Modeling Estuarine Bror Fredrik Jönsson Some Concepts of

Modeling Estuarine Bror Fredrik Jönsson Some Concepts of
Estuarine
Modeling
Some Concepts of
Bror Fredrik Jönsson
Some Concepts of Estuarine Modelling
Estuarine
Modeling
Some Concepts of
Bror Fredrik Jönsson
c
Bror
Fredrik Jönsson
Duvbo Tryckeri AB
Stockholm 2005
ISBN 91-7155-052-6
Contents
Introduction: Some Concepts of Estuarine modelling
7
Paper I: Turnover-Time Estimates in a Tropical Estuary
with High Spatial and Temporal Variability.
Accepted for publication in Estuarine Coastal and Shelf Science
33
Paper II: Fjord seiches in the Gulf of Finland and
their relationship to the global Baltic seiches.
Submitted to Journal of Geophysical Research
53
Paper III: Using Lagrangian Trajectories to Study
a Gradually Mixed Estuary.
Published as DM report number 93.
77
Paper IV: Baltic Sub-Basin Turnover Times Examined
Using the Rossby Centre Ocean Model.
Published in Ambio, v. 33 (4-5), pp 257-260, 2004
97
Some Concepts of Estuarine Modeling.
Bror Jönsson
Abstract
If an estuarine system is to be investigated using an oceanographic modeling approach, a decision must be made whether to
use a simple and robust framework based on e.g. mass-balance
considerations, or if a more advanced process-resolving threedimensional (3-D) numerical model are necessary. Although the
former are straightforward to apply, certain fundamental constraints must be fulfilled. 3-D modeling, even though requiring significant efforts to implement, generates an abundance of
highly resolved data in time and space, which may lead to problems when attempting to specify the ”representative state” of the
system, a common goal in estuarine studies.
In this thesis, different types of models suitable for investigating estuarine systems have been utilized in various settings. A
mass-balance model was applied to investigate potential changes
of water fluxes and salinities due to the restoration of a mangrove
estuary in northern Colombia. Seiches, i.e. standing waves, in
the Baltic Sea were simulated using a 2-D shallow-water model
which showed that the dominating harmonic oscillation originates from a fjord seiche in the Gulf of Finland rather than
being global. Another study pertaining to the Gulf of Finland
used velocity-fields from a 3-D numerical model together with
Lagrangian-trajectory analyses to investigate the mixing dynamics. The results showed that water from the Baltic proper is
mixed with that from the river Neva over a limited zone in the
inner parts of the Gulf. Lagrangian-trajectory analysis was finally
also used as a tool to compare mass-balance and 3-D model results from the Gulf of Riga and the Bay of Gdansk, highlighting
when and where each method is applicable.
Introduction
Some Concepts of Estuarine modelling.
From the present thesis it can be concluded that the above described estuarine-modeling approaches not only require different levels of effort for their implementation, but also yield results
of varying quality. If oceanographic aspects are to be taken into
account within Integrated Coastal Zone Managment, which most
likely should be the case, it is therefore important to decide as
early as possible in the planning process which model to use,
since this choice ultimately determines how much information
about the physical processes characterizing the system the model
can be expected to provide.
8
Introduction
1
Introduction
Even though the coastal zone accounts for only 20 percent of
the global land area , this region is one of the most important,
and indeed fragile, in a socio-economic context. World-wide,
the number of people living within a distance of 100 km from
the coast increased from roughly 2 billion in 1990 to 2.2 billion
in 1995, the latter figure constituting around 40 percent of the
world’s population This fraction is estimated to reach 80 percent
by 2010. The number of people who are affected by and affecting
the conditions of the coastal systems is even larger than the actual
population living in these regions. This is due to inland pollutants
being delivered to the littoral by rivers, and the fact that coastal
ecosystems are a major source of food, particularly in developing
countries. In Senegal, e.g., fisheries and aquaculture provide 75
percent of the animal-protein diet of the population.
As coastal and inland populations continue to grow, their impacts in terms of pollutant loads and the development and conversion of coastal habitats can be expected to increase as well. This
state of affairs has created an awareness among scientists and managers around the world for the need to address coastal-zone problems in an interdisciplinary manner. Many extensive programs
and organizational frameworks have been initiated to integrate
socio-economical, ecological and physical aspects of coastal zone
development, this in order to establish sound management practices for these regions. Efforts of this type are normally lumped
together under the term ”Integrated Coastal Zone Management”
(ICZM).
Due to a number of factors, ICZM projects and programs have
often focused more on biological and socio-economical factors
than on oceanographical descriptions of the systems under consideration. This can, perhaps, be explained in terms of different
traditions dominating within separate scientific disciplines as well
as a fully understandable urge to give priority to the most critical problems in each region under study. Still, to fully comprehend a complex system, it is necessary to address all aspects of the
problem, and perhaps in particular the spatial links and temporal
scales characterizing different processes. It is in the light of these
circumstances that the projects underlying the present thesis have
been initiated. This can be seen as a first trembling step towards
exploring some different oceanographical techniques useful for
ICZM purposes, and to furthermore discern under which condi-
9
Introduction
Some Concepts of Estuarine modelling.
tions these methods may be applicable. To narrow the scope in
order to increase the chances of success, the focus of this thesis is
on estuarine systems. The rationale behind this limitation is that
many coastal areas that can be dealt as well-defined systems have
estuarine characteristics. The general problems which arise when
boundaries are imposed at an open coastline are the same from
an oceanographic perspective as within an ICZM framework. In
what follows, attention will be directed towards various aspects of
estuarine modelling rather than on discussing ICZM matters any
further, even though the work underlying this thesis has been
undertaken with these issues in mind.
2
2.1
Estuaries
Topographic classifications
The classic oceanographic definition of an estuary as ”a semienclosed coastal body of water which has a free connection with
the open sea and within which sea water is measurably diluted
with fresh water derived from land drainage” is due to Pritchard
(1967). This distinguished pioneer in estuarine research also introduced a classification of estuaries based on their topographic
features. His scheme (Pritchard, 1955) distinguishes between
coastal-plain estuaries (drowned river valleys), fjords and barbuilt estuaries.
Coastal-plain estuaries, showing little sedimentation, were
formed by river-valley flooding subsequent to a sea-level rise.
Generally, the ancient river valley still determines the estuarine
topography. These shallow estuaries usually have depths below
30 m, and are mostly located in the temperate climate zones.
River flow is normally small compared to the tidal prism (viz. the
change of water volume between high and low tides) and sedimentation has not kept pace with inundation. These estuaries are
characterized by extensive mud flats with a sinuous, deeper central channel. Coastal-plain estuaries are generally found in temperate latitudes. A pertinent example from the Baltic is provided
by the river Vistula as it debouches into the Bay of Gdansk.
Fjords represent river valleys which deepened due to glacier
action during the last ice age. This scouring of the valley floor
resulted in very deep estuaries, sometimes exceeding 800 m in
depth. A characteristic feature of fjords is the presence of a shallow sill at the mouth formed by accumulated rock at the glacier
10
Introduction
front. These sills can be as shallow as 4 m, but depths between
40 m and 150 m are more common. Shallow sills of this type can
restrict the free exchange of ocean and estuary waters, in some
cases producing a small tidal prism with respect to river flow.
True fjords are only found at higher latitudes, but fjord-like estuaries can also occur elsewhere. Although not as spectacular as the
Norwegian fjords, a well-known Swedish example is Gullmaren
on the west coast.
Bar-built estuaries are inundated river valleys with a high sedimentation rate. They are generally very shallow with depths of
only a few metres, frequently branching out towards the mouth
to form a system of shallow waterways (lagoons). The riverine
entrance velocities can be quite high but quickly diminish as the
estuary widens. The sediment accumulates in the vicinity of the
mouth of the estuary and ultimately forms a bar, where the water
depth becomes even more shallow. Bar-built estuaries are a characteristic of the subtropics and tropics, but can be found wherever
the coastal zone is characterised by deposition of sediment. In the
Baltic, the mouth of the Oder is an example of such a system.
2.2
Phenomenological classifications
Classifications based on topography can be useful for general purposes, but do not take into account oceanographic conditions
such as circulation and mixing. These aspects of estuarine behaviour are very important, since a majority of the estuaries described in the scientific literature fall within the costal plain category, where large differences in the circulation patterns, density stratification and mixing processes are encountered. Consequently, a better classification would be one based on the salinity distribution and flow characteristics within the estuary. Such
schemes were introduced by Pritchard (1955) and Cameron and
Pritchard (1963), who identified four main classes: highly-stratified (or salt-wedge), fjord-like, partially-mixed, and vertically
homogenous estuaries. The last type can be subdivided into laterally inhomogenous and sectionally homogenous estuaries. This
classification scheme can, in simplified terms, be described as
a division between systems that are dominated by either freshwater input or tidal mixing (possibly being in an intermediate
state). These considerations, together with a quantification of
the stratification and circulation, led to a scheme (Hansen and
Rattray, 1966) based on two dimensionless parameters, one of
11
Introduction
Some Concepts of Estuarine modelling.
10
Salt Wedge
No Mixing
4
δs/<s>
1
3b
1b
Fjords
10-1
1a
3a
10-2
2a
Well Mixed
10-3
1 1.5
10
10
2
us/uf
10
3
10
4
10
5
Fig. 1. Visualization of the different mixing regimes in the Hansen and Rattray
(1966) classification, based on circulation and salinity stratification.
which is a stratification parameter δs/hsi defined as the ratio between the surface-to-bottom difference in salinity δs and the vertically or cross-sectionally averaged salinity hsi. The other is a
circulation parameter us /uf , defined as the ratio of the net surface current to the mean cross-sectional velocity, which serves as
a measure of the importance of the river flow in relation to the
mean fresh-water flow augmented by a contribution due to eddyentrainment. The scheme is shown in diagrammatic form in Fig.
1 and summarized in table 1. This classification can be useful as
an aid when deciding which model approach is most appropriate
for a certain estuary.
2.3 Classifications used in management
In addition to classifications focusing on physical factors, the need
for efficient management tools has led to the development of
more general indices pertaining to water quality, pollution and
other factors affecting human well-being. With regard to these
schemes, Ferreira (2000) distinguishes between three main approaches:
• Defining classes corresponding to concentration ranges for indicator parameters such as dissolved nutrients.
• Using synthetic approaches (such as the saprobic index) which
12
Introduction
employ faunal diversity or some other indicator as a representation of the quality of the sediment and water column.
• Applying composite approaches linking several indicator parameters such as e.g. sediment contamination and species composition.
Ferreira pursues this analysis one step further by aggregating different parameters and approaches to an integrated quality index
based on vulnerability, water quality, sediment quality, and trophodynamics,
cf. Ferreira (2000) which is also a highly useful reference source
in the field of estuarine quality indices.
3
Modelling of estuaries
Taking into account the diversity and variability of estuaries, in
addition to the wealth of different research objectives, it is not
surprising that a number of modeling approaches have been developed. In the present exposé, the methods have been grouped
in what might appear to be an arbitrary fashion, viz. by the number of spatial dimensions considered. This entity, however, serves
as a link between different paradigms as well as being a serviceable proxy indicating the level of complexity associated with each
model approach. In what follows, papers I-IV constituting the
present thesis will be used as case studies to illustrate various aspects of this dimensional classification.
3.1
Low-dimensional models
In many cases, the most important factors to estimate for an estuary are the exchange fluxes between the system and adjoining water-masses, from which the turnover time of the estuary
can be deduced. These characteristics are frequently used when
estuarine processes and biogeochemical properties are discussed
(see e.g. Jay et al., 1997). It is generally useful to describe the
time-integrated properties of an estuary, this in order to specify
the ”typical state” of the system. When the evolution in time is
of importance, this is generally related to changes of state, e.g.
man-induced eutrophication or an altered hydrological regime.
In cases such as these, a simple mass-balance budget approach is
often suitable. This method, first derived by Knudsen (1900) to
calculate the turnover time of the Baltic, is based on the principle of mass conservation. Due to the inherent properties of the
13
Weak
Internal Friction
Seaward flows at all Depths
Medium
-
2B
”Diffusion” & Circulation
Medium
Weak
-
2A
Strong
Medium
Fjord
3B
Flow reverses at depth
Inertia
Circulation
Weak
3A
Table 1
Estuarine classification based on circulation and salinity stratification (Hansen and Rattray, 1966) .
Notes
Force Balance
-
1B
Medium
Uppstream Salt Flux — ”Diffusion”
Velocity Stratification
Weak
Description
Density Stratification
1A
Type
Internal Friction
”Diffusion” & Circulation
Weak/Medium
Strong
Salt Wedge
4
Introduction
Some Concepts of Estuarine modelling.
chemical compounds constituting sea salt, no natural processes
”produce” or ”consume” salinity. Since ions do not evaporate
to any significant extent, the salinity of an estuary can be fully
explained in terms of advection to and from the ambient water
bodies. If the freshwater influx to the system is known, it is pos-
14
Introduction
sible to use the salinity of the system as a proxy for judging the
exchange fluxes affecting the estuary. (These fluxes are normally
quite difficult to measure in the field due to a number of practical
problems.) It must, however, be kept in mind that mass balance
models only are applicable for certain systems and furthermore
require some auxiliary assumptions, the latter to be described in
what follows.
3.1.1 A steady state
Mass-balance modelling generally presupposes a constant volume
of water in the system. Tides, seiches, “windstau”, and variable
river discharges may, however, alter the volume of estuaries and
lagoons, cf. Officer (1983). These processes, acting over different
time-scales, will in the long run tend to even out, and it may be
reasonable to assume quasi-stationary conditions. The best way
to achieve such a “steady state” is thus to average data over a
period considerably exceeding the time-scales of the dominant
processes.
Also the salinity as well as the nutrient concentrations are assumed to remain constant in the system. This state of affairs can,
similarly, be approximated by averaging data over time. Normally, the main interest when formulating a budget is to examine
the long-term changes and dynamics characterizing the system,
hereby reducing the importance of rapid fluctuations of volume
and salinity. For long-term extrapolations, it is crucial to first determine any trends in the data set. If a trend is well described, it is,
under certain conditions, possible to formulate a non-stationary
budget (Gordon et al., 1996).
3.1.2 A well-mixed system
Salinity and nutrient concentrations are not only assumed to remain unchanged over time, but also in space, implying that the
system must be well-mixed. It is feasible to formulate a two-layer
model with a separating pycnocline, but stratified systems are less
straightforward to model since several additional assumptions are
required. A system with variations of salinity can, nevertheless,
be modeled, provided that the salinity difference between the system and the adjacent water mass is much larger than the internal
variations characterizing the system (Gordon et al., 1996).
15
Introduction
Some Concepts of Estuarine modelling.
3.1.3 Physical constraints on the fluxes
Since the mass-balance approach for estuaries is based on the
salinity difference between the system and the adjacent ocean,
such a discrepancy must be at hand. The definition of an estuary
implies some kind of constraint on the fluxes to and from the system, and, for a mass-balance model to work, the topographical
limitations of the system must be distinct and not gradual. Such
constraints can be exerted by sills, sounds, or other well-defined
morphological features (Ketchum, 1983).
3.1.4 Quantification of freshwater influx to the system
The effects on the system-volume V of the freshwater fluxes to
and from a system can be described as
dV
= Qp − Qe + Qr + Qm + Qn + Qi + Qo ,
dt
(1)
where Qp is the precipitation, Qe the evaporation, Qr the river
influx, Qm the groundwater influx, and Qn the influx from anthropogenic sources. It is normally a straightforward to estimate
evaporation and precipitation using climatological data, whereas
the river discharge to the system is more complicated to quantify. If there are only a few major rivers entering the estuary
and at least some of them are monitored with gauges, it may be
feasible to estimate the effects of the remaining ones by interpolating the existing data (cf. Paper I). In systems with significant sheet/groundwater inflow, or with discharge from numerous minor tributaries, the freshwater influx must, however, be
estimated using more indirect methods, e.g. the one proposed by
Kjerfve (1990) where the watershed runoff is calculated as
r
∆f
Qr = AW
.
