considering a balance
equation for the mean wave energy. Such a balance merely states that the change
in energy is due to the net energy transported into the area, and the generation and
dissipation of energy in that same area. Augmented with transport among spectral
components (due to refraction and currents), and formulated in wave number space k,
it can be written in the form of the Radiative Transport Equation (RTE, Willebrand,
1975; Komen et al., 1994)
∂t E + cx · ∇x E + ck · ∇k E = S,
(2.1)
with transport velocities cx (k, x), and ck (k, x) in, respectively, geographic x-space
and spectral k-space, and where S denotes the various source terms. Since its initial
inception, much of the development has gone into the improvement of the source
terms on the right side of equation (2.1), but the left side of (2.1), the energy balance, has remained unchanged. In deep water, the source terms on the right side
of (2.1) include wind generation (Phillips, 1957; Miles, 1957; Cavaleri and Rizzoli,
9 WAve
Modelling.
12
2. Deterministic and stochastic wave models
1981; Snyder and Elliott, 1981; Komen et al., 1984), dissipation due to white capping (Hasselmann, 1974; Komen et al., 1994), and four-wave nonlinearity (Philips,
1960; Hasselmann, 1962). The nonlinear term, and the introduction of (economic)
approximations thereof (Hasselmann and Hasselmann, 1985), is a cornerstone to the
successful development and rapid evolution of this class of so-called third generation
models.
Following the success of these models on oceanic scales, a push was made to extend
the applicability of stochastic models to the coastal zone. This involves accounting for
shallow water processes such as depth-induced breaking (e.g., Salmon et al., 2014b,
for an overview), bottom friction (Hasselmann et al., 1973), and the effects of triad
interactions (Eldeberky, 1996; Becq-Girard et al., 1999), resulting in the development
of the shallow-water stochastic wave models such as SWAN10 (Booij et al., 1999)
and TOMAWAC11 (Benoit et al., 1996). In general, this approach has been quite
successful in predicting bulk statistics of the wave field (Wise Group, 2007), but
further improvement of these models in shallow water has primarily focussed on
development of the source terms, whereas the underlying premise that the wavefield
has near-Gaussian and homogeneous statistics has remained unchanged.
In the deep ocean, where the wave field principally evolves under the action of
wind, white-capping and quadruplet interactions, the effects of dispersion and a fairly
homogeneous medium, ensures that the conditions of Gaussianity and homogeneity
are often reasonable. However, in shallow water, the effects of medium variations
and transition to weakly dispersive wave motion, can create spatial coherency (e.g.
Janssen et al., 2008), and the representation of nonlinear effects requires higher-order
(three-wave) correlations (e.g. Agnon and Sheremet, 1997; Herbers and Burton, 1997;
Eldeberky and Madsen, 1999; Herbers et al., 2003). To capture such coherent structures in the wave field, be it introduced through medium variations or nonlinearity,
requires the evaluation of additional correlators, and cannot be described by the
evolution of auto-covariance contributions alone.
Efforts have been made to describe the evolution of non-Gaussian statistics for
special cases, such as the evolution of nonlinear statistics of forward propagating
waves over one-dimensional topography (e.g. Agnon and Sheremet, 1997; Herbers
and Burton, 1997; Eldeberky and Madsen, 1999; Herbers et al., 2003; Janssen et al.,
2008), and inhomogeneous effects in narrow-band waves in deep water (e.g. Alber,
1978; Janssen, 1983; Stiassnie et al., 2008). Invariably, the evolution of non-Gaussian
statistics involves accounting for the transport of third-order cross-correlations, i.e.
the bispectrum, combined with an (often heuristic) closure assumption. Through
additional assumptions (e.g. bandwidth, direction, 1D topography) such models are
limited to special cases, and equations describing the 2D (isotropic) evolution of
the bispectrum over topography are at present not available, which fundamentally
hampers the inclusion of nonlinear effects in shallow water. Instead, spectral evolution is approximated by sometimes rather crude approximations (Eldeberky, 1996;
Becq-Girard et al., 1999; Booij et al., 2009; Toledo and Agnon, 2012) to capture the
principal nonlinear evolution, but without accounting for the evolution of third-order
10 Simulating
WAves Nearshore.
Operational Model Addressing Wave Action Computation.
11 TELEMAC-based
2.2. Stochastic wave models in shallow waters
13
correlations.
Evolution of inhomogeneous statistics over topography
Although it is clear that in regions of strong medium variations, which are common in coastal areas, the wave field can develop inhomogeneity, the classic quasihomogeneous radiative transport approximation does not account for this, and despite many efforts to extend wave modelling to coastal areas, this has received very
little attention thus far. To date (to our knowledge), the only stochastic model to
include linear interference (diffraction) is based on an angular spectrum decomposition (Janssen, 2006; Janssen et al., 2008), and is therefore restricted to forward wave
propagation over weakly two-dimensional topography. A principal issue in the development of an isotropic description which includes heterogeneous effects is that there
appears to be no general conservation principle for the complete second-order wave
correlation matrix. The energy balance implied by the radiative transfer equation
only applies to the energy carrying (variance) components of the correlation matrix,
but the inhomogeneous effects associated with the evolution of wave cross-correlations
cannot be accounted for in this manner.
The issue can be illustrated by considering the classic case of a narrow-band
wave train in intermediate to shallow water (kd < 1, with k and d a characteristic
wave number and depth, respectively), propagating over a submerged shoal (see
Fig. 2.1) in an otherwise flat region. The effect of the shoal, reminiscent of a lens
refracting light, is to focus the waves in the convergence zone down-wave of the
shoal. In the region up-wave of the shoal, the wave field can be characterized by its
complex amplitude a0 , and wavenumber k0 , so that the spectrum E(k, x) takes the
form of a single peak near k0 , representing the variance h 21 a0 a∗0 i (where ∗ denotes
the complex conjugate). If we consider the directional wave spectrum in a point
behind the shoal, say in a point P along the central ray (see Fig. 2.1), we expect
three separate contributions (km , with m = 1 . . . 3), one peak (or wave component)
associated with each wave ray. However, since the three rays originate from the
same coherent wave front, and are not independent, additional information is needed
to fully characterize the second-order statistics of the waves. In addition to the
variance (or auto-correlations) denoted as h 21 am a∗n i with m = n, the cross-variance
contributions (m 6= n) are required also. In this case, it is obvious that such crosscorrelations emerge through the interaction with the topography, but it is not clear
how to represent their generation and propagation in a statistical sense.
In the present thesis (Chapter 5) we set out to derive a consistent extension of
the radiative transport equation to include such cross-variance contributions, where
we follow analogous developments in the context of quantum mechanics and optics
(e.g. Bremmer, 1972; Torre, 2005). Essentially, the radiative transfer equation (RTE)
represents a particle-like description of the wave field, where each ’wave-energy particle’ evolves along its own characteristic path, and is not affected by the presence
of the other ’particles’ in the field (at least in the linear approximation). The effects of nonlinearity (i.e. quadruplet wave-wave interactions) are included through
a particle ’collision’ analogy, and although this allows for particle interaction, their
behavior is still strictly particle-like. In essence, by deriving transport equations for
14
2. Deterministic and stochastic wave models
Incident Wave Front
cau
sti
c li
ne
Incident spectrum
Interference
Region
P
Spectrum at P
Wave Ray
Figure 2.1: Sketch of a possible ray pattern induced by the refraction of a monochromatic, unidirectional wave impinging on a submerged shoal (dashed lines). The shoal focuses the waves so that on
the leeward side, across the caustic line, an interference pattern occurs. Radiative transport models
correctly predict a singly peaked spectrum in the region before the shoal, but fail to account for the
interference that occurs between the peaks in the interference region behind the shoal.
the cross-variances in the wave field, and thus resolving coherent structures in the
wave field, wave-like behavior is re-introduced. As a consequence, the total variance
at any given location is no longer simply the sum of the variances associated with
the individual particles, but depends on constructive and/or destructive coherent
interference (wave-like behavior). By including wave-like features to the statistical evolution, the evolution model can account for coherent interference effects and
diffraction on the wave statistics.
Principally, if the underlying deterministic equation for the random wave variable,
such as the complex-valued free-surface ζ (so that η = Re{ζ}), can be written in the
form
∂t ζ(x, t) = −iΩ(i∇x , x)ζ(x, t),
(2.2)
where Ω is a linear (pseudo-differential) operator, the evolution for the complete
second-order statistics can be obtained by multiplication of (2.2) evaluated at x1
with ζ ∗ evaluated at x2 , adding the conjugate equation evaluated at x2 multiplied
with ζ1 = ζ(x1 , t), and ensemble averaging the result, which gives
∂t hζ1 ζ2∗ i = −i [Ω(i∇x1 , x1 ) − Ω∗ (i∇x2 , x2 )] hζ1 ζ2∗ i.
(2.3)
This equation captures the evolution of the full second-order statistics of the field,
including cross-correlations – as governed by the deterministic Eq. (2.2). However,
the cross-correlator is expected to oscillate rapidly as a function of the spatial co-
2.2. Stochastic wave models in shallow waters
15
ordinates, which makes solving (2.3) directly untenable. Instead, introducing a coordinate transformation, and exchanging a spatial coordinate for a wavenumber coordinate through a Fourier transformation12 (see Chapter 5 for details), Eq. (2.3)
can be transformed into an evolution equation for a distribution function referred
to as the Wigner distribution (Wigner, 1932), denoted as E(k, x, t). This distribution represents an intermediate form between a spectral, and a spatial description
of the second-order statistics, effectively assigning (cross-) variance contributions a
definite position and wavenumber, much like the wave packets of the variance density
spectrum. Furthermore, under quasi-homogeneous conditions, it becomes a slowly
varying function in space and reduces to the variance (or energy) density spectrum.
Moreover, irrespective
of the spatial variability of the statistics, the marginal dis´
tribution (i.e.
Edk) always represents the variance (or average potential energy
density) of the wave field.
However, for heterogeneous wave fields E does not represent an energy density
function, nor does the resulting evolution equation reduce to a radiative transfer
equation. After all, the cross-variance components in the wave field are not strictly
conserved, and can generally fluctuate between positive (constructive interference)
and negative (destructive interference) values. For example, in a node of the interference pattern behind the shoal in Fig. 2.1, where due to destructive interference
the surface is stationary, three additional interference peaks occur that are of equal
magnitude, but opposite sign to those predicted by Quasi-Homogeneous theory; conversely, at point P these same peaks contribute constructively, reflecting the enhanced
sea state at the focal point.
Naturally, if such formalism is to be useful, it requires a diligent definition of the
operator Ω(i∇x , x), so that (2.2) represents progressive wave motion over topography.
For linear wave motion in a uniform medium (constant depth and currents), Ω can be
expressed as an integral (or alternatively pseudo-differential) operator derived from
the dispersion relation. The extension to variable topography is not trivial (see also,
e.g. Van Groesen and Andonowati, 2011), and in the present work this is obtained
by an appropriate operator association argument (see Chapter 5). Defined in this
way the model reproduces classic geometric optics for wave evolution over slowly
varying topography, and the resulting equation for E reduces to the RTE under
quasi-homogeneous conditions (see Chapter 5).
The result is thus what appears to be a natural extension of the radiative transfer
equation for regions where the statistics undergo potentially rapid (intra-wave scales)
variations due to the presence of inhomogeneities (coherent wave interference) in
the wave field, and reduces to quasi-homogeneous theory when the statistics change
slowly (infra-wave scales). In the present work we explore the potential of a stochastic
wave model derived with this formalism, discuss the implications of the underlying
assumptions, and investigate the potential for the use in operational wave models.
´
E(k, x, t) = (2π)−2 Γ(ξ, x, t) exp [−ik · ξ] dξ where Γ denotes the covariance function that is defined as Γ(ξ, x, t) = hζ(x + ξ/2, t)ζ ∗ (x − ξ/2, t)i.
12 Specifically:
3
Depth-induced wave breaking
in a non-hydrostatic,
nearshore wave model
The energy dissipation in the surfzone due to wave breaking is inherently accounted for in shock-capturing non-hydrostatic wave models, but this requires
high vertical resolutions. To allow coarse vertical resolutions a hydrostatic front
approximation is suggested. It assumes a hydrostatic pressure distribution at
the front of a breaking wave which ensures that the wave front develops a vertical face. Based on the analogy between a hydraulic jump and a turbulent
bore, energy dissipation is accounted for by ensuring conservation of mass and
momentum. Results are compared with observations of random, unidirectional
waves in wave flumes, and to observations of short-crested waves in a wave
basin. These demonstrate that the resulting model can resolve the relevant
nearshore wave processes in a short-crested wave-field, including wave breaking
and wave-driven horizontal circulations.
3.1. Introduction
Of all the physical processes active in the nearshore, wave breaking often dominates the hydrodynamics in the surfzone. It controls wave setup (e.g. LonguetHiggins and Stewart, 1964), drives long-shore currents, rip-currents and undertow
(e.g. Longuet-Higgins and Stewart, 1964; Longuet-Higgins, 1970; Svendsen, 1984;
MacMahan et al., 2006) and is involved in the generation (or release) of infra-gravity
waves (e.g. Symonds et al., 1982; Battjes et al., 2004). It is therefore of paramount
importance to accurately include the macro-scale effects of wave breaking into coastal
wave models describing nearshore hydrodynamics.
This chapter has been published as: Smit, P.B., Zijlema, M. and Stelling, G.S., 2013 Depthinduced breaking in a non-hydrostatic nearshore wave model. Coast. Eng., 76, 1–16.
17
18
3. Wave breaking in a non-hydrostatic wave model
At present, nearshore, nonlinear wave models of a phase-resolving nature, are usually based on either a Boussinesq-type formulation or a non-hydrostatic approach.
Boussinesq models are well established (e.g. Madsen et al., 1991; Nwogu, 1993; Wei
et al., 1995) and have been very successful in applications in nearshore regions. However, to increase accuracy, these models have grown quite involved, thereby complicating the numerical implementation. The non-hydrostatic approach is more recent
(e.g. Stelling and Zijlema, 2003; Yamazaki et al., 2009; Ma et al., 2012) and uses
an implementation of the basic 3D mass and momentum balance equations for a
water body with a free surface. The resulting Euler equations can be supplemented
with second-order shear-stress terms when required (resulting in the Navier-Stokes
equations). The basic difference with conventional Navier-Stokes models is that the
free-surface is described using a single-valued function of the horizontal plane. When
compared to more involved methods (e.g. Volume of Fluid or Smoothed Particle
Hydrodynamics Hirt and Nichols, 1981; Dalrymple and Rogers, 2006), this allows
non-hydrostatic models to efficiently compute free surface flows.
However, neither Boussinesq models nor non-hydrostatic models can be directly
applied to details of breaking waves, since in both models essential processes such
as overturning, air-entrainment and wave generated turbulence, are absent. But, if
only the macro-scale effects of wave breaking are of interest, such as the effect on the
statistics of wave heights, details of the breaking process can be ignored. By observing
that both spilling and plunging breakers eventually evolve into a quasi-steady bore,
where the entire front-face of the wave is turbulent (Peregrine, 1983), a breaking wave
becomes analogous to a hydraulic jump (e.g. Lamb, 1932). Consequently, its integral
properties (rate of energy dissipation, jump height) are approximately captured by
regarding the breaking wave as a discontinuity in the flow variables (free surface,
velocities). Proper treatment of such a discontinuity in a non-hydrostatic model
(conservation of mass and momentum) can therefore be used to determine the energy
dissipation of waves in the surfzone (e.g. Hibberd and Peregrine, 1978).
However, compared to the vertical resolutions which are (1–3 layers) sufficient to
describe the wave physics outside the surfzone (e.g. refraction, shoaling, diffraction,
nonlinear interactions), such a representation of dissipation due to wave breaking
requires a disproportional high vertical resolution (∼10–20). At low resolutions (<5
layers) the initiation of wave breaking is often delayed when compared to observations, and dissipation in the surfzone is underestimated. Such high resolutions are,
at present, not feasibly attainable for extensive horizontal domains (say 10 × 10 wave
lengths). Hence, in such cases, an alternative, more efficient approach is required.
To this end we adopt a method which is akin to the approach of Tonelli and Petti
(2010) in their Boussinesq model. By enforcing a hydrostatic pressure distribution at
the front of a wave, we can locally reduce a non-hydrostatic wave model to the shallow
water equations. The wave then rapidly transitions into the characteristic saw-tooth
shape and, consistent with the high resolution approach, dissipation is captured
by ensuring momentum conservation over the resulting discontinuity. Compared to
more involved wave breaking models (e.g. eddy viscocity and surface roller models,
Schäffer et al., 1993; Kennedy et al., 2000; Cienfuegos et al., 2010, e.g.), such an
approach requires fewer additional parameters (controlling the onset and cessation
3.2. Non-hydrostatic modelling
19
of wave dissipation) and is easily extendible to two horizontal dimensions.
In the present work we will demonstrate that: (a) with a sufficiently high vertical
resolution non-hydrostatic models can properly determine the dissipation of breaking waves without additional model assumptions; and (b) that similar results – at
significantly reduced computational cost – can be obtained with a more practical low
vertical resolution, if locally the non-hydrostatic model is reduced to a hydrostatic
model. Furthermore, by comparison to experimental data (Dingemans et al., 1986),
we will show that, in contrast to a high resolution model, the latter can be feasibly applied to situations with short-crested waves over two-dimensional topography,
including the wave generated mean currents.
This paper is organized as follows: §2 introduces the basic equations that govern
the non-hydrostatic model SWASH1 (Zijlema et al., 2011b) and briefly addresses
the relevant details of its numerical implementation. In §3 the hydrostatic front
approximation is introduced and in §4 the parameter that controls the onset of wave
breaking is estimated from experimental data. Section §5 compares significant wave
heights and mean periods obtained from computations using high and low vertical
resolutions (using the hydrostatic front approximation) with measured data from
flume experiments. Subsequently, in §6 the approximate model is compared to the
experiment by Dingemans et al. (1986). Finally, we discuss our results and summarize
our main findings in §7 and §8.
3.2. Non-hydrostatic modelling
Governing equations
The non-hydrostatic model SWASH (Zijlema et al., 2011b), is an implementation
of the basic 3D mass and momentum balance of a free surface, incompressible fluid
with constant density. For reasons of exposition, we present these equations for
the 2D vertical plane (the extension to full 3D is straightforward). In terms of
Cartesian coordinates x,z (defined in and normal to the still water level, with z
positive upwards, respectively) and time t , these equations are
1 ∂(ph + pnh ) ∂τxz
∂τxx
∂u ∂uu ∂wu
+
+
=−
+
+
∂t
∂x
∂z
ρ
∂x
∂z
∂x
∂w ∂uw ∂ww
1 ∂pnh
∂τzz
∂τzx
+
+
=−
+
+
∂t
∂x
∂z
ρ ∂z
∂z
∂x
∂u ∂w
+
= 0,
∂x
∂z
(3.1)
(3.2)
(3.3)
in which u(x, z, t) and w(x, z, t) are the horizontal and vertical velocity, respectively;
ρ is density; ph and pnh are the hydrostatic and non-hydrostatic pressures, respectively, and τxx , τxz , τzz , τzx are the turbulent stresses. The water column is vertically restricted by the moving free-surface ζ(x, t) and stationary bottom d(x) (measured positive downwards), defined relative to the still water level z0 . Furthermore,
the hydrostatic pressure is explicitly expressed in terms of the free-surface level as
1 Simulation
WAves till SHore, freely available at http://swash.sourceforge.net/
20
3. Wave breaking in a non-hydrostatic wave model
ph = ρg(ζ − z) so that ∂z ph = −gρ (where ∂z is short for ∂/∂z) and ∂x ph = ρg∂x ζ
(with g = 9.81m/s2 the gravitational acceleration). Finally the time evolution of
the surface elevation is obtained by considering the mass (or volume) balance for the
entire water column
ˆ ζ
∂
∂ζ
u dz = 0
(3.4)
+
∂t
∂x −d
For waves propagating over intermediate distances (say O(10) characteristic wave
lengths), in the absence of strongly sheared currents, the influence of (turbulent-)
stresses on the wave motion proper is relatively small (e.g. Mei et al., 2005, Section
9) and can – to a good approximation – be neglected. However, to allow the influence
of bottom friction to extend over the vertical (important for low frequency motions),
some vertical mixing is introduced by means of stress terms based upon a turbulent
viscosity approximation (e.g. τxz = νv ∂z u, with νv the vertical mixing coefficient).
This also introduces additional vertical coupling, thereby increasing numerical stability. Depending on how well the vertical is resolved the eddy viscosity is estimated
with either a constant value, or obtained from a more elaborate k − ε closure model
(see Appendix A). Finally, to allow for lateral mixing in horizontal circulations due
to sub-grid eddies, lateral stresses are added (see Appendix A).
Equations (3.1)–(3.4) are solved for kinematic and dynamic boundary conditions
at the surface and at the bottom. The kinematic boundary conditions are that no
particle shall leave the surface and no particle shall penetrate the (fixed) bottom,
giving
∂ζ
∂ζ
+u
∂t
∂x
∂d
w = −u
∂x
w=
at
z = ζ(x, t)
(3.5)
at
z = −d(x)
(3.6)
The dynamic boundary condition at the surface is a constant pressure (zero atmospheric pressure, ph = pnh = 0) and no surface stresses. Since bottom friction is
important for low frequency motions and during wave run-up, a stress term at the
bottom boundary τb is added based on a quadratic friction law
τb = cf
U |U |
h
(3.7)
where U is the depth averaged current, h = d + ζ the water depth and cf is a
(dimensionless) friction coefficient. A correct estimation of cf is important in case of
steady horizontal circulations, where the bottom friction balances the forcing by the
radiation stress. Here we obtain the friction coefficient from the Manning-Strickler
1/3
formulation as cf = 0.015 (k/h) , where k is an (apparent) roughness value. In
the presence of waves this apparent roughness value is not necessarily related to the
bottom roughness. In particular in the surfzone, observations have shown that the
apparent roughness (or friction coefficient) can be substantially higher (e.g. Feddersen
et al., 2003), because the near bed vertical flux of horizontal momentum is influenced
by the breaking induced turbulence. For simplicity, nearshore circulation models
often do not separately account for this. Instead, they assume a constant cf , which
21
3.2. Non-hydrostatic modelling
is obtained by comparison to observations (e.g. Özkan-Haller and Kirby, 1999; Chen
et al., 1999; Tonelli and Petti, 2012, among many others). In the present study – for
flume-type situations – k is estimated based upon the surface properties. However,
in case of a mean horizontal circulation, we likewise obtain the apparent roughness
after calibration.
We assume that the domain is bound horizontally by a rectangular boundary that
consists of four vertical planes. Waves are generated by prescribing the horizontal
particle velocities normal to the boundary over the vertical. These velocities are
either obtained from time-series for each point on the boundary, or by imposing a
wave spectrum along the boundary (see Appendix B). Additionally, to allow low
frequency wave energy to leave the domain a vertically uniform particle velocity due
to this low-frequency motion is included (see Appendix B). At closed boundaries
(e.g. vertical walls) the normal velocity is zero and parallel to the boundary a freeslip condition is imposed. If a shoreline is present within the model the boundary is
formed by the moving shoreline.
(Standard)
(Box)
z
x
Figure 3.1: The layout of the velocities u, w (indicated by arrows) and the pressure p (indicated by
dots) for a vertical cell in case of the standard scheme (on the left), and when the Keller Box is
used (on the right). The standard scheme uses a conventional staggered layout in both directions
(x and z), whereas for the Keller Box scheme w and p are both located on the layer interfaces.
Numerical implementation
The numerical implementation of SWASH is based on an explicit, second order accurate (in space and time) finite difference method that conserves both mass and
momentum at the numerical level. The computational grid consists of columns of
constant width ∆x and ∆y in x- and y-direction respectively, vertically discretized
with a fixed number of layers of equal thickness between the fixed but spatially varying bottom and the moving, free surface. Horizontally, a staggered grid is employed
for the coupling between velocity and pressure. Consequently, the horizontal velocity
u is defined at the central plane of each layer and at the centre of each lateral face of
the columns (see Fig. 3.1). The vertical are velocities computed at the interfaces of
the layers, at the centre axis of each column. The pressures are also located at the
22
3. Wave breaking in a non-hydrostatic wave model
centre axis of each column, but vertically are located at the central plane (standard
layout) or at the layer interface (box scheme).
Depending on the vertical layout of the pressure points, the pressure gradients
in the vertical momentum equations are approximated by means of the Keller Box
scheme (e.g. Lam and Simpson, 1976) or central differences (e.g. Casulli and Stelling,
1998). At very low vertical resolution (one or two layers), the Box scheme gives
good dispersive properties (For two layers rougly 1% error relative to the linear
dispersion relation up to kd ≈ 7 where k is a representative wave number Zijlema
et al., 2011b), basically because the pressure is defined at both the surface and
the bottom (implying a vertical variation, even in a one-layer approach). At high
vertical resolutions the standard layout is preferable because it appears to be more
robust while its dispersion characteristics are then usually sufficiently accurate. In
the present study simulations with 5 layers or less were done with the Keller-Box
scheme, while for the other simulations (typically 15–20 layers) the standard layout
was employed.
In situations where the wave transforms into a bore, momentum is still conserved
on the macro scale of the motion (in nature) whereas energy is not. Preference should
then be given to a numerical method that conserves momentum (as in SWASH). Such
schemes are conventionally referred to as shock-capturing schemes or approaches
(e.g. Hirsch, 2007). To achieve this the advective terms are handled in the manner of
Stelling and Duinmeijer (2003). Their approach is momentum conservative, allows for
the use of a horizontally staggered grid (thus retaining their efficiency and accuracy,
e.g. Stelling, 1983), and is readily extended to three-dimensional, non-hydrostatic
flows. In the present study this method was employed in combination with central
approximations for the advective fluxes on the cell faces.
When a moving shoreline is present (flooding and drying) it is captured using
a simple explicit approach that conserves mass (Stelling and Duinmeijer, 2003). It
removes computational (velocity) points from the grid when the water depth h is
below a certain threshold (10−5 m in the present study). Finally, time integration
is performed using a time step that, for stability reasons, conforms to the
√ CFL
condition Cr = ∆t(c + u)/∆x ≤ 1. Here Cr is the Courant number and c = gh the
shallow water wave celerity. For efficiency (and stability) reasons, ∆t only remains
constant if the maximum of Cr in the domain remains between two pre-defined values
Crmin ≤ Cr ≤ Crmax , else the time step is either half- or doubled so that the new
value again lies within the defined range. For all simulations considered here we set
Crmin =0.2 and Crmax =0.6 and time steps indicated refer to initial values within that
range.
3.3. Wave breaking approximations
The above equations can be directly applied to nonlinear wave motion outside the
surfzone. In addition, they can be used to estimate the overall characteristics of a
quasi-steady breaking bore in the surfzone, such as its energy dissipation, without
needing to resolve complex phenomena such as the overturning of the surface or the
wave generated turbulence.
In nature, as the forward pitching of the waves increases, the wave front at a
3.3. Wave breaking approximations
23
certain point becomes unstable (spilling breakers), or overturns (plunging breakers)
and the entire front face of the wave becomes turbulent. Even though the nonlinear processes that induced the steepening are still present, the downward transport
of momentum due to the turbulent stresses stabilize the bore front (Madsen and
Svendsen, 1983), and a quasi-steady bore develops. In the model the turbulent motions generated by the breaking process, and thus their stabilizing effects, are not
accounted for. Hence, the steepness of the front continues to increase and eventually a jump-discontinuity, with a jump-height equal to the local wave height H, will
develop. By enforcing strict momentum conservation across the discontinuity the
energy dissipation rate is then proportional to H 3 , similar to the dissipation rate
found in an hydraulic jump (e.g. Lamb, 1932). In the model the flow field in the
entire water column below the turbulent front (between trough and crest) is thus
considered as a sub-grid phenomenon. Such a treatment of wave breaking has the
advantage that it does not require any additional measures, for instance, to initiate
or stop breaking, nor a separate approximation to explicitly account for the energy
dissipation. However, it does require a high horizontal and vertical resolution.
For relatively small domains the required horizontal resolution is fairly constant,
and is typically 1/50 or 1/100 of the dominant wave length. However, the necessary vertical resolution can change considerably in the domain. For mildly nonlinear
waves (H/d ≤0.1–0.4) propagating from intermediate (kd ∼ 1) into shallow water
(kd 1), the vertical gradients in the pressure and particle velocities remain small.
An accurate description of the flow can then be achieved with a relatively low vertical
resolution (1–3 layers). But, due to the decreasing water depth and wave shoaling ,
the importance of nonlinearity increases (H/d becomes O(1)) and, concurrent with
the forward pitching of the waves, strong vertical gradients in the particle velocities
develop. In the vicinity of the point where the wave becomes unstable, flow velocities in the upper half of the water column, i.e., near the crest, approach the wave
celerity (e.g. Sakai and Iwagaki, 1978; Van Dorn, 1978), whereas in the lower half
velocities remain much smaller (u/c ∼0.1–0.4, with c the wave celerity). An accurate description of the flow in the region just prior to the breakpoint thus requires a
comparatively high vertical resolution (10–20 layers).
Once a turbulent bore is formed, vertical velocity gradients directly under the
front remain large (e.g. Van Dorn, 1978; Stive, 1980; Lin and Liu, 1998), but the
relative importance of the non-hydrostatic pressure gradient diminishes, and its influence on the depth integrated momentum balance becomes small (e.g. Madsen and
Svendsen, 1983). In fact, measurements (e.g. Stive, 1980) and computations with
more advanced models (e.g. Lin and Liu, 1998) indicate that near the bed the maximum deviation from hydrostatic presure at the bore front is in the order of 10 percent.
Moreover, in the regions before and after the turbulent front, the pressure is approximately hydrostatic, and the horizontal particle velocity is almost depth uniform.
Since, in the model the front is not resolved, a relatively low-vertical resolution could
be applied again.
24
3. Wave breaking in a non-hydrostatic wave model
Hydrostatic front approximation
A coarse vertical resolution can thus be applied almost everywhere in the nearshore,
except to predict the onset of wave-breaking. Here, a coarse resolution will result
in an underestimation of the horizontal velocities near the crest, and thus an underestimation of the nonlinear (advective) contributions. This underestimation implies that at low vertical resolution the influence of the non-hydrostatic pressure
gradient is overestimated. Consequently, the stabilizing dispersive effects (i.e. the
non-hydrostatic pressures) postpone the transition into the characteristic saw-tooth
shape and therefore also the onset of dissipation.
The subsequent dissipation is well described by assuming depth uniform velocities and a hydrostatic pressure distribution. In fact, these assumptions often form
the basis to derive dissipation formulations to account for depth induced breaking
in energy balance type models (e.g. Battjes and Janssen, 1978, among many others). Hence, prescribing a hydrostatic pressure distribution in the model around the
discontinuity should result in the correct bulk dissipation. The model then reduces
to the conventional nonlinear shallow water equations (NSWE), which previously
have been successfully applied in the surfzone (e.g. Peregrine, 1983), supporting this
conjecture.
ζ
ζ
HFA enabled
HFA enabled
t
∂ζ
∂t
αc
t
∂ζ
∂t
t
βc
t
Figure 3.2: Sketch of the free surface ζ (top panels) and the rate of surface rise ∂t ζ (bottom panels)
as a function of time. Grey patches indicate regions where the HFA is enabled based on ∂t ζ > αc
(left panels) or on the reduced criterion ∂t ζ > βc, if the HFA is active in a neighbouring point (right
panels).
There is no need to assume a hydrostatic pressure distribution if the vertical
resolution is sufficient. However, imposing a hydrostatic distribution resolutions at
low resolutions will ensure that, due the absence of dispersive effects, the front quickly
transitions into a bore like shape. Hence, it can be used to initiate the onset of wavebreaking, thus allowing for the use of low-vertical resolutions throughout the domain.
This approach, which we refer to as the Hydrostatic Front Approximation (HFA), is
akin to the disabling of the dispersive terms in the Boussinesq equations (Tonelli and
Petti, 2010; Tissier et al., 2012) and essentially regards the entire turbulent front as
3.3. Wave breaking approximations
25
a sub-grid feature of the flow. In practice this means that once a mesh-point is in
the front of a breaking wave, vertical accelerations are no longer resolved, and the
non-hydrostatic pressure is set to zero.
The boundary of the horizontal region in which the model should operate in
hydrostatic mode, as described above, varies in time and horizontal space. We assume
that this mode is valid if the local surface steepness ∂x ζ exceeds a predetermined value
|∂x ζ| > α. However, this criterion was found to be sensitive to numerical noise in the
surface elevation. As an alternative we therefore consider the evolution equation for
a shallow-water wave of constant form over a flat bottom (e.g. ∂t ζ + c∂x ζ = 0) and
re-express the steepness criterion in terms of the rate of surface rise, i.e. ∂t ζ > αc
(Kennedy et al., 2000,
√ see also Appendix C). The wave celerity is estimated as c =
c0 + ∆c, with c0 = gh the wave celerity in the absence of an ambient current. Here
∆c = Ū cos ∆θ is a correction to c due to the presence of an ambient current with
magnitude Ū at a mutual angle of ∆θ with the wave direction (e.g. for a strictly
following or opposing current ∆c = ±Ū ).
This criterion has the advantage that for a weak ambient current (Ū /c0 1) it
is based on local scalar quantities and independent of wave direction (as ∆c ≈ 0)
so that it can be used unmodified in either one- or two-dimensional situations. This
contrasts with criteria based on the wave profile (e.g. wave height, Tonelli and Petti,
2010), the extension of which to two horizontal dimensions is generally not trivial.
However a disadvantage is that when Ū /c0 ∼ O(1) such a criterion is Doppler shifted,
potentially resulting in premature or delayed activation of the HFA. In this case ∆c
should be included, requiring estimation of the the ambient current vector Ū and its
mutual angle to the local instantaneous wave direction. For a non-blocking current
these can be estimated in the manner as described in Appendix B.
A grid point is therefore labelled for hydrostatic computation if ∂t ζ > αc. Once
labelled, a point only becomes non-hydrostatic again if the crest of the wave has
passed. This is assumed to occur when ∂t ζ < 0. Furthermore, because grid points
only become active again when the crest passes (where w ≈ 0), vertical velocities
(essentially diagnostic variables when the pressure is hydrostatic) are set to w =
0 on the front. To represent persistence of wave breaking, we locally reduce the
criterion α to β if a neighbouring grid point (in x- or y-direction) has been labelled
for hydrostatic computation. In this case a point is thus also labelled for hydrostatic
computation if ∂t ζ > βc, with β < α. In all other grid points, the computations
are non-hydrostatic. The activation and deactivation of the HFA based upon these
considerations is illustrated in Fig. 3.2.
The range of maximum steepness (or α) values found in the literature is relatively
wide, ranging from α = 0.3 (Schäffer et al., 1993) to α = 0.6 (Lynett, 2006), because
its value is sensitive to how well the model can represent the wave shape for highly
nonlinear waves. Hence, to estimate α, we use experimental data (see §4) to obtain
its model value near observed breaking locations. This results in α = 0.6. The value
for the persistence parameter β also depends on the model behaviour in the surfzone,
and for the cases considered in the present work β = 0.3 appears to gives reasonable
results. Because we represent a breaking wave as a shock, this is substantially higher
then reported values of a similar parameter in Boussinesq models, which ranges from
26
3. Wave breaking in a non-hydrostatic wave model
0.15 to 0.18 (Schäffer et al., 1993; Kennedy et al., 2000; Lynett, 2006; Cienfuegos
et al., 2010).