(2)
2549 · 109
r
HereAW is the watershed area (m2 ), r is precipitation (mm),
and ∆f
is a non-dimensional runoff factor estimated by Sellers
r
(1965) as
∆f
≈ e−E0 /r ,
(3)
r
where Eo is the potential evapotranspiration (mm/month). This
latter quantity can be calculated using a straightforward empirical
16
Introduction
relationship (Holland, 1978) between potential evapotranspiration and air temperature:
E0 = 1 · 109 e−462/r .
(4)
This method yields good estimates of the total discharge, including groundwater and small-creek contributions, but encounters some problems if used for highly resolved time-series when
the discharge is zero under conditions of no precipitation.
If estimates of the river discharge to the estuary are available,
this shortcoming can be minimized by comparing the observed
river discharge with the run-off model discharge and, for each
time-step, using the highest value. This can lead to problems if
there are large reservoirs in the drainage area, a factor that must
be taken into account.
For notational convenience, all freshwater contributions are
generally combined to a total freshwater influx Qf .
3.1.5 Definition of salinity in the system
Even though a mass-balance budget approach requires that the estuary has a well-defined and homogenous salinity, this is seldom
the case in reality.
For large and heterogeneous systems, the conceptual framework due to Gordon et al. (1996) defines the salinity of the system as that in the vicinity of its outflow loci. Another possibility
would be to use the salinity in the center of the estuary as system salinity. The external-reservoir salinity could, furthermore,
be defined as either the earlier mentioned salinity in the mouth
of the estuary, or that of the ambient ocean. Paper I of this thesis
suggests that the best method may be to use the center salinity
for characterizing the system and the ocean salinity for reference
purposes.
3.1.6 Calculating mass and salt budgets.
Once the freshwater fluxes and salinities have been established,
it is possible to calculate the fluxes between the estuary and the
surrounding water body by using the equation
dSi
= Sf Qf + So Qi − Si Qo .
(5)
dt
(Note that by definition Sf , the salinity of the freshwater influx,
is equal to zero.) Assuming that both the volume V and the
salinity Si are stationary, Eqs. (1) and (5) simplify to
V
17
Introduction
Some Concepts of Estuarine modelling.
Qi =
Sc ·Qf
So −Sc
and Qo =
So ·Qf
So −Sc
.
(6)
Hereafter, the outflow Qo can be used to determine the turnover
time τ = V /Qo , where V is the volume of the system (Bolin and
Rodhe, 1973). Turnover time is comparatively easy to calculate,
provides a good indication of the characteristics of the system,
and furthermore has an “integrating” property, which to some
extent compensates for temporal variations.
3.1.7 Case study: Ciénaga Grande de Santa Marta
In paper I of the present thesis, a mass-balance model for the
Ciénaga Grande de Santa Marta estuary is presented. This large
bar-built system is located on the northern coast of Colombia.
C.A. Gosselmann, an early Swedish traveller who crossed the region in 1825, gave the following vivid glimpses of the estuary in
its pristine state:
Den väg, man måste tillryggalägga i denna farkost, går genom det slags skärgård,
hvilken förenar Magdalena-floden med hafvet, och som med ett gemensamt namn
kallas los quatro Bocas — de fyra utloppen. Man kommer först till en stor fjärd,
kallad la Cienaga de Santa Marta, på hvilken de oftast nyttja segel på sina båtar,
emedan den är temligen stor; derefter till ett trångt vatten vid namn Caño sucio
— smutsiga kanalen — hvilken också bär skäl för namnet. Hela denna skärgård
i allmänhet, men sistnämde kanal isynnerhet, är beryktad för sina muskiter. . .
The estuary was originally surrounded by mangrove, but due to
infrastructural projects during the last 40 years, most of these
forests have died off. In 1995, a major restoration project was
initiated to rehabilitate the forest to its original state by opening a
number of canals from the adjacent river Magdalena to the estuary. Paper I is an attempt to evaluate the changes of the estuarine
hydrological regime caused by the restoration. To judge from this
study, the wide-spread perception of the estuary as a more-or-less
stagnant basin with long turnover-times (and, consequently, also
a high susceptibility to eutrophication) must be somewhat redressed. Rather, Ciénaga Grande appears to be a comparatively
dynamic system with short residence times of only around 20
days. It is also evident that the largest effect on the hydrological
regime from the restoration project is an increase of the freshwater influx during the dry seasons, whereas the changes during
the wet seasons are small. Overall, it is possible to conclude that
the restoration project probably has not changed the hydrology
of the estuary significantly. Finally, this study presumably shows
18
Introduction
that a rather simple approach to modeling a costal water body can
provide a useful addition to our understanding of the system.
3.1.8 LOICZ
It may be noted that the most comprehensive recent application of
low-dimensional modelling has been registered within the LOICZ
(Land Ocean Interactions in the Coastal Zone) project. This is a
core project within the International Geosphere-Biosphere Programme (IGBP), with the primary task of understanding the role
of the coastal sub-system in the functioning of the total Earth
system. This includes the role of the coastal zone (defined as extending from 200 m below to 200 m above the mean sea level)
for the pristine as well as the anthropogenically perturbed cycles of carbon, nitrogen and phosphorous. The main approach
taken involves compiling results from local and site-specific research, together with regional and wider-scale results, this in order to assemble budgets for materials and energy in coastal systems. LOICZ thus attempts to facilitate a global understanding of
the fluxes between land and ocean, and to create a framework for
coastal management by linking natural science to socio-economic
considerations. One of the main challenges for LOICZ is the globally uneven distribution of field studies. Many parts of Europe and
North America have been subjected to extensive studies and large
sets of high-quality data are thus available, including long time
series as well as the results from numerous short-term investigations with high geographic resolution. This, unfortunately, does
not hold true for Africa, South America and parts of Asia, where a
dearth of in-depth studies has lead to a severe lack of understanding of many important bio-geosphere systems. Consequently one
LOICZ objective is to develop sampling and modelling techniques
suitable for areas where data are scarce and research resources are
limited, in which context the use of mass-balance budgets has
proved highly beneficial.
3.1.9 Complex low-dimensional models
To conclude this discussion of low-dimensional modelling, some
more elaborate methods for estuarine modelling which also are
based on a box-model approach should be mentioned. Gordon
et al. (1996) thus describe how a complex system can be divided
into several boxes, to each of which a mass-balance model is applied. It is furthermore possible to incorporate analytical models
handling various oceanographic processes into these models, this
19
Introduction
Some Concepts of Estuarine modelling.
to further improve the results. Some practical examples of this approach are provided by Hagy et al. (2000), Stigebrandt and Wulff
(1987), Engqvist and Andrejev (2003), as well as Kjerfve et al.
(1991). This latter study is based on applying a hybrid between
a mass-balance model and a model of the type to be presented
next, viz. a numerical 2-D model.
3.2
2-Dimensional Shallow-Water Models
If the prerequisites for applying a mass-balance budget are too
strict, or if the spatial resolution and temporal evolution of the
system are important, recourse is frequently taken to numerical
methods. These are generally based on the concept of solving the
hydrodynamic equations of motion by using finite difference approximations. The simplest form of such numerical models are
those of 2 dimensions and based on the barotropic shallow-water
equations. This class of models is highly useful for describing the
circulation and water-level changes in well-mixed systems with
a strong barotropic signal, e.g. tidally-dominated estuaries. Even
though it may be a straightforward matter to formulate a linear shallow-water numerical model, it often proves necessary to
include non-linear effects, which tend complicate the numerical
scheme. In these cases it is often advantageous to use an already
perfected modeling package. Other problems with implementing
these numerical models are how to define the boundaries of the
system and how to prescribe the physical forcing correctly.
Paper II of the present thesis is an example of how a barotropic
model can be used to study sea-level changes in an estuary. Various harmonic oscillations in the Baltic Sea were investigated,
and it proved possible to identify three different local oscillatory
modes. One is located in the Gulf of Finland, with the two distinct periods 23 and 27 hours, one in the Danish Belt Sea, with
a less distinct period in the range 23-27 hours, and one in the
Gulf of Riga, with a period of 17 hours. The most pronounced
mode was that in the Gulf of Finland. It was also apparent that
no clear indications of global seiches in the Baltic could be found
from these numerical experiments. On the basis of these simulations, it was proposed that the sea-level oscillations of the Baltic
are dominated by an ensemble of weakly coupled local oscillators. Each of these corresponds to a ”fjord mode“ or a ”harbour
mode” in a particular bay or sub-basin. These oscillations are not
proper eigenmodes, since their energy gradually leaks out to the
20
Introduction
rest of the Baltic Sea, resulting in radiation damping. Nevertheless, their resonance may in fact be sharper than that of the proper
global eigenmodes.
Alternatively, a 2-D estuarine model can be defined using the
vertical and longitudinal axes of the estuary. This is an appropriate technique for studying density-driven mixing in systems
which are dominated by freshwater forcing. In the present thesis,
this type of model is, however, considered to be a variety of the
3-dimensional model species, the characteristics of which are to
be examined next.
3.3
3-Dimensional models
The possibly most realistic method for studying oceanographic
phenomena is to apply 3-dimensional numerical models. This
class of models represents an extension of the 2-dimensional ones
described above, with the important difference that the full set of
governing equations is used. This means that in principle, all
physical processes affecting the system could be resolved. This
is, however, not practically feasible due to the computational resources such a task would require. As a consequence, all models
of this type tend to rely on a various approximations and parameterizations, among which the assumption of hydrostatic equilibrium generally is the most important, but in estuarine modeling,
also the most problematic. Processes such as friction, turbulent
mixing, and small-scale eddy transports normally need to be included in the model by parameterizations. These 3-D models
are not only useful for studying barotropic fluxes, but also include the baroclinic modes and, at least indirectly, do justice to
vertical fluxes. This implies that the models are required to simulate salinity and temperature in the system, properties which are
highly interesting from the point of view of estuarine research.
Although not a trivial matter, it is also possible to examine the
transport and dispersal of a tracer in an estuary, using the same
approach as for salinity and temperature.
Due to the complex nature of 3-D models, both from a numerical and a computational perspective, it is generally not an option to develop such a model for a specific investigation of limited
duration. Instead, publicly available model packages, of which
the great majority are indebted to the groundbreaking work of
Bryan and Cox (1969), can be used for this task. The most
common ones are the Modular Ocean Model (MOM), the Prince-
21
Introduction
Some Concepts of Estuarine modelling.
ton Ocean Model (POM), and the MIT Global Circulation Model
(MITgcm). All of these have advantages and drawbacks, as well
as more-or-less vocal supporters. The choice of program package
represents a trade-off between earlier experiences and modeling
requirements. Except in the more extreme cases, all packages are,
however, capable of modeling most systems reasonably well. It is
also important to keep in mind what the desired outcome of the
model study in fact is. Not infrequently a significant amount of
time is spent on aspects of the implementation that do not always
affect the results to any greater extent. One should always try to
establish the level of ”good enough”.
One practical application that bears a strong kinship to 3-D
estuarine modeling is represented by the RCO modeling activities
for the Baltic sea. The Rossby Centre Ocean model is based on
the MOM-code which has been adapted via the OCCAM project
for use within the SweClim framework, a program for Northern
European regional climate-change scenarios. The RCO model has
also been run in a hind-cast fashion producing 40-year velocity,
salinity and temperature fields for the Baltic sea. As will be seen,
papers III and IV of the present thesis both rely heavily on these
data-sets for studying the estuarine characteristics associated with
different Baltic sub-basins.
3.4
Lagrangian trajectories
As previously mentioned, 3-D numerical models have many attractive features, e.g. high spatial and temporal resolution, which
make it possible to also examine local processes within the estuary under consideration. Even though a detailed knowledge of
these features frequently is highly desirable, too much emphasis on minute details does not always facilitate an understanding of the system as a whole. The higher the resolution is, the
more information is generated, which in turn becomes more and
more complicated to digest. A pertinent example concerns how
to estimate fluxes to and from an estuary with the help of a 3-D
model. In principle it would be possible to calculate these fluxes
as the mass transport across the boundary between the estuary
and the adjacent sea. This procedure would, however, include
small-scale-eddies and other local currents, which tend to inflate
the fluxes to unrealistic magnitudes. Similarly, a high temporal resolution counteracts attempts to determine a ”characteristic
state” of the system. Both of these issues can be handled by using
22
Introduction
mass-balance budgets, this since the fundamental characteristics
of these techniques lead to representative results for the entire
estuary. Although the two modeling approaches outlined here at
first glance may appear to be mutually exlusive, a way to combine
the high spatial and temporal resolution of 3-D models with the
integrating characteristics of mass-balance budgets is provided by
the use of Lagrangian trajectories.
One of the first systematic practical applications of trajectory
analysis was conducted already during the mid-1940s when US
meteorologists reconstructed the paths of high-altitude balloons
equipped with incendiary devices which the Japanese armed forces
released into the newly discovered jet-stream with the aim of
causing destruction in the continental United States. Classical trajectory analysis, as primarily applied to the atmosphere, mainly
employed graphical techniques used in conjunction with meteorological common sense. Present-day trajectory methods are,
however, based on numerical circulation models where the advection of the trajectories can be achieved both ”on-line” in a
3-D model package, or ”off-line” using velocity fields obtained
from previous model runs.
Papers III and IV both make use of a trajectory scheme based
on results due to Döös (1995) as well as Blanke and Raynaud
(1997), the ”neutral-particle” paths being calculated using 3-D
velocity fields for the Baltic sea originating from the previously
described RCO model. The scheme calculates trajectory paths
analytically for a prescribed velocity field, which is regarded as
stationary at each instant. This permits sub-grid analyses of the
particle motion by interpolating the zonal and meridional velocities u and v defined in the corners of the grid-cell. The zonal
velocity is found by first averaging meridionally and hereafter interpolating zonally:
u(x) =
+
1
(ui−1,j + ui−1,j−1 )+
2
x − xi−1
(ui,j + ui,j−1 − ui−1,j − ui−1,j−1 ).
2∆x
(7)
By introducing u(x) = dx/dt, the following differential equation is obtained:
dx
+ αx + β = 0,
dt
23
(8)
Introduction
Some Concepts of Estuarine modelling.
in which expression α = (ui−1,j + ui−1,j−1 − ui,j − ui,j−1 ) /2∆x
and
β = (xi−1 (ui,j + ui,j−1 − ui−1,j − ui−1,j−1 )) /2∆x− 12 (ui−1,j +
ui−1,j−1 ). By specifying x(ta ) = xa and x(tb ) = xb , this inhomogenous differential equation is found to have the solution
β
β
xb = xa +
exp[−α(tb − ta )] − ,
(9)
α
α
where xb is the zonal displacement and tb the associated time.
The meridional and vertical velocities are calculated in an analogous fashion, wherafter the times tb are compared (Döös, 1995).
The shortest of these indicates where the trajectory will leave this
particular box, a result which serves as the starting point for the
subsequent set of calculations.
The velocity fields used for these calculations can, on the basis
of linear interpolation, be more highly resolved in time than the
circulation-model data. Since the trajectory calculations can be
run in an autonomous fashion, i.e. off-line, the analysis can be
carried through using only modest computer resources.
To achieve the desired integrating properties for characterizing the system, one can either ”tag” the influxes to the system
continuously during the numerical experiment, or initiate the
simulation with a certain concentration of marked particles in the
system. Either the number of trajectories remaining in the system
after a certain time or the decay of the number of trajectories can
now be used as a scalar indicator for describing the state of the
system.
An example of this procedure is given in paper III of the
present thesis, where mixing in the Gulf of Finland has been
investigated by tagging incoming water parcels. This made it
possible to estimate the time-evolution of the most pronounced
mixing zone between freshwater from the river Neva and saline
water from the Baltic proper. It is also proved feasible to establish a ”characteristic front” between these two water types. Both
results would have been difficult to achieve without using Lagrangian trajectories. This formalism was also used to construct
overturning stream-functions pertaining the two source waters,
a useful tool when analysing the dynamics of the Gulf. These
trajectory methods have hitherto only proved their worth in the
Gulf of Finland, but can probably be implemented for any estuary
amenable to a 3-D model description.
24
Introduction
4
Conclusions and outlook
When examining any type of natural phenomenon, it is of vital
importance to use the right tool in the right way. Formulating a
model, it is thus essential to define what the model is intended
to describe, which questions are to be answered, and, if so required, to specify the necessary spatial and temporal resolution.
Models are, however, frequently formulated without an explicit
question in mind, or used for other purposes than those originally intended. In cases like these, the outcome often does not
fulfill expectations. In the field of estuarine modeling, model approaches and program packages are sometimes chosen on the basis their availability, rather than after evaluating different options.
Even though it may not necessarily be a bad thing to utilize the
expertise and/or software at hand, rather than using the optimal
method, some words of caution are called for. The purpose of
the model as well as the desired outcome must be well defined,
with acceptable levels of error kept in mind. It is also important to be aware of the different characteristics, prerequisites, and
advantages associated with each model approach.
If, after a systematic evaluation of which method to use for
modeling a specific estuary, a general approach is preferred, the
choice will most likely stand between a mass-balance budget and
a 3-D model. These two conceptual frameworks are, as earlier described, fundamentally different, both as regards the amount of
information generated and the efforts required for implementing
the model. Paper IV of the present thesis represents an attempt to
establish a systematic procedure for comparing these modeling
approaches. For this investigation, two different Baltic sub-basins
were chosen as objects of study, this since earlier budget calculations were already at hand and sufficiently resolved velocity fields
were available from the RCO model. To establish a common diagnostic variable, the decay times obtained by tagging water parcels
were examined using the earlier described Lagrangian-trajectory
formalism. These time-scales were hereafter compared with theoretical decay times based on turnover characteristics determined
using mass-balance models. Even though these two different results had their roots in entirely different methodologies and sets
of empirical data, they proved to coincide remarkably well in the
case of the Gulf of Riga, a semi-enclosed system which fulfills
the mass-balance model prerequisites reasonably well. This indicates that the two model approaches in this case should yield
25
Introduction
Some Concepts of Estuarine modelling.
more-or-less equivalent results, and that it may be advisable to
employ the simplest method which still provides a sufficient understanding of the system. An analogous examination of the Bay
of Gdansk showed, however, that in this case the two different
approaches yielded entirely different results, with the calculated
decay time-scales differing by an order of magnitude. Since the
reliability of the Lagrangian-trajectory results should be the same
for both systems, it is, most likely, the mass-balance model which
is inadequate. The Bay of Gdansk represents a less constricted
system, where the requirements underlying the use of a massbalance model are not as likely to be met. In systems of this type,
it is probably not a viable alternative to use a simplistic but easily implemented mass-balance budget, since the results will be of
limited significance as regards the true behavior of the system. In
these cases, process models based on observations, or even a suboptimized 3-D model, will probably lead to much better results.
It is hoped that the present thesis, although limited in scope,
might serve as an aid when selecting estuarine models for specific purposes. There is, however, much more that needs to be
done in this field. Hence it would be highly interesting to carry
through comparisons between the different modeling approaches
in a more systematic manner. If possible, it would also be fruitful to combine studies of this type with a more general hydrodynamic approach, e.g. to formulate a rule of thumb for estimating fluxes through a specified topography in cases when the
flow is sub-critical. Another interesting field of study would be
to determine how much information is necessary to implement
a ”good enough” 3-D model for different settings. How well
do the boundaries have to be defined? How important are local currents? Is it possible to use climatological forcing, or are
detailed meteorological observations necessary? Systematic tests
along these lines could provide valuable support when deciding
which model approach to use and what level of expectations to
aim for.
This thesis summary can usefully be concluded by underlining that it is obvious, even from a limited investigation such as
the present one, that different estuarine modeling approaches not
only require different levels of effort for their implementation,
but also yield results of varying quality. If oceanographic aspects
are to be taken into account within an ICZM project, which most
likely should be the case, it is therefore important to as early as
possible in the planning process decide which model to use, since
26
Introduction
this choice ultimately determines how much information about
the physical processes characterizing the system the model can be
expected to provide.
5
Acknowledgements
It is impossible to mention everyone who have helped and supported my work, or just made my life much better during the five
years I have been working on this thesis. Some, however, need
extra recognition as without them this document would not exist.
Peter Lundberg believed enough in me to promote my application to the Phd-program at the Department of Meteorology even
when I utterly lacked the prerequisites. I hope I haven’t failed
you. During my work, he allowed me to take on the projects I
wanted, supporting me in every way even when when I strayed
way outside his research interests. Without his ”tough love” in
editing my writing, this thesis would probably be unreadable.
Kristofer Döös have also been fantastic in giving me slack with my
work but still giving me the support I needed. His strong opinions about everything but science have been a welcome distraction and improved my debate techniques. I hope I will continue
to cooperate with both for a long time to come. The academic
environment both at the Department of Meteorology and the department of Systems Ecology must be credited. Basically all senior
scientists take their time to help and discuss with students on all
levels without ever displaying their superiority by rank. We who
exploit your time and expertise all owe you much. I must mention the fantastic people who helped and befriended me in connection to my project at INVEMAR in Colombia. They gave me
everything I needed even with if their resources were limited. My
work at CTM and NRM have been an essential safety-valve when
the physics and science have been overwhelming from time to
time. Everyone I have been working with there have also made
my life much more interesting and fun. Without the administrative personal taking care of everything we scientists and students
excel in missing or misunderstanding, the department would go
down like the Titanic within days. You deserve all credits. My
family have been great during these years and would help me till
death if I needed to. Finally, all my friends that have made my
life so much easier by being around in sun and rain, by telling
me what I need to hear, by accepting that I fade out sometimes,
by coping with my excessive talking and sometimes to strenuous
27
Introduction
Some Concepts of Estuarine modelling.
discussions, and by existing. To the ones who’s life I have made
more difficult, my apologies. To all, my warmest and deepest
thanks. I love you all! (I also love Katarina even if she won’t
come to my dissertation. . . )
References
Blanke, B., Raynaud, S., 1997. Kinematics of the Pacific Equatorial
Undercurrent: a Eulerian and Lagrangian approach from GCM
results. Journal of Physical Oceanography 27, 1038–1053.
Bolin, B., Rodhe, H., 1973. A note on the Concepts of Age Distribution and Transit Time in Natural Reservoirs. Tellus 25,
58–62.
Döös, K., 1995. Inter-ocean exchange of water masses. Journal of
Geophysical Research 100 (C7), 13499–13514.
Engqvist, A., Andrejev, O., 2003. Water exchange of the Stockholm archipelago: a cascade framework modelling approach.
Journal of Sea Research 49 (4), 275–294.
Ferreira, J. G., 2000. Development of an estuarine quality index
based on key physical and biogeochemical features. Ocean &
Coastal Management 43 (1), 99–122.
Gordon, D. C., Boudreau, Mann, K. H., Ong, J. E., Silvert, W. L.,
Smith, S. V., Wattayakom, G., Wulff, F., Yanagi, T., 1996.
LOICZ Biogeochemical Modelling Guidelines. Vol. 5 of LOICZ
Reports & Studies. LOICZ, Texel, The Netherlands.
Hagy, J. D., Sanford, L. P., Boynton, W. R., 2000. Estimation of
Net Physical Transport and Hydraulic Residence Times for a
Coastal Plain Estuary Using Box Models. Estuaries 23 (3), 328–
340.
Hansen, D. V., Rattray, M., 1966. New dimensions in estuary
classification. Limnology and Oceanography 11, 319–326.
Holland, H. D., 1978. The Chemistry of the Atmosphere and
Oceans. John Wiley & Sons. New York.
Jay, D. A., Geyer, W. R., Uncles, R. J., Vallino, J., Largier, J.,
Boynton, W. R., 1997. A review of recent developments in
estuarine scalar flux estimation. Estuaries 20 (2), 262–280.
Ketchum, B. H., 1983. Estuarine Characteristics. In: Estuaries and
Enclosed Seas. Vol. 26 of Ecosystems of the world. Elsevier Scientific Publishing Company, Amsterdam, p. 209.
Kjerfve, B., 1990. Manual for Investigation of Hydrological Processes in Mangrove Ecosystems. Tech. rep., UNESCO.
Kjerfve, B., L B Miranda, L., Wolanski, E., 1991. Modelling wa-
28
Introduction
ter circulation in an estuary and intertidal salt marsh system.
Netherland Journal of Sea Research 28 (3), 141–147.
Knudsen, M., 1900. Ein hydrographischer Lehrsatz. Annalen der
Hydrographie und Maritimen Meteorologie., 316–320.
Officer, C. B., 1983. Estuaries and Enclosed Seas. Vol. 26 of
Ecosystems of the world. Elsevier Scientific Publishing Company, Amsterdam.
Pritchard, D. W., 1955. Estuarine Circulation Patterns. Proceedings of the American Society of Civil Engineers 81, 717–725.
Pritchard, D. W., 1967. Observations of Circulation in Coastal
Plain Estuaries. Estuaries 83, 37–44.
Sellers, W., 1965. Physical Climatology. University of Chicago
Press.
Stigebrandt, A., Wulff, F., 1987. A model for the dynamics of
nutrients and oxygen in the Baltic proper. Journal of Marine
Research 45 (3), 729–759.
29
I
Turnover-Time Estimates in a Tropical Estuary with High
Spatial and Temporal Variability.
Accepted for publication in Estuarine Coastal and Shelf
Science
Accepted for publication in Estuarine Coastal and Shelf Science
Turnover-Time Estimates in a Tropical Estuary with High Spatial
and Temporal Variability.
Bror Jönsson a,∗ Robert Twilley b J. Ernesto Mancera c
Edward Castañeda-Moya b
a Department
of Meteorology/Oceanography, Stockholm University, SE-10691
Stockholm, Sweden
b Wetland
Biogeochemistry Institute, Department of Oceanography and Coastal
Sciences, Louisiana State University, Baton Rouge, LA 70803, USA.
c INTROPIC,
Universidad de Magdalena, Carrera 32 No. 22-08 Santa Marta,
Colombia
Abstract
A hydrological mass-balance budget was established for the
Ciénaga Grande de Santa Marta lagoonal system, located on the
Caribbean coast of Colombia. Monthly climatological and oceanographical data records spanning ten years were used to determine
the turnover time in the estuary. These calculations resulted in
a turnover time of 20 days, indicating a more dynamical system
than earlier believed. It is also shown that a restoration project in
the area has not had significant impacts on the salinity regime in
the Ciénaga Grande proper. Finally, a slight modification of the
method proposed by Webster et al. (2000) for averaging timeseries to be used in mass-balance budgets is proposed.
1
Introduction
The coastal zones of the world are of critical importance, since
more than half of the global population is found less than 50 km
from the sea and most of the world’s largest cities are located by
the ocean. The near coast regions also provide sustenance for a
high percentage of the world’s population. Although much interest has been devoted to issues concerning coastal systems, for
Paper I
Turnover-Times in a Tropical Estuary
management as well as purely scientific reasons, emphasis has
primarily been on conditions in the temperate zone. Rather few
tropical systems have been comprehensively described, and the
methodology used to investigate coastal areas has not been fully
developed to take into account the special conditions characterizing tropical areas, particularly the high variability in precipitation and the high but stable temperatures . The present study
thus aims at describing the hydrological change a tropical estuary
has undergone due to a restoration project, and to, furthermore,
address some general questions of how to handle the seasonally
fluctuating data series typically encountered when attempting to
establish tropical mass-balance budgets.
Ciénaga Grande de Santa Marta, on the northern coast of Colombia, is the largest estuary of its type in the Caribbean area.
To the east and southeast, the system borders the Sierra Nevada
de Santa Marta. From this mountain range four main rivers,
with a yearly-average flow of 120 m3 s−1 , drain into the Ciénaga
Grande, which to the north is separated from the Caribbean Sea
by the Isla de Salamanca. East of this island a passage, Boca de La
Barra (approximately 120 m wide and 8 m deep), connects the
complex of lagoons to the sea. The flood plain of the Magdalena
River is located in the western and south-western parts of the system. Until the 1970s five main channels fed riverine fresh water
to the estuarine system across this plain.
The estuarine system, with a total area of around 1800 km2 ,
can be divided into two hydrological subsystems. The main water body is the Ciénaga Grande (450 km2 ) / Ciénaga de Pajarales (120 km2 ), two more-or-less well-mixed brackish estuaries
with a rapid exchange of water between them. The other subsystem is constituted by numerous smaller connected lagoons,
which until the 1960s were surrounded by approximately 500
km2 of mangrove forest. The region is arid, with six to seven dry
months a year and an annual water deficit, since evapotranspiration (1200 mm/yr) exceeds precipitation (870mm/yr). The seasons are well-defined, with a long dry season from December to
April and a short wet season in May and June, a short dry season
during July and August, and a long rainy season from September to November. The temperature regime is isomegathermic
with annual means between 27 and 28◦ C and daily amplitudes
of 8-9◦ C. The Caribbean tidal range is approximately 30 cm, the
effect being detectable all the way into Ciénaga Pajarales, where
the range is on the order of 3 cm.
34
Accepted for publication in Estuarine Coastal Shelf Science
74˚W
75˚W
Santa Marta
Golfo de Salamanca
Boca de la Barra
Isla de Salamanca
11˚N
Pajarales
Ciénaga
Grande
Fr
io
Ar
Rio Magdalena
ac
a
n
ció
da
la
vil
ca
Se
n
Fu
ta
Tu
c
ur
in
ca
Sierra Nevada
de Santa Marta
Fig. 1. Map of the Cienaga Grande de Santa Marta estuary system and the Sierra
Nevada de Santa Marta coastal mountain range.
Historically, Ciénaga Grande was the main fish and shellfish
source for the northern part of Colombia. In the late 1950s, a
highway was built along Isla de Salamanca, leaving only one connection between the sea and the lagoonal system and cutting off
most of the sheet flow. Around 1970, fresh-water flow from
the Magdalena River to the system was interrupted by dikes and
beams, built for irrigation purposes and also to prevent flooding. Somewhat later, a road was built parallel to the Magdalena
River, interrupting flood-period sheet flow from the river. Freshwater discharges from the rivers draining the Sierra Nevada de
Santa Marta have also decreased due to irrigation. These changes
have resulted in hypersalinization of mangrove sediment (Botero
et al., 1997; Cardona and Botero, 1998), leading to the extinction
of almost 270 km2 of mangrove forest .
In the early 1990s, the Colombian government initiated a
large-scale rehabilitation program to recover the fresh-water influx to the area, hereby facilitating man-promoted regeneration
of the mangrove forest. The original passages through the west-
35
Paper I
Paper I
Turnover-Times in a Tropical Estuary
ern boundaries of the Ciénaga have been reopened (yielding a
maximum transport capacity of approximately 150 m3 s−1 ) and a
partial re-connection of the system to the sea has been accomplished through box-culverts built under the highway (Botero
and Salzwedel, 1999). Concerns have, however, been raised
about the possibility that the Ciénaga Grande could switch to a
freshwater regime, with concomitant deleterious effects on the
fisheries.
2
Methods
To evaluate the hydrological regime of the Ciénaga Grande system, the water turnover time will be used as a diagnostic variable.
Turnover time is comparatively easy to calculate, provides a good
indication of the characteristics of the system, and furthermore
has an “integrating” property, which to some extent compensates
for temporal variations. The most common method of estimating
turnover times in estuarine systems is to use a mass-balance budget of the type originally introduced by Knudsen (1900). When
undertaking this, a volume budget is first formulated:
dV
= Qp − Qe + Qr + Qm + Qn + Qi − Qo =
dt
= Qf − Qo + Qi ,
(1)
where Qp is the precipitation, Qe the evaporation, Qr the river
influx, Qm the groundwater influx, and Qn the influx from antropogenic sources. All these contributions may be combined to a
total freshwater influx, Qf . Hereafter the volume fluxes to and
from the system, Qi and Qo , respectively, together with the internal and external salinities Si and So can be used to formulate
a salinity budget:
dSi
= Sf Qf + So Qi − Si Qo .
(2)
dt
(Note that by definition Sf , the salinity of the freshwater influx,
is equal to zero.) Under the highly simplified assumption that
both the volume V and the salinity Si are stationary, Eqs. (1)
and (2) yield
V
Qi =
Si ·Qf
So −Si
36
; Qo =
So ·Qf
So −Si
.