Finally, if the HFA is used some additional horizontal viscosity is included to
prevent generation of high frequency noise in the wave profile due to the discrete
activation of the HFA. These frequencies are introduced because the model has to
adapt to the enforced hydrostatic pressure distribution. Assuming that the strength
of such components is related to local gradients in the horizontal particle velocity,
the additional viscosity νhHFA , included once the HFA is activated, is expressed as
s
2 2
∂U1
∂U2
HFA
2
+
(3.8)
νh
= Lmix
∂x1
∂x2
where Lmix is a typical horizontal length scale over which mixing occurs which we
assume is a fraction µ of the local depth, i.e. Lmix = µh. Once sufficient dissipation
is present, the predicted wave height become relatively insensitive to a broad range
of values for µ (0.75 – 3, see next section). This insensitivity confirms that the
primary dissipation mechanism is due to the formation of a shock. The inclusion of
the additional viscosity is mainly intended to prevent spurious components, and we
primarily regard µ as a numerical parameter and therefore simply set µ = 1 without
further calibration.
3.4. Monochromatic wave breaking over a sloping
beach
Figure 3.3: The experimental setup from Ting and Kirby (1994)
To demonstrate that the estimation of dissipation by wave breaking is reasonably accurate in SWASH, and to estimate the steepness α that controls the onset
of dissipation, we consider two laboratory experiments by Ting and Kirby (1994).
In these experiments, mechanically generated, monochromatic, cnoidal-type waves
(wave height H=0.125 m; period T =2 s or 5 s) propagate in a flume (40 m long, 0.6
m wide and 1 m deep) over a 1/35 plane beach (see Fig. 3.3). In the experiment with
T = 2 s, the breaking waves are of the spilling type, whereas they are of the plunging
type in the experiment with T = 5 s. After steady conditions were reached, measurements of the free surface elevation were taken throughout the tank for a duration of
102 periods. Here SWASH is employed with a horizontal resolution of ∆x=0.025 m
27
3.4. Monochromatic wave breaking over a sloping beach
and an initial time step of ∆t=0.001 s. Simulations are performed with a high (20
layers) and low (2–5 layers) vertical resolutions. In the high resolution computations
the HFA is always disabled. At low resolutions, simulations with and without the
HFA are conducted. At the incoming boundary we impose a second-order cnoidal
wave solution for the velocities (e.g. Svendsen, 2006, section 9.5), including a mass
flux contribution due to the return flow.
a
25
b
2
20
1.5
H
Fr
15
1
10
0.5
5
0
−2
0
2
4
6
x (m)
8
10
12
0
−2
0
2
4
6
8
10
12
x (m)
Figure 3.4: Wave heights H for spilling breakers in the Ting
p and Kirby (1994) experiment on the
left (panel a) and maximum Froude number Fr (= umax / gd) in the upper most layer for different
vertical model resolutions on the right (panel b). Comparison between measurements (circle markers,
only for wave heights) and computations without HFA using two (solid black line), three (dashed
black line), five (dashed red line) and twenty (solid red line) layers. In the right panel dots indicate
the location of the maximum wave height in the respective simulations and the dotted vertical line
indicates the observed breakpoint that is located near the maximum observed wave height.
To illustrate the erroneous model behaviour at low vertical resolutions, we first
carry out computations without the HFA for the spilling breaker case using 2, 3, 5
and 20 layers. The resulting wave heights compared with observations are shown in
Fig. 3.4a. It is clearly seen that at the lower resolutions ( 2–5 layers), compared
with the observations, the onset of dissipation is postponed. Once the vertical resolution becomes sufficiently high (20 layers), wave heights correspond well with the
observations. Inspection of the particle velocities in the upper layer shows, that the
onset of breaking in the computations
approximately coincides with the occurrence
√
of supercritical flow, i.e. umax / gd > 1. Here umax is the maximum velocity that
occurs during a wave period (see Fig. 3.4b). Thus, wave breaking is initiated when
velocities exceed the wave celerity. When the vertical resolution is coarse, particle
velocities near the free-surface are simply underestimated (Fr < 1, see 3.4b), and the
formation of a shock wave – and therefore the onset of dissipation – is delayed.
In the region prior to the observed breaking location, which occurred where the
observed wave height attained its maximum (x = 6.45 m, Ting and Kirby, 1994),
computed wave heights are insensitive to the vertical resolution. This confirms that
outside the surfzone a coarse resolution (2 layers) suffices. To investigate if enabling
the HFA can improve results in the surfzone, we must first estimate the breaker parameter α. Hereto, we obtain the maximum value of ∂t ζ at the observed breaking
location (x = 6.45 m) from the previous calculations; this was found to be approximately ∂t ζ = (0.6 ± 0.04)c0 . Hence, α=0.6, which was also found by (Lynett, 2006)
28
3. Wave breaking in a non-hydrostatic wave model
plunging
spilling
25
20
20
15
15
10
10
5
5
H
25
0
−2
0
2
4
6
x (m)
8
10
12
0
−2
0
2
4
6
8
10
12
x (m)
Figure 3.5: Wave heights H for spilling (left panel) and plunging (right panel) breakers in the Ting
and Kirby (1994) experiment. Comparison between measurements (circle markers) and computations using two layers with HFA (black line), and twenty layers (thin line).
suggesting that model behaviour near the breaking point at lower resolutions is similar to his multi-layered Boussinesq model.
The wave height evolution, but now with HFA (using α = 0.6) at low resolutions
(2 layers), is shown in Fig. 3.5. In the experiment with spilling breakers, both high
and low resolution computations now predict the evolution of the wave height well.
The location of the outer edge of the surfzone, the maximum wave height there and
the decay inside the surfzone are well reproduced. In the experiment with plunging
breakers dissipation is initially overestimated. At that location the physics of a
plunging breaker are poorly described by a bore analogy, but deep in the surfzone a
quasi-stationary bore is formed, and the computational results converge towards the
observations (for x>10m). In the low-resolution computations, the over-estimation
of the dissipation starts just outside the surfzone. Again, deep in the surfzone the
computational results converge towards the observations (for x>10m).
Outside the surfzone the wave profile is well reproduced (x<6.4 m and x<7.9 m
for respectively the spilling breaker and plunging breakers experiments), although
the slope of the wave front is slightly too gentle for the spilling waves (see Fig. 3.6).
Inside the surfzone, the absence of the stabilizing effects of turbulence in the highresolution computations, results in a continued steepening of the front of both spilling
and plunging breakers until it is almost vertical in case of the plunging breakers.
This contrasts with the effects of the added horizontal viscosity in the low-resolution
computations. This viscosity, which was added for numerical reasons in the HFA,
counteracts wave steepening such that the wave fronts are slightly less steep than
observed.
The maximum wave-induced setup is predicted well (see Fig. 3.7), particularly in
the plunging breaker case where the error in the wave height is relatively large (i.e.,
20%). This indicates that the radiation stress gradient in SWASH is well predicted
even if the wave height is (slightly) under-estimated. It does appear that for spilling
breakers neither model is able to reproduce the observed jump in mean water level
(MWL) around x = 5m. This jump precedes the decrease in wave height, so that
its physical origins are not obvious. This jump is also absent in results that were
29
3.4. Monochromatic wave breaking over a sloping beach
spilling
plunging
ζ (cm)
20
20
X = 5.9 m
10
10
0
0
−10
−10
−20
0.4
0.6
0.8
1
1.2
20
1.4
ζ (cm)
−20
0.4
0.6
0.8
1
1.2
20
X = 7.9 m
10
1.4
X = 8.8 m
10
0
0
−10
−10
−20
0
0.2
0.4
0.6
0.8
20
1
−20
0.6
0.8
1
1.2
1.4
20
X = 9.1 m
10
ζ (cm)
X = 7.3 m
1.6
X = 10.4 m
10
0
0
−10
−10
−20
0.4
0.6
0.8
1
1.2
1.4
−20
0.8
1
1.2
t/T
1.4
1.6
1.8
t/T
Figure 3.6: Phase-averaged profiles of the free surface elevation ζ for spilling (left panel) and plunging
(right panel) breaking waves at the indicated locations. Comparison between measurements (circle
markers) and computations using two layers with HFA (black line), and twenty layers (thin lines).
plunging
ζ̄ (cm)
spilling
3
3
2
2
1
1
0
0
−1
−2
0
2
4
6
x (m)
8
10
12
−1
−2
0
2
4
6
8
10
12
x (m)
Figure 3.7: Mean water-level setup ζ̄ for spilling (left panel) and plunging (right panel) breakers in
the Ting and Kirby (1994) experiment. Comparison between measurements (circle markers) and
computations using two layers with HFA (black line), and twenty layers (thin line).
30
3. Wave breaking in a non-hydrostatic wave model
obtained by others (e.g. Cienfuegos et al., 2010; Tonelli and Petti, 2010), and
1.4
1.3
α = 0.8
µ = 0.25
1.2
α = 0.7
H/Hobs
H/Hobs
1.2
α = 0.6
1
α = 0.5
µ = 0.75
1.1
µ=1
α = 0.4
0.8
µ=2
1
µ=3
0.6
0.9
4
5
6
7
8
x (m)
9
10
11
12
4
5
6
7
8
9
10
11
12
x (m)
Figure 3.8: Sensitivity of computed wave heights (normalized with observations) in case of spilling
breakers to variations in the maximum steepness α (left panel) and relative mixing length µ (right
panel).
Finally, the sensitivity of computed wave heights to changes in α (0.4–0.8) and µ
(0.25–3) is shown in Fig. 3.8. Results are scaled with the observed wave heights to
better illustrate the differences, especially were wave heights are low. As anticipated,
variation of α either causes premature (α < 0.6) or delayed (α > 0.6) initiation
of dissipation, therefore resulting in over- and under-prediction of wave heights in
the surfzone. Sensitivity of results to µ appears to be relatively small. Increasing
µ from 0.25 to 3, a twelvefold increase, gives 30 percent difference in wave heights.
Best correspondence is obtained with µ=2, but differences mainly occur deep in the
surfzone where wave heights are small, and within the range µ =0.75–3 results are
therefore not significantly different.
In general, the low-resolution computations (the two-layer model) combined with
the Hydrostatic Front Approximation, reproduces wave heights and wave profiles
quite accurately inside and outside the surfzone, although slightly better for spilling
breakers than for plunging breakers. However, it must be emphasized that the low
resolution computation with HFA uses the α value estimated from this same experiment. Conversely, the high-resolution computations does not use any information
from the experiment. This demonstrates that the high resolution can inherently
describe dissipation due to breaking without additional measures.
3.5. Random waves breaking over barred topography
To verify the two approaches of the high (15 layers) and low (2 layer) resolution
with the estimated coefficient from the Ting and Kirby (1994) experiment, we compare model predictions with laboratory measurements that include depth-induced
breaking of broad-banded random waves over 1D barred topographies.
We consider laboratory experiments by Battjes and Janssen (1978), Dingemans
et al. (1986), Boers (1996) and Van Gent and Doorn (2000). In all experiments
(except one) a wave maker situated at the end of a 1D flume generated broad banded
waves that propagated over a bar-type topography (see Fig. 3.9) where a large portion
31
3.5. Random waves breaking over barred topography
0.2
0.2
a
b
z
0
−0.2
−0.4
0
−0.2
−0.4
BJ78
−0.6
B96
−0.6
0
5
10
15
20
25
30
35
0
5
10
5
10
15
20
25
30
15
20
25
30
0.2
0.2
b
a
0
z
0
−0.2
−0.4
−0.2
GD00
D86
−0.4
−30
−25
−20
−15
−10
−5
0
0
x (m)
x (m)
Figure 3.9: Layout of the different flume experiments depicting the bed level locations z (thick solid
line), still water level(s) (dashed lines) and wave gauges (diamonds).
Table 3.1: Wave parameters at the boundary for the different flume experiments. Listed are the
names as used here (Experiment) and as referred to in the original study (their case), the significant
wave height Hm0 , peak period Tp , relative depth kp d (with kp the wave number at the peak),
relative wave height Hm0 /d and still water level z0 .
Experiment
Battjes and Janssen (1978)a
Battjes and Janssen (1978)b
Boers (1996)a
Boers (1996)b
Boers (1996)c
Dingemans et al. (1986)a
Van Gent and Doorn (2000)a
Van Gent and Doorn (2000)b
their case
run 13
run 15
1A
1B
1C
me25
test 1.04
test 2.51
Hm0 (m)
0.147
0.202
0.157
0.206
0.103
0.1
0.11
0.15
Tp (s)
2.01
1.89
2.05
2.03
3.33
1.25
1.42/1.8
1.77
kp d
1
1.07
0.96
0.97
0.55
1.22
1.45
1.13
Hm0 /d
0.19
0.27
0.2
0.27
0.13
0.25
0.17
0.21
z0 (m)
0
-0.14
0.
0.
0.
0.
0.05
0.11
of the wave energy is dissipated by wave breaking. The exception is the experiment
by Dingemans et al. (1986) which was carried out in a 2D basin, but the variation
across the basin was sufficiently small that it can be treated as a 1D experiment.
With the exception of Van Gent and Doorn (2000), which used a double peaked
spectrum obtained from scaled-down version of field measurements during a storm
at the north sea coast of the Netherlands (Petten sea defence), wave spectra at the
boundary are JONSWAP spectra (Hasselmann et al., 1973). A summary of the wave
parameters at the wave maker in each of the experiments is given in Tab. 1.
In each case the numerical model is set-up along a one-dimensional transect of
length Lf (with horizontal resolution ∆x) in the centre of the flume (or basin), and
ran for a duration D (with initial time step ∆t). For case specific model parameters
we refer to Table 3.2. At the wave maker boundary a wave spectrum is imposed,
whereas at the opposite boundary waves energy is dissipated at a numerical beach.
After a spin-up time of Ds (≥ 4Lf /cg , using the group velocity cg (fp ) calculated
32
3. Wave breaking in a non-hydrostatic wave model
Table 3.2: Numerical setup used to simulate the flume experiments. Listed are the horizontal meshsize ∆x, initial time step ∆t, domain length Lf , duration D, output duration Do and sampling rate
fs .
Experiment
Battjes and Janssen (1978)a
Battjes and Janssen (1978)b
Boers (1996)a
Boers (1996)b
Boers (1996)c
Dingemans et al. (1986)a
Van Gent and Doorn (2000)a
Van Gent and Doorn (2000)b
∆x (m)
0.04
0.05
0.02
0.02
0.02
0.02
0.025
0.025
∆t (s)
0.005
0.005
0.002
0.002
0.002
0.005
0.0025
0.0025
Lf (m)
38.8
38.8
32.
32.
32.
2400
32.5
32.5
D (s)
1890
1890
1890
1890
2460
1350
1290
1320
Do (s)
1800
1800
1800
1800
2400
20
1200
1200
fs (Hz)
10
10
10
10
5
20
10
10
at the boundary), output is generated using a sampling frequency of fs ≈ 20fp
for a duration of Do seconds. Variance density spectra are generated using the
recorded free surface elevations (with 60 degrees of freedom and ∆f = 30/Do ). In
the present work
´ we primarily compare integral parameters derived from the moments mn (= f n E(f ) df ) of the measured variance density spectra E(f ) (with f
frequency). These include the significant wave height Hm0 (= 4m0.5
0 ) and the mean
zero-crossing period Tm02 (= (m0 /m2 )0.5 ). Prior to calculating the integral primary
wave parameters the spectra were band-pass filtered (pass-band 0.5fp ≤ f ≤ 4fp ).
This is done to exclude infra-gravity wave energy and prevent undue influence on
estimates of Tm02 by errors in the high frequency tail.
The computed significant wave heights are compared with observations in Fig.
3.10. It is obvious that the high- and the low-resolution computations predict the
development the significant wave height well. This implies that both types of computations correctly predict the onset of breaking, and the continued dissipation inside
the surfzone. In this respect the two types of computations are similar, although
the high resolution results are generally slightly better. Furthermore, not only is the
onset and rate of breaking well predicted, but also the reduction of breaking when
the waves enter deeper water between two bars or between a bar and the beach. This
is visible behind the bar in the Dingemans et al. (1986) experiment, and to a lesser
extent between the two bars (20 < x < 25) in the Boers (1996) experiments. Moreover, SWASH reproduces the wave height evolution reasonably well when breaking
is mild and therefore the wave decay gentle. For instance in (Boers, 1996)b, and to
a lesser extent (Boers, 1996)a, the wave conditions were such that a small fraction
of the waves (a: 0–5% percent, b: 10–15% percent) were ’breaking’ (white water
near the crest) on the gentle foreshore (x <17m) in front of the bar (Boers, 2005).
Since the criterion of breaking is identical for each individual wave, the difference in
intensity of breaking is due to the difference in the fraction of breaking waves.
The computed mean wave periods, in all cases (except one) defined as Tm02 , are
compared in Fig. 3.11 with the observed mean wave periods. The exception is the
Dingemans et al. (1986) experiment, where the comparison is between the mean
zero-crossing periods T0 obtained from a zero-crossing analysis of the time-series
(observations of Tm02 were unavailable). Furthermore, only 6 cases are shown, since
the required information is not available for the experiment of Battjes and Janssen
33
3.5. Random waves breaking over barred topography
Hm0
0.15
0.2
0.15
0.1
0.1
0.05
0.05 Battjes & Janssen (1978)b
Battjes & Janssen (1978)a
0
0
10
15
20
25
8
10
12
14
x (m)
16
18
20
22
24
x (m)
0.1
0.1
Hm0
0.08
0.06
0.05
0.04
0.02 Dingemans et al. (1986)
van Gent & Doorn (2000)a
0
0
5
10
15
20
−30
−25
−20
Hm0
x (m)
0.15
0.15
0.1
0.1
0.05
−10
−5
0
20
25
30
20
25
30
0.05
van Gent & Doorn (2000)b
Boers (1996)a
0
0
−30
−25
−20
−15
−10
−5
0
0
5
10
15
x (m)
x (m)
0.1
0.2
Hm0
−15
x (m)
0.15
0.05
0.1
0.05 Boers (1996)b
Boers (1996)c
0
0
0
5
10
15
x (m)
20
25
30
0
5
10
15
x (m)
Figure 3.10: Spatial variation of the significant wave height Hm0 in the various experiments. Comparison between measured (circle markers) and predicted values (low resolution model: black line,
high resolution model: thin line).
34
3. Wave breaking in a non-hydrostatic wave model
1.2
1.2
1
Tm02
T0
1.1
0.9
0.8
Dingemans et al. (1986)
0.8 van Gent & Doorn (2000)a
0.7
5
1
10
15
20
−30
−25
−20
−15
x (m)
−10
−5
0
20
25
30
20
25
30
x (m)
1.2
1.4
Tm0 2
1.6
Tm0 2
1.4
1
van Gent & Doorn (2000)b
1.2
1 Boers (1996)a
0.8
−30
−25
−20
−15
−10
−5
0
0
5
10
15
x (m)
x (m)
2.4
1.6
2.2
Tm0 2
Tm0 2
1.4
1.2
1.6
1 Boers (1996)b
0
5
2
1.8
Boers (1996)c
1.4
10
15
20
25
30
0
5
10
15
x (m)
x (m)
Figure 3.11: Spatial variation of the mean zero-crossing period T0 (Dingemans et al. (1986), top left
panel) or Tm02 (other panels) in the various experiments. Comparison between measured (circle
markers) and predicted values (low resolution model: black line, high resolution model: thin line).
Boers A
ζ̄ (cm)
1.5
Boers B
1.5
1
1
1
0.5
0.5
0.5
0
0
0
−0.5
−0.5
0
10
20
X (m)
30
Boers C
1.5
−0.5
0
10
20
X (m)
30
0
10
20
30
X (m)
Figure 3.12: Mean water-level setup ζ̄ in Boers (1996), case a–c. Comparison between measured
(circle markers) and predicted values (low resolution model: black line, high resolution model: thin
line).
35
3.6. Short-crested waves over 2D topography
(1978). Both high- and low-resolution models reproduce the observed periods well
outside the surfzone. Inside the surfzone discrepancies with the observed periods are
somewhat larger, resulting in for instance an underestimation of the mean periods
in Van Gent and Doorn (2000)b, but overall predictions for the mean periods are
reasonable for both models.
The computed wave induced setup is compared to observations from Boers (1996)
in Fig. 3.12. For the other cases the observed mean water levels were not available.
The high resolution model slightly underestimates the maximum setup near the shoreline, whereas the low-resolution model somewhat overestimates the setup on the bar
in cases A and B. However, in general, both models are able to predict the mean
water level setup due to the gradients in the radiation stress well.
3.6. Short-crested waves over 2D topography
a
10
y (m)
5
0
−5
8
13
16
20
9
14
18
21
10
15
19
22
b
c
d
−10
0
5
10
15
20
25
30
x (m)
d (m)
0.1
0.2
0.3
0.4
Figure 3.13: Depth contours at 5 cm intervals of the Dingemans et al. (1986) experiment (case
me35). Wave gauges and current meters are indicated with circle markers and crosses respectively.
The solid circles indicate selected gauges for which the wave spectrum is given in Fig. 3.16
The final test in this study is the application of SWASH to a case in the 2D basin
of Dingemans et al. (1986), but now with a submerged breakwater extending across
half the basin (see Fig. 3.13), so that in this case the situation is two-dimensional
with the waves inducing a net horizontal circulation. The selected case is of particular
interest as it includes short-crested waves, thus enabling us to ascertain how well the
HFA performs under field-type conditions. Furthermore, the observations encompass
roughly 1500 wave periods, which combined with the extensive horizontal extent of
the basin (17×13 wave lengths), implies that using high vertical resolutions is no
36
3. Wave breaking in a non-hydrostatic wave model
Figure 3.14: Three successive snapshots of the free surface taken 0.8 s apart (∼ one wave period).
The first snapshot, after roughly 31 minutes, has been randomly selected. White patches indicate
regions where the HFA is active. Depth contour lines (from 0.4 m to 0.1 m at 0.1 m intervals) depict
the location of the breakwater.
longer feasible.
The observations of Dingemans et al. (1986) were obtained in a 2D basin of
34×26.4 m2 surface area in which short-crested waves were generated along the
shorter side. A submerged bar, parallel with the wave maker, spanned half across
an otherwise flat basin (water depth was 0.4 m). Both bar and basin floor were
covered with smoothly finished concrete. Over the horizontal top of the submerged
bar, which was 2 m long in the wave direction, the still water depth was 0.1 m. The
approach slope was 1:20 and the back slope was 1:10. The side walls of the basin
were fully reflective and the back wall was a wave-absorbing gravel beach with a
1:7 slope. The incident wave spectrum was a standard JONSWAP spectrum with
significant wave height 0.097 m and peak frequency 0.80 Hz (i.e. wave length ∼2
m). The observed Hm0 varied slightly along the stations near the wave maker (by
less than 10%), and on average was approximately 0.105 m. The short-crestedness
was defined with a cosm θ-directional energy distribution at the wave maker (with
m=4, or a directional width of 25◦ Kuik et al., 1988) and mean direction normal
to the breakwater, i.e. θ0 =0◦ . The surface elevation was measured at 26 locations
(see Fig. 3.13) and currents were measured on a equidistant 9 × 9 grid (with 3 m
spacing), centred near the tip of the breakwater (x=14 m , y=0 m). It was assumed
that stationary conditions were reached after 11 minutes (Dingemans et al., 1986),
and measurements were taken for the subsequent 21 minutes.
The horizontal resolution is set to approximately 1/40th of a wave length in the
mean wave and 1/60th of a wave length in the lateral direction. This resulted in
700 × 880 × 2 mesh points with ∆x = 0.05 m and ∆y = 0.03 m. Default parameters
were used for the HFA (α = 0.6 and β = 0.3). Since the observed circulation
attains relatively high flow velocities over the crest of the bar (U ≈ 0.3 m/s, so that
U/c0 ≈ 0.3), we included the effect of an ambient current in our estimation of the
37
3.6. Short-crested waves over 2D topography
0.15
0.15
0.15
0.1
0.1
0.1
0.1
Hm0
0.15
0.05
0.05
0.05
transect a
0
0
0
0.05
transect c
transect b
10
20
0
x (m)
10
20
0
0
10
x (m)
transect a
20
0
x (m)
20
x (m)
transect c
transect b
1
10
1
transect d
1
Tm0 2
1
transect d
0
0.5
0.5
0.5
transect a
0
0
0
10
x (m)
0.5
transect c
transect b
20
transect d
0
0
10
20
0
0
x (m)
10
x (m)
20
0
10
20
x (m)
Figure 3.15: Cross-sections (see Fig. 3.13) of the significant wave height Hm0 (top panels) and mean
zero crossing period Tm02 (lower panels) in the Dingemans et al. (1986) experiment (case me35).
Comparison between measured (circle markers) and computed values (solid lines).
wave celerity (see appendix C). At the wave-maker boundary a JONSWAP spectrum
is imposed, including the observed variation in significant wave height along the
boundary. After calibration a roughness coefficient of k = 0.005 is found to give
a good correspondence between the measured and computed mean flow field. The
friction coefficient thus varies between cf = 0.005 (0.1 m depth) and cf = 0.003 (0.4
m depth) and is similar to the value of cf = 0.006 used by Chen et al. (1999) for
modelling a rip-current system. The initial time step is set to ∆t = 0.005 s and
after 660 s output is generated at a sampling frequency of 20 Hz. Variance density
spectra and derived quantities are obtained in a similar manner as in the previous
section. As the recorded signals of the surface elevation at the wave gauges are no
longer available2 , observed spectra are obtained from Ris et al. (2002).
Three successive snapshots of the free surface, overlaid with white patches to
indicate regions where the HFA is active, are depicted in Fig. 3.14. These snapshots,
in addition to others (not shown), suggest that the HFA is only active in the shallow
region near the crest of the bar. In general the HFA is activated on individual
crests approaching the bar in between the 0.2 m and 0.1 m depth contours, which as
expected roughly corresponds to the region where Hm0 /d ∼ 0.5–1. Activation starts
on separate sections of the crest, but once initiated the active region usually spreads
along the crest. The HFA then generally remains active on the front-face of the wave
until it passes the bar and enters deeper water.
For a quantitative assessment of model performance, we compare the computed
wave height Hm02 and period Tm02 with the observed values in Fig. 3.15. The predicted wave heights generally correspond well with the measurements. In accordance
with observations the wave decay is gradual in the transect (b) skirting the break2 personal
communication with M.W.Dingemans (2012)
38
3. Wave breaking in a non-hydrostatic wave model
E (cm2 /Hz)
25
gauge 8
25
gauge 13
25
20
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1.5
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0.5
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gauge 9
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gauge 14
8
E (cm2 /Hz)
gauge 16
20
0.5
0.5
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1.5
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gauge 18
8
gauge 20
0.5
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1.5
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gauge 21
8
20
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25
1.5
6
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gauge 10
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gauge 15
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E (cm2 /Hz)
25
20
0.5
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gauge 19
8
0.5
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1.5
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gauge 22
8
20
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15
0.5
1
1.5
f(Hz)
2
0.5
1
1.5
f(Hz)
2
0.5
1
1.5
f(Hz)
2
0.5
1
1.5
2
f(Hz)
Figure 3.16: Comparison between measured (circle markers) and computed (solid lines) variance
density spectra in the Dingemans et al. (1986) experiment. The mean incident wave spectrum is
indicated with a thin dashed line. Stations are ordered according to their geographic location, see
also Fig. 3.13.
water tip, whereas it is more abrupt in the transects that traverse the bar (c and
d). Furthermore, the model reproduces the slight increase in significant wave height
that occurs after breaking in section c, which is probably due to refraction, diffraction
and directional spreading. Mean wave periods are less well resolved and are generally
overestimated. This particularly occurs past the the crest of the bar, in transects b
and c. This implies that, even though the total wave energy (or wave height) is well
resolved, discrepancies occur in the spectral shape.
Close inspection of the spectra along transects (b),(c) and (d) (see Fig. 3.16) confirms this. On the forward slope (stations 8–10) and crest (stations 13–15) of the bar
computed spectra correspond well with observations whereas on the downward slope
(stations 16–18), and behind the bar (stations 20–21), the peak is slightly underestimated. Furthermore, energy densities above twice the peak frequency (f > 1.6Hz)
are somewhat lower then observed, in particular at station 21. Notwithstanding these
discrepancies, the model reproduces the primary characteristics of the wave spectra
well. Away from the breakwater tip (stations 19 and 22), where effects due to the
lateral heterogeneity of the topography are negligible, the peak is well resolved. Near
the tip the model captures the increase of the peak level of the spectrum that occurs
3.7. Discussion
39
between stations 14–21 which is likely due to refraction, diffraction and directional
spreading. Moreover, in accordance with observations the model reproduces the secondary peak at twice the peak frequency behind the bar. The development of such a
peak in shallow water is characteristic for triad wave-wave interactions (e.g. Freilich
and Guza, 1984).
Since dissipation – as observed and in the model computations – is concentrated
near the bar, locally large gradients in the wave induced momentum flux are present.
This forces a wave-induced current field the overall appearance of which, i.e. flow
directions and relative magnitudes, is primarily determined by the topography. Comparing model predictions with the observations the model faithfully reproduces the
mean flow pattern (see 3.17); the anti-clockwise circulation cell is centred near x = 20
m, y = 0 m and the high flow velocities on top of the breakwater. Quantitatively,
inspection of the flow directions and magnitudes in Figures 3.18 and 3.19 demonstrates that the correspondence with model predictions is very good. Especially the
flow magnitudes over the breakwater are well captured (transect along x = 14 m).
Even though we calibrated the friction coefficient, that this suggests that the wave
averaged momentum balance, and in particular the wave momentum flux gradient, is
well approximated. This conjecture is supported by: (1) the reasonable approximation of the wave height gradients over the bar; and (2) that the current field is well
approximated before and after the bar. These suggests that the net transfer of energy
from the wave field to the mean circulation, in addition to the energy dissipation in
the rest of the domain, is correctly represented.
3.7. Discussion
The above example demonstrates that the non-hydrostatic wave model SWASH combined with the HFA can resolve most of the relevant nearshore wave processes, including wave breaking and wave-driven circulations, in a reasonably accurate way. It
shows that for practical applications a coarse vertical resolution suffices in large parts
of the domain. Furthermore in regions where breaking dominates, which otherwise
would require high vertical resolutions, the HFA forms an effective and effient approximation. Indeed, when compared to high resolution computations the computed
integral wave parameters – as the flume experiments demonstrate – compare equally
well with observations while the computational time typically reduces by a factor
5–10 when the vertical resolution was lowered from 15 to 2 layers.
The primary disadvantage of using low vertical resolutions combined with the
HFA is that the model no longer intrinsically accounts for the transition into a turbulent bore. Instead, the HFA is initiated once the rate of surface rise exceeds a
pre-determined value α which thus is a model parameter. This in addition to the
persistence parameter β and the additional viscosity (controlled by µ). In the present
work we primarily focussed on the potential of this approach, without performing a
full sensitivity analysis for the empirical tuning coefficients. Instead, we estimated
α and β by comparison to observations and an ad-hoc choice for µ. That this leads
to reasonable results, even in a complex case, demonstrates the robustness of the
present method.
Similar results as obtained here – at least for the flume experiments – can most
40
3. Wave breaking in a non-hydrostatic wave model
Figure 3.17: Measured (right panel) and computed (left panel) wave induced current field. (Dingemans et al., 1986, case me35).
200
200
200
y=-6 m
100
θ (◦)
200
y=0 m
y=6 m
100
x=14 m
100
100
0
0
0
0
−100
−100
−100
−100
−200
−200
0
10
20
−200
0
10
x (m)
20
−200
0
10
x (m)
20
−10
0
10
y (m)
x (m)
Figure 3.18: Measured (circle markers) and computed (solid lines) current direction θ (θ in degrees,
with θ = 0◦ pointing in the x-direction) along indicated transects in the x– (first three panels) and
y–direction (last panel) in the Dingemans et al. (1986) (case me35) experiment.
y=-6 m
U (cm/s)
30
y=0 m
30
y=6 m
30
20
20
20
20
10
10
10
10
0
0
0
10
x (m)
20
0
0
10
x (m)
20
x=14 m
30
0
0
10
x (m)
20
−10
0
10
y (m)
Figure 3.19: Measured (circle markers) and computed (solid lines) current magnitude U along
indicated transects in the x– (first three panels) and y–direction (last panel) in the Dingemans
et al. (1986) (case me35) experiment.
3.8. Conclusions
41
likely also be achieved in SWASH by means of an adaptation of more conventional
breaking parametrisations, e.g. the roller or eddy viscosity models. However, the
advantage of the present approach is its simplicity. Compared to more elaborate
formulations (e.g. Cienfuegos et al., 2010; Schäffer et al., 1993) it requires less additional model parameters. Furthermore, the bulk dissipation due to wave breaking
is still accounted for using shock capturing numerics, which thus remains consistent
with the high resolution model. Most importantly the application to short-crested
waves, as demonstrated in the present study, is straightforward and does not require
additional modifications.
Finally, even when the HFA is used, the computational effort to simulate intermediate to large areas (say a harbour basin of ∼ 10×10 wave lengths) at time
scales of interest (say ∼ 2000 waves) remains formidable. The principle computational effort at each step arises from the need to solve a discrete Poisson equation to
obtain the non-hydrostatic pressure. This entails that at each step a large, sparse,
non-symmetric M × M matrix equation needs to be solved where for m × n horizontal points and k layers, M = mnk. The resulting system is solved iteratively in
SWASH, requiring roughly O(M 1.25 ) operations (for details we refer to Zijlema and
Stelling, 2005). For example, the simulations for the Dingemans et al. (1986) experiment, which were performed on the Lonestar computer at TACC (Texas Advanced
Computing Center) in Austin, USA, using 132 processors and where M = 1.2 × 106 ,
required roughly 10 hours to complete. Although – at present – this prohibits routine
application within an engineering context, it does indicate that for selected critical
scenarios application on the intermediate scale is feasible. Practical application is
then best performed in synergy with regional wave models (spectral shallow water
wave models, e.g. the SWAN3 model, Booij et al., 1999, from the same suite as
SWASH), where results obtained by the regional model can be used to determine
where and for which conditions more detailed computations are necessary.
3.8. Conclusions
In the present study we have introduced the hydrostatic front approximation as an
effective and efficient approximation to wave breaking in the non-hydrostatic phase
resolving model SWASH. It assumes a hydrostatic pressure distribution at the front
of a breaking wave once the rate of change of the free-surface exceeds a predetermined
threshold. The associated energy dissipation is accounted for by ensuring that mass
and momentum are conserved once the wave has transferred into bore-like shape.
The threshold, which is essentially a proxy for the maximum surface steepness, is estimated from experimental data. Simulations of flume experiments demonstrate that
model results correspondent well with observed wave heights and periods. Furthermore, model results, for both waves and currents, are in agreement with observations
of short-crested waves in a two-dimensional wave basin. This demonstrates that the
non-hydrostatic wave model SWASH including HFA can resolve most of the relevant
nearshore wave processes in a short-crested wave-field, including dissipation due to
wave breaking, and wave-driven circulations.
3 Simulation
WAves Nearshore, freely available at http://swanmodel.sourceforge.net/
42
3. Wave breaking in a non-hydrostatic wave model
Acknowledgments
The authors would like to extend their gratitude to Dr. Leo Holthuijsen who has had
a substantial influence on this paper. Furthermore, we thank Prof. Clint Dawson and
Dr. Casey Dietrich from the Institute for Computational Engineering and Sciences
at the university of Texas, Austin (USA), whom graciously allowed us to make use
of the computing facilities at the Texas advanced computing center.
Appendices
3.A. Turbulent stress approximations
In this section we will give the expressions used for the turbulent stresses used in the
full 3D formulation of the model. To this end we introduce the horizontal particle
velocity vector u, with components u1 and u2 directed along the horizontal x1 and
x2 (or x, y) axes and write the turbulent stress terms as
∂ui
∂uj
∂ui
τz,xi = νv
τxj ,xi = νh
+
,
(3.9)
∂z
∂xj
∂xi
∂w
∂w
τxj ,z = νh
τz,z = νv
.
(3.10)
∂xj
∂z
Here νv and νh are the turbulent viscosities related to the vertical and horizontal
exchange of momentum.
A vertical eddy viscosity is included to allow the effect of bottom stress to influence the entire water column and to increase numerical stability by enhancing the
vertical coupling between layers. At high vertical resolutions (>10 layers), the eddy
viscosity is obtained using the k − ε model (Launder and Spalding, 1974) as a closure
approximation, whereas at low resolutions a constant eddy viscosity (νv = 10−4 m2 /s)
has been used.
In addition, to allow for horizontal mixing due to sub-grid eddies in the mean
flow, a horizontal viscosity νh is included. It is estimated using a Smagorinsky-type
approximation (Smagorinsky, 1963)
v
u 2
u X
∂Uj
1 ∂Ui
2t
cur
2
νh = (Cs Ls )
2
Sij
+
.
(3.11)
Sij =
2 ∂xj
∂xi
i,j=1
Here, Cs (=0.1) is the Smagorinsky constant, U1 ,U2 are the depth-averaged horizontal
velocities and the typical length
p scale of the sub-grid eddies is related to the horizontal
mesh size (∆x, ∆y) as Ls = ∆x∆y.
3.B. Wave generating boundary conditions
To obtain the normal velocities un to the boundary at s (measured along the boundary) from a prescribed (spatially varying) variance density spectrum E(θ, f, s) the
spectrum is sampled N discrete frequencies from at ∆f intervals. In the present
work spectra were sampled from f0 = 1/2fp till fN ≤ fnyq so that fj = f0 + j∆f
and fnyq = 1/(2∆t) is the Nyquist frequency. To avoid repetition of the signal
we set ∆f = 1/D, with D the duration of the simulation. Each frequency band
43
44
3. Wave breaking in a non-hydrostatic wave model
corresponds
q to a´ long crested wave with direction θj , spatially varying amplitude
Aj (s)(= 2∆f Ej (θ, s)dθ) and random phase φj . For unidirectional waves θj is
equal to the mean wave direction θ0 (with θ = 0 in the direction normal to the boundary). For short-crested waves with a given cosm θ directional distribution (assumed
constant along the boundary), the direction of each component is drawn randomly
using the directional distribution as a probability density function. This ensures that
the wave variance along the boundary as obtained from the resulting signal is quasihomogoneous (Miles and Funke, 1989), but requires that the spectrum is sampled
using a large number of frequency components. The normal velocity at the boundary
is then given as
un (s, z, t) = ur (u, t) +
N
X
ûj cos(θj ) sin [kj cos(θj )s − 2πfj t + φj ] ,
(3.12)
j=1
where kj is the wave number; ûj (z) is the vertically varying velocity amplitude that
is related to Aj by means of linear wave theory; and ur (u, t) represents a contribution
to allow reflected long waves to leave the domain. To estimate ur (u, t) the reflected
waves are assumed to be small amplitude shallow water waves propagating perpendicular to the boundary. The surface elevation due to the reflected wave can be
estimated by subtracting the surface elevation due to the incident waves ζw from the
instantaneous surface elevation. Using a radiation condition (e.g. Vreugdenhil, 1994,
Section 5.2)
p and the continuity equation then gives the reflected wave contribution
as ur = g/d(ζ(y, t) − ζw (y, t)). This approximation is based on the assumption
that high frequency energy, for which this approach is not accurate, has dissipated
in the domain to the extent that it may be neglected when it leaves the domain.
3.C. Wave breaking initiation criterion
The criterion to initiate the HFA, which is similar to the criterion used in Kennedy
et al. (2000), applies equally well to one and two-dimensional situations. Only when
a strong ambient current is present do we need to take wave direction into account.
To substantiate this, and to show how to take the presence of an ambient current
into account, we note that for a wave of (nearly) constant form propagating (in the
direction θw ) over a flat bottom in a region with a uniform current Ū (with direction
θc ) we have
∂ζ
+ c0 + Ū · ∇ζ = 0,
(3.13)
∂t
where c0 = c0 [cos θw , sin θw ] is the wave celerity in the absence of a current and
Ū = Ū [cos θc , sin θc ]. Let us define a coordinate system with m parallel to the wave
crest, and n perpendicular to the wave crest. Upon assuming that along the wave
crest there are no gradients in the surface (∂m ζ 0 = 0), equation (3.13) transforms
into
∂ζ 0
∂ζ 0
+ (c0 + ∆c)
= 0,
(3.14)
∂t
∂n
where ζ 0 (m, n, t) is the free surface and ∆c = Ū cos(∆θ) is the correction to the wave
celerity due to the presence of a current at a mutual angle ∆θ = θw − θc with the
3.C. Wave breaking initiation criterion
45
local wave direction. Assuming that dissipation initiates when the surface slope |∂n ζ|
exceeds α, we thus consider that waves are breaking when ∂t ζ 0 is larger then α(c+∆c).
When U/c0 1, which is often the case in the nearshore, the correction ∆c to c0
due to the currents can be neglected.√Moreover, as the waves are in shallow water to
a good approximation
we have c0 ≈ gh, and obtain the relatively simple breaking
√
criterion ∂t ζ > α gh (since ∂t ζ = ∂t ζ 0 at any location). In such cases the trigger
is thus independent of wave direction, simplifying its application in two-dimensional
situations.
However, neglecting ∆c when Ū /c0 is O(1) can lead to premature (|∆θ| < 90◦ ,
following current) or delayed (|∆θ| > 90◦ , opposing current) initiation of breaking.
Furthermore, as it also affects the persistence criterion, it can also prolong (following
currents) or shorten (opposing currents) the duration of a breaking event. Hence,
in such cases it is desirable to include an estimate for ∆c. Hereto we assume that
c0 + ∆c > 0 (thus excluding wave blocking) so that ∂n ζ 0 = − sgn(∂t ζ)|∇ζ| in which
case ∆c can be estimated by
∆c = − sgn(∂t ζ)
∇ζ
· Ū ,
|∇ζ|
(3.15)
which, for robustness, is only calculated when |∇ζ| > 10−3 and set ∆c = 0 elsewhere. Finally, in order to estimate the mean current Ū from the instantaneous
depth average current U , we use a relaxation model
∂ Ū
1
=−
Ū − U
∂t
Tr
(3.16)
where Tr is a relaxation time scale which is taken to be in the order of five to ten
peak periods.
4
Non-hydrostatic modelling of
surfzone wave dynamics
Non-hydrostatic models such as Surface WAves till SHore (SWASH) resolve
many of the relevant physics in coastal wave propagation such as dispersion,
shoaling, refraction, dissipation and nonlinearity. However, for efficiency, they
assume a single-valued surface and therefore do not resolve some aspects of
breaking waves such as wave overturning, turbulence generation, and air entrainment. To study the ability of such models to represent nonlinear wave
dynamics and statistics in a dissipative surfzone, we compare simulations with
SWASH to flume observations of random, unidirectional waves, incident on a
1:30 planar beach. The experimental data includes a wide variation in the incident wave fields, so that model performance can be studied over a large range
of wave conditions. Our results show that, without specific calibration, the
model accurately predicts second-order bulk parameters such as wave height
and period, the details of the spectral evolution, and higher-order statistics,
such as skewness and asymmetry of the waves. Monte Carlo simulations show
that the model can capture the principal features of the wave probability density function in the surfzone, and that the spectral distribution of dissipation
in SWASH is proportional to the frequency squared, which is consistent with
observations reported by earlier studies. These results show that relatively
efficient non-hydrostatic models such as SWASH can be successfully used to
parametrise surfzone wave processes.
4.1. Introduction
In the nearshore region and surfzone, ocean waves undergo a dramatic transformation mostly due to nonlinear wave-wave interactions and breaker dissipation. These
This chapter has been published as: Smit, P.B., Janssen, T.T., Holthuijsen, L.H. and Smith,
J.M., 2014b Non-hydrostatic modelling of surf zone wave dynamics. Coast. Eng., 83, 36–48.
47
48
4. Non-hydrostatic modelling of surfzone wave dynamics
dynamics play a central role in nearshore circulation and transport processes, e.g. by
controlling wave setup (e.g. Longuet-Higgins and Stewart, 1964), driving nearshore
currents (e.g. Longuet-Higgins and Stewart, 1964; Longuet-Higgins, 1970; Svendsen,
1984; MacMahan et al., 2006), and causing morphodynamic evolution (e.g. Hoefel
and Elgar, 2003). Understanding these processes and the development of predictive
models is important for both scientific research and engineering in the coastal zone.
Since most coastal and coastline processes take place on much longer scales than
that of the individual waves, predictive models are generally used to estimate wave
statistics (e.g. significant wave height, mean period) and variations therein. However, modelling waves statistics in the nearshore is complicated both by the strong
influence of nonlinear processes and an incomplete understanding of dissipation of
wave energy in shoaling and breaking waves.
Stochastic (or phase-averaged) wave models for coastal applications are usually
based on some form of energy (or action) balance equation (e.g. The WAMDI Group,
1988; Komen et al., 1994; Wise Group, 2007), which assumes that the wave field is
(and remains) quasi-homogeneous and near-Gaussian. However, due to nonlinearity,
surfzone wave statistics are generally strongly non-Gaussian, and apart from variance,
higher cumulants (e.g. skewness and kurtosis) are required to completely describe
the wave statistics. This poses high demands on the model representation of the
nonlinear and nonconservative dynamics. In particular, for statistical models, the
representation of nonlinearity invariably requires some form of closure approximation
and involves evolution equations for higher-order correlations, both of which generally
render the model considerably more complicated and computationally intensive (e.g.
Herbers and Burton, 1997; Herbers et al., 2003; Janssen, 2006; Smit and Janssen,
2013b).
Deterministic (and phase-resolving) wave models can generally incorporate nonlinearity more easily, and naturally include full coupling to the wave-induced circulations. However, although a model based on the Reynolds Averaged Navier-Stokes
(RANS) equations can model surface wave dynamics in great detail, and resolve very
small scales of motion (e.g. Torres-Freyermuth et al., 2007), the computational cost
can become prohibitive, even for small-scale applications. For wave modelling of
most coastal-scale applications, and in particular for coastal engineering, more approximate but efficient models, such as so-called non-hydrostatic models or models
based on a Boussinesq approximation are generally more useable. Boussinesq-type
wave models have evolved from weakly nonlinear and weakly dispersive models (see
Peregrine, 1967), to nearly fully dispersive and highly nonlinear models (e.g. Nwogu,
1993; Wei et al., 1995; Madsen et al., 2002), at the expense however, of much increased complexity of the underlying model equations and numerical implementations. In contrast, non-hydrostatic models are essentially numerical implementations
of the basic conservation equations for mass and momentum (e.g. Stelling and Zijlema, 2003; Yamazaki et al., 2009; Ma et al., 2012), which can be directly used for
wave propagation problems if sufficient spatial resolution, in particular in the vertical, is provided. As a consequence, such models are relatively simple, and grid
resolution can be readily adapted to a particular application and allow propagation
of waves from deep to shallow water.
4.2. Model description
49
However, such non-hydrostatic models (and Boussinesq models for that matter),
do not model all aspects of surfzone waves. In particular, for the sake of efficiency,
these models assume a single-valued representation of the free surface in the horizontal plane, which implies that processes such as overturning, air entrainment, and
wave-generated turbulence are not resolved. Instead, integral properties of breaking waves (including energy dissipation rate) are estimated by treating the breaking
wave as a discontinuity in the flow variables (free surface, velocities) and maintaining momentum (and mass) conservation across the discontinuity (Smit et al., 2013).
Although this gives good results for the second-order bulk statistics (such as the significant wave height), it is not clear whether such an integral approach, where some
of the details of breaking waves are treated as sub-grid processes, can actually resolve
the nonlinear and dissipative processes in the surfzone, and thus predict the details
of the spectral evolution and nonlinear statistics there.
In the present work we set out to study these issues by comparing simulations
with the non-hydrostatic model Surface WAves till SHore (SWASH, Zijlema et al.,
2011b) to flume observations of random waves over a 1:30 planar beach (see Smith
(2004)). The motivation behind this work is to assess whether an efficient nonhydrostatic model such as SWASH can be a viable tool to study surfzone dynamics
and accurately capture the statistics of strongly nonlinear and breaking waves.
In §2 we present the model equations, numerical approximations, and breaker
modelling in SWASH. The laboratory experiments and specific model settings are
described in §3 and we present our results (model-data comparison) in §4. We discuss
and sum up our principal findings and their implications in sections 5 and 6.
4.2. Model description
The non-hydrostatic model SWASH (Zijlema et al., 2011b), is an implementation
of the Reynolds-averaged Navier-Stokes equations for a incompressible, constantdensity fluid with a free surface. In the present work we use this model to study
one-dimensional wave propagation in a flume. In Cartesian coordinates, with x and
z the horizontal and vertical coordinate respectively, and with z measured up from
the still-water level z0 , the governing equations can be written as
1 ∂(ph + pnh ) ∂τxz
∂τxx
∂u ∂uu ∂wu
+
+
=−
+
+
,
∂t
∂x
∂z
ρ
∂x
∂z
∂x
∂w ∂uw ∂ww
1 ∂pnh
∂τzz
∂τzx
+
+
=−
+
+
,
∂t
∂x
∂z
ρ ∂z
∂z
∂x
∂u ∂w
+
= 0,
∂x
∂z
(4.1)
(4.2)
(4.3)
where t is time, u(x, z, t) and w(x, z, t) are the horizontal and vertical velocities,
respectively, and ρ is the (constant) density. Further, the hydrostatic pressure ph =
ρg(ζ − z) (with g being gravitational acceleration) and pnh represents the the nonhydrostatic pressure contribution. The turbulent stresses ταβ are obtained from
a turbulent viscosity approximation (e.g. τxz = ν∂z u, with ν the kinematic eddyviscocity) using a standard k− model closure approximation (Launder and Spalding,
50
4. Non-hydrostatic modelling of surfzone wave dynamics
1974). The water column is vertically restricted by the (time-varying) free-surface
z = ζ(x, t) and immobile bottom z = −d(x). Here d is the still water depth and the
location of the free surface ζ is found from continuity, expressed as
ˆ ζ
∂ζ
∂
u dz = 0.
(4.4)
+
∂t
∂x −d
Equations (4.1)–(4.4) are solved for constant pressure at the free-surface (i.e. p = 0),
while accounting for the kinematic free-surface and bottom boundary conditions,
w=
∂ζ
∂ζ
+u
∂t
∂x
(z = ζ) ,
and
w = −u
∂d
∂x
(z = −d).
(4.5)
At the up-wave boundary, waves are generated by prescribing the horizontal velocity
u(z, t) at that location, whereas opposite of the wavemaker the boundary is formed
by the moving shoreline.
Since the objective is to study surfzone wave dynamics, we need to include motions
in the infragravity band, which are generated in the shoaling process and released
during breaking. These low-frequency components are much longer than the primary
waves and much more strongly affected by bottom friction. To incorporate this, we
include a bottom stress at the bottom boundary assuming a logarithmic velocity
profile (Launder and Spalding, 1974) and a typical roughness height dr .
Numerical approximations
For non-breaking waves these equations describe nonlinear shoaling, and thus include
the energy transfers across different length scales in the wave spectrum due to triad
and higher-order nonlinearities. The accuracy with which these processes are represented in numerical models such as SWASH, depends on the numerical methods used
to approximate the governing equations, and the spatial (horizontal and vertical) and
temporal resolutions used in the simulation. Moreover, when extending such models
to the surfzone, careful attention must be paid to the conservation properties (in
particular of momentum) of the numerical method (see Zijlema et al. (2011b) for a
more general discussion of the numerical methods used in SWASH).
To accurately resolve wave motion in a phase-resolving model, the horizontal
resolution ∆x must be a fraction of the shortest wave length L that needs to be
resolved, i.e. L/∆x = O(10). Similarly, the timestep ∆t is usually a fraction of
the shortest wave period T , i.e. T /∆t = O(10). To allow for accurate, undamped
propagation over long distances, SWASH uses a staggered horizontal grid combined
with a second-order (in space and time), explicit, finite-difference method that is
neutrally stable (no numerical damping) for small amplitude (linear theory) waves.
The vertical resolution, and numerical approximations of the vertical pressure gradient, determines how well thepmodel approximates the linear dispersion relation for
surface gravity waves ω(k) = gk tanh(kd), which strongly affects propagation and
dispersive characteristics of the wave field.
Non-hydrostatic models often make use of a boundary-fitted vertical grid that
divides the instantaneous water depth h (= ζ + d) in a constant number of layers N ,
with variable vertical mesh-size ∆z = h/N . The required number of layers N then
4.2. Model description
51
generally depends on the deepest parts of the domain, and increases with the vertical
variability of the wave-induced velocity profile, for free-surface waves represented by
the relative depth kd. Hence, in shallow water (kd 1), a coarse resolution generally
suffices, whereas in deep water (kd 1) a vertical resolution similar to the horizontal
resolution is required, ∆z/∆x = O(1) (at least near the surface). Traditionally, this
has severely limited the application of these models to wave propagation, as the number of layers required to accurately capture dispersion (say a relative error smaller
than 1%) at high kd (O(10)) can become very large (N > 20), resulting in excessive
computational times. These constraints are significantly relaxed in SWASH due to
the use of an edge-based vertical grid combined with a compact numerical scheme
for the approximation of the vertical pressure-gradient (i.e., the Keller-box scheme;
see Lam and Simpson, 1976). Hence, for N = 6, the maximum relative error of the
numerical dispersion relation ωn , compared with ω (or alternatively the wave celerity), remains below 0.1% for kd < 40 (see appendix A). This numerical efficiency is
important in the surfzone where very short waves exchange energy through nonlinear
interactions, thus potentially imposing high demands on both horizontal and vertical
resolutions.
Wave breaking approximations
Since SWASH assumes the free surface to be a single-valued function ζ(x, t), it cannot
model processes such as overturning, air-entrainment, and the production of waveinduced turbulence after the wave form becomes unstable and breaking is initiated.
However, if we are not principally interested in the details of these fine-scale processes
associated with breaking waves, this approach can be very useful (and efficient) to
capture the larger-scale wave and current dynamics in the surfzone (e.g. Ma et al.,
2012; Smit et al., 2013).
In a non-hydrostatic model like SWASH, a breaking wave develops into a discontinuity (or hydraulic jump), which is similar to that predicted by the (non-dispersive)
shallow-water equations. If momentum conservation is maintained across the discontinuity by employing shock-capturing numerical methods (in SWASH this is done
using the method by Stelling and Duinmeijer, 2003) then, in analogy to hydraulic
jump dynamics, energy is dissipated at a rate proportional to the cube of the bore
height. In this way the entire turbulent front is essentially reduced to a sub-grid
phenomenon.
However, when waves approach breaking, wave steepening introduces strong vertical gradients in the flow variables, which requires a very high vertical resolution locally to accurately capture the bore dynamics using shock-capturing numerics. If the
resolution is insufficient, velocities are generally underestimated, and the initiation
of breaking is delayed (Smit et al., 2013). We avoid using a fine vertical resolution by
using a hydrostatic front approximation (HFA), inspired by similar developments in
Boussinesq models (e.g. Tonelli and Petti, 2012; Tissier et al., 2012), and described in
detail in Smit et al. (2013). The HFA method implemented here considers
the (non√
dimensional) rate of change of the free surface elevation, ζt0 = ∂t ζ/ gh and forces
the pressure at the wave front to be hydrostatic, i.e. pnh = 0, once this exceeds a
certain threshold (i.e. ζt0 > α). Effectively, near the bore front, the model reduces to
52
0
1
2
3 4 5 6 7 8 9 10
30 cm
z=0
62 cm
Wavemaker
4. Non-hydrostatic modelling of surfzone wave dynamics
1
30
5 x
1
10
15
20
25
30
35
40
x [m]
Figure 4.1: Layout of the flume experimental setup by Smith (2004). The diamond markers at
still-water level (z = 0) indicate measurement locations.
the nonlinear shallow-water equations, causing the front to assume a characteristic
sawtooth-like shape, and dissipate wave energy at a rate consistent with that of an
hydraulic jump. Once breaking is initiated, the breaking threshold α is reduced to
β (with β < α) in neighboring points to allow the breaker to more easily persist and
produce more realistic breaker dynamics.
The threshold value, α, at which the HFA is initiated, depends on the wave
dynamics leading up to breaking, and thus depends on the number of layers used
in the model. For two layers, Smit et al. (2013) found α = 0.6 by estimating the
maximum value of ζt0 at the observed breaking point in the experiments of Ting
and Kirby (1994). Recalibration for 6 layers (as will be used here) using the same
analysis method and data set as in Smit et al. (2013), resulted in α = 1. This value
produces similar model-data agreement for the Ting data set with the six-layer model
(not shown) as found in Smit et al. (2013) for a two-layer model. The persistence
parameter β was found much less sensitive to the number of layers used, and is
therefore set to the value of β = 0.3 (suggested by Smit et al., 2013).
Lastly, to prevent generation of high-frequency noise in the wave profile due to
the discrete activation of the HFA, some additional horizontal viscosity is introduced,
of the form
∂U 2
.
ν = Lmix (4.6)
∂x Here Lmix is a typical horizontal mixing length, set as a fraction µ of the local depth
(i.e. Lmix = µh), and U is the depth-averaged horizontal velocity. For 0.25 ≤ µ < 3,
this parameter supresses the generation of such noise, but has a marginal effect on
the bulk energy dissipation. To minimize its influence, we set µ = 0.25 in the present
study.
4.3. Experiment and model setup
We compare SWASH simulations to a series of laboratory observations performed by
Smith (2004) at the US Army Engineer Research and Development Center, Coastal
and Hydraulics Laboratory. The experiments were performed in a 0.45 m wide, 45.7
m long and 0.9 m deep flume (see Fig. 4.1), with glass walls and a smooth concrete
bottom. Waves are generated by a horizontally moving piston-type wave generator
53
4.3. Experiment and model setup
Table 4.1: Instrument location x (m) and still-water depth d (m) during experiments (see Smith,
2004).
x
d
1
6.7
0.62
2
28.7
0.3
3
30.3
0.24
4
31.3
0.21
5
32.2
0.18
6
33
0.15
7
33.9
0.12
8
34.8
0.09
9
35.4
0.07
10
35.9
0.05
located at one end of the flume (at x = 0 m), and propagate over a horizontal section
onto a 1:30 beach that starts at x = 19.3 m.
The surface elevation was measured at 10 locations (see Fig. 4.1), and sampled
at 5 Hz for a duration of 550 s. Data acquisition was started after 50 s, thus allowing
for spin-up time and prevent transient effects in the observations. The wave gauge
closest to the wavemaker, located on the horizontal section, is used as a boundary
condition for the model. The other gauges are placed on the slope, from a depth of
0.3 m to 0.05 m, at intervals varying from 0.5 to 1.6 m (see Table 4.1).
Irregular waves were generated using either a single- or double-peaked spectrum.
The individual spectral peaks are described by a parametric TMA shape (Bouws
et al., 1985), and the high-frequency peak of the double-peaked spectra always contained two-thirds of the total energy (see Smith (2004)). In our analysis, to avoid
ambiguity, we refer to the low-frequency peak as the primary peak, and to the highfrequency peak (if present) as the secondary peak; additional peaks that arise in the
spectrum due to nonlinear interactions are referred to as (higher) harmonic peaks,
or simply harmonics.
The experiments consider a range of incident wave conditions (see Table 4.2 for
an overview) by varying the TMA peakedness parameter γ, considering single and
(1)
double-peaked spectra, varying the peak period of the low frequency peak Tp =
(1)
1/fp , and including two different significant wave heights Hm0 . A total of 31
experiments, including 30 different wave conditions, were performed (case 11b was
repeated once). If a secondary peak is included in the incident wave spectrum, it is
(2)
generated at fp = 1 Hz. The Iribarren number, ξ (definition in caption to Table )
was below 0.5 (see Table 4.2), which suggests that the surfzone for the cases consisted
mostly of spilling breakers (see e.g. Holthuijsen, 2007). Since the wavemaker was not
equipped with either second-order control or reflection compensation, some spurious
second-order wave motion was generated, and long waves radiated back from the
beach were re-reflected into the flume.
(1)
(2)
refers to the primary peak period, whereas Tp
refers to the period
p´
k2 E(k) dk.
(2)
2
(1)
/2π the deep-water wave length. Listed values for kp d are calculated using the primary peak period and d = 0.62 m. All other
Hm0 /Lp , with Hm0 the significant wave height as listed, tan α = 1/30 the bottom slope and
Tp (s)
2.5
2.0
1.75
1.5
1.25
2.5
2.0
1.75
1.5
1.25
1.0
2.5
2.0
1.75
1.5
1.25
Case
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
(1)
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
(2)
TP (s)
9.0
9.0
9.0
9.0
9.0
9.0
9.0
9.0
9.0
9.0
9.0
6.0
6.0
6.0
6.0
6.0
Hm0 (m)
(a)
3.3∗
3.3
3.3
3.3∗
3.3
3.3∗
3.3∗
-
γ
(b)
20
20∗
20∗
20
20
20
20∗
20∗
20
20
20∗
20
20
20
20
20∗
(c)
100∗
100
100
100∗
100
100∗
100∗
-
Wave steepness
(a)
(b)
(c)
0.13 0.12 0.11
0.12
0.13 0.13 0.11
0.13
0.14 0.13 0.12
0.07 0.05 0.05
0.04
0.1 0.07 0.05
0.06
0.13 0.11 0.08
0.15 0.12 0.13
0.08
0.08
0.08
0.09
0.09
-
0.68
0.88
1.00
1.3
1.7
0.68
0.88
1.00
1.3
1.7
2.5
0.68
0.88
1.00
1.3
1.7
(1)
kp d
0.34
0.27
0.24
0.20
0.17
0.32
0.26
0.23
0.20
0.17
0.14
0.41
0.32
0.28
0.25
0.20
ξ
variables are defined as in the main text. Cases considered individually in figures or discussed in the main text are marked with an asterisk.
Lp = g Tp
The Iribarren number ξ is defined as ξ = tan α/
p
of the secondary peak (if applicable). Wave steepness is defined from the wave number spectrum E(k), with k the wave number, as
Table 4.2: Wave conditions considered in the experiments by Smith (2004). Tp
54
4. Non-hydrostatic modelling of surfzone wave dynamics
55
4.3. Experiment and model setup
The model domain extends from the first observation point at x1 = 6.7 m, to 45 m.
To ensure that the model accurately describes the wave characteristics over a wide
frequency range of at least 0 ≤ f ≤ 4fp (with fp the peak frequency1 ), which includes
the high-frequency tail, the horizontal resolution is set at ∆x = 0.01 m. Such high
resolution is needed to resolve the higher frequencies with wave lengths 0.1 to 0.4 m,
and to accurately propagate these components over O(102 ) wave lengths. Moreover,
to correctly describe linear dispersion at high kd values (maximum value of kd at
4fp is approximately 40), the vertical resolution is set to 6 layers, so that within the
prescribed frequency range the relative error in wave celerity stays below 0.1%. From
a nonlinear perspective, this also ensures that the resonant mismatch is well predicted
for interactions below 4fp (see Appendix A). Time integration was performed for a
duration of 600 s with a time step of ∆t = 0.005 s (so that 4fp ∆t ≥ 50 in all
cases). The bottom roughness height dr was set to a value of 4.5 × 10−4 m, which
is representative for smooth concrete (e.g. Chow, 1959). Each individual simulation
(each case) takes approximately 30 minutes to complete on a 6-core Intel Xeon 3.2
GHz CPU desktop computer.
Wave forcing
The model is forced with the measured free-surface records at the first wave gauge,
which implies that the model-data comparisons are essentially deterministic. To
directly force the model using the data, we relate the time-varying, layer-averaged
horizontal velocity un (t) for each vertical layer n (with n = 1 . . . N , and n = 1
denoting the bottom layer), required to drive the model at the up-wave boundary,
to the measured free-surface elevation at that location. Therefore we consider the
Fourier sum of the free-surface elevation and horizontal velocity,
ζ(t) =
J
X
ζbj exp [i2πfj t],
j=−J
u(z, t) =
J
X
u
bj (z) exp [i2πfj t],
(4.7)
j=−J
where fj = j∆f , ∆f = 1/T and i2 = −1. The horizontal velocity (using linear wave
theory) is found from
sinh [kj (z + d)]
(4.8)
u
bj (z) = 2πfj ζbj
sinh(kj d)
where kj is the wavenumber related to fj by the linear dispersion relation. The
layer-averaged velocity un (z, t) is then obtained by integration over the nth layer,
and subsequently dividing by the layer thickness,
un (z, t) =
J ˆ zn
1 X
u
bj (z, t) exp [i2πfj t] dz,
∆z
zn−1
(4.9)
j=−J
where zn = n∆z − d and ∆z = d/N .
1 Note
(1)
(2)
that fp and fp specifically refer to the locations of the incident waves, whereas we will
use fp to refer to the local peak frequency. We will use the same convention for peak wave periods
and wave numbers.
56
4. Non-hydrostatic modelling of surfzone wave dynamics
Although presumably, due to the lack of second-order control, some spurious
and second-order low-frequency motion is generated at the wavemaker, most of the
low-frequency energy present at the offshore gauge consists of low-frequency energy
generated in the breaking process through subharmonic triad interactions, reflected
off the beach, and subsequently re-reflected off the wavemaker. To model this consistently, the model is forced with the high-pass filtered observed time series (for
(1)
f > fp /2) at the offshore gauge (gauge 1), and the offshore boundary is set to be
fully reflective (to mimic the reflecting wavemaker). In this way, the low-frequency
motion is thus not forced into the model at the wavemaker, but allowed to develop
in the surfzone and reflect back into the model domain (as happens in the flume).
4.4. Results
H m0 (m)
6a / 1a
8b / 3b
(b)
11b / 16b
(c)
(d)
0.1
0.1
0.1
0.1
0.05
0.05
0.05
0.05
0
T m02 (s)
2
28 30 32 34 36
(e)
0
2
28 30 32 34 36
(f )
0
2
28 30 32 34 36
(g)
0
2
1.5
1.5
1.5
1.5
1
1
1
1
0.5
Sk
2
28 30 32 34 36
(i)
1
0
0
As
6c / 1c
(a)
0.5
2
28 30 32 34 36
(j)
1
28 30 32 34 36
(m)
0
0
0.5
2
28 30 32 34 36
(k)
1
28 30 32 34 36
(n)
0
0
0.5
2
28 30 32 34 36
(o)
0
0
−0.5
−0.5
−0.5
−1
−1
−1
−1
28 30 32 34 36
x [m]
−1.5
28 30 32 34 36
x [m]
−1.5
(h)
28 30 32 34 36
(l)
1
−0.5
−1.5
28 30 32 34 36
28 30 32 34 36
x [m]
−1.5
28 30 32 34 36
(p)
28 30 32 34 36
x [m]
Figure 4.2: Comparison between modeled (lines) and observed (symbols) values of significant wave
height (panels a–d), mean period (e–h), skewness (i–l) and asymmetry (m–p). In each column a
single-peaked (black solid line/circles) and double-peaked (red dashed line/diamonds) case with the
(1)
same Hm0 Tp , and γ are shown (the only difference between these cases is the single- or doublepeakedness of the spectrum). Case numbers (single/double) are indicated above each column.
57
4.4. Results
To asses overall model performance we first consider several bulk parameters.
√
In particular,
wave height Hm0 = 4 m0 and mean period
p we consider significant
´ n
Tm02 =
m2 /m0 , where mn = f E(f ) df , and higher-order bulk statistics of
skewness Sk and asymmetry As, which are defined as
Sk =
hζ 3 i
,
hζ 2 i3/2
As =
hζh3 i
.
hζ 2 i3/2
(4.10)
Here ζh is the imaginary part of the Hilbert transform of ζ (from which the mean
contribution is subtracted) and h. . . i denotes a time average.
The significant wave height and mean period are second-order bulk statistics
which are a measure of the total amount of energy, and a measure of period, respectively. The mean period Tm02 also provides some insight in the distribution of wave
energy across frequencies. Skewness and asymmetry are third-order bulk statistics,
measuring the wave asymmetry, around a horizontal and vertical plane, respectively.
Positive skewness is associated with steeper, higher peaks and flatter troughs (e.g.
second-order Stokes waves), and negative asymmetry is associated with the forward
pitching, saw-tooth like appearance of waves in the surfzone (Elgar and Guza, 1985).
Since for a Gaussian wave field both skewness and asymmetry are zero (as are all
cumulants higher than the second), the comparison between observed and modeled
skewness and asymmetry values measures the accuracy of the nonlinear dynamics in
the model, and - in the surfzone - the interplay between nonlinear and dissipative
processes (Chen et al., 1997).
All spectral estimates (for calculations and observations) are obtained from detrended and windowed Fourier transforms of 18.3 s length segments with 50% overlap,
which are subsequently ensemble averaged to yield estimates for E(f ) with a resolution of ∆f = 0.055 Hz and approximately 110 degrees of freedom.
The spatial evolution of the significant wave height and the mean period in the
surfzone is generally well captured by the model as illustrated by the eight cases
shown in Fig. 4.2. These results are representative and similar results were found
for the other cases (not shown). It can be seen that the location where intense
breaking starts (abrupt decrease in wave height), and the spatial variations in mean
period through the surfzone, are accurately resolved by the model. Only at gauge 8
(x = 34.8 m third observation point from the right), the model systematically underpredicts the observed wave heights. Variations in Tm02 are, apart from dissipation,
also strongly affected by nonlinearity. In the shoaling region, i.e. outside the surfzone
proper, Tm02 is reduced by the development of higher harmonics in the wave spectrum
(to be treated below, see Fig. 4.4), whereas inside the surfzone short waves are rapidly
dissipated and nonlinearity drives the generation of low-frequency infragravity waves,
thus resulting in the observed (and modeled) increase in Tm02 .
On the horizontal section, the waves are weakly nonlinear, with low asymmetry
and skewness values, generally consistent with Stokes second-order waves. On the
slope, skewness increases as the waves become more ’peaked’, and asymmetry takes
on larger negative values, indicative of the pitch-forward shapes developing as the
wave approach breaking (or are breaking). Near the shore, the magnitude of skewness
and asymmetry generally reduces resulting in statistics that are closer to Gaussian.
58
4. Non-hydrostatic modelling of surfzone wave dynamics
The variability of these third-order statistics is quite accurately reproduced by the
model (see Fig. 4.2). For the narrow-band case, 6c, there is somewhat less good correspondence between observed and modeled surfzone wave asymmetry than for the
other cases (see Fig. 4.2n, black line/markers). This is likely due to the hydrostatic
bore approximation which generates near-vertical fronts in breaking waves, whereas
in reality wave-induced turbulence would stabilize the front toward a more moderate
slope of the face of the breaking wave (e.g. Madsen and Svendsen, 1983). This exaggeration of wave asymmetry by the hydrostatic bore approximation, mostly affects
narrow-band waves; in wider-band wave fields (e.g. case 6a shown in panel m of Fig.
4.2), which we are more likely to encounter on natural beaches, the intermediate steep
bores ride on a background of irregular smooth waves, which apparently smoothes
the statistics, and produces model results in very close agreement with observations
(see Fig. 4.2).
Considering all 31 cases, the comparison between observed and modeled bulk
statistics shows excellent agreement (see Fig. 4.3), with R2 values larger than 0.9.
The agreement is best for the significant wave height and period. For the wave
heights, there is however a clustering of points that lie below the main diagonal.
This cluster is associated with observations made by gauge 8, which (as seen before
in Fig. 4.3a) recorded consistently slightly higher values than model predictions for
all cases and wave conditions, and which stands out relative to other (surrounding)
observations (see Fig. 4.3a). Moreover, the fact that there are no such differences
in the results for Tm02 , the skewness or asymmetry, led us to believe that these
differences are due to a slight gauge calibration inconsistency (experimental error),
rather than a systematic model error.