(3)
Accepted for publication in Estuarine Coastal Shelf Science
The outflow Qo is used to determine the turnover time τ =
V /Qo where V is the volume of the system (Bolin and Rodhe,
1973). It should be noted that although the terms turnover time
and residence time have different formal definitions, the calculated results normally prove to be the same.
The procedure outlined above assumes a constant volume of
the system. Tides, seiches, “windstau”, and variable river discharges in fact cause the volume of estuaries and lagoons to fluctuate somewhat, an upper bound generally being determined by
the morphological characteristics of the system. In the long run,
however, these changes in volume will tend to even out, and it
may be reasonable to assume a quasi-stationary volume of the
system. To achieve such a ”steady state”, data is averaged over a
considerable time-span.
The salinity is also assumed to remain constant in the system. In view of the strong contrasts between the wet and dry
seasons typical of the tropics, use of the stationarity assumption
may be somewhat questionable in this climatic zone. As will be
demonstrated in what follows, this problem can be investigated
systematically by evaluating the time derivatives dQi /dt arising
when various averaging techniques are used.
The salinity is not only assumed to be constant over time, but
also in space, which implies that the system must be well-mixed.
It is possible to formulate a two-layer model with a separating pycnocline, but stratified systems are less straightforward to model
since several auxiliary assumptions are required. A system with
variations of salinity can, nevertheless, be modeled, provided that
the salinity difference between the system and the adjacent water
mass is much larger than the internal variations characterizing the
system (Gordon et al., 1996).
2.1
The Ciénaga Grande System
The mass-balance procedure discussed above was employed to
calculate the turnover times for the combined water bodies of
Ciénaga Grande and Ciénaga Pajaral (for brevity henceforth denoted comprehensively as the Ciénaga Grande), and also to determine the fluxes through Boca de la Barra, the inlet from the sea.
Smaller lagoons, creeks and mangrove swamps were not taken
into account. To establish the budget, a number of assumptions
and simplifications had to be made: The freshwater inputs (of
zero salinity) were ascribed to solely three sources; direct pre-
37
Paper I
Paper I
Turnover-Times in a Tropical Estuary
cipitation, influx from the Sierra-side watershed and influx via
canals from the Magdalena region. All water exchange between
the ocean and the Ciénaga Grande was assumed to take place
through the Boca de la Barra. Sheet flow from Magdalena River
was neglected since roads cut it off, and Isla de Salamanca was furthermore assumed to be impermeable. Box culverts through the
island were ignored in the calculations since they probably only
serve as an extra outlet for water from the Caño Clarin, and thus
do not permit direct interchange with the Ciénaga Grande proper.
Precipitation and temperature in the Sierra-side watershed were
taken to be the same as in the estuary. (Possibly this may not be
the case for the entire watershed, but no data for high elevations
in the area are available, and most of this watershed is, furthermore, in the same climatological regime as the Ciénaga Grande.)
The estuary was assumed to be well-mixed, vertically as well as
horizontally, which it most likely is on a monthly timescale. No
water that leaves the Ciénaga was assumed to reenter the system,
since there is a strong external coastal current in the area (Wiedemann, 1973). Finally, a high exchange rate between the Ciénaga
Grande and Ciénaga Pajaral was assumed, as indicated by salinity
data as well as from the Pajaral tidal records.
A conceptual description of the currents has been provided
by Wiedemann (1973), suggesting the presence of a counterclockwise gyre in the Ciénaga Grande proper, fed by freshwater
influxes from various rivers and channels. This reinforces the
belief that we are dealing with a well-mixed system, where most
of the water experiences the same turnover time before leaving to
the Golfo de Salamanca.
In the forthcoming calculations, a comprehensive set of monthly data from 1988 to 2000 was used. Salinities in the Ciénaga
Grande, obtained during a number of different surveys normally
measured with a hand salinometer, are thus available from a database assembled by INVEMAR. (No salinities were measured in
the Boca de la Barra between 1988 and 1992 and furthermore
1997 lacks measurements in the whole system, why this year has
been excluded from the forthcoming budget calculations.) Universidad Nacional provided some measurements of salinity in the
Golfo de Salamanca from 1996 and 1997 (Camillo Garcia, pers.
comm.). These time-series are shown in monthly averaged form
in figure 2 . For large and heterogeneous systems, the LOICZ
framework defines the salinity of the system as that in the vicinity
of its outflow locations (Gordon et al., 1996). In our case this
38
Freshwater fluxes (m3s-1)
Accepted for publication in Estuarine Coastal Shelf Science
Paper I
a
800
600
400
200
0
1988
1989
1990
1991 1992
1993
1994
1995
1996
1997
1998
1999
2000
40
b
Salinity
30
20
10
0
1988
1989
1990
1991 1992
1993
1994
1995
1996
1997
1998
1999
2000
Fig. 2. Salinity records from Golfo de Salamanca (weak solid line), the center of
Ciénaga Grande (heavy solid line ), and Boca de la Barra (dashed line).
would be the exit salinity measured in the Boca de la Barra. Another possibility would be to use the salinity in the center of the
Ciénaga Grande proper as system salinity. The external-reservoir
salinity could, furthermore, be defined as either the earlier mentioned salinity in the Boca de la Barra, or that in the Golfo de
Salamanca. To compare the effects of applying these different
operational definitions, calculations have, as will be seen, been
undertaken using all three possible combinations of the salinity
measurements.
Precipitation and temperature in the region were acquired
from a meteorological station at 11o 00’ N , 74o 14’ W maintained
by IDEAM, which agency also provided discharge measurements
from a number of the larger rivers in the Sierra region. The rivers
Tucurinca, Sevilla, and Frio have been gauged for most of the
study period. Due to the fact that Rio Aractaca and Fundacion
were only studied during shorter time periods, discharge timeseries for these rivers were calculated using observations from Rio
Tucurinca and Frio respectively, assuming a linear relationship in
discharges between the rivers. Finally, CORPAMAG provided data
for calculating the influx from the canals on the Magdalena side
of the system. Where data were missing, interpolation based on
decadal monthly averages was used.
The Sierra-side watershed runoff was estimated by using the
river discharge Qs as well as the flux Qr from a climatic runoff
39
Paper I
Turnover-Times in a Tropical Estuary
River discharge from the Sierras
1000
900
800
700
m3s!1
600
500
400
300
200
100
0
1988
1989
1990
1 9 91
1992
1993
1994
1995
1996
1997
1998
1999
2000
Fig. 3. Calculated water discharge from the Sierra side based on a runoff model
(dashed line) and measured river discharge from the Sierra side of the Ciénaga
Grande (solid line).
Freshwater fluxes (m3s-1)
900
700
500
300
100
1988
1989
1990
1991 1992
1993
1994
1995
1996
1997
1998
1999
2000
Fig. 4. Freshwater discharge from the Magdalena side (heavy line), and total
freshwater influx used in the budget (weak line). Note the effects of the opening
of the canals between Rio Magdalena and Ciénaga Grande in 1995.
model (Kjerfve, 1990):
r
Qr = AW
2549 · 109
40
∆f
r
.
(4)
Accepted for publication in Estuarine Coastal Shelf Science
Paper I
HereAW is the watershed area (m2 ), r is precipitation (mm),
is a non-dimensional runoff factor estimated by Sellers
and ∆f
r
(1965).
∆f
≈ e−E0 /r ,
(5)
r
where Eo is the potential evapotranspiration (mm/month). This
quantity was calculated using a straightforward empirical relationship (Holland, 1978) between potential evapotranspiration
and air temperature:
E0 = 1 · 109 e
−4.62·103
r
.
(6)
For each month, the freshwater inputs Qs and Qr were compared and the highest value was selected in order to minimize
the shortcomings of each method. Qr comprises all discharges,
including groundwater and small-creek contributions, but is zero
when there is no precipitation. The river discharge Qs , on the
other hand, is a true value of the minimum freshwater input from
the Sierra region, even if sheet flows and groundwater discharge
have been omitted. Problems with this method of combining
fluxes may arise if there are large reservoirs in the area, but since
there are no dams or significant lakes in the Sierra watershed, the
method should give a reasonably good estimate of the freshwater influx to the Ciénaga Grande. The discharge Qm from the
canals connecting Rio Magdalena and the Ciénaga Grande was estimated using the Rio Magdalena water level in conjunction with
an empirical conversion factor (Roberto Montiel, pers. comm.).
Finally, the direct net precipitation Qp −Qe was calculated by using evapotranspiration, precipitation and the area of the lagoon.
The freshwater fluxes discussed above are shown in monthly averaged form in Figs. 3-4.
By applying the previously outlined calculational procedure
to differently averaged varieties of the salinity and freshwaterflux data sets discussed above, the transports through the Boca de
la Barra, together with turnover times, were determined. These
results will next be discussed, with particular emphasis on the
effects of applying different time-averaging techniques as well as
using the three possible definitions of the salinities Si and So .
41
Paper I
Turnover-Times in a Tropical Estuary
3
Results
The analysis was initiated by examining the freshwater fluxes to
the system. The times series of the precipitation as well as the
different freshwater fluxes to the system (Figs. 3, 4) demonstrate
a distinct pattern of intra-annual variations between dry and rainy
seasons. From these diagrams it is also obvious that 1999 was an
unrepresentative year in view of the extreme rainy-season precipitation caused by an unusually severe el Niño.
Fig. 4 also shows that the increased freshwater transport from
the Magdalena river side after the canals had been opened did
not affect the freshwater input to the system system significantly
during the rainy season. During the dry season, however, the
minimal freshwater influx was raised to a higher level. Salinity
profiles measured in the canals and soil indicate that this freshwater increase effectively prevents hypersaline conditions in the
mangrove substrata, cf. Rivera-Monroy et al. (2001b). This study
also shows that most of the mixing between fresh and estuarine
water takes place in the canals, rather than in the Ciénaga Grande
proper, during the dry season. Both the freshwater-influx and
salinity time series (Figs. 2, 4) indicate that the system is dominated by marine conditions during dry seasons, but is close to a
fresh riverine system during the rainy seasons. The only time the
lagoonal system was totally freshened out, however, was during
the pronounced el Niño conditions in 1999.
When carrying through a budget analysis, the standard procedure (suggested already by Knudsen (1900)) has been to average
all data over time and then undertake one sole mass-balance calculation. This procedure, however, does not only eliminate temporal variations, but, as Webster et al. (2000) have shown, can
give rise to misleading results in a system characterized by high
intra-annual variations. These latter authors suggested that the
fluxes instead be calculated month by month and hereafter averaged. Applying this technique to the Ciénaga Grande data set, the
resulting Qi proved to be 490 m3 s−1 , compared to a Qi of 400
m3 s−1 when using the standard method of first averaging all data.
Although the Webster et al. (2000) methodology may be theoretically well-founded, there remain some practical problems. If the
system, as e.g. in the present case, is affected by tides and only
a limited number of salinity observations are available, spurious
effects may arise if Si and So have been sampled in more-or-less
the same water masses. In cases like these, the calculated fluxes
42
Accepted for publication in Estuarine Coastal Shelf Science
tend to be unrealistically high. A third approach, which minimizes the possibility of unrealistic extreme fluxes and does not
give rise to the fundamental problems discussed by Webster et al.
(2000), is to average the data for each month over all years, calculate the fluxes and hereafter average these. Adhering to this
latter procedure resulted in Qi = 390 m3 s−1 .
1988-2000
1988-1995
1996-2000
(m3 /s)
390
410
350
Total flux (m3 /s)
590
560
650
Influx, Boca de la Barra
Turnover time (days)
20
22
Table 1
Results of budget calculations of the fluxes to and from CGSM
19
As mentioned earlier, one method to systematically compare
different averaging techniques is to evaluate the time derivatives
dQi /dt resulting from each of the methods. Due to the specific
calculations used for each of the cases, the resulting time-series
are of different duration. When calculating dQi /dt using the
Webster et al. (2000) method, a 155-month time-series arises
with a mean of 780 m3 s−2 , cf. Fig. 5a. From this diagram it is
furthermore seen that the derivative at times assumes exceedingly
large values, hereby to some extent undermining the stationary
assumption. Our suggested method, on the other hand, results in
a 12-month time-series with an average value of dQi /dt of 173
m3 s−2 . These results can be compared with the time derivative
of the freshwater flux, which has a mean value of 136 m3 s−2 .
Fig. 5b shows these three time derivatives with the freshwater
flux method and the Webster’s method results averaged monthly
over the roughly 13-year period. Both this figure and the earlier
presented results clearly show that our method tends to give a
smaller mean time derivative than that resulting from the Webster et al. (2000) method. The time derivative of our proposed
method is furthermore closer to that of the freshwater flux. Taking into account that the model is linear and that the dynamics
(i.e. salinity) ultimately depend on the freshwater flux, one could
argue that this reasonably close correspondence with the freshwater method derivative reinforces confidence in the averaging
method proposed here.
Another important aspect when establishing mass-balance budgets is how to define Si and So spatially. As earlier mentioned,
measurements have been taken in the center of the estuary, at the
exit to the ocean, and in the open ocean. The procedure consis-
43
Paper I
Paper I
Turnover-Times in a Tropical Estuary
10000
a
dQ0/dt (m3s-2)
5000
0
-5000
-10000
-15000
1988
1989
1990
1991 1992
1993
1994
1995
1996
1997
1998
1999
1500
2000
b
dQ0/dt (m3s-2)
1000
500
0
-500
-1000
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Fig. 5. Diagrams showing calculated dQi /dt obtained using the various averaging
procedures. Panel (a) shows this derivative calculated using each observation in
the data set. In panel (b), the solid line represents monthly averages of the data
presented in panel (a), hereby permitting a comparison with dQi /dt calculated
using fluxes based on already monthly averaged data (denoted by plus signs). The
dotted line shows the time derivative of the monthly averaged freshwater flux.
tent with LOICZ standards would be to use exit data as Si and
the Golfo de Salamanca data as So . In an earlier calculation of the
turnover times for Ciénaga Grande (Rivera-Monroy et al., 2001a),
Si was defined as the salinity centrally in the estuary, and So as
that at the exit to the sea. We have used these two definitions
with a third, where Si is defined as the salinity in the center of
the ciénaga and So as the salinity in the Golfo de Salamanca. Fig.
6 shows Qi (the inflow through the Boca de la Barra which will
be used to estimate the turnover time of the system) calculated for
all three sets of definitions using data from each month without
any smoothing or averaging. It is obvious that the resulting fluxes
for the two first definitions are too high to be physically realistic.
The main reason for these spurious magnitudes of Qi is, most
likely, that the estimates of the salinity within the system used
for the calculations are based on observations undertaken in recently entered water-masses. The third definition proposed here
gives rise to much less problems with such extreme values, and
the resulting yearly mean value of Qi = 390 m3 s−1 , was used to
calculate the turnover times of the system as reported in Table 1.
The calculated values of the total inflow can be compared
to direct current measurements conducted by CORPAMAG at the
44
Accepted for publication in Estuarine Coastal Shelf Science
Paper I
20000
15000
Water fluxes (m3s-1)
10000
5000
0
-5000
1988
1989
1990
1991 1992
1993
1994
1995
1996
1997
1998
1999
2000
Fig. 6. Calculated fluxes through Boca de la Barra based on using different definitions of salinity inside and outside the system. The heavy line represents the
results obtained using observations from the Golfo de Salamanca and the center
of the Ciénaga Grande (cCG) as So and Si , respectively. The weak line is based
on records from Boca de La Barra as Si and Golfo de Salamanca as So . The dotted
line, finally, arises using cCG and Boca de la Barra data as Si and So , respectively.
Boca de la Barra in November 1993 and in August 1999, where
total fluxes of 576 m3 s−1 and 286 m3 s−1 , respectively, were
recorded. Taking into account the floods in 1999 as well as the
hydrological changes caused by the opening of the canals, it is
probably most accurate to compare these direct measurements
with the calculated values for 1988-95. When this is done, the
November current measurement corresponds reasonably well to
the calculated fluxes. (The August survey was conducted during a
neap-tide period when tidal fluxes are small, which probably lead
to the comparatively low transport value.) Another method to
validate the results is to apply a simple tidal-prism model. Taking
into account the physical and climatological conditions characterizing the system (size, wind regime, depth &c.) it thus appears
likely that the main forcing for the water exchange between the
Ciénaga Grande and the ocean is provided by the predominately
semi-diurnal tide. A tidal range of 4 cm, an area of 700 km2 and
a tidal period of approximately 12 hours implies an exchange of
400-500 m3 s−1 This is well in agreement with the calculated values, particularly when disregarding the extreme floods of 1999.
In order to estimate which of the input variables is most critical when undertaking the calculations, a sensitivity analysis was
45
Paper I
Turnover-Times in a Tropical Estuary
Precipitation
Evapotranspiration
Freshwater influx, Run-off model
River discharge, Sierra side
River discharge, Magdalena side
Salinity, Center of CGSM
Salinity, Golfo de Salamanca
80
70
Turnover time (days)
60
50
40
30
20
10
0
0.10
0.16
0.25
0.40
0.63
1.00
1.58
2.51
3.98
6.31
10
Scaling factor
Fig. 7. Sensitivity analysis of the Ciénaga Grande mass-balance budget. The figure
has been constructed by scaling each of the input variables used in the calculations
by a factor ranging from 0.1 to 10 times the original value, while retaining other
variables constant. One can see that the turnover time of the system is more
susceptible to changes of salinity in the Cienaga Grande (heavy solid line) or in
the Golfo de Salamanca (heavy dotted line), than to changes of the other variables.
Evapotranspiration (weak dotted line) can not be scaled higher than the level of
fluxes from precipitation (weak solid line). The other fluxes used in the sensitivity
analysis are river discharge from the Magdalena side (dashed line), river discharge
from the Sierras (dash-dotted line) and discharge from the Sierras calculated using
a run-off model (dash-dot-dotted line).
also performed. Each variable was in turn multiplied with a factor
ranging from 0.1 to 10 and the resulting turnover time was used
as a diagnostic variable. The ensuing graph (Fig. 7) shows that
the salinities are the most critical variables for a correct estimate
of turnover times, consistent with well-established characteristics
of mass-balance budgets. Furthermore, it can be noted that the
salinities undoubtedly must be regarded as the most robust input variable, this due to the high precision with which they are
measured.
4 Discussion
Due to a dearth of quantitative hydrological studies of the Ciénaga
Grande de Santa Marta estuarine system, a general perception has
previously been that this estuary can be regarded as a stagnant
pool with long turnover-times and, consequently, also a high sus-
46
Accepted for publication in Estuarine Coastal Shelf Science
ceptibility to eutrophication. The present study alters this picture
somewhat, suggesting a more dynamic system with short residence times for water masses and, by implication, nutrients. A
turnover-time on the order of 20 days indicates that the system is
flushed regularly and instead should be described as an intermediary stage between the rivers supplying freshwater to the system
and the ocean. It is not likely that the high primary production
characterizing the system remains in situ long enough to settle and
create bottom sediments; rather the detritus is flushed out to the
Golfo de Salamanca. This conjecture is further supported by ongoing INVEMAR surveys showing low organic content in the sediment.
In this context, an even more important factor than the turnover-time may be the seasonal variation of the precipitation. Most
of the nutrient load probably occurs in the beginning of the wet
season, when flushing rates have a maximum. During the wet
season, the system should act more as a riverine extension than
as a “true” estuary, particularly as regards the biogeochemical
processes. This constitutes a marked difference in comparison
to most temperate systems, which generally show much lower
variations of turnover-times between the seasons. It is also evident that the largest effects from the restoration project manifest
themselves during dry seasons. Fig. 3 shows a significant increase of the minimum freshwater influx to the Ciénaga Grande
during these periods, whereas a similar increase during the wet
seasons has little, if any, effect. It is also clear from Table 1 that
the turnover-times have not changed to any greater extent by the
opening of the canals.
The present study has furthermore demonstrated the validity of the assumption that the Ciénaga Grande system can be regarded as hydrologically divided into two distinct parts, with the
mangrove swamp being affected by the increased influx of freshwater, this in contrast to the Ciénaga Grande proper where only
marginal effects can be detected. Thus the system has most likely
responded favorably to the hydrological changes, consonant with
the aims of the restoration project.
The results obtained here could also serve as a reference point
for the methodological discussion of how to average highly varying temporal data in mass-balance budgets initiated by Webster
et al. (2000). These authors compare two different methods of
averaging data for the budget calculations: pro primo using the average of the products in Eqs. 2-4 based on each of the time-series
47
Paper I
Paper I
Turnover-Times in a Tropical Estuary
or, pro secundo, making use of the product of the averaged timeseries. These two different approaches yield Qi equal to 490
and 400 m3 s−1 , respectively. However, the high value arising
from the method of Webster et al. (2000)) is probably caused
by the small differences between Si and So , which may occur
when sparse, discrete measurements are used in the calculations.
We therefore suggest a compromise between the two methods,
where averages for each month are calculated and the results used
in the budgets. Hereafter a final averaging resulted in a Qi of
390 m3 s−1 . Since most of the variations discernible in the datasets are of an intra-annual nature, this approach to some extent
overcomes the problems arising when the total means of the full
time-series are used (Knudsen, 1900). At the same time, the proposed averaging method is advantageous compared to the one
suggested by Webster et al. (2000), since it does not result in
unrealistically high values of the fluxes. A further advantage is,
as previously demonstrated, that the time derivatives only assume
moderately large magnitudes, hereby not invalidating the stationarity assumption underlying the analysis.
Finally, this study presumably shows that a rather simple approach for modeling a costal water body can provide a useful
addition to our understanding of the system.
5
Acknowledgments
This study could not have been completed without outstanding
help and support from several people in Colombia. We especially want to thank the Colombian agencies INVEMAR, CORPAMAG, and IDEAM, and the Universidad Nacional de Colombia. The project was funded by the Swedish International Aid
Agency (Sida) as a Minor Field Study. Funding by COLCIENCIAS
(code no. 2105-13-080-97) contributed to the preparation of
the manuscript. Finally we want to thank prof. Fredrik Wulff,
Stockholm University, who supervised the Minor Field Study and
helped in many ways.
References
Bolin, B., Rodhe, H., 1973. A note on the Concepts of Age Distribution and Transit Time in Natural Reservoirs. Tellus 25,
58–62.
48
Accepted for publication in Estuarine Coastal Shelf Science
Botero, L., Montiel, M., Estrada, P., Villalobos, M., Herrera,
L., 1997. Microorganism removal in wastewater stabilisation
ponds in Maracaibo, Venezuela. Water Science and Technology 35, 205–209.
Botero, L., Salzwedel, H., 1999. Rehabilitation of the Cienaga
Grande de Santa Marta, a mangrove-estuarine system in the
Caribbean coast of Colombia. Ocean and Coastal Management
42, 243–256.
Cardona, P., Botero, L., 1998. Soil characteristics and vegetation structure in a heavily deteriorated mangrove forest in the
Caribbean Coast of Colombia. Biotropica 30, 24–34.
Gordon, D. C., Boudreau, Mann, K. H., Ong, J. E., Silvert, W. L.,
Smith, S. V., Wattayakom, G., Wulff, F., Yanagi, T., 1996.
LOICZ Biogeochemical Modelling Guidelines. Vol. 5 of LOICZ
Reports & Studies. LOICZ, Texel, The Netherlands.
Holland, H. D., 1978. The Chemistry of the Atmosphere and
Oceans. John Wiley & Sons. New York.
Kjerfve, B., 1990. Manual for Investigation of Hydrological Processes in Mangrove Ecosystems. Tech. rep., UNESCO.
Knudsen, M., July 1900. Ein hydrographischer Lehrsatz. Annalen
der Hydrographie und Maritimen Meteorologie., 316–320.
Rivera-Monroy, V. H., Jonsson, B. F., Twilley, R., Casas-Monroy,
O., Castañeda, E., Montiel, R., Mancera, E., Troncoso, W.,
Daza-Monroy, F., 2001a. Cienaga Grande de Santa Marta
(1993-1999): a tropical coastal lagoon in a deltaic geomorphic setting. Tech. rep., LOICZ.
URL
http://data.ecology.su.se
/mnode/South/cienegagrande/
cienegagrandebud.html
Rivera-Monroy, V. H., Mancera-Pineda, J. E., Twilley, R. R., CasaMonroy, O., Castañeda, E., Restrepo, J., Daza-Monroy, F., Perdomo, L., Reyes-Forero, P., Campos, E., Villamil, M., Pinto,
F., 2001b. Estructura y Función de un Ecosistema de Manglar
a lo Largo de una Trayectoria de Restauración en Diferentes
Niveles de Perturbación: El caso de la Ciénaga Grande de Santa
Marta. Final report 2105-09-13080, Center for Ecology and
Environmental Technology, University of Louisiana-Lafayette,
Lafayette, Louisiana USA and Instituto de Investigaciones Marinas y Costeras- Colombia.
Sellers, W., 1965. Physical Climatology. University of Chicago
Press, Ch. Evaporation and evapotranspiration, pp. 156–180.
Webster, I. T., Parslow, J. S., Smith, S. V., 2000. Implications of
49
Paper I
Paper I
Turnover-Times in a Tropical Estuary
Spatial and Temporal Variation for Biogeochemical Budgets of
Estuaries. Estuaries 23 (3), 341–350.
Wiedemann, H., 1973. Reconnaissance of the Ciénaga Grande de
Santa Marta: physical parameters and geological history. Mitt.
Inst. Colombo-Alemn Invest. Cient. Santa Marta, 85–119.
50
II
Fjord seiches in the Gulf of Finland and their relationship to
the global Baltic seiches.
Submitted to Journal of Geophysical Research
Submitted to Journal of Geophysical Research
Fjord seiches in the Gulf of Finland and their relationship to the
global Baltic seiches.
Bror Jönsson Kristofer Döös Jonas Nycander Peter Lundberg
Department of Meteorology/Oceanography, Stockholm University, SE-10691
Stockholm, Sweden
Abstract
A linear shallow-water model was used to study different harmonic oscillations in the Baltic Sea. The model was initialized
using a linear sea-surface slope from east to west, and was hereafter run without forcing. In our results, we could identify three
different local oscillatory modes: one in the Gulf of Finland, with
the two distinct periods 23 and 27 hours, one in the Danish Belt
Sea, with a less distinct period in the range 23-27 hours, and
one in the Gulf of Riga, with the period 17 hours. The most
pronounced mode is that in the Gulf of Finland. No clear indications of global seiches in the Baltic could be found from our simulations. These results were further corroborated by a frequency
analysis of sea-level observations from the Baltic. This shows an
amplification of the K1 and O1 tidal modes in the Gulf of Finland, but not of the M2 and S2 modes. No such amplification
was seen in the rest of the Baltic Sea. On the basis of our model
simulations, we propose the following physical picture of the
sea-level oscillations of the Baltic: a collection of weakly coupled
local oscillators. Each oscillator corresponds to a ”fjord mode“
or ”harbour mode” in a particular bay or sub-basin. These are
not proper eigenmodes, since their energy gradually leaks out to
the rest of the Baltic Sea, resulting in radiation damping. Nevertheless, their resonance may in fact be sharper than that of the
proper global eigenmodes.
Paper II
Fjord seiches in the Gulf of Finland vs global Baltic seiches.
1
Introduction
A proper knowledge regarding the characteristics of the Baltic seiches is of great practical importance, not least in view of possible
flooding of St Petersburg, located at the inner end of the Gulf of
Finland.
Ever since Forel in the 1870s described the standing oscillations of Lake Geneva, considerable research efforts have been
directed towards the prediction and analysis of seiches. Early
last century Chrystal (1905) developed a technique for solving
the standing-wave eigenvalue problem for realistic bathymetries,
in which variations of the width and depth along the ”channel”
could be dealt with analytically. It was particularly well adapted to
elongated basins, such as the lochs of his native Scotland. Witting
(1911) applied this methodology when investigating the tides of
the Baltic, and made particular note of the fact that the K1 and O1
diurnal tidal modes were amplified in the Gulf of Finland.
Neumann (1941) focused on the global seiche modes of the
entire Baltic, using sea-level data from a number of events when
particularly pronounced standing oscillations had been observed.
His theoretical analysis was based on computational methods due
to Defant (1918) and Hidaka (1936). The rapid post-war development of the digital computer, as well as the catastrophic NorthSea floods in 1953, provided a great impetus for this branch
of research, and in the late 1950s two important studies of the
Baltic were conducted. Svanssson (1959) presented the first twodimensional barotropic model of this almost land-locked sea, and
Lisitzin (1959) undertook a detailed study of 17 instances when
regular sea-level oscillations (encompassing at least four distinctly
recorded periods) had been observed at Haamina in the Gulf of
Finland. She determined the associated periods, finding that they
tended to be fairly constant at around 26.4 hours, and furthermore undertook a systematic discussion of the possible effects of
the Coriolis force on the oscillations of the Gulf of Finland. Krauss
and Magaard (1962) applied Chrystal’s technique to calculate the
periods of the Baltic Sea seiches, and these authors later subjected
spectra of the 1958 sea-level records from 35 tide gauges along
the coasts of the Baltic to a thorough investigation (Magaard and
Krauss, 1966).
The first numerical study properly incorporating the effects
of the earth’s rotation was conducted by Wübber and Krauss
(1979). They used a two-dimensional shallow-water model with
54
Submitted to Journal of Geophysical Research
a 10-km grid size and the connection to the Kattegat closed. The
model was forced by a prescribed pressure field during 30 hours,
and then run without forcing or friction. Three modes with approximate periods of 26, 22 and 20 hours, respectively, were
found in the study. Meier (1996) investigated the differences between simulated eigenmodes in a similar study where he found
eigenfrequencies for the Baltic which corresponded to those determined by Wübber and Krauss (1979).
Most of these investigations have had a common denominator: the assumption that global seiches are important in the
Baltic. Their primary aim has been to calculate the period of these
conjectured standing waves, rather than to examine the validity
of this approach. The existence of global seiches, is, however,
not entirely corroborated by observations. This is particularly
true of the lowest mode, involving oscillations of the entire Gulf
of Bothnia-Baltic proper system. Analysing observations over a
time-span of four years, Neumann (1941) thus found only very
few instances of such an oscillation, and even the one explicitly
dealt with in his study is not very distinct, comprising hardly
more than two periods. Its period was about 40 hours. This
agreed with his theoretically calculated value of 39 hours, but
this calculation neglected the Coriolis force, and, according to the
numerical calculations by Wübber and Krauss (1979), incorporating this effect reduces the period from 40 to 31 hours. In the
the water-level spectra calculated by Magaard and Krauss (1966)
a 31-hour oscillation was observed only in the Belt Sea. In a recent study, Metzner et al. (2000) could not find any evidence for
a global Belt Sea-Bay of Bothnia seiche oscillation using combined
tidal-gauge records and satellite-altimetric data.
Neumann found seven instances of a different mode, with its
axis from the Gulf of Finland to the Danish Belt Sea, bypassing
the Gulf of Bothnia. Its period was about 27 hours, in agreement
with his one-dimensional non-rotating calculations. However,
even these events encompassed at most four oscillation periods.
Judging from its spacial structure, this oscillation might correspond to either of the second, third or perhaps fourth eigenmodes
found by Wübber and Krauss (1979). In general, spectral analysis of tide-gauge observations show no clear peaks at any of the
theoretically calculated eigenfrequencies.
Why are the global seiches so rarely seen in the Baltic? A
possible reason is the complex coastline and bottom topography,
which divide it into several bays and sub-basins, separated by
55
Paper II
Paper II
Fjord seiches in the Gulf of Finland vs global Baltic seiches.
straits and sills. This probably leads to the global seiches being
strongly damped by bottom friction, nonlinear effects, excitation
of internal waves, and the connection to Kattegatt. Therefore, the
seiches may lose coherence too quickly to be seen in reality.
We propose a different physical picture of the sea-level oscillations of the Baltic: a collection of weakly coupled local oscillators. Each oscillator corresponds to a ”fjord mode” or ”harbour
mode” in a particular bay or sub-basin. These are not proper
eigenmodes, since their energy gradually leaks out to the rest
of the Baltic Sea, resulting in radiation damping. This damping “smears out” their spectral characteristics, widening the resonance peaks. Nevertheless, their resonance may in fact be sharper
that that of the proper global eigenmodes. We find three examples of such local oscillators: the Gulf of Finland, the Belt Sea, and
the Gulf of Riga.