The higher-order statistics of skewness and asymmetry are in excellent agreement, but have slightly more scatter. Skewness is generally slightly under-predicted,
whereas the modeled asymmetry is slightly more negative than observed. However,
these higher-order statistics strongly depend on both nonlinearity and a correct representation of wave dissipation, and the good agreement suggest that the model
captures these processes very well.
Evolution of wave spectra
Apart from dissipation and linear shoaling effects, the enhancement of skewness,
asymmetry, and the evolution of Tm02 in the surfzone (see Fig. 4.2), indicate that
redistribution of energy due to nonlinear triad interactions plays an important role
in the wave evolution. To investigate this in more detail we consider the evolution of
observed and modeled spectra.
Harmonic amplification due to triad interactions becomes more pronounced in
shallow water since the interactions approach resonance. The development of such
higher harmonic contributions is most pronounced for narrow-band waves where the
primary and harmonic peaks are well defined, with limited phase mixing. For in(1)
stance, for case 6c (kp d=0.68, γ = 100) some harmonic amplification already occurs
on the flat (see Fig. 4.4a), and at gauge 2 we can clearly distinguish – in both the
model and the observations – the first three harmonics of the peak (0.4 Hz), located
at 0.8, 1.2 and 1.6 Hz respectively (see Fig. 4.4a). In the surfzone (Fig. 4.4b–c),
59
4.4. Results
2
(a)
(b)
1.5
H m0
T m02
0.1
1
0.05
0.5
2
R 2 = 0.97
R = 0.97
0
0
0.05
0.1
0
0.15
′
H m0
2
0.5
0.5
(c)
1
′
T m02
2
0
As
1
−0.5
−1
0.5
R 2 = 0.93
0
1.5
(d)
1.5
Sk
0
0
0.5
1
Sk′
1.5
2
R 2 = 0.94
−1.5
−1.5
−1
−0.5
As′
0
0.5
Figure 4.3: Scatter plots of observed (primed variables) versus computed values for: (a) significant
wave height, (b) mean period, (c) skewness and (d) asymmetry. The solid grey line indicates oneto-one correspondence, whereas the dashed line is the best fit regression line (R2 values indicated
in each panel). Red crosses correspond to values obtained from observations at gauge 8.
the combined effects of three-wave interactions and dissipation due to wave breaking
strongly attenuates the spectral peaks, so that in the inner surfzone only the primary
peak and sub-harmonics remain. Qualitatively, the picture is similar for other cases
although for higher-frequency incident waves (e.g. case 11c, Fig. 4.4d–f) and more
broad-banded wave fields (e.g. case 1a, Fig. 4.4j–l) the harmonic development is less
pronounced.
There are several sources of long wave motion in the flume. In the shoaling
process, three-wave interactions amplify bound long-wave components (here roughly
defined as components with frequencies f < 0.5fp ), which are subsequently released
in the breaking process (e.g. Janssen et al., 2003; Battjes et al., 2004). Most of the
energy contained in these frequencies subsequently reflects at the beach and radiates
back out through the surfzone as free long waves. In the field such components would
continue to propagate offshore (unless refractively trapped), but in the flume they
are almost completely re-reflected by the wavemaker. Although some low-frequency
energy is associated with second-order bound waves, and some spurious wave mo-
60
4. Non-hydrostatic modelling of surfzone wave dynamics
(1)
(1)
f /fp
1 2 3 4 5 6 7 8 9
E [cm2 /Hz]
2
10
(a)
0
10
−2
0
1
E [cm2 /Hz]
10
2
2
3
0
−2
1
0
10
2
1 2 3 4 5 6 7 8 9
(g)
0
10
2
10
0
10
−2
0
2
10
0
10
3
(f )
0
case 11c
−4
10
2
4
0
2
4
2
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
10
(h)
(i)
0
10
−2
10
case 1c
−4
−4
10
10
2
4
0
2
4
0
2
4
2
2
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
10
10
(j)
(k)
(l)
0
0
10
10
−2
−2
10
−2
10
−4
−4
0
1
4
−2
10
−4
10
2
2
10
−2
10
3
2
10
−2
0
−4
10
4
0
(e)
10
2
E [cm2 /Hz]
2
2
2
−4
(c)
0
−2
case 6c
10
4
0
1 2 3 4 5 6 7 8 9
10
10
10
2
10
10
−4
10
(d)
−4
E [cm2 /Hz]
0
10
4
0
10
10
(b)
−2
2
10
1 2 3 4 5 6 7 8 9
10
−4
10
2
10
f /fp
10
10
10
(1)
f /fp
2
f [Hz]
10
4
0
10
case 1a
2
f [Hz]
−4
10
4
0
2
f [Hz]
4
Figure 4.4: Comparison between the observed (dashed red lines) and computed (solid black lines)
energy density spectra for four different cases (ordered from top to bottom) at gauge 2 (left panels),
gauge 6 (center panels), and and gauge 10 (right panels). The lower horizontal axis indicates the
(1)
absolute frequency, whereas the upper axis in each panel indicates the relative frequency f /fp .
Incident wave spectrum at the up-wave boundary is indicated by the thin grey line and the dash-dot
(1)
line indicates the upper frequency (fp /2) of the infra-gravity (ig) range.
4.4. Results
61
tion due to the first-order wavemaker control, by far the largest contribution to the
low-frequency energy is associated with these reflected and re-reflected free-wave
components generated in the shoaling and breaking process by nonlinear three-wave
interactions (see e.g. Battjes et al., 2004). The level of agreement between model
and observations suggests that the nonlinear surfzone wave-dynamics are captured
accurately. The remaining differences between model and observations could be due
to errors in the estimate for bottom friction, which is one of the primary mechanisms
(in addition to breaking of long waves, e.g. Van Dongeren et al., 2007) by which the
longer waves loose energy.
One aspect that makes the present experimental dataset (Smith, 2004) particularly interesting, is the fact that it includes several cases with double-peaked spectra.
These cases show a dramatic spectral evolution across the flume. Similar to previous
laboratory observations by Smith and Vincent (1992), the secondary peak, which
initially contains the bulk of the energy, diminishes rapidly on the slope, whereas
the low frequency peak appears to grow at its expense. This behavior, which is also
largely due to triad interactions, is clearly seen in the observed spectra for the double peaked cases, both for narrow or wide-band initial spectra, as exemplified here
in cases 1c and 1a, respectively (see Fig. 4.4, panels g–l). The model also reproduces
this reduction of the secondary peak (at 1 Hz), which is initially still visible at gauge
2 (panels g/j), but has virtually disappeared at gauges 6 (panels h/k) and 10 (panels
i/l).
Energy Fluxes
The effects of nonlinearity and dissipation on the various spectral regions can be
further illustrated by considering the spatial evolution of the linear energy flux,
F (f, x) = cg E(f, x) at distinct frequencies. Since for linear and conservative wave
propagation the wave energy flux is constant, changes in this flux indicate where
nonlinear effects and dissipation are present. To reduce the sensitivity of the results
to the details of the spectral analysis (e.g. such as the frequency resolution), we
consider the normalized flux integrated over a finite frequency band ∆fn ,
´
F (f, x) df
∆fn
,
(4.11)
∆Fn (x) =
Ftot (x1 )
´
where we normalized with the total energy flux Ftot (x) = F (f, x) df at the up–
wave boundary (located at x1 ). The frequency band ∆fn is defined as a narrow
frequency band around the nth harmonic frequency, i.e. ∆fn is defined as the interval
(1)
(1)
0.95(n + 1)fp < f < 1.05(n + 1)fp . In the following we consider the integrated
energy flux for the primary peaks (∆F0 ), and their respective first (∆F1 ) and second
harmonics (∆F2 ).
On the flat and at the start of the slope, the model shows near constant energy
fluxes (see Fig. 4.5), indicating that there is very little dissipation or nonlinear transfers before the waves reach the slope. On the slope, the energy flux for the primary
peak reduces, in part due to dissipation, and in part due to transfers to higher and
lower frequencies, consistent with the increase in energy flux at the harmonic peaks
at roughly the same location.
62
4. Non-hydrostatic modelling of surfzone wave dynamics
Case 7b
1
(a)
∆F0
Case 11a
1
(b)
0.5
Case 2b
1
(c)
0.5
0.5
Peak
∆F1
0
0.2
20
30
0.02
10
(e)
0.1
10
(g)
20
30
0
0.02
0.01
0
0
0.2
0.1
0
∆F2
10
(d)
20
30
x (m )
0
30
10
(h)
20
10
(f )
20
30
10
(i)
20
30
10
20
30
x (m )
0.1
30
0
0.02
2nd harmonic
10
0
0.2
1st harmonic
0.01
10
20
20
30
x (m )
0.01
0
Figure 4.5: Comparison between the observed (symbols) and computed (lines) energy flux ∆Fn
(defined in (4.11)) contained in a frequency band around the peak (panels a–c), the first harmonic
(panels d–f) and the second harmonic (panels g–i). Each column represents a different case (indicated on top). In the upper panels (a–c) the normalized total flux (= Ftot (x)/Ftot (x1 )) is included
(red lines/symbols) for reference.
At the start of the slope (x = 19 m, see Fig. 4.1), dissipation is still weak as the
total energy flux remains constant, so that for relatively long, narrow-band waves
(1)
(e.g. case 7b, with kp d = 0.88, γ = 20) the amplification of the higher harmonics
can be clearly distinguished (panels a,d, and g in Fig. 4.5). In case of relatively high(1)
frequency, broad-banded waves (e.g. case 11a, with kp d = 2.5, γ = 3.3) such amplification is less pronounced, as the waves are in relatively deeper water and three-wave
interactions therefore further from resonance. Moreover, for broad-banded waves,
each frequency participates in more triads, thus broadening the resulting secondary
peaks and resulting in less distinct spectral features than for narrow-band waves.
Further up the slope, for higher-frequency, broader-band cases, amplification generally occurs only in the very shallow part of the flume (e.g. panels b,e, and h in Fig.
4.5). For both broad and narrow-banded cases, once dissipation becomes dominant,
the energy flux around the peak and its harmonics is rapidly reduced. Overall, observed and modeled evolution of the primary and harmonic energy bands are in very
good agreement.
The difference between the evolution of a single and double peaked spectrum is
illustrated by comparing case 7b with case 2b, which differ only in their spectral
shape. The evolution of the total energy flux, and that of the primary peak is
relatively similar (viz. panel a and c in Fig. 4.5). However, for 20 < x < 30 no
amplification occurs at the secondary peak (see Fig. 4.5f), despite the fact that in
(2)
(1)
this case it coincides with the 1st harmonic of the primary peak (i.e. fp = 2fp ).
63
4.4. Results
1
0.3
(a)
0.25
0.8
(b)
0.05
0.2
(c)
0.04
∆F 2
∆F 1
∆F 0
0.6
0.15
0.03
0.4
0.2
R 2 = 0.98
0.5
∆F 0′
0.1
0.02
0.05
0.01
R 2 = 0.97
1
0.1 0.2
∆F 1′
R 2 = 0.97
0.3
0.02 0.04 0.06
∆F 2′
Figure 4.6: Comparison between the observed (∆Fn0 ) and computed (∆Fn ) normalized energy fluxes
at the primary peak (left panel), and its first (middle panel) and second (right panel) harmonic.
The grey line indicates one-to-one correspondence, whereas the dashed line is the best fit linear
regression line (R2 values given in panels).
In contrast, even before the onset of strong dissipation (which occurs for x > 30,
see Fig. 4.5c), the energy flux around the secondary peak is attenuated, presumably
due to nonlinear transfer to other frequencies. The regions where amplification and
(1)
attenuation of the energy flux around the 2nd harmonic (3fp ) occur, and the relative
magnitudes of the respective fluxes, are again very similar (viz. Fig. 4.5g and 4.5i).
The overall performance of the model with regard to ∆Fn , including contributions
up to the second harmonic, is summarized in the scatter plots in Fig. 4.6a–c. The
absence of outliers, and the high R2 values, confirms that the model captures the
development up to and including the second harmonic very well. It should be noted
that R2 values in this case are inflated due to the large range within the data, where
the highest and lowest ∆Fn can differ a factor factor 102 ; even for large relative
errors, small values in the data contribute little to the total variance of the error, but
significantly to the observed variance, thus somewhat inflating values of R2 .
Shape of the high-frequency tail in the surfzone
The wave spectra presented in Fig. 4.4 demonstrate that the strong amplification
of the spectral energies in the high-frequency tail (f > 2.5fp ), by at least an order
of magnitude when compared to the incident spectra, is reproduced by the model.
Furthermore, in all cases the model relaxes the high frequency tail into a fairly
featureless shape in the surfzone. The development of such an asymptotic form in
shallow water has been previously observed in both field and laboratory observations,
which led Smith and Vincent (2003) to propose a universal parametric tail in the
surfzone. Expressed in terms of the wave number spectrum E(k) (obtained form a
linear transformation), their parametric tail consists of two ranges: the Toba range
(kd > 1), and the Zakharov range (2kp d ≤ kd ≤ 1, with kp the peak wave number).
64
4. Non-hydrostatic modelling of surfzone wave dynamics
0
−2
n =-5/2
−2.5
−1
−3
n
n
−3.5
−4
n=−2
−2
−3
−4.5
−5
n=-4/3
(a) Toba range
0.1
0.2
d (m)
0.3
−4
(b) Zakharov range
0.1
0.2
d (m)
0.3
Figure 4.7: The slope n of best fit line to the log of the wavenumber spectrum E(k) within the
Toba (kd > 1) and Zakharov (kp < kd < 1) range. In both panels the mean n as calculated from
observations (symbols) and from model results (line) is plotted as a function of depth. The region
that lies within one standard deviation from the mean is indicated by bars for the observations,
and as a shaded area for the model simulations. The arrow indicates that the wave direction in the
figure is towards shallow water.
For the Toba range, Smith and Vincent (2003) found that the spectrum scales as
E(k) ∼ k −5/2 . For the Zakharov range they found that the spectrum scales as
E(k) ∼ k −4/3 , similar to the tail in shallow water proposed by Zakharov (1999). As
noted by Kaihatu et al. (2007), referring to these ranges as the Toba and Zakharov
range is somewhat questionable for strongly nonlinear, breaking surfzone waves. Toba
proposed his asymptotic form based on observations of wind-driven, deep-water waves
(which obviously does not apply here), and the strongly nonlinear conditions found
in the surfzone are outside the range where the theory by Zakharov (1999) can
be applied. Regardless of whether the eponymous naming of these spectral ranges
is entirely justified, the observations considered by Kaihatu et al. (2007) generally
supported the results of Smith (2004) at the edge of the surfzone, although they also
found that the spectrum has a clear E(k) ∼ k −2 asymptote in very shallow water.
Because in the present data set the Zakharov range only exists at the most shoreward location (elsewhere 2kp d > 1), we extended – following Kaihatu et al. (2007) –
this range to kp d ≤ kd ≤ 1. For each case we transformed the spectra to wavenumber space using linear theory, and determined the exponent, n, of the shape function
αk n that best fitted the data from the slope of the linear regression line between
log[E(k)] and log(k). This analysis was performed for the Zakharov and Toba ranges
separately, yielding a value for n in either range at each separate gauge for all of the
31 experiments. Subsequently, at each gauge the mean value and the standard deviation were determined, yielding a mean value nZak and nToba , and standard deviation
σZak and σToba , for the Zakharov and Toba range respectively, the results of which
are shown in Fig. 4.7.
In very shallow water (d < 0.15 m), the mean value of nToba compares well with
the earlier reported values (see Fig. 4.7a); it tends to -5/2 with relatively small
65
4.4. Results
scatter around the mean in both the observed or computed data (σToba < 0.1 for
0.05 < d < 0.12 m). For increasing depth, values for ntoba obtained from computed
spectra are at some variance with the observations (d > 0.2 m), with increased scatter
around the mean (σToba > 0.3), but generally larger scatter in the computed than
in the observed results. This larger scatter in the spectral shape for the Toba range
in the deeper parts of the flume, and also the lesser agreement with observations,
are likely due to being in the vicinity to the wavemaker, so that the spectrum is
still adjusting from the artificial wavemaker input, and has not yet developed into
an equilibrium shape. A better correspondence with the observed tail most likely
requires that the numerical wavemaker exactly reproduces the high-frequency tail,
including possible nonlinear contributions.
For the Zakharov range, the agreement with observations, both in the mean
trend and in the scatter, is excellent (Fig. 4.7b), with good correlation (R2 = 0.97)
between results obtained from observations and computations (not shown). However,
although the averages in the surfzone are close to theoretical value of nzak = −4/3
(they are -1.3 and -1.41 for observations and model, respectively), neither the model
nor observations appear to converge to this value when nearing the shoreline. In
general, the scatter is quite large, but near the shore-line (between a depth of 0.05
to 0.1 m) σZak this scatter is significantly reduced, both in the observations (from 1.3
to 0.3) and computations (from 1.3 to 0.1). As d → 0 the model seems to converge
to nzak = −1.9 with σZak = 0.08, in accordance with E(k) ∼ k −2 asymptote in very
shallow water as found by Kaihatu et al. (2007).
Nonlinear energy transfer
In shallow water, the dominant nonlinear energy transfer is associated with nearresonant triad interactions, the strength of which generally depend the coupling coefficient, energy content of the components involved in the interaction, and how close
the interaction is to resonance. To compare transfer rates inferred from observations
and model results therefore requires the evaluation of a nonlinear source term Snl
that depends on the bipectrum B(f1 , f2 , x), and which has the general form (e.g.
Herbers and Burton, 1997; Becq-Girard et al., 1999; Janssen, 2006)
ˆ
Snl (f ) = 4
0
f
C(f 0 , f − f 0 )Bim (f 0 , f − f 0 )df 0
ˆ ∞
−8
C(f + f 0 , −f 0 )Bim (f, f 0 )df 0 , (4.12)
0
where C is a (real) coupling coefficient and Bim denotes the imaginary part of the
bispectrum. The bispectrum captures the phase relationship between a triad of
waves f1 , f2 and f3 = f1 + f2 , and describes the third-order statistics of the wave
field. For instance, skewness and asymmetry can be obtained from the real or
imaginary part of the integral over the bispectrum, respectively, i.e. Sk + iAs =
˜
3/2
B(f1 , f2 ) df1 df2 /m0 . The source term Snl (f ) includes all sum and difference contributions to frequency f . For second-order Stokes waves on a horizontal bottom
Bim (f1 , f2 ) = 0, and no energy transfer occurs.
66
4. Non-hydrostatic modelling of surfzone wave dynamics
(1)
(1)
f /fp
S nl [cm2 /Hz/s]
2
1
−2
3
2
4
−2
0
2
4
−4
0
2
1
2
4
3
−4
0
2
4
−2
1
2
4
3
4
−2
(e) gauge 10
−4
2
0
(d)
4
4
(c)
0
−2
3
−2
2
0
2
f [Hz]
2
(b) gauge 5
1 2 3 4 5 6
0
1
0
(a)
−4
2
f /fp
0
2
S nl [cm2 /Hz/s]
2
1 2 3 4 5 6
0
−4
(1)
f /fp
0
2
f [Hz]
(f )
4
−4
0
2
f [Hz]
4
Figure 4.8: The nonlinear source term Snl computed at two different gauges (ordered from top to
bottom) for case 3b (left panels), case 10a (middle panels), and case 10c (right panels). The source
term is computed from the bispectrum obtained from the observations (markers) and computations
(solid black line). In addition the linear flux gradient Fx (dashed red line) estimated from model
results is included. The lower horizontal axis indicates the absolute frequency, whereas the upper
(1)
axis in each panel indicates the relative frequency f /fp .
Since the observation points are in shallow water (kp d < 1) we use Boussinesq
theory (Herbers and Burton, 1997; Herbers et al., 2000) to evaluate Snl , so that
C(f1 , f2 ) = 2π(f1 + f2 )/d. The use of Boussinesq theory only affects the interaction
kernel, which will be the same for both model results and observations (the bispectrum is estimated from observations and model results directly), so that the
inter-comparison is consistent. We approximate the integrals over the bispectrum
using the trapezoidal rule with an upper-frequency of 6 Hz in the second term on
the RHS of Eq. (4.12). Bispectra are obtained using the same spectral analysis of
observed and computed time series as before.
Results for three representative cases, including a wide (10a), narrowband (10c),
and double-peaked (3b) incident spectrum, are shown in Fig. 4.8. The transfer rates
computed from the model simulation results are in good agreement with the values
computed from the observations, with overall better correspondence in the energetic
part of the spectrum (f < 2fp ), and slightly worse correspondence in the tail. The
nonlinear interactions transfer energy from the primary peak(s) of the spectrum to
the higher (and lower) frequencies. For instance, in case 10c at gauge 5 (Fig. 4.8c),
energy is transferred from the spectrum peak (f = 0.8 Hz), to its first (f = 1.6 Hz),
4.5. Discussion
67
second (f = 2.4 Hz) and (presumably) higher harmonics. In the case of more broadbanded irregular waves, the great number of interactions that take place result in a
more uniform shape of Snl at the frequencies above the peak (e.g. Fig. 4.8a/c). In
the inner surfzone (gauge 10), energy is mostly transferred from 0.5fp ≤ f ≤ 2fp ,
toward higher frequencies (f > 2fp ), regardless of the width of the incident spectrum
(viz. case 10a with case 10c in Fig. 4.8e/f, respectively).
In the absence of dissipation Snl approximately balances with Fx , and large differences between Snl and Fx are therefore indicative of dissipation. The gauge spacing
(varying from 0.5 to 1.6 m) is too large to obtain estimates for Fx from the observations directly in the highly dynamic surfzone. However, such estimates can be
readily obtained from the more finely spaced model results using second-order finite
differences,
F (f, x + ∆x) − F (f, x − ∆x)
(4.13)
Fx (f, x) ≈
2∆x
where ∆x is the computational mesh-size. Near the peak of the spectrum (see Fig. 4.8
panels a-c) changes in Fx are approximately balanced by Snl , suggesting that for f <
2fp the evolution of the spectrum is primarily determined by nonlinearity whereas the
dissipation rates – even in the inner surfzone – remains relatively small in this spectral
range (e.g. panels d—f). For f > 2fp the evolution of Fx is minimal, suggesting that
dissipation and nonlinear transfers are nearly in balance. This (model) behavior
shows that energy is not dissipated in the energetic range of the spectrum, but
instead is transferred from the peak to the higher frequencies, where it is subsequently
dissipated, consistent with observations by other researchers (e.g. Herbers et al.,
2000).
4.5. Discussion
The results presented thus far are quasi-deterministic, in the sense that all results
were derived from a single, relatively short realization using deterministic boundary
conditions. The differences between model and observations can therefore be ascribed to modelling inaccuracies (we ignore measurement noise or errors, except for
the possible gauge calibration issue at gauge 8) and are not due to uncertainty in
statistical estimates, which would be inevitable for a finite-length time series. Since
the deterministic comparison showed excellent agreement between model and data
for all relevant surfzone wave processes studied here, we expand on this here to study
the surfzone statistics and dissipation characteristics from Monte Carlo simulations
with the model. The Monte Carlo simulations consist of 10 realizations for each case
considered. Each realization is forced with randomized initial conditions (from the
observed spectrum with added random phases). This extends the total number of
data points N tot from 1.2 × 104 to 1.2 × 105 .
For these 2DV simulations (1D wave propagation), the complete Monte Carlo
simulation (10 realizations) takes about 300 minutes per case on a 6-core Intel Xeon
3.2 GHz CPU. When considering 2D wave propagation, computational times are
considerably higher but feasible on larger multi-processor systems, in particular since
even a more coarse vertical grid (i.e. 2 vertical layers) can still yield reliable estimates
68
4. Non-hydrostatic modelling of surfzone wave dynamics
for the bulk parameters and wave spectrum up to three times the peak frequency
(see e.g. Smit et al., 2013).
Free-surface statistics
0
10
0
case 10b gauge 2
−1
−1
10
0
case 10b gauge 6
10
case 10b gauge 10
−1
10
p(ζ ′ )
10
10
−2
10
−3
10
−2
10
−3
−2
10
−3
10
10
−5−4−3−2−1 0 1 2 3 4 5 −5−4−3−2−1 0 1 2 3 4 5 −5−4−3−2−1 0 1 2 3 4 5
ζ′
ζ′
ζ′
√
Figure 4.9: Probability density functions (pdf) for the normalized free surface ζ 0 = ζ/ m0 estimated
from the observations (circles) and from the Monte Carlo simulations (solid black line), compared
with a Gaussian distribution (dashed red line) and a two-term Gram-Charlier (Eq. (4.14)) series
(grey line) using the skewness values obtained from the Monte Carlo simulations.
The probability density function for Case 10b, characterized by relatively short
(1)
incident waves (kp d = 1.7) with a fairly narrow-band spectrum (γ = 20), is shown
in Fig. 4.9. The theoretical distribution in that figure is a two-term Gram-Charlier
expansion (Longuet-Higgins, 1963), which can be expressed as
P (ζ 0 , Sk)
1
p(ζ 0 ) = √
exp − (ζ 0 )2 .
(4.14)
2
2π
For P = 1 this is the normalized Gaussian distribution, whereas for the nonlinear
pdf the polynomial P (ζ 0 , Sk) depends only on variance and skewness (for details of
P , see Longuet-Higgins, 1963).
At the edge of the surfzone (left panel of Fig. 4.9) the pdf is strongly skewed,
which increases at the locations further inside the surfzone (centre and right panel
of Fig. 4.9). The deviation from the Gaussian distribution shows that the waves are
nonlinear, with relatively sharp and tall peaks, and shallow and elongated troughs,
which shows that a nonlinear wave model is required to reliably estimate surfzone
statistics. Although a direct comparison with the observed pdf for |ζ 0 | > 3 is difficult
due to the relatively short time series (and thus relatively low data density to populate
the tails of the distribution), the agreement between the observations, Monte Carlo
simulations, and the theoretical distribution is very good. Comparisons for other
cases showed similar agreement (not shown). This shows that nonlinear effects are
important for surfzone statistics, but that knowledge of the lowest two moments,
variance and skewness, suffices to capture the principal characteristic of the pdf. As
shown in the model results, SWASH can accurately model and predict these moments.
69
4.5. Discussion
D ′ /E ′
Spectral distribution of dissipation
60
60
60
40
40
40
20
20
20
2.0f 2
0
Gauge 6
0
2
4
f (Hz)
6
1.6f 2
0
Gauge 8
0
2
4
f (Hz)
6
1.5f 2
0
Gauge 10
0
2
4
6
f (Hz)
Figure 4.10: Mean (over all 31 cases) of the distribution function scaled with the normalized energy,
D0 (f )/E 0 (solid black line), compared with a least squares fit αf 2 shape (dashed line). The patched
area indicates the region within one standard deviation of the mean.
Energy dissipation due to wave breaking is arguably the most important, and yet
the least understood process in the surfzone (e.g. Peregrine, 1983). In general, bulk
dissipation rates are reasonably well estimated by semi-empirical formulations based
on a bore analogy (e.g. see Salmon et al., 2014b, for an extensive overview), however, spectral models require a spectral distribution of the dissipation, which is not
available from theory or observations. As a consequence, spectral breaker dissipation
functions D(f ) are invariably expressed´ as D(f ) = D0 D0 (f ), where D0 (f ) is an unknown distribution function, for which D0 (f )df = 1 and D0 is the bulk dissipation.
Eldeberky (1996) assumed that the distribution function D0 (f ) is proportional to the
normalized spectrum, D0 (f ) = E 0 (f ), where E 0 (f ) = E(f )/m0 , so that dissipation
is stronger in the more energetic ranges of the spectrum. Other studies found that
better results for third-order bulk statistics are obtained by weighting the dissipation
function toward higher frequencies using D0 (f ) = f 2 E 0 (f ) (e.g. Kirby and Kaihatu,
1996; Chen et al., 1997), or a linear combination of the two shapes (Mase and Kirby,
1992). Weighting the dissipation toward higher frequencies is consistent with the
observation that dissipation takes place mostly at the higher frequencies but is not
very strong in the energy-carrying ranges (see e.g. Section 4.4 of the present work or
e.g. Herbers et al., 2000).
In SWASH, we do not impose any distribution as wave breaking is handled in
the time domain, but the resulting spectral signature can be estimated from the
model results. Assuming that dissipation is the term responsible for closing the
balance between Fx and Snl , i.e. D(f, x) = Fx (f, x) − Snl (f, x), D0 (f ) can be directly
estimated from the Monte Carlo simulation. In Fig. 4.10, we show the mean and
standard deviation from the 31 Monte Carlo simulations for the ratio D0 (f )/E 0 (f ),
which measures the dissipation rate at each frequency relative to the amount of
energy. In the presence of strong dissipation in the inner surfzone, the mean of D0 /E 0
strongly emphasizes higher frequencies (see Fig. 4.10). At deeper, more offshore,
70
4. Non-hydrostatic modelling of surfzone wave dynamics
locations (i.e. gauge 2–5) dissipation is weak, so that D0 (f )/E 0 (f ) is more variable
and noisy, and no clear pattern can be discerned (not shown). For f < 2 Hz the
computed ratio can become slightly negative, which would seem to imply ’positive
dissipation’, but is likely caused by small relative errors in the estimation of Snl
and Fx , which dominate the balance in this region (although dissipation is small
here, it results as the difference of two large, but opposing, terms). Overall, the
model-predicted dissipation rate D0 (f ) agrees fairly well with the earlier suggested
f 2 weighting (see Fig. 4.10). Since in SWASH D0 (f ) is not prescribed but rather
follows from the Monte Carlo data directly, this result appears to corroborate the
use of an f 2 weighting for breaking dissipation. More generally, this also illustrates
how a fairly detailed non-hydrostatic model can be a valuable tool in developing or
testing parameterisations uses in operational (e.g. Tolman, 1991; Booij et al., 1999)
or research spectral wave models.
4.6. Conclusions
In the present work we considered modelling of wave dynamics in the surfzone using the non-hydrostatic model SWASH. Detailed comparison to flume observations
shows that a relatively efficient model such as SWASH, in which the details of the
breaking process such as overturning and turbulence are not resolved, can reliably
predict surfzone (non-Gaussian) wave statistics. Our results show that even without calibration or fine-tuning, the model accurately predicts both second-order bulk
parameters such as wave height and period, higher-order statistics, including skewness and asymmetry of the waves, and the details of the spectral evolution (up to
10 times the peak frequency). The generally excellent agreement between the model
results and the observations, demonstrates that the model accurately captures the
macro-effects of the dominant nonlinear and dissipative processes in the surfzone,
in particular the triad wave-wave interactions and the dissipation due to breaking.
These results show that for the predominantly spilling breaker conditions considered,
a non-hydrostatic model with a single-valued representation of the free surface, can
provide an accurate presentation of the wave statistics in the surfzone. Hence, the
representation of the free-surface as a single-valued function appears not to prevent
an accurate representation of the wave statistics in the surfzone, at least in case of
spilling breakers. The pdf of the free-surface, estimated by Monte Carlo simulations,
compares well with a theoretical nonlinear pdf that depends on the first two moments,
variance and skewness, both of which can be reliably estimated from SWASH simulations. From the energy balance we derived that the wave dissipation in SWASH is
proportional to a frequency-squared distribution function, which is consistent with
observations in other studies. Although the present study considers one-dimensional
wave propagation in a flume, we note that triad nonlinear and dissipation processes
are not fundamentally different for 2D surfzones with short-crested waves, so that our
conclusions are probably also valid under such conditions. Overall, the findings of
this study suggest that SWASH is a viable tool for modelling wave and wave-driven
dynamics in a nonlinear, dissipative surfzone.
4.6. Conclusions
71
Acknowledgments
This research is supported by the U.S. Office of Naval Research (Littoral Geosciences
and Optics Program and Physical Oceanography Program) and by the National
Oceanographic Partnership Program.
Appendix
4.A. Frequency dispersion in SWASH
(b)
3
2
5
4
Relative error ∆ω (%)
Relative error ω (%)
(a)
1
7
6
0.5
0
0
2
10
10
10
3
2
4
7
6
5
0
0
2
10
kd
5
10
kd
Figure 4.11: Absolute relative error (in percent) in (a) the angular frequency ω(k) , and (b) resonant
mismatch for self-self interactions ∆ω(k, k) = 2ω(k) − ω(2k), when using the dispersion relation
obtained from an N layer system, ωN compared to using the linear dispersion relation, ω, as a
function of the relative depth kd. Depicted are the error curves for N between 2 to 7, which shows
that for N = 6 (thick solid line) the error remains below 0.1 and 5 percent (horizontal dashed lines),
respectively, for kd < 40 (vertical dashed line).
To a large extent, the magnitude of the nonlinear transfers is determined by the
resonant mismatch, ∆ω(k1 , k2 ) = ω(k1 ) + ω(k2 ) − ω(k1 + k2 ) among a triad of waves
with wavenumber k1 , k2 , k3 so that k3 = k1 +k2 . A correct representation of nonlinear
interactions therefore not only requires a good approximation to linear dispersion of
the three interacting components, but also of their resonant mismatch. In general, for
non-hydrostatic models, the horizontal scales are well resolved, whereas the vertical
resolution is relatively coarse. Hence the accuracy of the linear dispersive behavior
of the numerical model is largely determined by the vertical resolution, and can
therefore be estimated from the semi-discrete system obtained by discretization of
the vertical. In the case of SWASH, the linear, semi-discrete system corresponding to
Eq. (4.1) to (4.4), for N vertical layers, is given by application of an edge-based grid
in the vertical combined with the Keller-box approximation for the vertical pressure
73
74
4. Non-hydrostatic modelling of surfzone wave dynamics
gradients,
∂ζ
1 ∂pnh
1 ∂pnh
n−1
n
+
+
∂t
∂x 2 ∂x
2 ∂x
pnh − pnh
∂wn
∂wn−1
n−1
+
+2 n
∂t
∂t
∆z
∂un− 12
wn − wn−1
+
∂x
∆z
N
X
∂un− 21
∂ζ
+ ∆z
∂t
∂x
n=1
∂un− 12
+g
= 0,
n = 1..N,
(4.15)
= 0,
n = 1..N,
(4.16)
= 0,
n = 1..N,
(4.17)
= 0.
(4.18)
Here, the nth pressure and vertical velocity components, pn and wn (with n ∈
{0 . . . N }) are located on the layer interface, whereas the layer averaged velocity un− 12
(with n ∈ {1 . . . N }) are located in the central plane, where {. . . }n = {. . . }(x, zn , t)
with zn = n∆z − d, with ∆z = d/N . From the dynamic boundary condition at the
free surface we have pnh
N = 0. Moreover, if we assume a horizontal bottom, we obtain
from the kinematic boundary condition at the bed that w0 = 0. If we then consider
the initial value problem on an infinite domain, we can associate ∂t ζ → −iω ζ̂(k, ω),
and ∂x → ik ζ̂(k, ω), and similarly for un− 21 , wn , to obtain for each mode ω, k a
(3N + 1) × (3N + 1) linear system A(k, ω) in terms of the Fourier amplitudes
h
i
ŷ = û 21 , . . . , ûN − 12 , ŵ1 , . . . , ŵN , p̂0 , . . . , p̂N −1 , ζ̂ ,
(4.19)
so that Aŷ = 0. For this system to have solutions other than the trivial solution, A
must be singular, or Det(A) = 0, resulting in a polynomial relating ω and k, from
which the linear dispersion relation ωN (k) that corresponds to the N layer system
can be determined. As the calculations quickly become involved for N > 1, this is
done using a symbolic algebra system2 .
From ωN thus obtained, it can be seen that, assuming the horizontal scales are
well resolved, the number of layers required to accurately model dispersion mainly
depends on the maximum relative depth within the frequency range of interest (see
Fig. 4.11a). In the present work this corresponds to waves with f < 4 Hz, or kd < 40.
By using 6 layers, the maximum relative error for kd < 40 remains below 0.1%. A
similar analysis can be made for the frequency mismatch ∆ω(k1 , k2 ), when considering the mismatch for the self-self interactions, ∆(k, k), for different values of kd
(see Fig. 4.11b). This shows that for 6 layers, the relative error is less than 6% for
kd < 40.
2 implemented
using the Matlab Symbolic Toolbox.
5
The evolution of
inhomogeneous wave
statistics through a variable
medium
The interaction of ocean waves with variable currents and topography in coastal
areas can result in inhomogeneous statistics due to coherent interferences, which
affect wave-driven circulation and transport processes. Stochastic wave models, invariably based on some form of the radiative transfer equation (or action
balance), do not account for these effects. In the present work we develop
and discuss a generalization of the radiative transfer equation that includes the
effects of coherent interferences on wave statistics. Using multiple scales, we
approximate the transport equation for the (complete) second-order wave correlation matrix. The resulting model transports the coupled-mode spectrum (a
form of the Wigner distribution) and accounts for the generation and propagation of coherent interferences in a variable medium. We validate the model
through comparison to analytic solutions and laboratory observations, discuss
the differences with the radiative transfer equation and the limitations of our
approximation, and illustrate its ability to resolve coherent interference structures in wave fields such as those typically found in refractive focal zones and
around obstacles.
This chapter has been published as: Smit, P.B. and Janssen, T.T., 2013b The Evolution of Inhomogeneous Wave Statistics through a Variable Medium. J. Phys. Oceanogr., 43, 1741–1758.
75
76
5. The evolution of inhomogeneous wave statistics
5.1. Introduction
The dynamics and statistics of ocean waves are important e.g. for upper ocean dynamics (e.g. Craik and Leibovich, 1976; Smith, 2006; Aiki and Greatbatch, 2011),
air-sea interaction (e.g. Janssen, 2009), ocean circulation (e.g. McWilliams and Restrepo, 1999), and wave-driven circulation and transport processes (e.g. Hoefel and
Elgar, 2003; Svendsen, 2006). Modern stochastic wave models are routinely applied to a wide range of oceanic scales, both in open-ocean applications and the
nearshore, and either as stand-alone wave prediction models, or as part of coupled
ocean-atmosphere models for global circulation and climate studies (e.g. The WAMDI
Group, 1988; Tolman, 1991; Komen et al., 1994; Booij et al., 1999; Wise Group, 2007).
These so-called third-generation wave models are invariably based on some form of
the radiative transfer equation (or action balance)
∂t E + cx · ∇x E + ck · ∇k E = S,
(5.1)
which describes the evolution of the variance (or action) density spectrum E(k, x, t)
through time t, geographical space x, and wavenumber space k, with the transport
velocities cx and ck , respectively, and augmented with (parametrised) source terms
S(k, x, t) to account for non-conservative and nonlinear processes.
Continuing development of these models is generally through improvements of the
source term parameterisations on the right side of equation (5.1), but the left side, the
radiative transfer equation (RTE), has not changed since the early development of
these models (e.g. The WAMDI Group, 1988; Komen et al., 1994; Wise Group, 2007).
The RTE transports wave variance density through a slowly varying medium such
that wave energy (or action) is conserved, while assuming that the wave field is (and
remains) quasi-homogeneous and near-Gaussian. In the open ocean, where medium
variations are generally very weak, and wave statistics evolve principally through
the action of wind, dissipation (whitecapping) and third-order nonlinear effects, the
assumptions of homogeneity and Gaussianity are often easily met. However, on
continental shelves and in coastal regions, where wave fields travel through shallower
water, and medium variations are stronger (both currents and topography), the wave
field can develop and maintain inhomogeneities that strongly affect the wave statistics
(e.g. Janssen et al., 2008; Janssen and Herbers, 2009). For instance, the refraction
over coastal topography or currents (e.g. Berkhoff et al., 1982; Vincent and Briggs,
1989; Magne et al., 2007; Janssen et al., 2008), or diffraction around obstacles such as
breakwaters, reefs, or headlands (e.g. Penney and Price, 1952), can result in relatively
fast variations in wave statistics due to coherent wave interference patterns. The
effect of such coherent structures on the wave statistics are not accounted for by the
RTE (Vincent and Briggs, 1989; O’Reilly and Guza, 1991).
To account for the effects of coherent interferences on the wave statistics, more
general transport models for second-order wave statistics were developed in other
fields, such as optics and quantum mechanics (e.g. Wigner, 1932; Bremmer, 1972;
Bastiaans, 1979; Cohen, 2010). For ocean waves such models were developed for
special cases, including narrow-band waves (e.g. Alber, 1978) and forward-scattered
waves through a weakly two-dimensional medium (e.g. Janssen et al., 2008). In the
present work we apply the ideas developed in optics (e.g. Bremmer, 1972; Bastiaans,
5.2. Evolution of correlators
77
1979; Cohen, 2010), to derive a more general transport model for ocean wave statistics
in the presence of caustics and coherent interferences, which includes the RTE as a
special case.
Thereto we derive a general transport equation for the second-order correlation
matrix for linear waves in a slowly varying medium, and – using multiple scales –
derive a consistent, quasi-coherent approximation that includes coherent interferences
(§5.2). In §5.3, to illustrate the accuracy of our approximations and the differences
with the RTE, we compare a numerical implementation of the model to an analytic
solution for the evolution of coherent Gaussian wave packets, and compare model
simulations to observations of random wave propagation over a two-dimensional shoal
(Vincent and Briggs, 1989). We discuss (§5.4) the spectral distribution function
(the coupled-mode spectrum), the limits of the approximation, wave diffraction in
the quasi-coherent approximation, and show that the quasi-coherent approximation
includes earlier results as special cases (e.g. Alber, 1978).
5.2. Evolution of correlators
To study the generation and propagation of coherent structures in random ocean
wave fields propagating through a variable medium, we consider the statistics of
the free-surface elevation η(x, t), represented as a zero-mean random wave variable
and a function of the horizontal coordinates x = (x, y) and time t. We define a
complex variable ζ, such that η = Re{ζ} , its real and imaginary parts form a
Hilbert transform pair (see e.g. Mandel and Wolf, 1995), and its Fourier transform ζb
is defined by the transform pair1
ˆ
ζ (x, t) = ζb (k, t) exp (ik · x) dk,
(5.2a)
ˆ
1
ζ (x, t) exp (−ik · x) dx.
(5.2b)
ζb (k, t) =
2
(2π)
We assume that medium variations are relatively slow, varying O(1) over distances
l0 /, with l0 being a characteristic wave length and 1, so that plane wave
solutions are admitted and a dispersion relation of the form ω = σ(k, x) exists
locally. The free-surface elevation η is considered a superposition of slowly varying
wave packets ζj , each characterized by its position xj (t), wave number kj (t), and
angular frequency ωj (t). In the present work we consider medium variations due to
variations in depth, h(x),
p so that in absence of currents (and to O()) the dispersion
relation is σ(k, x) = gk tanh(kh), where k = |k| (e.g. Dingemans, 1997; Mei et al.,
2005).
Our starting point is the equation of motion for the transformed free-surface
b t), which we write as (e.g. Salmon, 1998; Bremmer, 1972)
variable ζ(k,
b t).
∂t ζb (k, t) = −iΩ(k, i∇k )ζ(k,
1 Unless
(5.3)
made explicit otherwise, integration limits on (Fourier) integrals are from −∞ to +∞, and
the transforms are assumed to exist in the context of generalized functions (Strichartz, 1993).
78
5. The evolution of inhomogeneous wave statistics
Here we use operator correspondence between conjugate variables: −iωj → ∂t ,
xj → i∇k and kj → k, to relate the local dispersion relation to an operator
σ(kj , xj ) → Ω(k, i∇k ), which is defined using the Weyl correspondence rule (see
Appendix A, and e.g. Agarwal and Wolf, 1970). It can be readily shown that the
wave equation (5.3) describes progressive waves (Appendix A), is exact in a homogeneous medium (Appendix A), and is consistent with WKB theory for slowly varying
waves (Appendix B). From
the
E wave equation (5.3), an evolution equation for the
D
second-order moments ζb1 ζb∗ is obtained in the usual way2 , and upon transforming
2
the coordinates to k = (k1 + k2 )/2 and u = k1 − k2 , and Fourier transforming the
result with respect to the difference wavenumber u, the transport equation for the
second-order statistics can be written as (see e.g. Bremmer, 1972; Bastiaans, 1997;
Cohen, 2010)
i
i
i
i
∂t E = −i Ω k − ∇x , x + ∇k − Ω k + ∇x , x − ∇k E,
2
2
2
2
(5.4)
ˆ D u u E
ζb k + , t ζb∗ k − , t exp [iu · x] du.
2
2
(5.5)
with
1
E (k, x, t) =
2
Here the h i denote ensemble averaging. The distribution function E represents
the complete second-order
statistics and carries the same information as the twoD
E
∗
b
b
point correlator ζ1 ζ2 , but with the wave number separation exchanged for a space
coordinate. In appearance it is similar to the widely used variance density spectrum,
and likewise, the local wave variance V (x, t) is found from the marginal distribution
ˆ
V (x, t) =
E (k, x, t) dk.
(5.6)
However, the distribution function E is generally not point-wise positive, and can
thus not be interpreted as a variance density function (see e.g. Janssen and Claasen,
1985). Only in the limit where the wave field is quasi-homogeneous, is the spectrum
E positive everywhere and reduces to a variance density function (which is thus a
special case). To distinguish E, as defined here, from the widely used variance density
spectrum, we refer to it as a Coupled-Mode (CM) spectrum.
The evolution equation (5.4) describes the evolution of the CM spectrum through
a variable medium, and is exact in the WKB sense in that it does not make any
assumption regarding the statistical homogeneity or the scales of variation of the
wave statistics. In other words, although the wave components themselves are slowly
varying (in accordance with WKB), the statistics can undergo rapid variations (on the
scale of individual wave lengths) through the development and propagation of crosscorrelations in the wave field, either through the interaction with medium variations,
or radiated in from the boundaries.
(5.3) for ζb1 = ζb(k1 , t) by ζb2∗ = ζb(k2 , t), and the equation for ζb2∗ by ζb1 , sum both relations
and ensemble average the result.
2 Multiply
79
5.2. Evolution of correlators
An approximate model for inhomogeneous wave fields
The transport equation (5.4) governs the evolution of the CM spectrum within a
slowly varying medium, but apart from certain special cases, the dynamical implications of the operators in (5.4) are not easily understood, and they cannot be
readily numerically evaluated. To derive a consistent approximation to these operators, we introduce the following coordinate scaling. We define two (independent)
spatial scales: a slow scale (Xm ) associated with the medium variations, and a scale
(X) that captures the spatial variations of the spectrum due to the wavenumber
mismatches in the cross-correlations, written as
Xm = x,
X = µx,
(5.7)
respectively. Here µ = ∆u/k0 (where k0 = 2π/l0 ), with ∆u a representative difference wave number for the cross-correlations and k0 a characteristic wavenumber of
the wave field. Since we consider cross-correlations developed through the interaction
with a slowly varying medium, we have µ 1 (and µ is generally of the same order
as ), so that X is effectively a slow scale (with O(1) variations on length scales of
l0 /µ). To make the width of the spectrum explicit, we consider the wave number
scale
K = δ −1 k,
(5.8)
where δ = ∆k/k0 , with ∆k representing a characteristic width of the spectrum. The δ
can be thought of as an inverse correlation length scale so that for a highly coherent
(narrow-band) wave field δ 1 (and K is a fast scale), whereas for moderate or
wide-band wave fields δ ∼ O(1). Lastly, we introduce the time scales
Tj = µj t,
j = 1, 2, ...N.
(5.9)
Using these scales, the dependent variable E and the dispersion relation σ become
E = E(K, X, T1 , T2 , . . . , TN ),
σ = σ(k, Xm ),
(5.10)
so that, to O(µN ), the governing equation (5.4) can be expressed as
i
i
µ ∂Tj E = −i Ω k − µ ∇X , Xm + δ ∇K
2
2
i
i
−Ω k + µ ∇X , Xm − δ ∇K
E. (5.11)
2
2
PN
Here the summation over repeated indices is understood (i.e. µj ∂Tj = j=1 µj ∂Tj ).
To make the magnitude of the various terms in the operators on the right side of
equation (5.11) explicit, we write the operators Ω as (see Appendix A)
j
i
i
Ω k ∓ µ ∇ X , X m ± δ ∇K =
2
2
i
i
exp ±β ∇X̃m · ∇K ∓ µ ∇k̃ · ∇X σ(k̃, X̃m )
, (5.12)
2
2
k̃=k,X̃m =X
80
5. The evolution of inhomogeneous wave statistics
where β = /δ. Physically, β thus measures the de-correlation length scale of the
waves (δ −1 ) relative to length scale of the medium variations (−1 ). If β 1 the
wave field de-correlates over distances short relative to the bottom variations, so
that regions separated by O(1) medium variations are statistically independent. In
fact, if O(µ) = O(β) = O() 1, equation (5.11) reduces (to lowest order) to the
well-known RTE (Bremmer, 1972)
∂E
+ ∇k σ · ∇x E − ∇x σ · ∇k E = 0,
(5.13)
∂t
where we dropped the scaling on the coordinates. From our analysis we see that the
RTE (5.13) is valid in a slowly varying medium only if the wave field de-correlates
on shorter scales than the scale of the medium variations. In other words, in this
limit, cross-correlations induced by medium variations are lost faster than they are
generated so that the wave system retains no memory of them and the wave field
remains effectively homogeneous (i.e. µ remains O()). The RTE is thus valid if
the spectrum of the medium variations is mostly confined to wavenumbers that are
smaller than a characteristic (wavenumber) width of the wave spectrum. Since for
most oceanic conditions this condition is easily satisfied, equation (5.13) is widely
used and - in one form or another - stands at the heart of most modern, large-scale,
stochastic models for ocean wind waves.
However, in coastal areas exposed to ocean swells, the interaction of waves with
the seafloor topography on the inner shelf (or coastal currents), or the interaction
of narrow-band wave field with coastal structures and headlands, can result in coherent interferences in the wave field that are visible even to a casual observer (e.g.
interference in a focal zone induced by currents or topography). In such regions, the
length scales of medium variations and decorrelation length scale of the waves can be
of similar magnitude, so that β = O(1), and the approximations implied in the RTE
are not valid. In this case, a truncated expansion in β is not a useful approximation,
but the general transport equation (5.11) can be alternatively approximated through
a Fourier integral representation of the operators as in (see Appendix A)
ˆ
µj ∂Tj E(K, X, T1 , . . . , TN ) = −i dQ exp [iQ · Xm ]×
i
exp − µ∇X · ∇k̃ σ
b(k̃, Q)
µj E (K − βQ/2, X, T1 , . . . , TN )
2
k̃=k
+ C.C.,
(5.14)
where Q = q/ and C.C. denotes the complex conjugate. On account of the slowly
varying medium, major contributions to σ
b(k, Q), and thus the integral, are limited to
the domain |Q|/k0 ≤ O(1) so that the integral in (5.14) can be efficiently numerically
approximated and, to O(µN ), the transport equation (5.14) becomes (in physical
coordinates)
∂t E (k, x, t) =
ˆ
1
i
(N )
b
−i
Ω
k − ∇x , q E k − q, x exp [iq · x] dq + C.C. (5.15)
2
2
D
5.3. Evolution of coherent wave structures
81
Here D denotes the domain of integration such that |q|/k0 ≤ O(), and the kernel
b (N ) operating on E is defined as
Ω
|n| n
N
X
i
∂ σ
b ∂n
1
b (N ) k − i ∇x , q =
Ω
.
−
2
n!
2
∂kn ∂xn
(5.16)
|n|=0
The expressions (5.15) and (5.16) describe the evolution of the second-order wave
statistics while accounting for the generation and transport of coherent structures
in the wave field. In this case, cross-correlations can be generated (by medium
variations) faster than that they are destroyed, so that they can develop and persist,
and affect the wave statistics. Equation (5.16) is a central result of this paper, which
we will refer to as the N th -order Quasi-Coherent (QC) approximation, or QuasiCoherent approximation if the order of the approximation is understood. The RTE
(5.13) is thus a special case of equation (5.15) where O(β) = O(µ) = O() 1.
5.3. Evolution of coherent wave structures
To illustrate the implications of the Quasi-Coherent approximation (equation (5.15)
with (5.16)), and the differences with the RTE (equation (5.13)), we consider two
distinct cases where cross-correlations affect wave statistics. The first example considers the evolution of a group of wave packets through a homogeneous medium, where
the inhomogeneity is fully determined by the initial condition and then transported
through the domain. The second example considers the evolution of ocean waves over
a two-dimensional topographic feature (shoal); in this case, the incident wave field is
homogeneous and cross-correlations are generated through the interaction with the
variable medium and transported down-wave of the shoal.
Gaussian packets through a homogenous medium
We consider a wave field consisting of three coherent Gaussian wave packets propagating in deep water, for which the surface elevation at some arbitrary initial time
(t = 0) can be written as
3
X
2
ζ (x, 0) = exp −α |x|
Aj exp (ikj · x) .
(5.17)
j=1
Here the Aj are the (complex) packet amplitudes and kj is the carrier wavenumber.
The initial spectrum (at t = 0), Eb (k, u, 0), is then given by
Eb (k, u, 0) =
3
X
hAm A∗n i
×
32π 2 α2
m,n=1
"
2
2 #
kn
km 1 1 exp −
k−
−
−
u − um + un . (5.18)
2α
2
2
8α
82
5. The evolution of inhomogeneous wave statistics
Figure 5.1: Snapshots of normalized wave variance (normalization by 32 a2 ) of three-packet interference example. Normalized variance is shown at discrete times tI−V , starting at tI = 110Lp /vx
(left panels) increasing in time (from left to right) in intervals of ∆t = 20λ/vx . Top panel (a) shows
evolution for exact model; middle panel (b) shows the QC-approximation; bottom panel (c) shows
the evolution according to the RTE. The x0 and y 0 denote the horizontal coordinates normalized by
6Lp .
For a homogeneous medium, and in a reference
frame that moves with the mean
p
p group
velocity, the dispersion relation σ (k) = g |k| + k · v, where v = 12 k̄kp−1 g/kp and
k̄ is the mean carrier wavenumber, so that equation (5.4) has the exact solution
ˆ
b u, 0) exp [iu · x − iω∆ (k, u)t] du,
E(k, x, t) = E(k,
(5.19)
with
ω∆ (k, u) = σ(k + u/2) − σ(k − u/2).
th
(5.20)
The relations (5.19) and (5.20) are exact. The N -order QC approximation is obtained by substituting σ
b(k, q) = δ(q)σ(k) in (5.15), where δ(q) denotes the Dirac
delta function, Fourier transforming with respect to x, and solving the resulting ordinary differential equation. On applying the inverse transform with respect to u,
the result is again (5.19), but with ω∆ replaced by its N th -order Taylor series in u
around u = 0,
|n|≤N
X
un ∂ n σ
(N )
.
(5.21)
ω∆ (k, u) =
n! 2|n|−1 ∂kn
|n|=1,3,...
83
5.3. Evolution of coherent wave structures
(a)
0.5
0
−3
−2
−1
0
1
2
3
(b)
V0
1
0.5
0
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
(c)
3
1.5
0
−1.5
−1
−0.5
0
0.5
1
1.5
y0
Figure 5.2: Cross-sections (at x0 = 0) of normalized wave variance of three-packet interference
example. The wave variance is shown at discrete times tI−III , starting at tI (a) and increasing
in time (from left to right). Comparison between exact model (solid line), the QC-approximation
(circle markers) and the evolution according to the RTE (thin line). Normalization of vertical and
horizontal scales and discrete times as in Figure 5.1. Note that the horizontal and vertical range
can vary between panels.
In what follows, we consider three coherent packets of the same energy and carrier
wave length (|Aj | = a, |kj | = k), that propagate at angles of −20, 0, 20 degrees
1 −2
relative to the positive x-axis. We set α = 36
Lp , so that a characteristic length
scale of the packets is roughly six wave lengths (µ ≈ 1/6), and we consider the
evolution from t = −150Tp to t = 20Tp (with Tp = Lp /vx a characteristic period).
The QC approximation is initialized at t = −150Tp using the exact solution.
Since the simulation time is roughly Tp /µ3 , we use a third-order QC approximation (by truncating (5.21) after the second term). Each packet is calculated individually on a discrete equidistant k-mesh
√ centered at kmn using thirty points in each
direction with a mesh-size of ∆k = α/5. The Fourier integrals are approximated
using a√Fast Fourier Transform with u1 and u2 discretized as [−31 . . . 31] ∆u, with
∆u = α/5, and the result interpolated to a discrete 241 × 241 x-mesh centered at
the origin with mesh-size ∆x = ∆y = Lp /5.
The initial evolution of the wave system is characterized by convergence and interference of the wave packets (Figure 5.1a), followed by de-focusing and divergence,
84
5. The evolution of inhomogeneous wave statistics
Figure 5.3: Spectral evolution of the three-packet interference example (5.18) evaluated at (x0 , y 0 ) =
0 (left panels) and (x0 , y 0 ) = (0, 61 ) (right panels) for tI−III (times as defined in caption Figure 5.1).
Top panels (a) show the auto-variance contributions; (b) shows the cross-variance contributions;
(c) shows the resulting CM spectrum (sum of auto- and cross-variance contributions). The spectral
coordinates kx0 and ky0 are normalized with kp and spectra are normalized with the peak contribution
of an individual packet (4πα)−1 .
5.3. Evolution of coherent wave structures
85
after which the packets emerge unchanged and the initial state is recovered. The QC
approximation captures the principal dynamics of the wave evolution, including the
coherent interference (Figures 5.1b and 5.2), which confirms that the QC approximation accurately represents cross-correlations in the evolution of random waves
associated with the coherent interference of wave packets traveling at moderate angles. In contrast, the wave packet evolution as represented by the RTE (Figures 5.1c
and 5.2) is dramatically different from the exact result; in particular, the interference
pattern at t = tIII is not resolved since the RTE does not account for the transport
of cross-variance contributions (m 6= n).
The differences in evolution are apparent also from the spectra (left panels in
Figure 5.3). The variance density spectrum at x0 = 0 (where x0 = x/Lp ) contains a
single positive peak for tI and tII related to the central packet, and three peaks at
tIII when the three packets have converged (Figure 5.3a). Since the CM spectrum
accounts for inter-mode coupling, it contains additional interference peaks, which
travel along rays midway between the rays of the auto-variance contributions involved
in the interference. For the case considered here, at x0 = 0, the cross-variance at tI
consists of a single interference peak (Figure 5.3b), which represents the coherence
between the outer packets, and travels along the ray y 0 = 0. Coherence between the
centre and outer packets travels along different rays and only manifests itself at tII
and tIII where these rays cross through y 0 = 0. The CM spectrum is the sum of the
interference contributions and (auto-)variance density spectrum (Figure 5.3c). The
spectral interference terms capture the rapid spatial oscillations of the wave statistics
due to wave interference in the focal zone (Figure 5.1c). This fast-scale variability is
also seen in the spectral domain, when the spectra are evaluated at a slightly offset
location (compare left and right panels in Figure 5.3). Note that the wave packets in
this example do not ’interact’ with each other in the usual sense, and the coherent
interference is completely determined a-priori by the inclusion of interference peaks
in the initial condition. No cross-correlations are generated (or destroyed) in the
course of the evolution.
Coherent interference patterns induced by topography
To consider the generation and persistence of coherent interference patterns through
the interaction with a slowly varying medium, we compare model simulations to laboratory observations of waves traveling over a submerged shoal (Vincent and Briggs,
1989). In these experiments, an elliptic shoal with its crest 15.24 cm below still-water
level was placed in a wave basin with a uniform depth of h=45.72cm (See Figure 5.4).
We consider a case with monochromatic waves (M2) and random waves with a TMA
spectrum (N4); for the latter, the peak enhancement factor γ=20, directional spreading is approximately 10◦ (as defined by Kuik et al., 1988), and the (significant) wave
height and (peak) period are 2.54 cm and 1.3 s, respectively. For additional details
we refer to Vincent and Briggs (1989).
We compare model simulations with a first-order QC model and the RTE to
the laboratory observations. Since the observational data is sparse, we also include
comparison to simulations with a deterministic hydrodynamical model (SWASH),
which has been independently validated (see Zijlema et al., 2011b).
86
5. The evolution of inhomogeneous wave statistics
10
B
C
y [m]
5
0
A
−5
−10
0
5
10
15
20
25
x [m]
Figure 5.4: Plan view of the experimental setup by Vincent and Briggs (1989) including a ray-traced
solution (thin grey lines) for unidirectional monochromatic waves (period 1.3 s, incident direction
θ0 = 0◦ ). Depth contours (drawn at 0.15–0.45 m, at 0.1 m intervals) are indicated by black solid
lines; instrumented transects (at y = 0 m, x = 12.19 m and x = 15.24 m) are indicated with dashdot
lines.
The spectral models are numerically evaluated on a rectangular spatial (30×25
m2 ) and spectral domain (4kp × 3kp , starting at k = [−0.5kp , −1.5kp ]), uniformly
discretized with mesh sizes ∆x = ∆y = 12.5 cm and ∆k/kp = [18, 20]. We use slope
limited second-order finite difference approximations for the spatial (and spectral)
gradients, in combination with a explicit first-order time stepping. The integral in
equation (5.15) is approximated by a second-order numerical quadrature on the domain D delimited by |q| < 2kp . At the wave maker, the incident spectrum is imposed,
and periodicity is assumed in the lateral direction. Combined with a radiation type
boundary condition opposite to the wave-maker, the solution is then marched in time
until a steady state is reached.
Over the shoal, the incident monochromatic waves (case M2) are refracted in
different directions, resulting in fast lateral variations in wave variance behind the
shoal, due to the coherent interference of the crossing wave components. The QC
model captures the variations in wave energy induced by these interferences and
agrees well with observations and the deterministic model (5.5). In contrast, the
RTE predicts much stronger focusing and lower wave heights in the shadow zone (see
Figure 5.6), and does not resolve the fine-scale structure in wave energy associated
with wave coherency.
For the random incident wave field (case N4), due to the increased dispersion
87
5.3. Evolution of coherent wave structures
4
A
H
3
10
20
A
2.5
0
2
C
1
0.5
0.5
0
2
1.5
1
3
H
2.5
1.5
1
3.5
B
2
2
0
3
−5
0
0
5
B
1.5
−5
0
5
C
1.5
2
1
1.5
1
1
0.5
2.5
0
10
20
A
H
2
0.5
2
−5
0
B
1.5
−5
0
5
C
1.5
1.5
1
1
1
0.5
0.5
5
0
10
x [m]
20
0.5
−5
0
y [m]
5
0.5
−5
0
5
y [m]
Figure 5.5: Shown are normalized wave heights along transects across (panels marked with A) and
behind the shoal (marked with B and C, see Figure 5.4 for locations) as considered by Vincent
and Briggs (1989) for case M2 (top panels), case N4 (lower panels) and the additional case N40
considered in the present work (middle panels). Comparison is between the QC1 approximation
(solid black line), observations (circle markers, when available), the deterministic model SWASH
(crosses), and the RTE (dashed red line).
88
5. The evolution of inhomogeneous wave statistics
Figure 5.6: Plan view of modeled (normalized) wave heights for the experimental set-up as considered by Vincent and Briggs (1989) for case M2 (top panels), case N4 (lower panels) and the
additional case N40 considered in the present work (middle panels). Comparison between the QC1
approximation (left panels) and the RTE (right panels) shows that the QC1 approximation, in contrast to the RTE, resolves the fine-scale interference pattern in the focal zone of a topographical
lens.
5.4. Discussion
89
(mostly directional), the wave field decorrelates faster, resulting in a smoothing of
the wave statistics behind the shoal (see Figure 5.6). In this case, the QC results are
qualitatively more similar to the RTE result, although several differences remain. In
the region directly behind the shoal, wave heights predicted by QC theory are consistent with observations and those obtained with the deterministic model, whereas the
wave heights predicted by the RTE are approximately 20% lower. If the directionality
(and thus dispersion) is reduced, say to 3◦ (N40 ) a well-defined interference pattern
emerges again and the QC model provides a much more realistic presentation of the
wave statistics than the RTE3 . These cases, which can represent the propagation
of narrow-band swell waves over coastal topography, show that including coherent
effects can be significant for situations that are of practical interest. The emergence
and persistence of coherent interferences in narrow-band random waves over coastal
topography can be important for regional wave statistics and thus affect wave-driven
circulation and transport processes in such areas.
5.4. Discussion
Coherent wave interference patterns are common in the coastal ocean, for instance
due to the interaction with submerged topography, currents, islands, headlands, or
coastal structures. Statistical models based on the RTE do not account for such
interference patterns, and this can result in systematic differences between observed
variations in wave statistics and model predictions. In this work we introduce a new
transport model for what we refer to as a Coupled-Mode (CM) spectrum, which
includes the effects of coherent interference on the wave statistics. The concept
of a CM spectrum to describe the statistical evolution of inhomogeneous random
processes is not new, but has been developed independently across various fields, such
as the Wigner distribution in Quantum mechanics and Optics (e.g. Bastiaans, 1997;
Bremmer, 1972; Wigner, 1932), the concept of generalized radiance in radiometry
(Walther, 1968), and the Wigner-Ville distribution in signal processing (Ville, 1948;
Cohen, 1989). Here we apply these ideas to ocean waves traveling through a variable
medium and in the presence of caustics, such as are commonly found in e.g. coastal
areas and other regions characterized by relatively strong two-dimensional medium
variations.
Interference terms
Apart from energy (or variance) contributions, the coupled-mode spectrum carries
cross-correlation and cross-phase information on coherent interferences between noncollinear wave components in the wave field. For example, if we revisit the correlated
wave packet example in §5.3, and - for convenience - reduce it to two packets propagating at some equal but opposite angle with respect to the positive x-axis, so that
k1 = (κ, λ), k2 = (κ, −λ), and Aj = aj exp iφj , with aj = |Aj | and φj = arg Aj , the
3 This
case was not considered by Vincent and Briggs.
90
5. The evolution of inhomogeneous wave statistics
CM spectrum can be written as
E(k, x) =
1
2
ˆ
2
−2α|x| b u) exp [ik · u] du = e
E(k,
ha21 iG11
4πα
+ ha22 iG22 + +2ha1 a2 iG12 cos (2λy + hφ1 − φ2 i)] , (5.22)
b u) is the same as in equation (5.18) and
where the E(k,
"
2 #
ki
kj 1 k−
− .
Gij = exp
2α 2
2
(5.23)
The last term in (5.22) is a coupled-mode contribution that represents the contribution from the coherent interference between the two packets. This contribution is
located midway between the associated auto-variance contributions (Hlawatsch and
Flandrin, 1997), can become negative and, since it does not directly correspond to a
field component, does not carry energy itself.
Instead, this contribution determines how the energy of the wave field is distributed between kinetic and potential energy. After all, in a statistically homogeneous (and linear) wave field, the energy is equipartitioned between potential and
kinetic energy so that knowledge of either potential or kinetic energy suffices. On
the other hand, in a wave field that undergoes coherent interference, information on
the distribution of potential and kinetic energy is required to fully characterize the
wave field statistics. This information is provided by the cross-contributions in the
coupled-mode spectrum. In our example (equation (5.22)), the interference results
in a coherent standing wave motion along the y-coordinate where the wave packets
alternately interfere constructively and destructively; the coupled mode thus contributes negatively in the nodes (where more energy is kinetic) and positively in the
anti-nodes (where more energy is potential).
Aperture limitation on wave interferences
The CM spectrum is a general representation of the complete second-order statistics,
including interferences between wave component with arbitrary mutual orientation,
and without inherent aperture limitation. However,
the QC transport equation,
which is based on a series expansion of σ
b k ± u2 , q in u (equation (5.33)), is fundamentally more restricted, and cannot accurately transport cross-correlations between
waves that enclose an angle greater than π/2. This limitation can be understood from
b (N ) (equation (5.16)) is obtained by Fourier transforming
the fact that the kernel Ω
(with respect to u) the series expansion of σ
b k ± u2 , q (equation (5.33)). Before
truncation (for N → ∞), this is exact if the series approximation to σ
b k ± u2 , q
converges ∀u. However, the dispersion relation for ocean waves depends on |k|, so
that ∇k σ is singular at k = 0, and the radius of convergence of the series expansion for σ(|k|) around k0 is |k0 |. As a consequence, the operators are generally
approximations and, since in the QC approximation the expansions are in 12 u, the
maximum enclosed angle is π/2 (k+ · k− = 0, k± = k ± u2 ), when both wave components are on the circle |u| = 2k (see Figure 5.7). Therefore, if k0 is a typical wave
91
5.4. Discussion
ky
u = 2k
k+
u
k
Singularity
k-
kx
Figure 5.7: Sketch of wave interference geometry on the radius of convergence. For a dispersion relation Ω(|k|) (where |k| is singular at the origin, as indicated in the Figure), the radius of
convergence of the QC
approximation (for a particular k) is given by |ũ| = 2k. Consequently,
components k+ , k− with average wavenumber k = (k+ + k− )/2, which are located on the limit
circle k+ − k− = 2 |k| , propagate perpendicular to each other, as shown in the Figure (solid
black and striped gray lines).
number, the smallest spatial scale that can be resolved by the QC approximation is
limited to π/k0 . Such aperture restrictions only apply to the coherent interference
terms; the spectrum of the auto-mode distributions (variance-carrying contributions)
is arbitrary.
Wide-angle diffraction
The propagation of ocean waves around thin barriers and obstructions results in coherent interferences, associated with diffraction. From a statistical viewpoint, such
diffraction patterns are a coherent interference pattern originating from the interaction with the barrier. Although these effects are ignored by the RTE, they can be
readily accounted for in the QC approximation.
To illustrate this, we apply the QC approximation to the propagation of waves
through a gap (extending over −G1 < y < G2 ), in an otherwise rigid, but fully
absorbing barrier (situated at x = 0). Apart from the obstacle, the medium is
uniform and the dispersion relation of the form σ = σ (|k|). The incident wave
field is homogeneous, unidirectional, and normally incident on the barrier with a
known variance density spectrum S0 (ω). Using the Kirchhoff, or physical optics,
approximation (see Appendix C), and a third-order QC approximation, we obtain
the E spectrum just behind the barrier (x = 0+ ), E0 (k, y).
Behind the barrier, the coupled-mode spectrum is given by (5.19), and for a
stationary solution we have ω∆ (k, u) = σ(|k + u/2|) − σ(|k − u/2|) = 0 (see (5.20)),
b u) = E(k,
b uy )δ(ux +
so that |k+u/2| = |k−u/2| and k·u = 0. Therefore we have E(k,
92
5. The evolution of inhomogeneous wave statistics
10
10
x0
15
x0
15
5
0
−6
5
−4
−2
0
y0
2
4
0
−6
6
−4
−2
0
10
10
4
6
2
4
6
x0
15
x0
15
2
y0
Semi-infinite breakwater
5
0
−6
5
−4
−2
0
y0
2
4
6
0
−6
−4
−2
0
y0
breakwater gap
0.3
0.6
0.9
1.2
1.5
H0
p
Figure 5.8: Contours of normalized wave height H 0 =
V/V0 (where V0 is the variance of the
incident waves) behind a semi-infinite breakwater (top panels) and a breakwater gap (bottom panels). Left panels show the analytic solution (Penney and Price, 1952); right panels show the QC
approximation. The incident wave field consists of unidirectional waves, normally incident onto
the breakwater, with a peak angular frequency of ωp = π rad/s, kp h = 1.2 and a narrow-band
1
Gaussian-shaped frequency distribution with characteristic width of ∆ω = 100
ωp . The horizontal
coordinates (x0 , y 0 ) are normalized with the peak wave length.
uy ky kx−1 ) for kx > 0 (and zero elsewhere), and (from (5.19)) find that
ˆ
b uy ) exp iuy y − iuy ky kx−1 x duy .