We examine this hypothesis by using a numerical shallowwater model. The model is run forward in time without forcing,
using a large-scale initial condition capable of exciting all different oscillations in the main basin of the Baltic. However, we commence the investigation by analysing a set of tide-gauge records
from the Gulf of Finland as well as the Baltic proper.
2
Observed water-level spectra
In the 1960s, Magaard and Krauss (1966) undertook a thorough
study of the water-level spectra from 35 stations along the coast of
the Baltic, using sea-level data from 1958. To their surprise, they
did not find any indications of seiche oscillations at the periods
calculated in their previous study (Krauss and Magaard, 1962).
Instead, the first four tidal modes M2 (12.42h), S2 (12.00h), K1
(23.93h) and O1 (25.82h) were clearly visible in the spectra.
A 30-hour oscillation in the Belt Sea (but not in the rest of the
Baltic) was also seen in the spectra. It was explained as a harmonic
of a 120-hour oscillation in the atmospheric forcing.
We undertook a spectral analysis of data from the same tidal
stations in the Gulf of Finland (Hamina, Helsinki, and Hanko) as
used by Magaard and Krauss (1966). Our investigation was based
on four-year records (1997-2000) generously put at our disposal
by the Finnish Institute of Marine Research (FIMR). Each annual
record was analyzed separatel. Our procedure differed from that
previously employed (Magaard and Krauss, 1966) in that a higher
resolution without any lag window was used. The resulting spec-
56
Submitted to Journal of Geophysical Research
Paper II
Location of water−level stations
61 oN
40’
20’
Hamina
60 oN
Helsinki
40’
Hanko
20’
59 oN o
18 E
21oE
o
27 E
24oE
o
30 E
Fig. 1. Location of three Finnish water-level stations in the Gulf of Finland
tral distributions were hereafter averaged, and standard deviations
were calculated. (The small sample without any well-established
distribution precluded the use of confidence intervals.)
When examining the resulting spectra shown in Figure 2,
we first confirm the earlier results due to Magaard and Krauss
(1966) that no peaks at the predicted global seiche periods are
discernible. The most pronounced features of these spectra are
instead peaks coinciding with the diurnal K1 and O1 tidal periods. This state of affairs may appear somewhat surprising since
the semi-diurnal M2 and S2 tides are subjected to stronger astronomical forcing than the diurnal component, but, as already
recognised by Witting (1911), the K1 and O1 tides are in approximate resonance with the gravest-mode “fjord seiche” in the
Gulf of Finland. (On the basis of the crude predictions provided
by Merian’s formula, cf. Defant (1960), the fundamental mode
of the oscillation should have a period T = 4L(gH)−1/2 , where
L is the length of a rectangular fjord of depth H. The Gulf of Finland has a length on the order of 400 km from St. Petersburg
to Hanko, and its average depth is around 40 m, which yields
a gravest-mode period of approximately 23 hours.) This ”fjordmode interpretation” of the sea-level records in Figure 2 is further
reinforced by noting that the spectral amplitude of the pertinent
peaks is highest at Hamina and decreases towards the entrance of
the gulf.
3
Numerical simulations of the seiches
The model is a further development of the one originally formulated by Döös (1999). The linear barotropic shallow water
equations are
57
Fjord seiches in the Gulf of Finland vs global Baltic seiches.
4
Amplitude (cm)
Hamina
3
2
1
0
0
5
10
15
20
25
30
35
40
45
50
5
10
15
20
25
30
35
40
45
50
5
10
15
20
25
Period(hours)
30
35
40
45
50
4
Amplitude (cm)
Helsinki
3
2
1
0
0
4
Hanko
Amplitude (cm)
Paper II
3
2
1
0
0
Fig. 2. Spectral diagrams of water-level time-series sampled at the three stations
in the Gulf of Finland 1997-2000. The curves show averages of the four annual
spectra, while the shaded areas represent +/- one standard deviation.
∂u
+ f k × u = −∇gη + A∇2 u − Γu,
(1)
∂t
∂η
+ ∇ · (hu) = 0,
(2)
∂t
where u is the horizontal velocity vector, f ≡ 2Ω sin φ the Coriolis parameter, k is the unit vector normal to the plane, A and
Γ the coefficients of viscosity and Rayleigh friction, respectively,
and ρ the average water density. The total depth of the water
column is h (λ, φ, t) = H (λ, φ) + η (λ, φ, t), where η is the seasurface elevation, and H is the bathymetric water depth relative
the geoid. The model is based on spherical coordinates with λ
denoting longitude and φ latitude.
Equations (1)-(2) were discretised on a C-grid with the energyconserving scheme proposed by Sadourny (1975). The model has
previously been implemented (in baroclinic form) for the Tropical Atlantic by Döös (1999). The horizontal resolution of the
model is two nautical miles and a realistic bathymetry provided
by Seifert et al. (2001) was employed. A sponge zone was used
to handle the open boundary located in the North Sea (cf. the
model domain in Figure 3).The energy in the model was dissipated by using a low viscosity (A = 10−3 m2 s−1 ), and a very
58
Submitted to Journal of Geophysical Research
Paper II
66 oN
63 oN
60 oN
57 oN
54 oN
10 oE
15oE
o
20 E
o
25 E
o
30 E
Fig. 3. 2 nautical-miles bathymetry of the Baltic sea. The contour curves represent
50m. The transect used for a detailed representation of harmonic amplitudes is
shown as a dashed line.
low Rayleigh friction of Γ = (1year)−1 in the interior but increasing significantly at the open-boundary sponge zone. Most of
the energy was dissipated in the sponge zone.
The numerical runs were initialised by tilting the sea surface
linearly in the east-west direction with respect to its equilibrium
position, from -1 m at the western extreme of the Skagerrak to
+1 m at St. Petersburg. (As will be seen, this somewhat artificial initial condition not only induces predominantly longitudinal oscillations in the basin but also, to some extent, gives rise
to transversal motion. The excited oscillatory motion was found
to encompass a wide range of wave-numbers.) The model was
hereafter run without any forcing for 480 hours.
3.1
Large-scale oscillations
From the model runs briefly described above, frequency spectra
have been calculated at each gridpoint of the domain using FFT.
The maps in Figure 4 show the resulting distribution of the spectral amplitude for the periods from 15 to 37 hours in two-hour
intervals. The most striking results from this suite of graphs are
the what at first sight appear to be global oscillations with periods
59
Paper II
Fjord seiches in the Gulf of Finland vs global Baltic seiches.
15 Hours
17 Hours
19 Hours
21 Hours
23 Hours
25 Hours
27 Hours
29 Hours
31 Hours
33 Hours
35 Hours
37 Hours
Fig. 4. Contour plots of the harmonic amplitudes for different periods, calculated
from a single model simulation. Each isoline represents 5 cm.
of around 23 and 27 hours. Both of these have a large amplitude
in the Gulf of Finland, indicating resonance with the local fjord
mode. At the other extreme of the Baltic basin, viz. the Danish
Belt Sea, the 23-hour oscillation only shows weak resonance effects, in contrast to the 27-hour oscillation, which in this region
has a large amplitude.
To demonstrate that this behaviour is not an artifact arising
from the particular initialisation, runs were also made with a different initial condition: the surface sloping linearly from +1 m at
Tornio, located at the northern extreme of the Gulf of Bothnia, to
-1 m at Wladyslawowo on the southern coast of the Baltic. The
results were similar to those obtained using the previous initial
condition. The main difference was that the local 17-hour seiche
60
Submitted to Journal of Geophysical Research
Paper II
35
10
30
10
10
20
20
Period (hours)
25
20
15
Depth (m)
10
100
200
500
1000
Transect (km)
1500
2000
2500
Fig. 5. Harmonic amplitude along the transect in Figure 3. The same data are used
as in Figure 4, where the harmonic amplitude in the whole Baltic Sea is shown for
the periods here indicated by arrows. Normal bathymetry. Each isoline represents
5 cm.
in the Gulf of Riga was not excited, which is probably because
the main entrance to the gulf, the Irbe Strait, stretches in the EastWest direction. The seiche in the Gulf of Riga will be further
discussed in subsection 3.3.
To study the large-scale response in greater detail, the results
from a transect following the ”Talweg” from St. Petersburg to the
northern Kattegat, cf. Figure 3, were also examined. A contour
plot of the spectral amplitude as a function of the distance along
the transect and the oscillation period is shown in Figure 5. We
see the same overall pattern as previously described. The most
prominent features are the strong 27-hour oscillation in the Gulf
of Finland and the Belt Sea, and the 23-hour oscillation with only
weak resonance in the Belt Sea. It is also noteworthy that the
spectral peaks are considerably sharper in the Gulf of Finland than
in the Belt Sea.
Finally, it might be of interest to mention that a distinct pattern of possible Kelvin waves was seen to propagate from the
mouth the Gulf of Finland into the Baltic and along the coastline. The period of these waves was estimated to be about 24
hours, which coincide well with the two main peaks in earlier
mentioned spectral diagrams.
61
Paper II
Fjord seiches in the Gulf of Finland vs global Baltic seiches.
35
10
30
5
15
Period (hours)
25
20
15
5
Depth (m)
10
100
200
500
1000
Transect (km)
1500
2000
2500
Fig. 6. Harmonic amplitude along the transect in Figure 3. Bathymetry with the
Gulf of Finland closed. Each isoline represents 5 cm.
3.2
Numerical experiments with a modified bathymetry
To determine whether the oscillations seen in the previous simulations are best explained as local resonance effects or as global
seiches, versions of the model with artificially modified bathymetries were set up. In the first one the Gulf of Finland was closed
off from the Baltic proper by a solid wall. In Figure 6 one can
see that, as expected, a normal seiche oscillation develops in the
closed Gulf of Finland. The period of 10 hours is consistent with
an analytic estimate.
In the Baltic proper, on the other hand, the global seiche patterns disappear. The only remaining oscillation of any significance is a local phenomenon in the Danish Belt Sea, with a period
of about 25 hours. This result agrees well with that due to Magaard and Krauss (1966) of a 30-hour mode primarily found in the
Belt Sea. The difference in period probably depends on the poor
resolution of the Belt-Sea bathymetry in our simulation. It must
be noted that even small deviations from the real bathymetry are
likely to change the period of this kind of local oscillation.
To study the fjord seiche in the Gulf of Finland, a version
of the model with an even more distorted bathyhmetry was used.
Only the shoreline and adjacent coastal bathymetry of Finland and
62
Submitted to Journal of Geophysical Research
Paper II
66 oN
63 oN
60 oN
57 oN
54 oN
10 oE
15oE
o
20 E
o
25 E
o
30 E
Fig. 7. Bathymetry where Sweden and Denmark have been replaced with a 90 m
deep basin. The distance between each line represent 25m. The transect used for
a detailed representation of harmonic amplitudes is shown as a dashed line.
the Baltic states down to Poland were retained, while Sweden and
Denmark were replaced by a rectangular basin having a uniform
depth of 90 m as shown in Figure 7. The results in Figure 8 show
that the oscillations in the Gulf of Finland are only weakly affected
by this drastic change of the bathymetry.
These experiments show that the Belt Sea and the Gulf of Finland are both capable of supporting local modes. A probable explanation of the apparent global seiche pattern in Figure 5, for
a realistic bathymetry, is that it is a superposition of these local
modes. The two oscillators are only weakly coupled, as shown by
each of them being only slightly affected when the other one is
completely removed.
The most important result of the coupling between the two
local modes seems to be that the resonance frequency is split into
two close but distinct resonance frequencies, corresponding to
the periods 23 and 27 hours. This is seen particularly clearly in
the Gulf of Finland, cf. Figure 5. When only one of the two
oscillators is present, there is only one resonance frequency, cf.
Figures 6 and 8.
Such frequency splitting occurs very generally when two os-
63
Fjord seiches in the Gulf of Finland vs global Baltic seiches.
10
Paper II
35
30
10
20
30
Period (hours)
25
10
20
15
Depth (m)
10
100
200
500
1000
1500
2000
Transect (km)
2500
Fig. 8. Harmonic amplitude along the transect in Figure 3. Bathymetry in Figure
7 used. Each isoline represents 5 cm
cillators with close frequencies are coupled. Describing the oscillators by the simple equations
d2 ξ1
+ ω12 ξ1 = αξ2 ,
dt2
d2 ξ2
+ ω22 ξ2 = αξ1 ,
dt2
we obtain the eigenfrequencies
ω 2 + ω22
±
ω = 1
2
2
"
ω12 − ω22
2
#1/2
2
+α
2
.
For ω12 = ω22 this reduces to ω 2 = ω12 ± α, illustrating the frequency splitting.
3.3
Harbour seiche in the Gulf of Riga
One interesting feature is the indication of an oscillation with a
period of 17 hours in the Gulf of Riga, as seen in Figure 4. The
fact that the period is fairly long, although this gulf is small, is a
consequence of the narrow straits that connect it to the rest of the
Baltic.
64
Submitted to Journal of Geophysical Research
Such a low-frequency mode is often referred to as a Helmhotz
mode, and is well known in harbour seiching (Miles, 1974). The
Helmholtz mode in the Gulf of Riga has recently been studied in
detail by Otsmann et al. (2001). They constructed a simple theoretical model which was calibrated using current observations
in the Suur Strait, at the northern extreme of the Gulf, and concluded that the main resonance period of the system is 24 hours.
Oscillations with this period are also the most prominent feature
of the currents observed in the Irbe Strait, the main passage connecting the Gulf of Riga to the Baltic proper.
The difference between this value and our result of 17 hours
is probably explained by the resolution of the model bathymetry.
To describe the Helmholtz mode accurately it is essential to have
a good resolution in the deeper parts of the straits connecting
the sub-basin to the rest of the sea, which is not the case in our
model. Even the coupling to the tiny Väinameri basin just north
of the Gulf of Riga was found to increase the resonant period by
one hour (Otsmann et al., 2001).
4
Oscillations in a long channel
To explain and interpret oscillations in a closed basin, it is customary to compute the global eigenmodes. However, in a basin
with a complex coastline and bottom topography, like the Baltic
Sea, an interpretation in terms of local ”quasi-modes” may be
more useful. These are localised modes that couple weakly to the
rest of the basin, gradually losing their energy by wave radiation.
The resulting damping is described by the imaginary part of the
eigenfrequency. Harbour seiches (Miles, 1974) are an example
of such quasi-modes.
In the present section we illustrate the relation between these
two points of view by examining the modes of oscillation in a
long and narrow basin, which will be treated as a one-dimensional
channel. This example is not chosen to be a realistic model of the
Baltic Sea. It is rather an easily solvable conceptual model, intended to illustrate the basic ideas.
The channel consists of two sub-basins, a short and shallow
one with the length L1 and the depth H1 , and a long and deep
one of length L2 and depth H2 . Conceptually, we can think of
the short sub-basin as analogous to the Gulf of Finland, while the
long sub-basin is analogous to the Baltic proper. Neglecting the
Coriolis force, the equation for barotropic waves is
65
Paper II
Paper II
Fjord seiches in the Gulf of Finland vs global Baltic seiches.
∂
∂x
∂h
gH
∂x
+ ω 2 h = 0,
(3)
where H(x) is the equilibrium depth and h the free-surface perturbation, and we have assumed that the time variation is proportional to exp(−iωt). The boundary condition at the ”coastal
points” x = −L1 and x = L2 is the usual one of no normal
flow, implying ∂h/∂x = 0. At the step in bottom topography at
x = 0 we must require continuity of the surface elevation h, and
of the mass flow H∂h/∂x.