E (k, x) = E(k,
(5.24)
The wave statistics for x > 0, including coherent interferences associated with diffracb uy ) at x = 0
tion, are thus entirely determined from the boundary condition E(k,
(the gap). No new information is added as the waves travel behind the barrier, and
the diffraction effects commonly seen in harbors or from areal photography of waves
around islands, are - from a statistical point of view - simply the manifestation of
cross-correlations determined by the up-wave boundary conditions, transported by
(5.24).
In Figures 5.8 and 5.9 we compare simulation results to analytic solutions (Penney
and Price, 1952) for a monochromatic incident wave field impinging on a semi-infinite
L
breakwater (G1 = 2y and G2 = 0, with Ly the lateral extent of the domain), and
93
5.4. Discussion
1.5
1.5
x 0 = 4.0
x 0 = 15.0
H0
1
H0
1
0.5
0
−6
0.5
−4
−2
0
y0
2
4
0
−6
6
−4
−2
1.5
0
2
4
6
y0
Semi-infinite breakwater
1.5
x 0 = 4.0
x 0 = 15.0
H0
1
H0
1
0.5
0
−6
0.5
−4
−2
0
y0
2
4
6
0
−6
−4
−2
0
2
4
6
y0
breakwater gap
p
Figure 5.9: Cross-sections of normalized wave height H 0 =
V/V0 (where V0 is the variance of the
incident waves) along x0 = 4 (left) and x0 = 15 (right) behind a semi-infinite breakwater (top) and
a breakwater gap (bottom). Shown are the analytic solution (solid line Penney and Price, 1952)
and the QC approximation (circle markers). Incident wave field and coordinates as in Figure 5.8.
a barrier gap (G1 = G2 = 2.65Lp Ly ). The QC approximation is in excellent
agreement with the analytic solution for distances greater than about 4 wave lengths
behind the barrier (x/Lp > 4), which confirms the accuracy of this approach in
the intermediate to far-field. Differences in the near-field are principally due to
the omission of evanescent modes (especially in the region x/Lp < 1), and the use
of an approximate (Kirchhoff) boundary condition (see e.g. Stamnes, 1986). These
examples illustrate that the QC approximation, despite its fundamental π/2 aperture
restriction on interference terms, can accurately represent wide-angle coherencies due
to diffraction, as implied by the good agreement with the analytic solutions.
Relation to other transport equations
In the derivation of the stochastic model (5.4) we started, following Bremmer (1972),
from the wave equation (5.3). This approach is quite general, and makes no explicit
assumptions regarding the bandwidth of the wave field. Alternatively, if we restrict
our derivation to narrow-band waves from the outset, we could, following Wigner
(1932) and Alber (1978), start with specific evolution equations for a narrow-band
wave train. To show that our approach and resulting transport equations are indeed
consistent with earlier results for narrow-band waves, we consider the free-surface
94
5. The evolution of inhomogeneous wave statistics
elevation ζ of a zero-mean, narrow-band wave field in deep water, given by ζ (x, t) =
A (x, t) exp (ik0 · x − iω0 t), where k0 = [k0 , 0]T is the principal wavenumber, ω0 =
σ (k0 ) the angular frequency, and A (x, t) denotes the slowly changing wave envelope.
Assuming that the bandwidth ∆k/k0 is O (δ) and the steepness is small (A0 k0 δ,
with A0 a typical wave amplitude), the linear evolution of the envelope, up to O δ 3 ,
is governed by the linear part of the Dysthe equation (Dysthe, 1979)
i (1) ∂ 2
i (2) ∂ 2
∂
(1) ∂
−
+ σk0
− σk 0
σ
∂t
∂x 2
∂x2
2k0 k0 ∂y 2
∂3
1 1 (3) ∂ 3
(2)
(1)
− 2 k0 σk0 − σk0
A = 0. (5.25)
− σk0
2k0
∂x∂y 2
6
∂x3
The procedure for obtaining the evolution equations for the CM spectrum from the
governing equation (5.25) is essentially the same as that followed in Section (5.2), and
equivalent to the procedure followed by Alber (1978) and Cohen (2010). We evaluate
equation (5.25) at two different locations for A1 = A (x1 , t) and A∗2 = A∗ (x2 , t); we
multiply the first equation with A∗2 , the second with A1 , sum the two resulting
equations and ensemble average the result. After introducing the spatial average and
difference coordinates x = (x1 + x2 ) /2 and ξ = x1 − x2 , we obtain an evolution
equation for the product ρ = 12 hA1 A∗2 i. Upon applying the Fourier transform with
respect to ξ, we can express (5.25) in terms of the CM spectrum
"
∂
∂
(2)
(1)
+ σk0 + (kx − k0 ) σk0
∂t
∂x
ky (1) ∂
1 (3) ∂ 3
1
2 (3) ∂
− σk0
+ (kx − k0 ) σk0
σk0
k0
∂y 24
∂x3
2
∂x
!#
2
3
ky ∂
1
∂
1 ∂
(2)
(1)
− 2 k0 σk0 − σk0
−
− (kx − k0 ) ky
E = 0, (5.26)
k0
8 ∂x∂y 2
2 ∂x
∂y
+
where we used that E (k, x) = ρb (k − k0 , x), with ρb denoting the transform of ρ with
respect to ξ. The first three terms inside the brackets on the left side of (5.26)
correspond to the linear spectral evolution equation of Alber (1978). Moreover,
equation (5.26) is a narrow-band approximation (around k0 ) of the third-order QC
approximation in a uniform medium.
This can be seen if we consider the Taylor
b (3) k ∓ i ∇x , q in k around k0 which, when retaining terms
approximations for Ω
2
up to O(δ 3 ), reads
i
i
1
(1)
(2)
2 (3)
(3)
b
Ω
k ∓ ∇x , q = ∓ δ(q) σk0 + (kx − k0 ) σk0 + (kx − k0 ) σk0
2
2
2
!
∂
∂
ky2 ky (1) (kx − k0 ) ky (2)
(1)
(2)
(1)
+ 2 k0 σk0 − σk0
+
σk0 +
k
σ
−
σ
0 k0
k0
2k0
∂x
k0
k02
∂y
#
(3)
∂3
σ
∂3
1 (2)
(1)
− k0
−
k
σ
−
σ
. (5.27)
0
k0
k0
24 ∂x3
8k02
∂x∂y 2
5.5. Conclusions
95
If the expression (5.27) is substituted in the transport equation (5.15), equation
(5.26) follows, which shows that our results are consistent with earlier results for
narrow-band wave fields, and contain the linear part of the Alber equation as a
special case.
5.5. Conclusions
We have presented a new transport model for the statistics of inhomogeneous wave
fields of arbitrary bandwidth propagating through a variable medium. The model
accounts for the generation and transport of coherent interferences between wave
components that enclose angles smaller than π/2 radians. The theoretical framework
presented here is a natural extension of the radiative transfer equation, and valid for
arbitrary spectral width. We show that in the limit of narrow-band waves, the
transport equation reduces to the linear Alber equation, which is thus a special case.
Moreover, for homogeneous waves with an arbitrary spectrum, our result is consistent
with the radiative transfer equation for the transport of the variance (or action)
density spectrum. Comparison with analytic solutions for wave packet interference,
and with observations of random surface wave propagation over a two-dimensional
bottom feature, confirm that the Quasi-Coherent (QC) approximation accurately
represents both the generation and transport of cross-correlations in the wave field,
and resolves the fine-scale interference patterns associated with crossing waves. The
effects of diffraction on statistics of waves around and behind obstacles and barriers,
can be accurately modeled by including appropriate boundary conditions on the
QC approximation. These results show that the application of QC theory to ocean
waves can resolve some of the restrictions of quasi-homogeneous theory (the radiative
transfer equation) in areas characterized by two-dimensional medium variations and
caustics. This is likely to be of particular importance for wave-driven circulation and
transport processes in coastal areas and inlets.
Acknowledgment
This research is supported by the U.S. Office of Naval Research (Coastal Geosciences
Program and Physical Oceanography Program) and by the National Oceanographic
Partnership Program. The authors thank Leo Holthuijsen whose continuing support
made it possible for PBS to pursue this research.
Appendices
5.A. Operator definition
We assume that the wavefield consists of a large number of progressive wave-packets,
and each packet j has a location xj (t), slowly varying wavenumber kj (t), and angular
frequency ωj (t), which are related by a dispersion relation, ωj = σ(kj , xj ). To obtain
b t), we associate
an evolution equation for the transformed free-surface variable ζ(k,
b
the (dependent) wave variables xj , kj , ωj with operators on ζ(k),
i.e. kj (t) → k,
xj (t) → i∇k and ω(t) → i∂t . So that the dispersion relation defines a linear operator
b written as
Ω on ζ,
b t).
∂t ζb (k, t) = −iΩ(k, i∇k )ζ(k,
(5.28)
Although equation (5.28) can be justified by the analogy between the ray equations of
geometric optics and the canonical equations of Hamilton (e.g. Salmon, 1998), the definition of the operator Ω requires particular consideration due to non-commutability
of the operators (e.g. Torre, 2005). Here we follow the Weyl correspondence rule
(e.g. Agarwal and Wolf, 1970) to uniquely define the operator Ω so that the resulting
linear operator is Hermitian with real eigenvalues (i.e. the angular frequencies), and
orthogonal eigenfunctions. Thereto we expand σ in terms of its Fourier integral,
¨
σ(k, x) =
σ
b(p, q) exp [ip · k + iq · x] dqdp,
(5.29)
where σ
b(p, q) denotes the Fourier transform of σ(k, x) with respect to (p, q). After
Taylor expanding the exponential function, applying the operator correspondence and
associate any products with the sum of all its possible permutations (e.g. kj xj →
i
i
2 k∇k (. . . ) + 2 ∇k (k . . . )), we have
¨
Ω(k, i∇k ) =
σ
b(p, q) exp [ip · k − q · ∇k ] dqdp.
(5.30)
Here the exponential operator is defined in terms of its Taylor series expansion
exp [ip · k − q · ∇k ] =
∞
X
1
n
(ip · k − q · ∇k ) ,
n!
(5.31)
|n|=0
and n is a multi-index. With these definitions in place, equation (5.28) describes
periodic and undamped wave motion over topography and is consistent with WKB
theory (See appendix B). Moreover, for an ocean of constant depth (5.28) is exact
since we have σ
b(p, q) = σ(k)δ(p), so that Ω(k, i∇k ) reduces to a Fourier multiplier
σ(k) .
97
98
5. The evolution of inhomogeneous wave statistics
To demonstrate that equation (5.12) and subsequently (5.14) follow from the
definition (5.30), we introduce the sum and difference coordinates k and u in (5.30)
to obtain
i
1
b u) =
Ω k ± u, i∇u ± ∇k E(k,
2
2
¨
h
1
u i
b u)
exp ∓q · ∇k dqdp E(k,
exp [−q · ∇u ]b
σ (p, q) exp ip · k ±
2
2
ˆ
u 1
b u). (5.32)
= exp [−q · ∇u ]b
σ k ± , q exp ∓q · ∇k dq E(k,
2
2
We replace σ
b k ± u2 , q by its Taylor series expansion in u around k,
∞
X
1 ∂nσ
b (±u)n
b (∞) k ± u , q =
,
Ω
2
n! ∂kn 2|n|
(5.33)
|n|=0
so that, upon Fourier transforming with respect to u, (5.32) can be written as
Ω± E(k, x) =
ˆ
b (∞) k ∓ i ∇x , q exp ∓q · 1 ∇k dq E(k, x), (5.34)
exp [iq · x]Ω
2
2
b k ± u2 , q is analytic, the Taylor
where we used Ω± = Ω k ∓ 2i ∇x , x ± 2i ∇k . If σ
series approximation (5.33) converges and equation (5.34) is exact. If the series only
converges on some sub-domain of u, equation (5.34) formally only applies to contributions to E that originate from contributions in Eb on that domain; the implications
of this in the context of ocean waves are considered in §4b.
After integration with respect to q, equation (5.34) can be written in the form of
equation (5.12), written here as
i
i
(∞)
k̃ ∓ ∇x , x̃ Ω E(k, x) = exp ± ∇x̃ · ∇k Ω
E(k, x)
2
2
k̃=k,x̃=x
i
i
= exp ± ∇x̃ · ∇k ∓ ∇k̃ · ∇x σ(k̃, x̃) E(k, x). (5.35)
2
2
k̃=k,x̃=x
±
Alternatively, if we observe that exp − 21 q · ∇k operating on E is equivalent to a
Taylor series of E(k − 12 q, x), we can write equation (5.34) as
Ω± E(k, x) =
ˆ
i
1
dq exp [iq · x] exp ∓ ∇k̃ · ∇x σ
b(k̃, q) E k − q, x , (5.36)
2
2
k̃=k
which is the operator in equation (5.14).
5.B. Relation to geometric optics
99
5.B. Relation to geometric optics
To show that the wave equation (5.3) (or (5.28)), combined with the definition (5.30),
is consistent with WKB theory (to O()), and describes progressive ocean waves in
a slowly varying medium, we rewrite equation (5.28) in the spatial domain, as
∂t ζ(x, t) = −iΩ(−i∇x , x)ζ(x, t).
(5.37)
We assume that the wave field is characterized by a carrier wavenumber k0 , and
frequency σ0 (k0 , x0 ), and write the operator Ω as
¨
Ω(−i∇x , x) =
σ
b(p, q) exp [−p · ∇x + iq · x] dqdp
¨
i
=
σ
b(p, q) exp − p · q exp [iq · x] exp [−ip · ∇x ] dqdp
2
i
, (5.38)
= exp − ∇k̃ · ∇x̃ exp [−i∇k̃ · (∇x − k0 )]σ(k̃, x̃)
2
k̃=k0 x̃=x
where the extra factor exp − 2i p · q in the second line appears because the products
in the arguments of the exponentials do not commute, i.e. ∇x (x . . . ) 6= x · ∇x (. . . )
(see e.g. Mesiah, 1961, p. 442). We introduce the slow coordinates T = t, X = x,
substitute the ansatz ζ(X, T ) = A(X, t) exp [iS(X, T )] in (5.37), and assume that
the amplitude and phase are real and can be expanded as A = A0 + A1 + . . . and
S = −1 S0 + S1 + . . . , respectively. On expanding the exponential operators in
a Taylor series and retaining terms up to O(1 ), while dropping the scaling of the
coordinates, we obtain the eikonal equation for the phase function
∞
X
1
n
∂t S0 = −
[(∇x S0 − k0 ) · ∇k ] σ(k, x)
n!
k0 ,x
|n|=0
= −σ(∇x S0 , x). (5.39)
Similarly, the amplitude evolution equation becomes
∂t A0 = −
∞
X
1
n
[(∇x S0 − k0 ) · ∇k ] ×
n!
|n|=0
1
1
∇x A0 + A0 ∇2x S0 ∇2k + A0 ∇x · ∇k σ(k, x)
2
2
k0 ,x
1
= − ∇k σ · ∇x + ∇k · ∇x σ A0 , (5.40)
2
where the derivatives of σ are evaluated at (∇x S0 , x). If we multiply equation (5.40)
by ρgA0 , with ρ the density, and define the wave energy as E = 21 ρgA20 , we find
∂t E + ∇x · (∇k σE) = 0.
(5.41)
Equations (5.39) and (5.41) are the usual geometric optics approximations for waves
in a slowly varying medium (e.g. Dingemans, 1997; Mei et al., 2005), which shows
that the wave equation (5.3) is consistent with WKB theory (to the order considered).
100
5. The evolution of inhomogeneous wave statistics
5.C. Boundary condition for wide-angle diffraction
To derive a boundary condition for the diffraction example, we use the expression
derived by Janssen et al. (2008), based on a forward-scattering assumption and a
Kirchhoff approximation (e.g. Born and Wolf, 1999, p. 422). To transform the
b uy ) used in (5.24), we express
mutual spectrum used by Janssen et al. (2008) to E(k,
the frequency as a function of both k and u, i.e. ω = ω(k, u). In a homogeneous
medium (this case), this can be done by noting that each contribution to the CM
b ± ) exp [−iω ± t], with wavenumbers
spectrum E involves two coherent waves ζ± = ζ(k
±
±
±
k = k ± u/2 and frequencies ω = σ(k ). Associated with these components
we then have a beat frequency ω + − ω − , related to the slow scale changes in time
of the variance, and a mean frequency ω + /2 + ω − /2, associated with the fast scale
oscillations; the former corresponds to ω∆ (k, u) whereas the latter serves as our
definition of ω(k, u). To obtain a consistent QC approximation, we replace (as before)
ω(k, u) with its Taylor approximation, so that
|n|≤N
ω
(N )
X
(k, u) =
|n|=0,2,...
un ∂ n σ
.
n! 2|n|+1 ∂kn
(5.42)
(3) Using a third-order approximation, the CM spectrum becomes Eb (k, uy ) = S dω
dkx ,
for kx > 0 (and zero elsewhere), where
S ω
(3)
, kx , uy
2
|k| + u2y 1 + ky2 kx−2
1
= 2 S0 ω (3) exp (−iuy G∆ )
π
kx2 − 14 kx−2 ky2 u2y
sin ((−ky − uy /2)Gm ) sin ((−ky + uy /2)Gm )
, (5.43)
(ky + uy /2)
(ky − uy /2)
with S0 (ω) the incident, unidirectional frequency spectrum and G∆ = (G2 − G1 ) /2,
Gm = (G1 + G2 ) /2.
To approximate (5.24) numerically, we consider the solution on an equidistant
1
Lp (where Lp is the peak
Cartesian grid with horizontal resolution ∆x = ∆y = 10
wave length), and define the k-mesh as ki,j = (i, j) ∆k, with i = 1 . . . 100, j =
−200 . . . 200 and ∆k = 0.011kp (where kp = 2π/Lp ). For each wave number ki,j
the Fourier integral in (5.24) is approximated using a Fast Fourier Transform on an
equidistant discrete array [−N . . . N ] ∆u for uy with mesh-size ∆u = 2πL−1
y and Ly =
(2N + 1) ∆x. Since the accuracy of the approximate transformation deteriorates for
u/k > 1.5 the contributions for these high wavenumbers are set to zero.
6
The evolution of narrow-band
wave statistics over complex
nearshore topography
The statistical evolution of narrow-band waves in a variable medium can result in fast-scale variations in the (mean) wave statistics, which cannot be
resolved using conventional third-generation stochastic wave models based on
the radiative transport equation. In the present work we therefore consider
the application of a recently developed quasi-coherent wave theory that can
account for inhomogeneous statistics on the wave length scale. Hereto, we derive a consistent model formulation and present a numerical implementation
suitable for large-scale application over complex topography. We show that the
generalized stochastic model can be written in the form of an inhomogeneous
radiative transport equation, including a scattering term that accounts for the
generation of coherent contributions to the wave statistics. Comparison with
a laboratory and field observations of swell waves propagating over topography demonstrates that the quasi-coherent model allows for a more complete
description of the evolution of the second-order statistics over complex topography compared to the traditional radiative transport equation.
6.1. Introduction
The evolution of wind-driven surface waves on the open ocean is largely determined
by the action of wind (Phillips, 1957; Miles, 1957), dissipation (white capping, Hasselmann, 1974) and third-order nonlinear effects (or quadruplet wave-wave interactions,
The work in this Chapter has been presented in Honolulu, USA, as: Smit, P.B., Janssen, T.T. and
Herbers, T.H.C., 2014a Refractive focusing of coherent waves. In Paper presented at the Ocean
Sciences Meeting, Honolulu, USA, 23–28 February. It forms the core of a journal manuscript to
be submitted later in 2014 to Ocean Modelling.
101
102
6. Narrow-band wave statistics over nearshore topography
Hasselmann, 1962). In the shallower waters over the continental shelves, ocean waves
are affected by refraction, shoaling, bottom friction, and eventually - near the coast second-order nonlinearity (or triad wave-wave interactions, Eldeberky, 1996; Janssen
et al., 2006), and depth-induced breaking (Hasselmann et al., 1973; Komen et al.,
1984; Battjes and Janssen, 1978; Salmon et al., 2014b). Although the instantaneous
action of some of these processes is highly nonlinear and difficult to quantify (e.g.
wave breaking, white-capping), on average their effect on the mean energy is weakly
nonlinear and well described in a stochastic framework formulated in terms of the
directional variance density spectrum (Komen et al., 1994; Holthuijsen, 2007).
As a result, for quasi-homogeneous wave fields, in which the mean wave statistics
(e.g. variance) varies on spatial scales much longer than a characteristic wave length,
the evolution of the variance density spectrum E(k, x, t) through time t, geographical
space x, and wavenumber space k is described by the radiative transport equation
(RTE),
∂t E + cx · ∇x E + ck · ∇k E = S.
(6.1)
The left side of (6.1) represents the conservation of wave energy in a slowly varying
medium, with cx and ck denoting transport velocities through geographic and spectral space, respectively (Willebrand, 1975). The forcing term, S(k, x, t), on the right
of eq. (6.1) represents source term contributions to account for non-conservative and
nonlinear processes. Apart from the quadruplet interactions (Hasselmann, 1962), the
source terms are mostly empirical and accompanied by large uncertainties. For this
reason, continuing development of the stochastic description has generally focussed
on improvements of the source term parameterisations on the right side of the RTE.
With proper tuning of the various source terms, and on larger scales, the RTE (eq.
(6.1)) often provides a very good approximation to the evolution of the mean wave
characteristics (Wise Group, 2007).
Apart from the usual WKB assumptions regarding the slow variations of the
medium, the statistical description of wave evolution using the RTE to propagate
the variance density spectrum E, implies the assumption that wave components are
statistically independent (Komen et al., 1994). In deep water, where medium variations are generally very weak, and the evolution is mostly dominated by the source
term balance, the premise of quasi-homogeneous (and Gaussian) statistics often holds
very well. However, in progressively shallower water, interaction with the slowly varying topography (or currents) becomes increasingly more important, and scattering of
narrow-band wave fields can cause coherent interferences in coastal wave fields, which
can result in fast scale oscillations of the variance as the wave components move in
and out of phase with one another (Janssen et al., 2008). For broad-banded waves
such coherency effects - although still present - are suppressed in the mean statistics, which are thus well described by the RTE. (Vincent and Briggs, 1989; O’Reilly
and Guza, 1991). In contrast, for narrow-band waves, coherent interferences can
have an O(1) effect on the mean statistics, in particular when the topography varies
on scales shorter than the coherent length scale the wave field (Smit and Janssen,
2013b, SJ13 hereafter). In the latter case, since the wave field de-correlates slower
than the medium varies, it retains a memory of the coherency introduced by the
medium scattering. The RTE keeps no track of wave field coherence, and to account
6.2. Evolution of inhomogeneous wave fields
103
for these effects SJ13 derived a new evolution equation, which they referred to as the
Quasi-Coherent Approximation (QCA) and represents a generalization of the RTE
(left side of (6.1)) to account for the generation and propagation of inhomogeneous
effects in the wave field.
In the present work we revisit the theoretical result from SJ13, develop a consistent numerical implementation for this model, validate the model against laboratory
and field observations, and explore the use of the additional information available in
the coupled-mode spectrum (as opposed to the variance density spectrum). Thereto
we briefly summarize the principal results from SJ13 (see §2), derive and discuss a
consistent approximation for medium variations based on the wave field de-correlation
length scale (§2.2), and present simulations with the new model of laboratory flume
experiments (§3) and field observations of ocean waves across a nearshore submarine
canyon (NCEX, see Magne et al., 2007).
6.2. Evolution of inhomogeneous wave fields
To describe the evolution of inhomogeneous surface wave statistics in a variable
medium, we consider the transport of what we refer to as the Coupled-Mode (CM)
spectrum (see SJ13), defined as
E(k, x, t) = Fξ,k {Γ(ξ, x, t)},
(6.2)
where Fξ,k {. . . } denotes the Fourier transform from spatial lag ξ to wavenumber k
(see Appendix A), and the correlation function Γ is defined as
Γ(ξ, x, t) =
1
hζ(x + ξ/2, t)ζ ∗ (x − ξ/2, t)i.
2
(6.3)
Here ζ(x, t) is a complex, zero-mean, Gaussian variable, of which the real part
is the surface elevation η(x, t) = Re{ζ(x, t)} and the imaginary part its Hilbert
transform (see e.g. Mandel and Wolf, 1995), and h. . . i denotes the ensemble average. Since Γ is symmetrical with respect to the separation distance ξ, it follows
that Γ(ξ, x, t) = Γ∗ (−ξ, x, t), and thus E is real. Although the Coupled-Mode
(CM) spectrum E is generally not pointwise positive, and can thus generally not
be interpreted
´ as a variance density function (see SJ13), the marginal distribution
V (x, t) = E (k, x, t) dk does represent the bulk wave variance. The CM spectrum represents the complete second-order wave statistics, including cross-variance
contributions, which can produce negative contributions to the spectrum. Such contributions are omitted in the variance density spectrum, which only accounts for
variance (and not cross-variance) contributions.
Under the assumption that the wave field consists of progressive plane surface
gravity waves propagating through a slowly varying medium, such that the wavenumber and frequency are related by a linear dispersion relation, an evolution equation
for the coupled-mode spectrum, can be derived (see SJ13)
ˆ
b (k, q) E k − 1 q, x + ∗,
(6.4)
∂t E (k, x, t) = −i dq exp [iq · x]Ω
2
104
6. Narrow-band wave statistics over nearshore topography
b operates only on E, and is
where ∗ denotes the complex conjugate, and the kernel Ω
defined as
i
b (k, q) = σ
Ω
b− σ
bk k̃ · ∇x .
(6.5)
2
Here σ
b(k, q) denotes the spatial Fourier transform (x0 → q) of the dispersion relation,
such that σ
b(k, q) = Fx0 ,q {σ(k, x0 )}. Furthermore, in equation (6.5), σ
bk is short for
∂k σ
b, the wavenumer q = [q1 , q2 ] is associated with spatial variations of the medium,
and k̃ = k/k (with´k = |k|).
˜ For brevity, here and in the remainder of this paper, we
use the shorthand dq = dq1 dq2 . In the following, we assume that the dispersion
relation σ and σk are given by the relations obtained from linear theory as
p
σ 1
kh
σk =
+
(6.6)
σ = gk tanh (kh)
k 2 sinh (2kh)
with g(=9.81m2 /s) the gravitational acceleration, and h(x) the mean depth.
Combined, the equations (6.4) and (6.5) summarize the principal theoretical result
from SJ13 and represent the starting point of this work. Notably, (6.4) includes the
RTE as a special case, which we will show in what follows.
Medium variations in a coherent wave field
To develop (6.4) into a form that is more easily related to the RTE, and to derive
a consistent numerical implementation, we consider some properties of this equation
that require attention.
From the Fourier transform on the right side of eq. (6.4) it is clear that in this
form the evolution of the Coupled-Mode spectrum depends on medium variations
throughout the entire spatial domain. Not only is this impractical from a numerical
viewpoint, it is also unexpected from physical considerations. After all, the wave
field has a finite decorrelation length scale so that medium variations outside this
coherent radius should not affect the local statistics.
In the following we consider that the slow medium variations are characterized by
the small parameter 1, so that the medium varies O(1) over distances L0 /, with
L0 being a characteristic wave length. Inhomogeneities in the wave field, induced
through medium variations, cause O(1) variations in the wave field statistics on the
scale L0 /µ, with µ 1, being a measure of the wavenumber mismatch between
coherent components. The width of the spectrum is measured by the parameter
δ = ∆k/k0 , where ∆k is a characteristic width and k0 = 2π/L0 . The latter is used
to define a coherent length scale as ξc = L0 /δ so that Γ(ξ, x) → 0 as |ξ| → ξc .
For narrow-band waves δ 1, implying that the wave field remains correlated over
many wavelengths. To relate the coherent radius to the variations in the medium,
we consider the ratio β = /δ, such that for β 1 changes in topography occur over
distances much larger than the decorrelation scale, whereas for β = O(1) significant
changes occur within the coherent radius of the waves. The latter parameter, β, is
of great importance in making suitable approximations to the stochastic model.
To illustrate how the coherent radius constrains the effect of medium variations,
and thus derive a consistent and local approximation, we write (6.4) as
∂t E (k, x, t) = G(k, x, t) + ∗,
(6.7)
105
6.2. Evolution of inhomogeneous wave fields
where
−i
G=
(2π)4
¨
1
Ω (k, x0 ) Γ (ξ, x) δ(x + ξ − x0 ) exp [−ik · ξ] dξdx0 .
2
(6.8)
−1
b
Here we substituted Fx0 ,q {Ω(k, x0 } for Ω(k,
q) and Fξ,k
{Γ(ξ, x, t)} for E(k, x, t).
For waves with a finite coherent radius ξc , we have that Γ (ξ, x) = 0 for ξj > ξc so
that the integration over ξ can be limited to |ξj | ≤ ξc , which in turn implies that
|x0j − xj | ≤ ξc /2, and (6.8) can be written as
−i
G=
(2π)4
˚
1
0
Ω (k, x + x ) Γ (ξ, x) exp iq ·
ξ − x − ikξ dξdx0 dq.
2
0
|x0j |≤ξc /2
|ξj |≤ξc
(6.9)
In (6.9) the evolution of the statistics is thus only affected by medium variability
within the region of statistical dependence, as would be expected on physical grounds.
Although here we make such approximations explicitly, the same approximation is
implicit in the RTE. After al, if we assume that the wave field is relatively broadbanded with a small coherent radius (relative to the medium variations), we have
that β 1. In this case, the wavefield decorrelates before significant changes in the
topography occur and it is reasonable to express the medium variability as a local
Taylor expansion around x0 = 0, so that Ω → ΩRTE , with
Ω(k, x + x0 ) ≈
ΩRTE (k, x, x0 ) = σ + x0 · ∇x0 σ +
i
(σk + x · ∇x σk ) k̃ · ∇x . (6.10)
2
Substituting the local approximation (6.10) into eq. (6.9), we can show that (6.7)
reduces to the RTE as in
rte
∂t E = GRT E + ∗ = −crte
k · ∇k E − cx · ∇x E
(6.11)
rte
rte
where crte
k = −∇x σ and cx = ∇k σ. Note that ck is determined by the slope and
rte
cx by the intercept of the (local) plane approximations to σ and ∂k σ, respectively
(see equation (6.10)). The RTE is thus a special case of the transport equation (6.7),
which is obtained when assuming that the wave field decorrelates on a much faster
scale than the medium varies (β 1), in which case only the local bathymetry (depth
and bottom slope) are relevant to the evolution of the wave field statistics. In reality,
wave fields described by the RTE have a finite spectral width and coherent radius
and whether the assumptions implied by the RTE are reasonable depends entirely
on the nature of the medium variations and the width of the spectrum.
To write the general transport equation (6.7) in a form similar to the RTE, but
with an additional source term that accounts for the development of inhomogeneities
by medium variations (e.g. coherent interferences), we write
Ω(x + x0 ) = Ωplane (x, x0 ) + ∆Ω(x, x0 ),
(6.12)
106
6. Narrow-band wave statistics over nearshore topography
so that ∂t E = Gplane + GQC + ∗, and thus
∂t E + ck · ∇k E + cx · ∇x E = GQC + ∗
(6.13)
lop
e
Analogous to ΩRTE , Ωplane represents a plane approximation but with the slope
and intercept defined somewhat more general to ensure that ck and cx represent
the topographical variations within the coherent radius (instead of a strictly local
approximation). Such variations, due to small local disturbances – even when β 1
– might differ significantly from a strictly local first-order Taylor expansion (see
Fig. 6.1). Instead, to achieve better correspondence for the plane approximation
throughout the coherent footprint, we determine the plane coefficients for Ωplane by
means of a least-squares fit to σ and ∂k σ of which the slope and intercept then
define ck and cx , respectively (see Fig. 6.1). Note that the definition of Ωplane only
influences the relative significane of Ωplane compared to ∆Ω, it has no influence on
the resultant operator Ω.
(a) QC-Remainder Least squares
Lo
ca
ls
σ
Least squares
non-windowed
windowed
(b) QC-Remainder Local slope
non-windowed
Coherent zone
windowed
x
Figure 6.1: Sketch of a local slope and a least squares fit to σ (or σk ) within the predefined coherent
zone. Panel (a) and (b) illustrate the remainder which is absorbed into the QC approximation
when using a windowed (black) or non-windowed (gray) least-squares (a) or a local slope (b) plane
approximation. The least squares approximation likely leads to smaller differences and a less abrupt
relaxation to the plane approximation.
The source term GQC is approximated through discretizing the integral over q
such that q = [mq1 ∆q, mq2 ∆q], with ∆q = 2π/ξc , so that the complete coherent
footprint is included, and GQC can be written as
¨
−i(∆q)2 X
b q (k, x) Γ (ξ, x) exp iξ 1 q − k dξ.
GQC =
∆
Ω
(6.14)
(2π)4 q
2
Since the truncation of the integration domain implies that ∆Ω is periodic, we apply
a Tukey window so that ∆Ω̂q = F̄x0 ,q {W ∆Ω} to prevent any errors due to possible
6.3. Wave deformation by an elliptical shoal
107
jump discontinuities. At the extremities of the domain (where the wave field is nearly
decorrelated), Ω is thus relaxed to the Ωplane . Note that now the the definition of
Ωplane is significant, as it determines how the far-field representation of Ω is included.
Upon substituting (6.14) into (6.7), and upon applying the Fourier integral with
respect to ξ in (6.14), we thus have
X
b q (k, x) E (k − q/2, x) + ∗.
∂t E + ck · ∇k E + cx · ∇x E = −i
∆Ω
(6.15)
q
The equation (6.15) is a consistent approximation of the Quasi-Coherent theory developed by SJ13 for variable bathymetry, and can be readily numerically evaluated. In
what follows we will refer to (6.15) as the Quasi-Coherent approximation. Although
the equation (6.15) closely resembles the RTE with a source term to incorporate
inhomogeneous effects, there are some subtle, but important, differences. After all,
the transported variable E is generally a coupled-mode spectrum, and not a variance
density, and the transport velocities ck and cx are not strictly local but defined as
the slope and intercept of a least-squeares fit to σ and ∂k σ, respectively.
In the simulations presented in this work, we consider steady state solutions to
equation (6.15) (such that ∂t E = 0) computed on a discrete grid by means of a finite
difference approximation using an algorithm similar to that found in (Booij et al.,
1999). For details on the implementation we refer to appendix B.
6.3. Wave deformation by an elliptical shoal
A monochromatic, unidirectional wave field can be considered as the archetype of
coherent wave fields – albeit mostly of academic interest. It also represents a rather
severe test for the stochastic model, since formally the QC approximation is based
on the premise that the wavefield has a finite coherent radius (or finite ∆q). For a
strictly monochromatic and unidirectional wave field we would formally have ξc → ∞
so that ∆q → 0, whereas in the stochastic model we will by necessity assume a
narrow, but finite-width spectrum (and thus a finite coherent radius). Despite this, a
monochromatic unidirectional example will most clearly show the differences between
the RTE and a QC approximation.
Therefore we consider the wave basin experiment by Berkhoff et al. (1982), where
monochromatic (period 1 s), unidirectional waves (wave height H = 0.0464 m) were
generated at the wavemaker (at x = −10, depth 0.45 m) and propagated towards a
shoal (crest located at x = 0 m, depth of 0.135 m) situated on a 1:50 slope (see Fig.
6.1). Wave heights were measured along 8 transects at regular intervals, of which we
consider 6 in the present work (see Fig. 6.2).
The spectral models are numerically evaluated on a rectangular spatial (20×20
m2 ) and spectral domain (10 × 10 rad2 /m2 , starting at k = [−0.05, −5] rad/m), uniformly discretized with mesh sizes ∆x = ∆y = 5 cm and ∆kx = ∆ky = 0.1 rad/m.
Moreover ∆q = 0.2 so that ξc = 31.4 m, and we include components q up to kp /2,
where kp is the peak wave length. At the boundary, the wave spectrum E is assumed
Gaussian, with the spectral peak at kp = 4.21 rad/m and a width of 0.2 rad/m; this
effectively corresponds to a narrow-band longcrested wavefield with a width of 0.1 Hz.
and 1.5◦ in frequency and directional space, respectively.
108
6. Narrow-band wave statistics over nearshore topography
Figure 6.2: Plan view of normalized wave heights obtained using the QC approximation, the RTE
and the ’linear’ SWASH model for the experimental setup by Berkhoff et al. (1982). Grey lines
indicate bottom contours (drawn between 0.4 m till 0.1 m at 0.05 m intervals), while the solid lines
correspond to a ray-traced solutions. The bottom right panel indicates the instrumented transects.
Although no wave breaking was observed in the experiment, the wave height to
depth ratio along the observed transects is large (H/d ≈ 0.5), in particular in the focal
region behind the shoal, so that nonlinear dynamics significantly influence the pattern
behind the shoal (see also results by Janssen, 2006). Because our focus is on linear
interference effects, we in include model simulations with the deterministic model
Surface WAves till SHore (SWASH Zijlema et al., 2011b), which effectively solves
the 3D Euler (or RANS) equations for a free-surface fluid of constant density. This
model reproduces the observations very accurately (see, e.g. Stelling and Zijlema,
2003; Smit, 2008), and from it – by reducing the incident wave height to H = 0.01
m – near-linear results are obtained. For brevity, we shall therefore refer to the
deterministic model forced with observed wave heights as the “nonlinear” SWASH
model, and the results obtained with the reduced wave height as the “linear” SWASH
model.
The SWASH model used the same model domain and horizontal resolution,
whereas the vertical was subdivided into two layers. A duration of 1 min was simu-
109
6.3. Wave deformation by an elliptical shoal
3
3
3
Transect 2
Transect 3
1
H′
2
H′
2
H′
2
1
0
−5
0
x (m)
1
0
−5
5
3
0
x (m)
0
−5
5
3
Transect 7
H′
2
H′
1
1
0
5
y (m)
10
5
Transect 8
2
H′
2
0
x (m)
3
Transect 6
0
Transect 4
0
1
0
5
y (m)
10
0
0
5
y (m)
10
Figure 6.3: Normalized wave height H 0 (normalized with incident wave height) along the indicated
transects. Comparison between observations (symbols), and relative wave heights obtained with the
QC-approximation (solid line), the RTE (dashed line), ’linear’ and ’nonlinear’ Swash (Crosses and
grey solid lines, respectively)
lated (using a time step of 0.01 s) of which the first 40 s where discarded to remove
transient effects pertaining to the model initialization. For the final 20 s the situation
was assumed stationary (with regard to mean parameters), and ζ was recorded at
20 Hz from which mean wave heights were obtained from a zero-crossing analysis (see
e.g., Holthuijsen, 2007).
The refractive focussing of the waves introduces a lateral interference pattern in
the wake of the shoal (Fig. 6.2). The fine-scale pattern is reproduced by the QC
model, and normalized wave heights correspond well with observations (Fig. 6.3). In
contrast, the quasi-homogeneous model (RTE) underestimates wave heights along the
central transect 7, and does not reproduce the lateral interference patterns evident in
transects 3 and 4 resulting in the dramatic difference in the pattern behind the shoal
(Fig. 6.2). Marked discrepancies between observations and the QC model do occur in
transect 8, which we attribute to the omission of nonlinear effects. This is confirmed
by comparison with the deterministic results, where the nonlinear model reproduces
the node near y = 8 m in transact 8, and produces overall better correspondence with
measurements, whereas the results from the linear model are consistent with those
of the QC model.
110
6. Narrow-band wave statistics over nearshore topography
Wave spectra and coherence functions
Figure 6.4: Coupled mode (QC and SWASH, left and right panels) or variance density (RTE, centre
panel) spectra at points A (top panels) and B (bottom panels) located down wave of the shoal (see
Fig. 6.2).
To compare the results in more detail, we further consider the wavenumber spectra
as calculated by the different models. Although we cannot verify this directly by
comparison to observations (as observed time series are not available), these can be
compared to spectra obtained from the linear SWASH results. To obtain the wave
spectrum from the SWASH results, we first approximate Γ(x, ξ) with the ensemble
mean, and then approximate the continuous CM spectrum with
E(k, x) ≈
F̄ξ,k {Γ(ξ, (x))}
,
∆kx ∆ky
(6.16)
where ∆kx and ∆ky are the wavenumer intervals in the discrete Fourier transform.
The spectrum is evaluated at two points A and B (located at xA = [7, 0] and xB =
[7, 2], respectively, see Fig. 6.2).
At point A, the ray-traced solution indicates that geometric optics predicts three
separate contributions within the spectrum (marked I,II and III), which are indeed
faithfully reproduced in the RTE model (see Fig. 6.4). Correspondence is not exact,
since the spectral representation of monochromatic waves – and numerical diffusion
– introduce widening of the peaks, but significant variance is located around those
6.4. Swell over submarine canyons
111
wavenumers also found in the ray-traced solution. Yet these results stand in stark
contrast with those from SWASH and the QC model, which predict a much richer
structure due to the occurrence of coherent interference. For instance, there is a large
positive band between I and II, related to the constructive interference behind the
shoal. At point B, this band becomes negative, indicative of the destructive interference and the rapid reduction in wave height that occurs at point B (see also Fig
6.3, transect 4 at y=2 m). More obscure are the alternating positive and negative
contributions that emerge at both points right of the main lobe, at wavenumber magnitudes well beyond those that occur in the geometric optics solution. Presumably,
these are related to changes in the mean statistics along the wave direction. Their
net contribution to the local variance is however small, as the alternating contributions tend to cancel one-another. Qualitatively, the QC solution agrees well with
those from SWASH, as the main features are all present, although the QC solution
is generally smoother.
Since the CM spectrum contains cross-variances, it is not a strictly local variable,
which can make it more difficult to interpret. However, the CM spectrum carries
additional information and this can be explored by considering Γ, which is related
to the directly observable covariance between two points. Although at large mutual
separations, such dependence cannot be accounted for within the QC model (or the
RTE), the real part of the co-variance function Γ should approximate the covariance
between η(x + ξ/2, t) and η(x − ξ/2, t) at small mutual separations. Moreover, the
modulus of the covariance function (|Γ|) removes the fast scale oscillations and thus
effectively captures the slow scale changes (or envelope) of the covariance function –
that is, it represents the changes in the mean statistics.
For a strictly monochromatic wave field, the wave statistics do not decouple anywhere in the basin, which is reflected in the envelope of the normalized covariances
function |Γ0A/B (ξ)|, where
Γ0A/B (ξ) =
Γ(ξ, xA/B )
,
ΓSWASH (0, xA/B )
(6.17)
which has covariance contributions at arbitrary spatial lags away from the centre1 .
If we consider cross-sections along the coordinate axis (Fig. 6.6), we can clearly
discern that |Γ0A/B (ξ)| → 0 in the far field (ξ/L 1) for both the RTE and the QC,
whereas the deterministic model can even predict far field covariance contributions
exceeding those at ξ = 0 (e.g, Transact 2, Point B). Despite this damped behaviour,
which reflect the finite coherence length inherent within stochastic models, it is evident that the QC approximation can resolve the second-order correlation matrix up
to ξ/L ≈ 5, including the fast oscillations due to the wave motion.
6.4. Swell over submarine canyons
Offshore of the coastline just north of San Diego, stretching from Black’s Beach
up until La Jolla point, in the Southern California Bight (see Fig. 6.7), the seafloor
1 Whereas
in a homogeneous field Γ thus normalized would be strictly contained between -1 and 1,
this is not the case in a spatially inhomogeneous field
112
6. Narrow-band wave statistics over nearshore topography
Figure 6.5: Normalized modulus of Γ0 (ξ, x) (normalized with local variance) at points A (top panels)
and B (bottom panels, see Fig. 6.2 for locations) as predicted by the QC approximation, a RTE
based model and SWASH (left to right). The spatial-lag coordinates ξx and ξy are scaled with the
local wave length L.
Transect 1
Γ′
1
0
−1
−1
2
4
6
8
10
point B
2
0
0
−2
−2
2
4
6
ξ/L
8
10
2
4
6
8
10
point B
2
0
0
point A
1
0
0
Γ′
Transect 2
point A
0
2
4
6
8
10
ξ/L
Figure 6.6: Normalized covariance functions |Γ0 | (shaded regions, QC: Dark, RTE: red, SWASH:
light) and Re{Γ0 } (lines, QC: solid line, RTE: dashed line, SWASH: markers), both normalized with
local variance. The functions are evaluated at points A (top panels) and B (bottom panels) along
the indicated transects. The spatial-lag coordinate ξ is scaled with the local wave length L.
113
6.4. Swell over submarine canyons
40
40
34
n
Ca
160
0
12
32
14
15
25
31
1
29
0.5
13
28
26
17
16
30
27
10
20
1
36
37
40
y (km)
ps
rip
Sc
yon
20
24
22
33
La
−5
x (km)
0
n
−10
0
0
0.5
80
o
any
La Jolla Point
5
aC
Joll
−1
−2
Black’s Beach
160
0
40
80
160 200
120
80
1
0
12
160
4
3
2
23
2
1.5
6
5
35
510
y (km)
2
20
80
200 20
1260001
3
20
4
7
80
105
40
120
5
8
2.5
80
0
20
6
3
20
7
120
Torres Peynes
160
8
1
1.5
x (km)
2
2.5
Figure 6.7: Bathymetry near Scripps and La Jolla canyon (left and right panel), and locations of
observation points (right panel). Contourlines drawn at depths of 5, 10, 20, 40, 80, 120, 160 and
200 m. The origin is located at a latitude and longitude of 32◦ 49.70 N and 117◦ 21.90 W. with the
positive coordinate axis orientated along the North (y-axis) and East (x-axis) direction.
bathymetry is characterized by two steep submarine canyons, Scripps canyon (∼150 m
deep and ∼250 m wide) and La Jolla canyon (∼120 m deep and ∼350 m wide). Along
these canyons (see Fig. 6.7), which extend to 200 m from the shore, strong wave
refraction occurs due to steep slopes along the canyon walls (locally exceeding 45◦ ).
Specifically for swell waves, refraction can introduce large spatial gradients in wave
height in these circumstances, so that locally coherent effects such as diffraction may
become important.
In the fall of 2003 the Nearshore Canyon Experiment (NCEX) was conducted
to study wave transformation over the canyons, with a particular focus on Scripps
Canyon. Pressure sensors, Waverider directional buoys and NORTEK vector current
meters (PUV2 ) were deployed around the canyons, of which a subset of 18 pressure sensors, 5 buoys and 12 PUV’s are used in the present study (see Fig. 6.7b
for locations). The offshore wave conditions were recorded by the permanently deployed Outer Torres Peynes directional Waverider buoy, that is located approximately
12.5 km offshore at 549 m depth (see Fig. 6.7a).
For the pressure sensors and PUV’s, variance density spectra are obtained from
the de-trended 3 h time-record which is subdivided in windowed segments with 50%
2 So
called because they measure Pressure, the two horizontal particle velocities ’U ’ ’V ’ and in
addition the vertical component W .
114
6. Narrow-band wave statistics over nearshore topography
Table 6.1: Model parameters used for the different nested grids I to III. The spectral resolution was
set to ∆k = 0.01 rad/m in all cases.
Grid
I
II
III
x
(km)
-11
-8
0
y
(km)
-25
-2
0
∆x
(m)
50
25
25
Nx
(-)
180
120
100
Ny
(-)
660
360
200
kx
rad/m
-0.0205
-0.0205
-0.0205
ky
rad/m
-0.0605
-0.1005
-0.1505
Mx
(-)
121
121
171
My
(-)
121
201
301
overlap. These are Fourier transformed and ensemble averaged to yield estimates of
the pressure spectrum (with 50 degrees of freedom and frequency resolution ∆f =
0.0025 Hz). Subsequently, the band-passed (0.03 Hz< f < 2 Hz) free surface spectrum is obtained using a transfer function from linear theory. For the wave buoys,
the spectrum is estimated by averaging six spectra, each obtained from 26 min-long
records, with ∆f = 0.0025 Hz, yielding approximately 50 degrees of freedom. The
directional spectrum, needed to force the models at the offshore boundary, is estimated from the measured first four directional Fourier moments using a Maximum
Entropy Method (Lygre and Krogstad, 1986). The spectral models are numerically
evaluated on a set of nested rectangular spatial grids (see Tab. 6.1). The coarsest
grid (I) is forced by the buoy data, and extends well beyond the southern border
of Fig. 6.7a to ensure that southern incident swell is well reproduced. The finest
grid (III) is focused near the NCEX site, and coincides with the area depicted in
6.7b. Moreover, going from I to III, the spectral domain is successively enlarged to
ensure that the wave spectrum falls within the computational domain. The RTE was
employed on grid I, because those parts that affect the subsequent nested grids are
mostly located in sufficiently deep water, so that differences between the RTE and
QC are expected to be small. For the remaining domains, results where obtained
using both the RTE and the QC model.
From the 3-month period during which the NCEX experiment was conducted,
we selected two cases to compare the QC and RTE with observed wave conditions.
Because coherent effects are expected to be most dominant for directionally narrow
fields (Smit and Janssen, 2013b), we consider two cases in which narrow swell waves
are incident from either the west or the south.
Southern swell
On November 16, 2003, the wave field at the NCEX site, was composed of a distinct
swell peak from the south, and a mixed sea-swell system incident from the west (see
Fig. 6.8). The south swell had a mean (nautical) direction of 197◦ , a peak period of
18.2 s, significant wave height of Hm0 = 0.45 m and a narrow directional spreading
of 7◦ (defined according to Kuik et al., 1988)). The sea-swell system was more
energetic (Hm0 = 1.26 m) and comprised of shorter more directionally spread waves,
with the swell and sea peaks located at 11.8 s and 5.8 s, respectively. In the present
analysis we only consider the southern swell peak at the up-wave boundary. This
peak is distinctly located in the directional spectrum E(f, θ) between 145◦ to 235◦
degrees for f < 0.07 Hz, the other contributions to the wave spectrum are discarded.
6.4. Swell over submarine canyons
115
Figure 6.8: Directional wave spectra (top panels) and integrated frequency spectra (bottom panels)
as observed at the Torres Peynes outer buoy on the 16th and 30th of November, 2003. The arc in
the directional spectra indicates the swell fields considered in the present work. Frequency spectra
are integrated over the indicated directional range (solid line) or over all directions (dashed lines).
Observed spectra are also band-pass filtered for 0.025 < f < 0.07 to remove the
contributions of the mixed sea-swell system, and to remove low-frequency ig-wave
energy generated by nonlinear interactions.
For southern swell, the QC model predicts that a significant part of the energy is
refracted towards the coast before it arrives at the NCEX site due to the relatively
shallow region south of the canyons (see Fig. 6.7). Hence, wave energy is already
much reduced as it arrives at the NCEX site (see Fig. 6.9). Because these are long
period waves, they subsequently reflect strongly over the steep canyon walls, which is
clearly visible in the bands of enhanced mean wave height along the south part of La
Jolla, and the Northern part of Scripps canyon. In between the canyons, the convex
shape of the topography focusses wave energy so that a mild focal zone emerges. In
between this focal zone and either of the two canyons virtually no wave energy is
able to penetrate.
Qualitatively, differences between the QC and RTE predictions appear to be minor. The only significant difference in pattern is found north of the tip of Scripps
Canyon. To compare the models in more detail with each-other, and with observa-
116
6. Narrow-band wave statistics over nearshore topography
Figure 6.9: Normalized wave heights (normalized with incident wave height) near the NCEX site
on 16 November, as predicted by the QC model (Left panel) and by the RTE (right panel).
1.5
1.5
D =50 [m]
D =15 [m]
1
0.5
0.5
H′
1
QC
RTE
Obs.
0
0
1
2
3
4
0
5
1.5
0
1
2
3
4
1.5
15 10
D =10 [m]
H′
y (km)
1
0.5
0
5
0
1
2
3
s (km)
4
5
50
1
10
50
0.5
0
50
15
0
0.5
1
1.5
x (km)
2
2.5
Figure 6.10: Normalized wave height H 0 along the indicated depth contours on 16 November. The
coordinate s represents the along-contour distance measured from the starting point (S = 0, red
dot) of the contour lines as indicated in the lower right panel.
6.4. Swell over submarine canyons
117
tions, we consider transects of the wave height along different depth contours (Fig.
6.10). Overall correspondence between the observations and both models is very
reasonable, specifically for the 50 m and 15 m contour lines. In these, there is very
little to distinguish between the RTE or the QC approximation. Only along the
10 m contour line, starting at the eastern end of Scripps Canyon (around s = 2.5 km)
do significant differences between the observations and the models, and interestingly
in-between the models themselves, show-up.
In this region waves that are refracted out of the canyon in a Northern direction
cross waves that travel in a roughly south-western direction. Because for narrowband swell the wave field remains coherent over significant distances, they interfere
with one-another, which results in the fairly rapid oscillations found in the observed
wave heights. This pattern is qualitatively reproduced (at least near s = 3 km) in
the QC model, though the size of the oscillations is underestimated. In contrast, the
RTE model merely shows an enhanced sea state, but fails to reproduce oscillatory
behaviour.
Western swell
On November 30, 2003, a westerly swell (mean direction 272◦ ) with a peak period
of 15.4 s, significant wave height Hm0 = 0.77 m, and a directional spreading of 9◦
(defined according to Kuik et al., 1988), was measured at the Torres Peynes outer
buoy. For this case, we band-pass filter the boundary signal and the observations
between 0.025 < f < 0.1.
This system, owing to the steep drop-off west of the NCEX site, arrives at the
eastern boundary with wave heights similar to those off-shore. The subsequent pattern is again dominated by the local geometry of the Canyons. Similar as before,
waves are refracted out of the canyons so that along the canyon walls we again see
a band of enhanced wave energy, and also a focal region in between the two canyons
(Fig. 6.11). However, due to the incidence angle of the waves, and the larger directional distribution off-shore, more wave energy is able to penetrate the regions
between the focal zone and the respective canyons.
Qualitatively, differences are again small, but more pronounced then before. The
QC solution, in particular in the focal region, is fairly smooth, whereas the RTE
produces a more banded pattern of energy. This difference is likely because the RTE
reacts more volatilely to local disturbances in the topography; a local steep slope can
introduce strong refraction. In the QC model, which accounts for the topography
within the entire coherent footprint, such a local topographical feature becomes less
significant for the evolution of the wave statistics (see also Fig. 6.1).
This difference in behaviours is also visible when considering cross-sections of
the wave height along different contour lines. Again, both models are very similar
around the 50 m contour. However, in this case the RTE shows rapid oscillations in
the region between the canyons (s < 2.5) along the 15 m contour line, associated with
the banded patterns mentioned above. The QC appears to be in better agreement
with observations as neither the QC nor the observations reproduce these oscillations.
Though, the magnitude of the oscillations is small, and the RTE results are still in
relatively good agreement with the observations.
118
6. Narrow-band wave statistics over nearshore topography
Again, on the 10 m contour the most significant difference occurs North of Scripps
canyon, where the observations indicate the occurance of a single peak followed by
fairly homogeneous conditions north of the canyon (s > 3 km). This behaviour is
faithfully reproduced by the QC, though wave heigts are somewhat overestimated
for (s > 3 km). However, in this case the RTE produces two distinct peaks, where
the erroneous secondary peak (at s = 3 km) locally overestimates the wave heights
by 20%. Further north, where the topography has almost parallel depth contours,
the QC and RT are yet again virtually indistinguishable.
6.5. Discussion and conclusions
In the present we considered the application of the quasi-homogeneous wave theory
for random wave propagation in a varying medium as presented by Smit and Janssen
(2013b). To derive a consistent model suitable for large scale application over complex topography, we first demonstrated that for wave fields with a finite coherence
length, only the topographical variation within the coherent footprint influences the
evolution of the wave statistics. Subsequently, on the premise of a finite coherence
length we derived an augmented form of the radiative transport equations (RTE),
which includes a scattering term accounting for the generation of coherent contributions within random wave fields, and which we refer to as the Quasi-Coherent (QC)
approximation. Comparison with a monochromatic wave experiment of wave scattering over a submerged shoal, demonstrated that, even when the wave field remains
correlated over large distances, the resulting model is able to capture not only the
bulk parameters, but also provides a good approximation to the wave spectrum and
to its Fourier transform, the covariance function. Moreover, hind-casts of swell wave
conditions that occurred during the ONR NCEX experiment show that wave heights
predicted by the QC agree with observations, and are a slight improvement on those
predicted by the RTE.
The modelling paradigm introduced in the present study extends conventional
stochastic models to include the effects of coherent interference. This not only gives
a more complete description of the bulk second-order statistics in case of complex topography, but also describes the covariance structure of the waves – which cannot be
predicted using quasi-homogeneous theory alone. In particular for narrow-band swell
conditions, focussing of the waves migth introduce coherent interference patterns in
the wave field that require an extended description of the covariance structure to
fully capture the fast spatial variations in the wave field. Moreover, an extended description of the second-order statistics is not only important for predicting the bulk
wave parameters, but due to the resulting fast scale gradients in the momentum flux
(or radiation stress), also influences the mean circulation and through it nearshore
transport processes. For these reasons, extension of existing third-generation wave
models to include the QC scattering term, can help improve the prediction of the
wave statistics, and possible wave-driven circulations, in complex topographical enviroments.
119
6.5. Discussion and conclusions
Figure 6.11: Normalized wave heights (normalized with incident wave height) near the NCEX site
on 30 November, as predicted by the QC model (Left panel) and by the RTE (right panel).
1.5
1.5
D =50 [m]
D =15 [m]
1
0.5
0.5
H′
1
0
0
1
2
3
4
0
5
1.5
0
1
2
3
1.5
y (km)
H′
QC
RTE
Obs.
0.5
0
0
1
2
3
s (km)
4
5
50
1
50
0.5
0
5
15 10
D =10 [m]
1
4
0
0.5
50
10
15
1
1.5
x (km)
2
2.5
Figure 6.12: Normalized wave height H 0 along the indicated depth contours on 30 November. The
coordinate s represents the along-contour distance measured from the starting point (S = 0, red
dot) of the contour lines as indicated in the lower right panel.
120
6. Narrow-band wave statistics over nearshore topography
Acknowledgments
This research is supported by the U.S. Office of Naval Research (Littoral Geosciences
and Optics Program and Physical Oceanography Program) and by the National
Oceanographic Partnership Program. We would like to thank dr. M. Zijlema, who
supplied the initial version of the BiCGSTAB solver used in the present work, and
dr. L. Holthuijsen, who, through his continuous support of PBS, made the present
work possible.
Appendices
6.A. Fourier transform operators
As we make frequent use of continuous and discrete Fourier transforms, it is convinient to introduce the following operators. Let ζ be a dummy continuous function,
b
b
for which we denote the conjugate pair as ζ(x), ζ(k),
such that ζ(k)
= Fx,k {ζ(x)}
−1 b
and ζ(x) = F {ζ(k)}, with
k,x
ˆ
1
ζ(x) exp [−ik · x] dx
Fx,k {ζ(x)} =
(2π)2
ˆ
−1 b
b exp [ik · x] dk
Fk,x
{ζ(k)} = ζ(k)
(6.18)
(6.19)
In a similar fashion, we define the discrete conjugate Fourier transform pair for a
periodic function ζ(x) with period L, as ζ(x), ζbk where
ˆ L/2
1
ζ(x) exp [−ik · x] dx
(6.20)
ζbk = F̄x,k {ζ(x)} = 2
L −L/2
X
−1 bk
ζ(x) = F̄k,x
{ζ } =
ζbk exp [ik · x].
(6.21)
k
From these definitions we see that, for a function that is stricly zero outside a certain
rectangular domain centred at the origin, and with sides smaller than L, the sampled
b
values of ζ(k) at k = [m1 , m2 ]∆k (with ∆k = 2π/L) are equal to ζ(k)
= ζbk /(∆k 2 ).
6.B. Discrete model
To consider the numerical solution of the QC approximation, we consider the solution
on a discrete regular rectangular mesh in both geopgraphical and wavenumber space.
For the spatial and wavenumber mesh we set
xmx1 ,mx2 = x0 + [mx1 ∆x1 , mx2 ∆x2 ],
kmk1 ,mk2 = k0 + [mk1 ∆k1 , mk2 ∆k2 ],
(6.22)
with mxj ∈ {0 . . . Mjx }, mkj ∈ {0 . . . Mjk } and where x0 ,k0 denote the coordinates
of the lower left corner of the geographical and wavenumber grid, respectively. For
brevity, we denote the discrete coupled mode spectrum evaluated at xmx1 ,mx2 , kmk1 ,mk2
as E k,x (t), where the dependency on the subscripts mxj , mkj is implied. With these
definitions in place the spatially-discrete, stationary, version of the Quasi-Coherent
approximation can be expressed as
X ik̃
q
k,x
k,x
q
ck · D k E
+ cx · D x E
= −i
∆b
σ − ∆b
σk · Dx E k−q/2,x + ∗ (6.23)
2
|q|≤qmax
121
122
6. Narrow-band wave statistics over nearshore topography
Here Dx and Dk denote linear finite difference operators which approximate ∇x or
∇k , respectively, by means of second-order upwind approximations. The operator
Dx E k,x is defined as
j
Dxj E k,x = sj
3E k,x − 4E k,x−∆x̃ + E k,x−2∆x̃
2∆xj
j
(6.24)
with s = sgn(k) and ∆x̃j = sj [δ1j ∆x1 , δ2j ∆x2 ] where δij denotes the Kronecker
delta. The operator Dk Ek,x is defined analogously, but then along the spectral dimensions
j
j
3E k,x − 4E k−∆k̃ ,x + E k−2∆k̃ ,x
k,x
Dkj E
= s˜j
(6.25)
2∆kj
where s̃ = sgn(cx ) and ∆k̃j = s̃j [δ1j ∆k1 , δ2j ∆k2 ].
To exclude interactions between waves and topographical variations on the infrawave scale, which are excluded at the order O() considered, the sum over q is
restricted to |q| ≤ qmax . Here, qmax is the minimum of |k|/2 or a pre-described
maximum bottom wavenumber component. When solving for the RTE, we disregard
this sum altogether.
At the geographic boundary (along the lines mx1 = 0, mx1 = M1x and mx2 =
0, mx2 = M2x ) the wave spectrum is prescribed for wavenumbers directed into the
computational domain. For the spectral boundary (along the lines mk1 = 0, mk1 = M1k
and mk2 = 0, mk2 = M2k ) we assume that it is located at wavenumbers which are
effectively deep water waves, so that the interaction with components outside the
computational domain (assumed to be zero) can be neglected. Moreover, at points
adjacent to the geographic or spectral boundary (e.g., along the line mx1 = 1) firstorder approximations are used if eqs. (6.24) or (6.25) reference points outside the
computational domain. Here the discrete operators Dxj and Dkj are locally reduced
to
j
Dxj E k,x = sj
E k,x − E k,x−∆x̃
,
∆xj
Dkj E k,x = s˜j
E k,x − E k−∆k̃
∆kj
j
,x
,
(6.26)
respectively.
Due to the second-order approximations, the resulting numerical model does not
preserve monotonicity in the solution, which might give erroneous results near sharp
spatial (or spectral) gradients in the spectral densities. For the QC model, which
generally produces smooth results, this is not an issue problem. However, in case
of the RTE, which does introduce strong spatial gradients in focal regions (e.g., as
in Fig. 6.2), this can introduce numerical artefacts in the solution, such as negative
spectral densities. In principle, this can be addressed by for instance introducing
total variation diminishing approximations for the discrete operators (e.g., Hirsch,
2007). However, these approximations are nonlinear, and therefore would introduce
additional complexity in the iterative solution algorithm introduced below. In the
problems considered here, because this is only an issue in a few points of the domain,
this is considered overly complex. Instead, we avoid negative values in the RTE by
simply reducing the discrete operator to first order approximations whenever negative
densities occur in the iteration process described below.
123
6.B. Discrete model
Coefficients
When considering the RTE, the geographic and spectral propagation velocities are
defined as cx = k̃σk and ck = −∇x σ and ∆σ̂ q = ∆σ̂kq = 0. However, for the QCapproximation, ck , cx in eq. (6.23) are obtained from a local least-squares approximation to σ, σk , whereas the remainder is used to calculate ∆σ̂, ∆σ̂k . To calculate
these, we define a local geographic grid x0 over the coherent footprint centred at x,
and its conjugate set of wavenumers q as
0
0
x0mx0 ,mx0 = [mx1 ∆x1 , mx2 ∆x2 ],
1
2
qmq1 ,mq2 = [mq1 ∆q1 , mq2 ∆q2 ],
(6.27)
0
with mxj , mqj ∈ {−Mjq /2 . . . Mjq /2}, Mjq = 2qmax /∆qj , and ∆xj = 2π(Mjq ∆qj )−1 .
To avoid interpolation it is convenient to ensure the sum in eq. (6.23) over q coincides
with the k grid, so we set ∆qj = 2∆kj , assuming that ∆kj ≤ 2π/ξc .
With these definitions in place, we define a plane approximation on x0 to ck , cx
as
plane
σ(k)
(k, x0 , x) = α(k) + β(k) · x0 ,
(6.28)
where the optional subscript (k), is omitted to obtain the expression for σ plane ,
or replaced with k to denote σkplane . Furthermore, the slope and intercept coefficients α(k) , β(k) are obtained from least squares approximations to σ(k, x + x0 )
and σk (k, x + x0 ). From these, the propagation speeds are defined from the slope
and intersect as ck = −β and cx = αk k̃, so that the QC remainder, defined as the
difference between the plane approximation and the exact expressions, becomes
plane
∆σ(k) (k, x0 , x) = σ(k) (k, x + x0 ) − σ(k)
(k, x0 , x)
(6.29)
Arguable, ∆σ̂, ∆σ̂k can now be obtained by a discrete Fourier transform. However, to
avoid errors due to jump discontinuities between the non-smoothly matching domain
borders in a periodic extention of ∆σ(k) , we define the transform as
∆σ̂(k) = F̄x0 ,q {W1 (x01 )W2 (x02 )∆σ(k) }
(6.30)
where Wj (x0j ) are window functions that smoothly transistions to 0 near the edges
of the domain, and for which in the present work we use a tapered cosine (Tukey)
window which is given by
 1 1
if x0j > (1 − γ)lj
 2 + 2 cos π (x0j − lj + γlj )/(γl
j)
0
1
1
0
Wj (xj ) =
if x0j < γlj
.
(6.31)
2 + 2 cos π xj /(γlj ) − 1

1
elsewhere
Here, lj = 2π/∆qj is the length of the j th side of the x0 domain, and γ is a dimensionless width parameter controlling the length of the transitional area where W → 0
and which is set to γ = 0.1.
Iterative solution technique
The resulting set of equations involve M = M1x M2x M1k M2k variables so that solving
for the steady state involves inverting a large sparse M × M matrix. Solving this
124
6. Narrow-band wave statistics over nearshore topography
system directly is difficult, as even for a moderate number of grid point in each of the
for dimensions the total number of points quickly becomes large. However, because
propagation principally occurs in geographic space, and is dominated by the LHS
of (6.24), the system can be solved iteratively in a marching fashion, similar to the
method employed in Booij et al. (1999), explained in detail in (Zijlema and van der
Westhuysen, 2005).
Hereto, k space is subdivided into four quadrants, each bounded by the cartesian
axes, which we will number 1 till 4 in a counter-clockwise fashion, where quadrant
1 is the set Q1 = {k, x|k1 > 0 ∧ k2 ≥ 0}. During a single Gauss-Seidel iteration,
each of the four quadrants of the spectral domain is visited consecutively using four
sweeps per iteration. During each sweep, only points that belong to the quadrant are
updated. For example, during iteration step n, in the mth sweep (nm ) the quadrant
m is considered, and for all points P = [k1 , k2 , x1 , x2 ] where P ∈ Qm , we substitute
the unknown EPnm in eq. (6.23) ∀P . Conversely, for all points R = [k1 , k2 , x1 , x2 ]
where R ∈
/ Qm are approximated by the values from the most recent update at nm−1
n
(with n0 = (n − 1)m ), and for these we therefore substitute the known values ERm−1 .
Spatial Grid
x2
Legend
Spectral Grid
Fill
To be calculated
Visited during sweep
Not visited during sweep
Outline
Not coupled
Coupled
Spatial coupling
Coupling points in Q 1
Coupling points not in Q 1
x1
Figure 6.13: Illustration of the four directional sweep method used in the present study. Because
the spatial coupling between computational points in Q1 is only in the up-wave direction, points in
Q1 can be calculated successively by marching in the wave direction. Contributions due to points
in the other directional sectors are included by using the latest available estimate.
Not only does this reduce the number of unknowns per sweep, the structure of
the resulting matrix is such that spatial dependencies involving EPnm only occur in
the downwave direction. Hence, for the first sweep, when starting at the spatial
point with indices mx1 = 1 and mx2 = 1 (denoted P1,1 , ∀k such that P1,1 ∈ Q1 ), all
downwave information is known (from the boundary), and to solve for E nm ∀k in that
125
6.B. Discrete model
point we therefore only need to invert a M1k M2k by M1k M2k matrix involving those
wavenumers k in the Quadrant, whereas the dependencies on the other quadrants
are included explicitly from the last sweep over that quadrant. Similarly, with P1,1
known, when marching forward, all downwave information for points P1,2 or P2,1 is
available, and the solution is again found by inverting a small local matrix. Hence,
when marching forward in this way, starting at the appropriate corner for each sweep,
down-wave points of the quadrant have been visited previously during the sweep (or
are known from the boundary), and can be included in a Gauss-Seidel manner, and
we only need to solve a linear system containing the spectral mesh-points in the
quadrant under consideration as unknowns (see Fig. 6.13). Nevertheless, if the
topography is captured with Mq2 fourier modes, this can give a dense linear system
containing M1q M2q diagonals. However, given that for slowly varying topography the
off-diagonal contributions are small, the resulting system can still be solved relatively
fast using the BicGStab method with an ILU-preconditioner (Van der Vorst, 1992).
In the cases considered, the resulting algorithm always converged to a solution
within relatively few iterations (n < 10), where the solution was considered to be
converged at after a complete iteration n4 , when the following criterion was met
n4
k (Ek,x
P
(n−1)4 2
− Ek,x
n4 2
k (Ek,x )
P
)
< α2 ,
, ∀x.
Here α is the convergence criterion, set to α = 10−4 in the present study.
(6.32)
7
Conclusions and outlook
In the present work we considered advances in the prediction of two complementary
type of models for simulating wave conditions on the continental shelf up until the
coastline. We investigated this in the context of advancing predictive capability and
understanding of wave dynamics, and in particular the evolution of wave statistics, in
shallow water. Hereto, two separate (though related) paths were pursued. First, we
considered an evolutionary approach within deterministic nearshore wave modelling
(Chapters 3 and 4), where we considered an efficient and accurate way of approximating dissipation due to wave breaking in non-hydrostatic models (in the SWASH
model in particular). We explored how well this reproduces not only bulk statistics,
but also higher-order wave-statistics in a dissipative surfzone. Next, on the premise
of dispersive wave motion over slowly varying topography, we derived a generalisation of the radiative transport equation that governs the evolution of the complete
second-order statistics (Chapters 5 and 6), including coherent interference. In what
follows, we will summarize the main results, and provide an outlook on possible
future developments and applications.