We first consider a quasi-mode in the shallow sub-basin, taking the deep sub-basin to be infinitely long (analogous to the
open ocean). Thus, instead of using the boundary condition at
x = L2 , we require the wave in x > 0 to be purely rightwardpropagating, making the following ansatz:

 a eiωx/c1 + a e−iωx/c1 , x < 0
+
−
h(x) =
(4)
 b eiωx/c2 ,
x>0
+
The phase velocity in the two regions is given by c1 = (gH1 )1/2
and c2 = (gH2 )1/2 , respectively, and we assume that ω is positive, so that the terms proportional to a+ and b+ represent rightward propagating wave components, while the term proportional
to a− represents a leftward propagating component. The continuity conditions at x = 0 give the reflexion coefficient at the
bottom step:
a−
c1 − c2
=
.
a+
c1 + c2
(5)
Note that if H1 /H2 → 0, so that c1 /c2 → 0, then |a− /a+ | = 1,
i.e. a wave travelling toward the step from the shallow sub-basin
into the deep one is totally reflected. We then get a standing wave
in the shallow sub-basin, a ”fjord mode”.
The boundary condition at x = −L1 gives another relation
between a+ and a− . Combining it with eq. (5) we obtain the
dispersion relation for the quasi-mode:
e−2iωL1 /c1 =
c1 − c2
.
c1 + c2
(6)
If we assume that H1 H2 , the right-hand side of this equation
is close to −1, and we obtain approximately
66
Submitted to Journal of Geophysical Research
c1
ω=
L1
π
c1
+ nπ − i
2
c2
Paper II
n = 0, 1, 2....
(7)
These eigenfrequencies represent damped oscillations. The real
part is the same as the eigenfrequency of the fjord modes, and
the imaginary part describes the damping due to wave radiation.
To see how this quasi-mode relates to the global eigenmodes,
we will then solve the problem with a finite value of L2 . The
ansatz (4) is then replaced by

 a eiωx/c1 + a e−iωx/c1 , x < 0
+
−
h(x) =
(8)
 b eiωx/c2 + b e−iωx/c2 , x > 0
+
−
and we must also use the boundary condition at x = L2 . After
some standard calculations, the following dispersion relation can
be obtained:
tan
c2
ωL2
ωL1
= − tan
.
c1
c1
c2
(9)
It has two sets of roots. If c2 /c1 is large, the first set is approximately given by cos(ωL1 /c1 ) = 0, i.e. the frequencies of the
fjord mode in the shallow sub-basin. The second set is approximately given by sin(ωL2 /c2 ) = 0, corresponding to the eigenmodes of the deep sub-basin.
To understand how these global eigenmodes relate to the
quasi-mode, it is useful to calculate the ratio between the energy
densities in the shallow and deep sub-basins. The energy density
of a harmonic oscillation is given by E = (ω 2 + Hk 2 )|h|2 /2 =
ω 2 |h|2 . Using the fact that |a+ | = |a− | and |b+ | = |b− |, the
ratio between the energy densities in the two sub-basins can then
be calculated as E1 /E2 = cos2 (ωL2 /c2 )/ cos2 (ωL1 /c1 ). With
the help of eq. (9) this can be written as
1
E1
=
.
2
E2
cos (ωL1 /c1 ) + (H1 /H2 ) sin2(ωL1 /c1 )
(10)
This function is shown in Figure 9, setting H2 /H1 = 25. For
H2 > H1 its maximum value is E1 /E2 = H2 /H1 , and the
maximum points are at ω = (c1 /L1 )(π/2 + nπ), n = 0, 1, 2, ...
. If H2 /H1 is large, the maxima are narrow resonance peaks,
with the half-width ∆ω ≈ (2H1 /H2 )(c1 /L1 ). At these peaks,
67
Paper II
Fjord seiches in the Gulf of Finland vs global Baltic seiches.
E1/E2
20
10
0
0.0
0.5
L2=20L1
L2=50L1
ω
1.0
Fig. 9. Ratio of the energy density in the shallow sub-basin to that in the deep
one as a function of frequency. The curve is eq. (8), and the dots represent the
global eigenfrequencies for two different values of L2 .
the global eigenmodes are in resonance with the fjord mode in
the shallow sub-basin.
In Figure 9 we also show the energy ratio for the eigenfrequencies obtained from eq. (9) for two different values of L2 .
The points all lie on the curve defined by eq. (10). Thus, the only
effect of increasing the value of L2 is that the eigenmodes sample
this curve more densely, while the curve itself is independent of
L2 . Its shape entirely mirrors the properties of the quasi-mode
in the shallow sub-basin: the location of the resonance peaks is
equal to the real part of the eigenfrequency of the quasi-mode,
and their width is determined by the imaginary part, i.e. by the
reflexion coefficient at the bottom step.
With the parameters used in Figure 9, the frequency of the
fjord mode coincides with one of the eigenfrequencies of the
deep sub-basin. This causes frequency splitting, as discussed in
the previous section, which gives two symmetrically situated global eigenfrequencies near the resonance peak.
68
Submitted to Journal of Geophysical Research
Paper II
1.0
0.8
0.6
0.4
h
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
0
5
10
15
20
25
x
30
35
40
45
50
Fig. 10. Time development of the free surface, with one time unit between each
curve. The elevation is decreasing in the left half of the figure and increasing in
the right half. The shallow sub-basin is in −1 < x < 0.
We have also solved the time-dependent shallow-water equations corresponding to eq. (3) numerically, using a uniformly
sloping surface h as initial condition. The result is shown in Figure 10. It has the appearance of a local damped oscillation in the
shallow sub-basin, with the period T=4 given by eq. ( 7) with
n = 0, superimposed on a global oscillation.
How should we understand the local oscillation in Figure 10?
One answer is that it is a superposition of those global eigenmodes that have a frequency near the resonance peak of Figure
9, and therefore have a large amplitude in the shallow sub-basin.
There are two (or perhaps four) such modes for the parameters
used in the simulation. The damping of the local oscillation is
then a result of these modes gradually going out of phase.
Another answer, which is mathematically equivalent but physically more natural, is that this is a local quasi-mode. This point
69
Paper II
Fjord seiches in the Gulf of Finland vs global Baltic seiches.
of view also gives a natural explanation of the waves that can be
seen to radiate away from the shallow sub-basin in Figure 10.
They qualitatively resemble the Kelvin waves that were seen to
radiate away from the Gulf of Finland in our simulations.
The basic condition for the existence of a distinct quasi-mode
is that the relexion coefficient for waves entering the main basin
from the small sub-basin is close to unity. In the simple channel
model considered here, the reflexion is caused by a large step in
the depth. This does not apply to the Baltic Sea and the Gulf of
Finland. In that case the the reflexion is instead caused by the
rapid widening of the water body at the mouth of the gulf. (This
is a close analogue of the resonance in a trumpet,or other openended wind instruments.) The reflexion coefficient approaches
unity (corresponding to total reflexion) as the ratio between the
width and the length of the gulf goes to zero. However, the
calculations required to solve such a two-dimensional problem
are lengthy and complicated, and since the basic mechanism is
the same in the simple one-dimenisonal channel problem, we
instead chose to use that as illustration.
5
Discussion
In our shallow-water simulations of the Baltic Sea, we could identify three different local oscillatory modes: one in the Gulf of
Finland, with the two distinct periods 23 and 27 hours, one in
the Danish Belt Sea, with a less distinct period in the range 23-27
hours, and one in the Gulf of Riga, with the period 17 hours.
The most pronounced mode is the one in the Gulf of Finland.
This agrees with the frequency analysis of sea level observations,
which shows that the amplitude there is highest for periods in
the range 23-30 hours, and also with the fact that the tidal components K1 and O1 are much stronger in the Gulf of Finland than
elsewhere, cf. Figure 2.
The local Helmholtz mode in the Gulf of Riga also exists in
reality; however, as shown recently in the detailed study by Otsmann et al. (2001), its real period is 24 hours rather than 17
hours. The discrepancy is most likely caused by insufficient resolution of the bathymetry. In order to describe this mode accurately, one must have a better resolution of the straits connecting
the Gulf of Riga to the Baltic proper than in our model.
We are also uncertain about the accuracy with which the local
70
Submitted to Journal of Geophysical Research
Belt-Sea mode is described in our simulations. The Belt Sea is
characterized by a complex bathymetry with narrow straits, and
is moreover likely to be affected by the open boundary in the
North Sea, which is a potential problem in any model. Possibly
this could be the 31-hour oscillation described by Magaard and
Krauss (1966).
It has sometimes been asked whether the sea level oscillations
observed in the Gulf of Finland are caused by a local fjord mode or
by global eigenmodes (Neumann 1941). As illustrated in section
4, these two alternatives are really two sides of the same coin.
Mathematically speaking, a local ”fjord mode” is a superposition
of several global eigenmodes with close frequencies.
However, the interpretation in terms of a local mode focuses
on the most robust aspect of the problem. If, for example, the
bathymetry is modified outside of the Gulf of Finland, the local mode there remains almost the same, and so does the temporal evolution of a sea-level perturbation in the gulf. Yet the
global eigenmodes of which this quasi-mode consists may change
strongly.
The most pronounced non-local effect seen in our simulations
is the frequency splitting that appears to be caused by a coupling
between the local modes in the Gulf of Finland and the Belt Sea.
In the analysis of observed oscillatory events by Neumann (1941)
there are in fact indications of a similar double peak, but this cannot be seen in the spectra of sea-level observations in the Gulf of
Finland that we have analysed. There are several reasons why one
would expect this double peak to be less pronounced in observations than in our model. One is that this splitting is an effect
of these two modes coincidentally having almost the same resonant periods, and, as already remarked, we are not fully confident about the accuracy of our value of the period of the Belt Sea
mode. Another reason is that some effects, such as nonlinearities
and coupling to internal waves, that would tend to decorrelate the
global modes, are not present in our model. Also note that the
oscillatory events found by Neumann (1941) encompass at most
four oscillation periods, which indicates a spectral width comparable to the difference between the two periods we observed the
Gulf of Finland, and also comparable to the difference between
the periods of the first few global eigenmodes found by Wübber
and Krauss (1979).
It is curious that the periods of all the three most distinct local
modes are so similar (using the value 24 hours in the Gulf of
71
Paper II
Paper II
Fjord seiches in the Gulf of Finland vs global Baltic seiches.
Riga, as given by Otsmann et al. 2001). This of course enhances
the interaction between them. Theoretically this should lead to
frequency splitting, but in practice the result is most likely that
the response is broadened, so that no distinct resonance at all is
observed.
This also means that if one wants to compute the global eigenmodes with periods in this range, one must have a good description of all these local modes. In particular, it is necessary to have
a good resolution in the straits connecting the Gulf of Riga to
the Baltic proper. In other words, the local modes are also the
computationally robust aspect of the problem, in contrast to the
global eigenmodes.
Acknowledgements
This work was carried out under the auspicies of the SWECLIM
programme. We thank Barry Broman at SMHI and Kai Myrberg
at FIMR for generous assistance.
References
Chrystal, G., 1905. On the hydrodynamical theory of seiches.
Trans. Roy. Soc. Edinburgh 41, 599–649.
Defant, A., 1918. Neue Methode zur Ermittlung der Eigenschwingungen (Seiches) von Abgeschlossenen Wassermassen
(Seen, Buchten usw.). Ann. Hydrographie 46.
Defant, A., 1960. Physical Oceanography. Vol. 2. Pergamon, New
York, pp. 154–244.
Döös, K., 1999. The Influence of the Rossby waves on the
Seasonal Cycle in the Tropical Atlantic. J. Geophys. Res.
104 (C12), 29591–29598.
Hidaka, K., 1936. Application of Ritz’s variational method to the
determination of seiches in a lake. Kobe.
Krauss, W., Magaard, L., 1962. Zum System der Eigenschwingungen der Ostsee. Kieler Meeresforsch. 18, 184–186.
Lisitzin, E., 1959. Unimodal seiches in the oscillation system
Baltic proper, Gulf of Finland. Tellus 4, 459–466.
Magaard, L., Krauss, W., 1966. Spektren der Wasserstandsschwankungen der Ostsee im Jahre 1958. Kieler Meeresforsch.
22, 155–162.
72
Submitted to Journal of Geophysical Research
Meier, H. E. M., 1996. Ein regionales Modell der westlichen Ostsee mit offenen Randbedingungen und Datenassimilation.
Metzner, M., Gade, M., Hennings, I., Rabinovich, A. B., 2000.
The observation of seiches in the Baltic Sea using a multi data
set of water levels. J. Marine Syst. 24, 67–84.
Miles, J. W., 1974. Harbor seiching. Ann. Rev. Fluid. Mech. 6,
17–35.
Neumann, G., 1941. Eigenschwingungen der Ostsee. Arch.
Dtsch. Seewarte u. Marineobs. 61, 1–59.
Otsmann, M., Suursar, Ü., Kullas, T., 2001. The oscillatory nature
of the flows in the system of straits and small semienclosed
basins of the Baltic Sea. Continental Shelf Res. 21, 1577–1603.
Sadourny, R., 1975. The dynamics of finite-difference models of
the shallow water equations. J. Atmos. Sci 32, 680–697.
Seifert, T., Tauber, F., Kayser, B., 2001. A high resolution spherical grid topography of the Baltic Sea. - revised edition. In: Proceedings of the Baltic Sea Science Congress, Stockholm 25-29
November 2001. Baltic Sea Science Congress, Stockholm.
Svanssson, A., 1959. Some Computations of Water Heights and
Currents in the Baltic. Tellus 2, 231–238.
Witting, R., 1911. Tides in the Baltic Sea and the Gulf of Finland
(in Swedish). Fennia 29.
Wübber, C., Krauss, W., 1979. The two-dimensional seiches of
the Baltic. Sea. Oceanol. Acta 2, 435–446.
73
Paper II
III
Using Langrarian Trajectories to Study a Gradually Mixed
Estuary.
Published as DM report number 93.
Published as DM report number 93.
Using Lagrangian trajectories to Study Mixing in a Gradually
Varying Estuary
Bror Jönsson
Department of Meteorology/Physical Oceanography, Stockholm University,
SE-10691 Stockholm, Sweden
Abstract
When using modeling techniques to investigate estuarine systems, a decision must be made whether to use a simple and robust approach based on mass-balance considerations, or if more
advanced process-resolving three-dimensional (3-D) numerical
models are to be employed. Although the former are easy to apply, they require certain fundamental constraints to be fulfilled.
whereas the latter require significant efforts to implement. An
example of a system where such a trade-off has had to be taken
into account when choosing the mode of analysis is provided by
the Gulf of Finland, a gradually mixed estuary in the Baltic Sea.
In the present study, Lagrangian-trajectory analyses has been
applied to to minimize the disadvantages associated with both
of the model approaches. Hereby it has been possible to show
that the main part of the Gulf of Finland is dominated by water from the Baltic proper, the most pronounced mixing with
Neva water taking place over a limited zone in the inner parts of
the Gulf. The Lagrangian formalism has also been used to construct overturning stream-functions pertaining the two source
waters, a useful tool when analysing the dynamics of the Gulf.
It deserves to be underlined that these trajectory methods have
not only proved their worth in the Gulf of Finland, but can be
implemented for any estuary previously described using a 3-D
model.
Paper III
Using Lagrangian Trajectories to Study a Gradually Mixed Estuary.
6 2 oN
6 1 oN
6 0 oN
Hanko
Stockholm
5 9 oN
Moshnyi
Island
Helsinki
Neva
St Petersburg
Tallin
Hiiuma
5 8 oN
1 8 oE
o
21 E
o
24 E
o
27 E
o
30 E
o
33 E
Fig. 1. Map of the Gulf of Finland where the thin black line represent the model
coastline. The heavy grey line indicates where the trajectories are released to the
Gulf and the heavy black line where they are removed.
Introduction
When undertaking oceanographic analyses, one is frequently interested in following the paths of distinct water types taking part
in the circulation. In the context of numerical modeling, trajectory analysis has proved to be a valuable tool for determining the
origin as well as the fate of specific water masses. This holds
true when examining processes ranging from the global scale (
e.g. the thermohaline circulation) down to such local-scale phenomena as the dispersion of pollutants from a point source. The
present study will focus on intermediate-scale processes, specifically those in the north-eastern part of the Baltic, an area of considerable ecological interest due to its proximity to large population centers.