7.1. Conclusions
Non-hydrostatic modelling
Conceptually, non-hydrostatic models resolve the Reynolds-averaged Navier-Stokes
equations using a single-valued approximation for the free surface, and as such implicitly account for the relevant wave dynamics (e.g. shoaling, refraction, diffraction,
wave-current interaction, etc.) for intermediate to shallow-water wave propagation
over relatively small distances (O(10) wave lengths). Their accuracy is in principal
only restricted by model resolution in space and time. Arguably, their most attractive feature is that, when combined with shock-capturing numerics, the dissipation
due to wave breaking is implicitly accounted for. Unfortunately, as demonstrated in
this thesis, correctly resolving the location of incipient breaking requires fine vertical resolutions (see, for instance, Fig. 3.4), much finer than required to resolve the
127
128
7. Conclusions and outlook
nonlinear wave dynamics found in the shoaling region. At coarse vertical resolutions
flow velocities near the crest are underestimated so that the kinematic instability
associated with wave breaking (particle velocity exceeds wave celerity) is postponed,
and wave shoaling continues past observed experimental breaking locations.
Consequently, the requirement of fine vertical resolutions in a small, but dynamically important, region introduces prohibitive computational costs for spatial and
temporal scales in the order of O(10) wave lengths and O(100) periods, even on large
scale multi-processor machines. Hence, problems including 2D wave propagation
over extended nearshore topography, or the generation of a large ensemble to estimate nonlinear wave statistics, cannot be resolved in this manner. In this context,
inspired by prior research demonstrating that the evolution of the bore is well described by the NSWE (Hibberd and Peregrine, 1978), we introduced the hydrostatic
front approximation to wave breaking to make model application to such problems
feasible.
The method, referred to as the Hydrostatic Front Approximation (HFA), enforces
a hydrostatic pressure distribution under the front of a wave. This initiates the transition into a near-vertical sawtooth like wave, so that the associated energy dissipation
can be accounted for by ensuring that mass and momentum are conserved over the
resulting shock – in-line with the analogy between a breaking wave and a hydraulic
jump. The hydrostatic pressure √
is activated point-wise once non-dimensional rate of
change of the free-surface (∂t ζ/ gd) exceeds a predetermined threshold α, and the
full non-hydrostatic computations are restored once the wave crest has passed, or
when ∂t ζ < 0. The threshold, which is essentially a proxy for the maximum surface
steepness, is obtained from computed values of α(=0.6) at observed breaking locations of monochromatic wave breaking on a planar beach (Ting and Kirby, 1994).
Moreover, to represent persistence of wave breaking, the criterion α is reduced to
β(=0.3) in locations where in neighbouring points the HFA is active; this ensures
that wave breaking continues until the steepness of the front becomes small, and the
wave reforms.
Comparison with several laboratory observations of random waves breaking over
barred topography (Battjes and Janssen, 1978; Boers, 1996; Dingemans et al., 1986;
Van Gent and Doorn, 2000) demonstrates that the HFA – without further calibration
– is able to predict not only bulk parameters, such as significant wave height and mean
zero-crossing period, but also the wave-induced mean set-up, using a relatively coarse
resolution in the vertical (2 layers). Furthermore, the obtained results are virtually
indistinguishable from those obtained from more involved computations using a high
resolution model. Comparison to observed wave propagation in a 2D wave basin
(Dingemans et al., 1986) confirms that the method is equally applicable to shortcrested waves; computed wave heights and wave spectra compare favourably with
observations, and the measured wave-induced circulation is faithfully reproduced in
both a qualitative and quantitative manner.
Nonlinear phase-resolving models are of interest because they can account for
strongly nonlinear wave dynamics in the surfzone; this includes not only predicting
the bulk parameters, but also higher order statistics, and the details of the spectral evolution. To ascertain whether a relatively efficient model such as SWASH, in
7.1. Conclusions
129
which the details of the breaking process such as overturning and turbulence are not
resolved, can reliably predict surfzone (non-Gaussian) wave statistics, we performed
a comparison to a large number of flume observations of random waves (Smith,
2004). Because interest included the high-frequency tail, this required a higher vertical resolution (6 layers) compared with previous simulations in this thesis to resolve
dispersion for such relatively deep water waves; nevertheless, inclusion of the HFA
prevented the need for an even finer mesh size in the surfzone.
In accordance with previous results the model accurately predicts second-order
bulk parameters such as wave height and period; this without requiring calibration.
In addition, bulk higher-order statistics, i.e. skewness and asymmetry, also agree well
with observations, although in the case of narrow-band spectra asymmetry values are
somewhat exaggerated in the surfzone. This is likely due to the exclusion of breakinginduced turbulence, which in reality stabilizes the bore front to a finite steepness;
for wider-band fields, the statistics are smoothed, providing better agreement with
observations.
Yet, the most profound result is that development of spectral levels is well reproduced throughout the surfzone: from the attenuation of the spectral peak(s) and the
amplification of higher harmonics, to the increase of energy in the high frequency tail
by as much as two orders of magnitude. This demonstrates that a non-hydrostatic
model with a single-valued representation of the free surface can provide an accurate
presentation of the concomitant action of the dominant nonlinear and dissipative processes that govern the evolution of the spectrum in the surfzone. The former is further
corroborated by agreement between nonlinear transfer rates estimated from observed
and computed bispectra. More so, agreement extends well beyond the energetic part
of the spectrum, to up to ten times the peak frequency, and the resulting shape of
the spectral tail conforms with speculations in the literature that in shallow water
the tail approaches an asymptotic form (Smith and Vincent, 2003; Kaihatu et al.,
2007). Whether or not an asymptote of the tail in the surfzone exists is yet to be
conclusively demonstrated, but results and observations are in line with suggestions
presented by Smith and Vincent (2003) and Kaihatu et al. (2007).
The second-order statistics, as encapsulated by the variance density spectrum,
are thus well resolved in SWASH, whereas agreement with observations for the thirdorder moments suggests that the non-Gaussian statistics that occur in the surfzone
can be predicted by SWASH as well. This is further corroborated by comparison
of the probability density function (pdf) of the free surface, estimated by Monte
Carlo simulations, with the observed pdf, and a theoretical nonlinear pdf (LonguetHiggins, 1963). Though the observations are relatively short, so that the tails of
the estimated pdf from observations are not reliable, the predicted pdf agrees well
with observations, and with the theoretical pdf that only depends on the first two
moments, variance and skewness.
These findings suggested that the influence of fourth and higher order moments
on the statistics is relatively minor. Presumably the nonlinear wave dynamics are
therefore well approximated using second-order theory, i.e. the primary energy balance in the surfzone is between the flux gradient, triad nonlinear interactions and
dissipation. Because the flux gradient and the nonlinear term can be estimated from
130
7. Conclusions and outlook
second-order theory combined with calculated (bi)spectra from SWASH, we could
estimate the dissipation as the term that closes the balance. From this energy balance we derived that the wave dissipation in SWASH is proportional to a frequencysquared distribution function, which is consistent with observations in other studies.
Though not conclusive, it illustrates that SWASH can be used to investigate properties which are not easily accessible by observations, for instance to verify or obtain
better approximations for use in more economic models.
The SWASH model combined with the HFA as considered in the present study
has proven to be a remarkably robust, accurate and relatively efficient tool to predict surfzone dynamics. Although the detailed analysis in the present study primarily
considers one-dimensional wave propagation in a flume, we note that triad nonlinear and dissipation processes are not fundamentally different for 2D surfzones with
short-crested waves, so that our latter conclusions are probably also valid under such
conditions. This is somewhat corroborated by the 2D basin experiment we considered, though computational restrictions prevent more detailed analysis in that case.
Overall, the findings of this study suggest that SWASH is a viable tool for modeling
wave and wave-driven dynamics in a nonlinear, dissipative surfzone.
Stochastic modeling
The interaction of ocean waves with ambient currents and topography in coastal areas
can result in a wave focal zone where, due to coherent interferences, the mean wave
statistics (e.g. mean wave heights) become strongly non-homogeneous. This can
affect wave-driven circulation and transport processes, yet stochastic wave models,
invariably based on some form of the radiative transfer equation (or action balance),
do not account for these effects. In the present thesis, to extend the stochastic
framework to include effects of coherent interference, we therefore derived a new
transport model that accounts for the statistics of inhomogeneous wave fields of
arbitrary bandwidth propagating through a variable medium.
Our starting point was a linear description of the underlying dynamics, in which
the medium (i.e. the topography) is assumed to slowly change its properties compared
to the wave motion proper. On the premise that the frequency and wavenumber of
local plane wave solutions are then coupled by the linear dispersion relation, we used
an operator correspondence argument based on the Weyl association rule to define a
deterministic integro-differential equation that governs the linear wave motion. For
mild slopes this equation (to the order considered) reproduces geometric optics, yet
makes no a-priori assumptions on spectral bandwidth.
Based on the deterministic equation, we derive an evolution equation for the
second-order statistics which, by exchanging a wavenumber coordinate for a spatial
coordinate through an inverse Fourier transformation, is cast in terms of an evolution equation for the Wigner – or Coupled Mode (CM) – spectrum. This spectrum
accounts for the complete second-order correlation matrix, and occupies an intermediate position between a full spectral and spatial description of the correlation
function, much like the well-known variance density spectrum. In fact, when QuasiHomogeneous (QH) theory applies, the CM spectrum reduces to the variance density
spectrum, whereas it is a natural generalisation thereof in case of strongly heterogeneous statistics (e.g. in a wave focal zone). For narrow-band waves travelling in
7.1. Conclusions
131
a homogeneous medium the transport equation reduces to the linear Alber equation (Alber, 1978), which is thus a special case. Moreover, when quasi-homogeneous
theory applies, the transport equation reduces to the Radiative Transport Equation
(RTE) for the transport of the variance density spectrum.
Subsequently we demonstrate by means of a multiple scale analysis that the RTE
is only valid for small values of the ratio β between the coherence length scale, and
a scale on which the medium (topography) changes (β 1). If β is O(1), as is typical for strong wave focal zones, a truncated series expansion of the operator cannot
account for the ensuing fast spatial variations in the wave statistics, and because the
effect is non-perturbative, the series approximation must be replaced by an integral
representation which we refer to as the Quasi-Coherent (QC) approximation. To
derive a consistent model suitable for large scale application over complex topography, we demonstrated that for wave fields with a finite coherence length, only the
topographical variation within the coherent footprint influences the evolution of the
wave statistics. Subsequently, on the premise of a finite coherence length we derived
an augmented form of the radiative transport equations, which includes a scattering
term accounting for the generation of coherent contributions within random wave
fields. The resulting model accounts for the generation and transport of coherent
interferences between wave components that enclose angles smaller than π/2 radians, and the theoretical framework presented here forms a natural extension of the
radiative transfer equation.
To illustrate the difference between the traditional QH approach and the QCapproach we considered the convergence of multiple coherent wave packets, for which
an analytical solution can be derived. The lateral standing wave pattern that occurs in the convergence point is faithfully represented by the QC-approach, whereas
the QH approximation assumes mutual independence of the wave packets, and is
therefore unable to resolve the fine scale oscillations in the variance (or mean wave
height). Strong differences occur not only in the wave variance, but also in the wave
spectra where, due to the inclusion of coherent interference, negative values occur.
This prohibits the interpretation of the CM spectrum as an energy density function.
We argue that the appearance of these negative terms can be interpreted as a departure from the equal partitioning between the mean kinetic and potential energy
found under progressive wave. For instance, in the node of a standing wave pattern
there is no potential energy associated with the wave motion due to interference; in
the coupled mode spectrum this results in negative contributions of equal magnitude
but opposite sign to the variance (or energy) associated with the interfering waves
such that the surface variance disappears.
We further verified the QC theory, including its ability to generate coherency
due to the interaction with the topography, by comparison with observations of the
experiments by Vincent and Briggs (1989) and Berkhoff et al. (1982). These experiments considered the wave statistics, in particular wave heights, in the focal zone of
a submerged shoal. Especially for nearly unidirectional waves the QC approximation
is a marked improvement over the traditional RTE, and is not only able to capture
the bulk parameters, but also provides a good approximation to the wave spectrum
and to its Fourier transform, the covariance function. Differences become less pro-
132
7. Conclusions and outlook
nounced as directionality increases. Effectively, with increased directional spreading,
β decreases, implying that the RTE becomes an increasingly viable alternative. Nevertheless, for moderately spread waves (say 10◦ ), representing a realistic swell field,
the QC results are in better agreement with observations. Finally, we considered
field observations of swell propagation over submerged canyons as observed during
the ONR NCEX experiment. Model predictions of swell compare well with observations, and form a slight improvement on those predicted by the RTE. This confirms
that the QC approximation can be used to predict the wave statistics for complex
nearshore topography
Comparison with analytic solutions, and with observations of random surface
wave propagation in the laboratory and in the field confirm that the Quasi-Coherent
(QC) approximation accurately represents both the generation and transport of crosscorrelations in the wave field, and resolves the fine-scale interference patterns associated with crossing waves. The effects of diffraction on statistics of waves around
and behind obstacles and barriers can be accurately modelled by including appropriate boundary conditions on the QC approximation. These results show that the
application of QC theory to ocean waves can resolve some of the restrictions of
quasi-homogeneous theory (the radiative transfer equation) in areas characterized by
two-dimensional medium variations and caustics. This is likely to be of particular
importance for wave-driven circulation and transport processes in coastal areas and
inlets.
7.2. Outlook
The breathtaking speed at which computational capabilities evolve1 , and our increasing understanding of nearshore wave dynamics, likely implies that as time progresses
the more simple wave models will be increasingly abandoned in favour of physically
more complete approximations such as those considered here. However, within the
foreseeable future, it is unlikely that either of the models presented in the present
thesis will supplant spectral stochastic models for use in operational forecasting, or in
daily engineering practice as there are important fundamental and practical hurdles
that will prohibit replacement of such models for now. Nevertheless, both methods
show great promise, and can complement such models for select situations where
more accurate results are deemed necessary.
If non-hydrostatic models are to be applied on larger scales, generation by wind
and dissipation by deep water wave breaking (white capping) need to be accounted
for; both of which are not straightforward and require further research to include
within the non-hydrostatic modelling framework. It is even debatable whether this,
apart for research purposes (e.g. to derive better spectral representations), is worthwhile pursuing; for the open ocean present day spectral models are an order of magnitude more efficient, and are, given the often large uncertainties in the forcing, often
sufficiently accurate. Instead, it is in particular near and in the surfzone where the
potential lies for further development of the model – and for potential applications
1 The
recent introduction of GPU’s might well introduce another paradigm shift in scientific computing.
7.2. Outlook
133
thereof.
Arguably, non-hydrostatic models have reached a maturity that they can now be
used to fill gaps not yet covered, or are difficult to acquire, by observations in the
surfzone. Naturally, there are still areas that need further validation and improvement; for instance inclusion of wave-breaking induced turbulence can improve wave
shapes and may help account for enhanced bed friction coefficients found in the surfzone. The latter is particularly important for infra-gravity (ig) motions, which have
not featured prominently in the present work, but are well represented in SWASH
(including ig-wave breaking and shoreline reflection, see Rijnsdorp et al., 2014c) – if
bed friction is calibrated for. How well ig-waves are resolved in the field, in particular
when including short-crested primary waves, is still an open question, though preliminary results are encouraging2 . Nevertheless, a model like SWASH has the potential
to not only investigate the physics, but also to help derive parametrisations of certain
processes (e.g. the spectral distribution of dissipation) for use in more approximate
models, like SWAN.
However, significant challenges – and research opportunities – not only lie in the
wave hydrodynamics proper, but also in the coupling to other processes of interest;
for example the transport of (passive) scalar tracers3 , coupling to morphological
models, or the inclusion of layered density flows (e.g. extending Balkema, 2009).
This would allow application of the model to investigate the dispersion of chemicals
in the surfzone, the morphological development under strongly nonlinear forcing, or
the evolution of internal waves.
Whereas non-hydrostatic (and Boussinesq) models are pushed towards deeper water, encompassing increasingly longer temporal and larger spatial scales, the stochastic quasi-coherent model development presented in this thesis in a way represents an
analogous push into progressively shallower water, including increasingly fine scale
effects due to coherent interference. The great advantage of the QC being that it
retains compatibility with the existing stochastic models (e.g. SWAN), so that principally expressions for wave physics (accounting for generation by wind, dissipation,
and nonlinear interactions) applicable to those models can be used with minor modifications within the QC framework.
The principle problem remains that because the QC approach evaluates a convolution like integral over the spectrum, it is two orders of magnitude more computationally expensive compared with the RTE. Although still much more efficient than
time-domain models like SWASH, this represents a considerable increase in effort
that prohibits routine usage. Fortunately, the equations permit easy nesting within
operational large scale models. Furthermore, models can locally switch between the
RTE or QC approach when appropriate. Combined with a more flexible spatial
discretisation based on an unstructured grid, this would allow the model to include
fast-scale variations in the statistics when need be, but to switch to the more efficient
– and more approximative – RTE when possible.
Besides issues pertaining to the numerical implementation, there are two important theoretical extensions to the QC model that need to be further explored.
2 Personal
3 Under
communication with D.P. Rijnsdorp (2014), manuscript in preparation.
development in SWASH, personal communication with M. Zijlema (2014).
134
7. Conclusions and outlook
Foremost of these is inclusion of ambient currents and its effects on the wave dynamics (shoaling, refraction, etc.). Though this is seemingly straightforward to account
for in the derivation – one merely includes the effect of an ambient current in the
dispersion relation – the principle issue is to include the energy exchange between
the mean current and the waves (and vice versa). Presumably, much like the CM
spectrum generalizes the variance density spectrum, this can be achieved through
the transport of a generalized action variable which reduces to the wave action if QH
assumptions apply.
Secondly, extension towards shallower water into possibly the surfzone will require
the inclusion of wave breaking and the energy exchange due to three wave interactions (or triads). Expressions for breaking-induced dissipation are well established,
and can easily be incorporated, but triad interactions are at present only approximated in a very rudimentary way. Fundamentally, this requires an evolution equation
for three-wave correlators in the form of the bispectrum combined with a suitable
closure assumption. Even though what constitutes an appropriate closure assumption for higher-order correlators will likely remain controversial (e.g. see Janssen,
2006, Chapter 4, for a discussion), the same methodology followed in the present
work can also be used to derive evolution equations for the bispectrum over topography. This is of theoretical interest as earlier attempts restricted the propagation
direction, whereas this would constitute the first isotropic evolution equation. This
could help facilitate understanding of the evolution of the third-order statistics over
2D topography. Whether this is also a practical approach is debatable as this entails
the evolution of a six dimensional function through time, raising obvious concerns
regarding possible storage restrictions and the computational effort involved.
In a way this is indicative of the limitations of the spectral description; after all,
these models presume that the motion is weakly nonlinear, with a clear separation
of scales between the fast scale wave motion and the mean wave parameters. Even
though this approximately holds on the open ocean, within the surfzone the motion becomes strongly nonlinear and the distinction in scales is blurred. Under such
conditions, Monte-Carlo simulations using time-domain models likely provide a more
convenient, and potentially more accurate, description of the average wave dynamics.
Ultimately, this dichotomy between the most suitable application range for stochastic wave models, deep to intermediate water, and for deterministic non-hydrostatic
models, from intermediate to shallow water, illustrates that these approaches are not
mutually exclusive, but complementary. Hence, barring revolutionary developments
in scientific computing, the stochastic and deterministic frameworks are likely to continue to operate in synergy, with operational models based on the RTE – including
the QC corrections when deemed necessary – providing the boundary conditions for
deterministic models. Hence, within a general engineering or scientific context there
is no “better” approach to predict the wave statistics, though – given a particular
problem – there might be one which is “best suited”.
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P.B., 2013 Swan and its recent developments. In Conference Proceedings, 13th Int.
Workshop on Wave Hindcasting and Forecasting, Banff, Canada.
List of figures
2.1
Sketch of a ray pattern induced by the refraction of a monochromatic,
unidirectional wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1
The layout of the velocities u, w (indicated by arrows) and the pressure
p (indicated by dots) for a vertical cell. . . . . . . . . . . . . . . . . . .
Sketch of the free surface ζ (top panels) and the rate of surface rise
∂t ζ (bottom panels) as a function of time. . . . . . . . . . . . . . . . .
The experimental setup from Ting and Kirby (1994) . . . . . . . . . .
Wave heights and maximum Froude number for spilling breakers in
the Ting and Kirby (1994) experiment. . . . . . . . . . . . . . . . . . .
Wave heights H for spilling (left panel) and plunging (right panel)
breakers in the Ting and Kirby (1994) experiment. . . . . . . . . . . .
Phase-averaged profiles of the free surface elevation ζ for spilling and
plunging breaking waves. . . . . . . . . . . . . . . . . . . . . . . . . .
Mean water-level setup ζ̄ for spilling and plunging breakers in the Ting
and Kirby (1994) experiment. . . . . . . . . . . . . . . . . . . . . . . .
Sensitivity of computed wave heights in case of spilling breakers to
variations in the maximum steepness and relative mixing length. . . .
Layout of the different flume experiments. . . . . . . . . . . . . . . . .
Spatial variation of the significant wave height Hm0 in the various
experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spatial variation of the mean zero-crossing period T0 (Dingemans et al.
(1986), top left panel) or Tm02 (other panels) in the various experiments.
Mean water-level setup ζ̄ in Boers (1996). . . . . . . . . . . . . . . . .
Depth contours and experimental layout of the Dingemans et al. (1986)
experiment (case me35). . . . . . . . . . . . . . . . . . . . . . . . . . .
Snapshots of the free surface indicating where the HFA is active. . . .
Cross-sections of the significant wave height and mean zero crossing
period in the Dingemans et al. (1986) experiment. . . . . . . . . . . .
Comparison between measured and computed variance density spectra
in the Dingemans et al. (1986) experiment. . . . . . . . . . . . . . . .
Measured and computed wave induced current field for the Dingemans
et al. (1986) experiment. . . . . . . . . . . . . . . . . . . . . . . . . . .
Measured and computed current direction for the Dingemans et al.
(1986) experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measured and computedcurrent magnitude for the Dingemans et al.
(1986) experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
147
21
24
26
27
28
29
29
30
31
33
34
34
35
36
37
38
40
40
40
148
List of figures
4.1
4.2
Layout of the flume experimental setup by Smith (2004). . . . . . . .
Comparison between modeled and observed values of significant wave
height. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Scatter plots of observed versus computed values for the significant
wave height, mean period, skewness and asymmetry. . . . . . . . . . .
4.4 Comparison between the observed and computed energy density spectra.
4.5 Comparison between the observed and computed energy flux contained
in a frequency band around contained in a frequency band around the
peak. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Comparison between the observed and computed normalized energy
fluxes at the primary peak and its first and second harmonics. . . . . .
4.7 The slope of best fit line to the log of the wavenumber spectrum within
the Toba and Zakharov range. . . . . . . . . . . . . . . . . . . . . . . .
4.8 The nonlinear source term computed at two different gauges. . . . . .
4.9 Probability density functions for the normalized free surface estimated
from the observations and from the Monte Carlo simulations. . . . . .
4.10 Mean of the spectral distribution function of energy dissipation. . . . .
4.11 Absolute relative error in the angular frequency, and the resonant
mismatch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.1
6.2
6.3
6.4
6.5
6.6
6.7
Snapshots of normalized wave variance of the three-packet interference
example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cross-sections of normalized wave variance of the three-packet interference example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spectral evolution of the three-packet interference example. . . . . . .
Plan view of the experimental setup by Vincent and Briggs (1989)
including a ray-traced solution for unidirectional monochromatic waves.
Normalized wave heights along transects across and behind the submerged shoal as considered by Vincent and Briggs (1989). . . . . . . .
Plan view of modeled wave heights for the experimental set-up as
considered by Vincent and Briggs (1989). . . . . . . . . . . . . . . . .
Sketch of wave interference geometry on the radius of convergence. . .
Contours of normalized wave height behind a semi-infinite breakwater
and a breakwater gap. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cross-sections of normalized wave height behind a semi-infinite breakwater and a breakwater gap. . . . . . . . . . . . . . . . . . . . . . . .
Sketch of a local slope and a least squares fit to σ within the predefined
coherent zone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Plan view of normalized wave heights. . . . . . . . . . . . . . . . . .
Normalized wave height along the indicated transects. . . . . . . . .
Coupled mode and variance density spectra down wave of the shoal.
Plan view of the modulus of the normalized covariance function. . .
Normalized covariance functions along transects. . . . . . . . . . . .
Bathymetry near Scripps and La Jolla canyon, and locations of observation points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
52
56
59
60
62
63
64
66
68
69
73
82
83
84
86
87
88
91
92
93
106
108
109
110
112
112
. 113
List of figures
6.8
6.9
6.10
6.11
6.12
6.13
Directional and integrated frequency spectra as observed at the Outer
Torres Peynes buoy on the 16th and 30th of November, 2003. . . . .
Normalized wave heights near the NCEX site on 16 November. . . .
Normalized wave height along the indicated depth contours on 16
November. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Normalized wave heights near the NCEX site on 30 November. . . .
Normalized wave height along the indicated depth contours on 30
November. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Illustration of the four directional sweep method used in the present
study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
149
. 115
. 116
. 116
. 119
. 119
. 124
Acknowledgements
Being jet-lagged on a Friday afternoon, after returning from a conference, might
be considered a small price to pay compared with the fantastic experience gained
from a visit to the beautiful island of Hawai’i. However, from the perspective of
productivity, during the final months of my thesis – what surely must be the most
stressful moment in the life of a PhD student – it is downright inconvenient. That
said, somehow it provides for the perfect reflective mindset that is required for writing
the acknowledgements.
Looking back on 4+ years of research, the transition from a somewhat clueless
student, who thought he could understand everything, into a just as clueless academic
who knows there is always a next level of knowledge to master, has been alternately
rewarding, challenging and frustrating. During this process, I have managed to
stumble on many a pitfall – in research and in life – but was fortunate enough to
receive support by many, each of whom in one way or another contributed to help
me complete this dissertation.
From my personal perspective I feel that this thesis would not have existed without the tremendous support by Leo Holthuijsen and Tim Janssen. Leo always supported me along my endeavours, even when I managed to stray away from the initial research path. Although this implied we could not always directly cooperate,
I profited immensely from the feedback he provided, and from the example he set
in scientific rigor and most importantly, scientific integrity – thank you Leo! The
contribution by Tim to my scientific development can hardly be overestimated. Although we were mostly separated by the Atlantic Ocean – and the entire continental
United States to boot – his support, critical reading, ideas and feedback have been
invaluable. I tremendously profited from my short stays in San Francisco, where
during a few weeks I would learn more than during the previous few months, and
enjoyed meeting up with Anke, Erik, Roos, Pepijn and being viciously greeted by
Beans – thank you for everything Tim! To quote your own acknowledgements: “I
hope we can have some more of that!”.
Naturally my promoter – Professor Guus Stelling – also played a crucial role in my
development. I fondly remember his role during my MSc thesis, the collaboration
we had on developing non-hydrostatic version of XBeach (and SWASH), and the
many discussions and meetings we had. However, above all, I appreciate the trust
and freedom he gave me to pursue my own research path. I am also grateful for
the support I received from the staff of the section, and in particular I would like to
thank Dr Marcel Zijlema with whom I enjoyed collaborating on SWASH, and who
provided welcome feedback on this thesis. Furthermore, I wish to express gratitude
to Professor Wim Uijtewaal and Professor Julie Pietrzak for their support.
My co-conspirators in pursuit of a PhD (or otherwise engaged), at either the
second/third floor, or those hidden deep within the bowels (i.e. the ‘lab’) of the con151
152
Acknowledgements
crete monstrosity that is the civil engineering building, provided for a great working
climate over the years. I enjoyed meeting up with you over a coffee, a beer, and –
on those rare occasions I skipped working – lunch! Thanks André, Andres, Cynthia,
Matthieu, Melike, Miguel, Nicolette , Sierd, Steven, XueXue, Willem and all others
whom I worked with! However, those I shared a room with – be it at work or at a
hotel – deserve a special mention. Thank you James, I enjoyed travelling all over the
world accompanied by a cynical British travel guide; “Gu” and Victor, for raising
my spirits while drinking spirits, discussing economic policy and for your interior
decorating skills; Dirk, for our constant “foot fight” due to your ridiculously long
legs, and our collaboration; and Ocean, for our discussions on anything from Chinese
politics, printable folding crafts depicting “Angry Birds” to numerical schemes!
Without doubt, there are many more that deserve to be mentioned here, and
to those I inadvertently left out, I apologize. If you let me know, and if there ever
will be a second edition, I will surely include you! However, before wrapping up, I
want to express my gratitude towards my parents; I cannot even begin to list all the
things you have done – and are doing – for me! Nor can I express in full how much I
admire you. Moreover, I tremendously admire the way of life that my brother Xander
and his partner Sandra share, and the energy and ambition of my sister Marion and
her partner Michiel. Most of all I think all of you have been patient and kind to a
brother who has his personal quirks – and never checks or answers his mail. It is a
privilege to see ones nieces (Iris & Lotte), nephews (Danko & Aram) and children
of friends (Teije) being born, and grow up! To Dena, thank you for the good times,
and for showing me the wonder that is Iran. To my friends of old, Mirza, Johan,
Dirk, Martijn, Simke and Nicole (& partners); thank you for providing the necessary
distractions, and for helping me out when the currents of life were going against me.
Finally, despite the stress that accompanies the final days of any large project,
these last months my quality of life has improved exponentially, for which the blame
squarely lies with Mathilde. Thank you for tremendously enriching my life, you
wonderful creature . . .
Delft – May, 2014
Pieter Bart Smit
List of publications
Journal Articles
First author
Smit, P.B., Zijlema, M. and Stelling, G.S., 2013 Depth-induced breaking in a
non-hydrostatic nearshore wave model. Coast. Eng., 76, 1–16.
Smit, P.B. and Janssen, T.T., 2013b The Evolution of Inhomogeneous Wave
Statistics through a Variable Medium. J. Phys. Oceanogr., 43, 1741–1758.
Smit, P.B., Janssen, T.T., Holthuijsen, L.H. and Smith, J.M., 2014b Nonhydrostatic modelling of surf zone wave dynamics. Coast. Eng., 83, 36–48.
Co-author
Rijnsdorp, D.P., Smit, P.B. and Zijlema, M., 2014c Non-hydrostatic modelling
of infragravity waves under laboratory conditions. Coast. Eng., 85, 30–42.
Zijlema, M., Stelling, G.S. and Smit, P.B., 2011b SWASH: An operational
public domain code for simulating wave fields and rapidly varied flows in coastal
waters. Coast. Eng., 58 (10), 992–1012
Conference Proceedings, Workshops and Talks
First author
Smit, P.B. and Stelling, G.S., 2009 Accurate prediction of tsunami propagation
in complex bathymetries. In paper presented at Tsunamis and Geophysical Warnings: Workshop at the Lighthill Institute of Mathematical Science, London, UK,
17 December.
Smit, P.B. and Janssen, T.T., 2011 Coherent Interference and Diffraction in Random Waves. In Conference Proceedings, 12th Int. Workshop on Wave Hindcasting
and Forecasting, Hawai’i, USA.
Smit, P.B., Janssen, T.T. and Herbers, T.H.C., 2012b Topography-induced
focusing of random waves. In Conference Proceedings, 33rd Int. Conf. Coastal
Eng, Santander, Spain.
Smit, P.B., Janssen, T.T. and Herbers, T.H.C., 2012a Coherent interference
in random waves. In Paper presented at the Ocean Sciences Meeting, Salt Lake
City, USA, 20–24 February.
Smit, P.B., Zijlema, M. and Stelling, G.S., 2012c Depth-induced Wave Breaking
in SWASH. In paper presented at the 1st International SWASH workshop, Delft,
The Netherlands, 24–25 September.
153
154
List Of publications
Smit, P.B. and Janssen, T.T., 2013a Evolution of coherent interference in random
waves. In paper presented at Waves in Shallow Enviroments, Washington, USA,
21–25 April.
Smit, P.B., Janssen, T.T. and Herbers, T.H.C., 2014a Refractive focusing of
coherent waves. In Paper presented at the Ocean Sciences Meeting, Honolulu,
USA, 23–28 February.
Co-author
Rijnsdorp, D.P., Smit, P.B. and Zijlema, M., 2012 Non-hydrostatic modelling
of infragravity waves using SWASH. In Conference Proceedings, 33rd Int. Conf.
Coastal Eng., Santander, Spain.
de Bakker, A., Tissier, M., Ruessink, B.G., Smit, P.B. and Herbers, T.H.C.,
2014 Bispectral evolution over a laboratory beach. In Paper presented at the
NCK Days, Delft, The Netherlands, 27–28 March.
Rijnsdorp, D.P., Smit, P.B., Ruessink, B.G. and Zijlema, M., 2014b Recent
experiences with non-hydrostatic modelling of infragravity waves. In Paper presented at the NCK Days, Delft, The Netherlands, 27–28 March.
Rijnsdorp, D.P., Smit, P.B., Ruessink, B.G. and Zijlema, M., 2014a Modelling
of infragravity waves near Egmond aan Zee with a non-hydrostatic wave model.
In 34th Int. Conf. Coastal Eng, Seoul, South Korea, 15-20 June (to be presented).
Salmon, J., Holthuijsen, L.H., Smit, P.B., van Vledder, G. and Zijlema,
M., 2014a Alternative source terms for SWAN in the coastal region. In To be
presented at: 34th Int. Conf. Coastal Eng, Seoul, South Korea, 15–20 June (to
be presented).
Zijlema, M., Stelling, G.S. and Smit, P.B., 2011a Simulating nearshore wave
transformation with non-hydrostatic wave-flow modelling. In Conference Proceedings, 12th Int. Workshop on Wave Hindcasting and Forecasting, Hawai’i,
USA.
Zijlema, M., Van Vledder, G.P., Holthuijsen, L.H., Salmon, J.E. and Smit,
P.B., 2013 Swan and its recent developments. In Conference Proceedings, 13th
Int. Workshop on Wave Hindcasting and Forecasting, Banff, Canada.
Scientific Reports
Smit, P.B., 2008 Non-hydrostatic modelling of large scale tsunamis. Master’s thesis,
Delft, University of Technology.
Smit, P.B., Stelling, G.S., Roelvink, D., van Thiel de Vries, J., McCall,
R., van Dongeren A., Zwinkels, C. and Jacobs, R., 2009 XBeach: Nonhydrostatic model. Tech. rep., Delft University of Technology and Deltares.
Curriculum vitae
Born on the 17th of January, 1983, I grew up in Zoetermeer near the Hague in the
Netherlands. Starting in 1995, I received my secondary education (VWO) at the
Erasmus College in Zoetermeer, of which I graduated in 2001. After a brief spell at
Utrecht University studying Physics (2001-2002), I started my academic education
(BSc and MSc) at Delft University of Technology studying Civil Engineering. I graduated in November 2008 at the section of Environmental Fluid Mechanics under the
supervision of Prof. Guus Stelling on the thesis entitled “Non-hydrostatic modelling
of large scale tsunamis”. Following my graduation, I started as a full-time Ph.D.
candidate in 2008 at the TU Delft, during which I also served as a full time lecturer
for the graduate course Ocean Waves. The research performed during my doctoral
research has been documented in this dissertation, which I defend on the 6th of June,
2014. Hereafter I will continue working for two months at the Delft University of
Technology, after which I hope to receive a Rubicon grant from the NWO to continue research as a post-doctoral fellow at Stanford University under the guidance of
Professor Stephen Monismith.
155
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