The Gulf of Finland is an elongated estuary with a mean depth
of 37 m (maximum depth 123 m) and a length of around 400
km, cf. the map in Fig. 1. The western part of the Gulf is a
direct continuation of the Baltic proper, whereas the river Neva
debouches into the eastern extremity of the basin. Thus the Gulf
is not only vertically stratified, but also characterized by a pronounced horizontal salinity gradient (Witting, 1912) This state
of affairs (cf. Fig. 2, adapted from Jurva (1951)) has, as will be
seen, significant dynamic consequences. The water-masses originating from the two source regions also have markedly different
characteristics, not only as regards the salinity but also the degree
of anthropogenic contamination, why their ultimate destiny in
the Gulf is of considerable practical environmental interest. This
78
DM Report no. 93
Paper III
6 2 oN
6 1 oN
2.0
3.0 2.5
6 0 oN
5 9 oN
5 8 oN
>6.5
1 8 oE
o
21 E
6.0
5.5
5.0
3.5
4.5 4.0
o
27 E
o
24 E
2>
o
30 E
o
33 E
Fig. 2. Surface isohalines from observations in the Gulf of Finland. Adapted from
Jurva (1951).
general question has been dealt with in a number of investigations, most recently in one due to Andrejev et al. (2004) which
focused on the age and renewal time of the water masses in the
Gulf of Finland. The methodological limitations of this study,
however, had the consequence that the ultimate fate of the Baltic
and Neva source waters could not be fully ascertained by Andrejev
and co-workers.
To remediate this situation, the present investigation will attempt to resolve this important problem using trajectory analysis. In next section the technical details of this procedure are
outlined, whereafter section 3 deals with the results concerning
mixing and water-mass composition obtained for the Gulf of Finland. The study is concluded by a review of the overall outcome
of the analysis and a discussion of the large-scale circulation in
the Gulf based on the overturning stream-function, an important
diagnostic quantity which also is obtained using trajectory analysis.
Methods
Classical trajectory analysis, as primarily applied to the atmosphere, mainly employed graphical techniques used in conjunction with meteorological good sense and experience. Present-day
trajectory methods are, however, based on numerical circulation
models, the one used in this study being the Rossby Centre Ocean
(RCO) Baltic model (Meier et al., 2003). This well-proven (Meier
and Kauker, 2003) finite-difference model has 41 depth levels (at
intervals ranging from 3 m in the surface region to 12 m at greater
79
Paper III
Using Lagrangian Trajectories to Study a Gradually Mixed Estuary.
depths) and a horizontal resolution of 2 x 4 nautical miles. It is
run on an Arakawa B-grid and is capable of resolving the mesoscale eddies of importance for the dynamics of the Baltic. For
the present investigation with focus on the north-eastern Baltic,
the remotely applied boundary conditions along the HanstholmLindesnes transect at the border to the North Sea do not give rise
to any spurious effects. Given 0.2◦ x 0.2◦ gridded standard meteorological forcing (sea-level preassure, geostrophic 10-m wind
components, 2-m air temperature and relative humidity, precipitation, and cloud cover) provided by the Swedish Meteorological
and Hydrological Institute (SMHI) at 3-hour intervals, the circulation model yields the evolution in time of the velocity, temperature and salinity fields.
In the present study, a trajectory scheme based on results due
to Döös (1995) as well as Blanke and Raynaud (1997) was used
to compute ”neutral-particle” paths from the three-dimensional
velocity fields in the Gulf of Finland provided by the RCO model.
This algorithm calculates trajectory paths analytically for a prescribed velocity field (regarded as stationary at each instant), which
permits sub-grid analyses of the particle motion by interpolating
the zonal and meridional velocities u and v defined in the corners
of the grid-cell. The zonal velocity is found by first averaging
meridionally and hereafter interpolating zonally:
u(x) =
+
1
(ui−1,j + ui−1,j−1 )+
2
x − xi−1
(ui,j + ui,j−1 − ui−1,j − ui−1,j−1 ).
2∆x
By using u(x) =
ential equation
dx
dt ,
(1)
this expression can be written as the differdx
+ αx + β = 0,
dt
(2)
where
ui−1,j + ui−1,j−1 − ui,j − ui,j−1
,
2∆x
xi−1 (ui,j + ui,j−1 − ui−1,j − ui−1,j−1 )
β=
−
2∆x
1
− (ui−1,j + ui−1,j−1 ).
2
α=
80
(3)
(4)
DM Report no. 93
Paper III
Together with the boundary conditions
x(ta ) = xa ;
x(tb ) = xb ,
(5)
the inhomogenous differential equation (2) has the solution
β
β
xb = xa +
exp[−α(tb − ta )] −
(6)
α
α
or, equivalently,
"
xb +
1
tb = ta − log
α
xa +
β
α
β
α
#
,
(7)
where xb is the zonal displacement and tb the associated time.
The meridional and vertical velocities are calculated in the same
manner and the times tb are compared Döös (1995). In other
words: after a trajectory has entered a grid box the times needed
to reach a zonal, a meridional and a vertical wall of the box, respectively, are calculated. The shortest time tb of these indicates
where the trajectory will leave this particular box. This serves as
the starting point for the subsequent set of calculations.
The velocity fields used in the calculations are five-fold more
highly resolved in time than the circulation-model data, from
which they are generated by linear interpolation. The fact that
the trajectory calculations can be run in an autonomous fashion
with regard the circulation model, i.e. off-line, makes it possible
to carry through the analysis without having to take recourse to
excessive computer resources.
Convection has not been taken into account in the present
study, but the effects of this process have been examined in the
course of a previous investigation by assigning a water parcel a
random depth whenever it enters a convectively unstable water column. (Like the velocities used to calculate the trajectories, these convective events also originate from the circulation
model.) This study(Döös, 1995) showed, however, that the effects of convection did not affect the trajectories to any significant
degree.
To study the behavior of the system, all water parcels entering
the Gulf of Finland were ”tagged” in order to determine the origin of the water masses characterizing the system when in steady
state. Thus trajectories were continuously associated with water parcels entering the system from the river Neva and the Baltic
81
Paper III
Using Lagrangian Trajectories to Study a Gradually Mixed Estuary.
proper, the latter flux being defined over the Hanko-Hiiuma transect at the exit from the Baltic (cf. Fig. 1). Each trajectory represents a volume flux of 100 m3 s−1 , a ”concentration” deemed
to be sufficient in view of the magnitude of the transports entering the Gulf. The sensitivity of the overall results was tested
by stepwise decreasing the number of trajectories. The threshold when changes of the trajectory concentration proved to have
deleterious effects on the outcome was found to be well below
the ”density” introduced above. The numerical experiment was
run for 4000 days, which was judged as sufficient in view of the
estimated turnover time of the system (Andrejev et al., 2004) being 1-2 years . (The validity of this assumption was confirmed
a posteriori on the basis of a stable ”trajectory composition” over a
north-south transect half-way into the Gulf.) All trajectories leaving the system were removed from the model run at a meridional
transect located 5 grid-points west of the Baltic-proper boundary
where source water originally was tagged (cf. Fig. 1). The rationale behind this procedure was to reduce the computational costs
as well as to prevent trajectories from recirculating, which would
have entailed the risk of multiple tagging of water parcels. When
the model run had attained a steady state, the trajectory results
were analyzed using techniques to be outlined in what follows.
Results
Before proceding with a discussion of the larger-scale results from
the numerical experiments, we examine some specific features in
order to judge whether the model can be regarded as performing in an adequate fashion. The underlying rationale is that even
if a GCM yields more-or-less correct overall results, considerable
local irregularities may arise, in particular adjacent to the boundaries of the system. In the larger Baltic perspective, the Gulf of
Finland can be looked upon as a marginal area, why the modelers in the course of developing the RCO formalism did not pay
any specific attention to this region, even though the discharge
from the river Neva potentially could create anomalies here. The
model results were thus verified by collating the time-averaged
RCO-generated velocity fields with the results from earlier studies. One interesting area for comparisons of this type is in the
vicinity of Moshnyi Island in the inner part of the Gulf, where Andrejev et al. (2004) reported exceptionally long residence times
of the water. This state of affairs also manifests itself in the present
82
DM Report no. 93
Paper III
Year
1990
1985
1981
24°E
25°E
26°E
27°E
28°E
29°E
Fig. 3. Hovmøller diagram showing the time-evolution of the mixing between
the water masses from the river Neva (red) and those originating from the Baltic
proper (blue) following a longitudinal transect through the Gulf of Finland.
circulation-model results, assuming the form of a prevalence of
stable, highly persistent eddies in the area. In general, the timeaveraged RCO velocity fields employed for calculating the trajectories proved to be consonant with well-known transport patterns
(Palmén, 1930) characterizing the Gulf of Finland. Thus highsaline water from the Baltic proper enters the Gulf as a boundary
current adjacent to the Estonian coast, whereas the river Neva discharge tends to follow the Finnish coast.
The analysis of the trajectory behavior was undertaken by calculating the steady-state ratio between the number of trajectories
originating from each source. This was done for each cell in the
Gulf of Finland modeling domain, the value 1 taken to represent
a situation where all water originated from the river Neva, the
value 0 indicating the presence of only Baltic water. To visualize
the mixing gradient between the two sources along the main axis
of the Gulf of Finland, this distribution was averaged across the
Gulf, yielding a representation of where the mixing takes place
between the water-masses originating from the Baltic proper and
from the river Neva.
By also carrying through a vertical integration, it proves feasible to construct a Hovmøller diagram representing the temporal evolution of this mixing, cf. Fig. 3. From these results
it may be concluded that, given the supply of trajectories pre-
83
Paper III
Using Lagrangian Trajectories to Study a Gradually Mixed Estuary.
Mixingfront
Location
28˚E
Wind
m/s
10
27˚E
8
26˚E
6
River
Discharge
3
m /s
3000
2000
0
20
40
60
80
100
120
140
160
Fig. 4. Diagram showing the evolution in time of the location of the zone of
maximum mixing in the Gulf of Finland (heavy black line), the Neva freshwater
discharge (dashed line), and the observed wind at the Landsort meteorological
station south of Stockholm in Sweden (thin black line).
scribed in the numerical experiments, a steady state is achieved
within a year. Although the most pronounced mixing zone hereafter varies somewhat in position as well in extent, no systematic
tendencies are visible. The two most likely ”suspects” when attempting to assign responsibility for these irregularities are the
variations of the freshwater input from the Neva and the longerterm properties of the overall wind conditions characterizing the
Baltic region. Fig. 3 thus shows the evolution in time of the Neva
freshwater discharge, the observed wind at the Landsort meteorological station south of Stockholm in Sweden, and the position of
the mixing zone (defined as the location of the maximum gradient in Fig. 4). From this diagram it is appears as though, in contrast to commonly held prejudices, the river discharge does not
play a major role. The wind regime, however, tends to demonstrate the same temporal scales of variability as does the mixing
zone. Although the results do not show any pronounced correlations, it is relevant to underline that previous studies have suggested that the Baltic system may be quite sensitive to long-term
changes of the westerly winds (Meier and Kauker, 2003). These
questions, nevertheless, remain an unresolved issue and further
studies are needed.
Even if the mixing zone tends to wander somewhat, this fea-
84
DM Report no. 93
Paper III
Depth (m)
50
100
150
24°E
25°E
26°E
27°E
28°E
29°E
Fig. 5. Diagram showing the time-average (1980-1994) of the mixing between
waters from the river Neva (red) and those from the the Baltic proper (blue) along
a transect through the Gulf of Finland.
ture appears to be comparatively well-localized, with the water
masses originating from the Neva and the Baltic proper mainly
mixing in a rather narrow zone between 28◦ E and 29◦ E. The fact
that this takes place comparatively close to the exit of the Neva
also indicates the predominance of Baltic water in the largest part
of the Gulf, a state of affairs that is easily verified on the basis of
salinity records from the area (Jurva, 1951).
In Fig. 5, the the original laterally-averaged distribution ranging from 0 to 1 is instead represented as a time-mean over the
entire period from 1980 to 1994. This diagram confirms the picture of a stable mixing zone located in the inner part of the Gulf.
(Note that in these results, there is a distinct anomaly in the surface layer corresponding to the uppermost grid-box, most likely
due to the presence of an Ekman layer.) It may be argued that
the mixing zone represented here constitutes a distinct ”Gulf of
Finland Water mass”, the possible existence of which has been
subjected to considerable debate.
In a broader oceanographical context, however, the overall
circulation in the Gulf of Finland is also of a considerable interest. A convenient way (Blanke et al., 1999) of representing the
long-term circulation of an estuary is to use a Lagrangian streamfunction, which is calculated by summing over selected trajectories describing water pathways of particular interest. Hereby
85
Paper III
Using Lagrangian Trajectories to Study a Gradually Mixed Estuary.
one can isolate the behaviour of specific water masses by following sets of trajectories from one or more initial sections (in the
present case case the mouth of the river Neva and the HankoHiiuma transect). Each trajectory (identified by a number n) can
be associated with a volume transport Tn determined by the velocity, the area of the initial section, and the number of trajectories initiated for each grid box. The volume transport associated with each trajectory is hence directly proportional to the
number of trajectories that are ”released” from the initial section.
This latter quantity can be interpreted as a ”Lagrangian resolution”, which should be sufficiently high to ensure that the Lagrangian stream-function does not change as the number of trajectories is further increased. The trajectories are summed over
the model grid by recording when a trajectory passes a grid wall.
Northward, eastward and upward motion is registered as positive, southward, westward and downward as negative. In this
way a 3-dimensional field corresponding to the volume transport
associated with these trajectories is obtained. A trajectory entering a grid box is assumed to also make an exit, and hence the
transport field fulfills
∆Ty
∆Tz
∆Tx
+
+
= 0,
∆x
∆y
∆z
where Tx , Ty and Tz are the trajectory volume transports in the
three directions through the grid-box walls. Integrating this field
along a prescribed direction results in a two-dimensional nondivergent field that can be represented by a stream-function. Vertical integration of the transport yields a Lagrangian barotropic
stream-function ΨLB :
XX
∆ΨLB
=
Ty
∆x
n
or
k
XX
∆ΨLB
=−
Tx ,
∆y
n
k
whereas zonal integration results in the Lagrangian overturning
stream-function ΨLZ :
XX
∆ΨLZ
=−
Ty
∆z
n
i
or
XX
∆ΨLZ
=
Tz .
∆y
n
i
Here, the indices i and k pertain to the the zonal and vertical
discretizations, respectively, whereas n represents the trajectories
in each box.
86
DM Report no. 93
Paper III
50
Depth (m)
750
350
100
-350
-750
150
A) River Neva Meridional Stream-function
24°E
25°E
26°E
27°E
28°E
29°E
50
Depth (m)
4500
1500
100
-1500
-4500
150
B) Baltic Proper Meridional Stream-function
24°E
25°E
26°E
27°E
28°E
29°E
50
Depth (m)
4500
1500
100
-1500
-4500
150
C) Total Meridional Stream-function
24°E
25°E
26°E
27°E
28°E
29°E
Fig. 6. Lagrangian overturning stream-functions projected with depth as the vertical
coordinate in the diagrams. The upper panel (A) shows the behaviour of the
water debouching into the Gulf of Finland from the river Neva. The middle panel
(B) represents the motion of the Baltic water entering across the Hanko-Hiiuma
transect. The bottom panel (C) shows a combination of the results from the upper
two panels, and corresponds to the standard Eulerian stream-function.
87
Paper III
Using Lagrangian Trajectories to Study a Gradually Mixed Estuary.
Applying this formalism to the previously described velocity
fields from the RCO-model, the river Neva inflow to the Gulf of
Finland is first considered. In Fig. 6A, the resulting meridionally
averaged Lagrangian overturning stream-function along the Gulf
is shown as a function of the depth. The river discharge is seen to
progress towards the Baltic proper predominantly in the form of a
surface flow, although note the progressively deepening streamlines indicating an entrainment of Neva water into the depths of
the Gulf.
In Fig. 6B, the analogous set of results for the Baltic inflow to
the Gulf across the Hanko-Hiiuma transect is shown. The character of these streamlines is seen to differ markedly from that
associated with the Neva discharge. In Fig. 6B, the dominating
overturning cell is constituted by the incoming saline deep water
from the Baltic proper, which upwells in the Gulf as it mixes with
the river Neva water and returns westwards as much fresher water in the surface layers. This is the Gulf of Finland contribution
to what has been termed the Baltic haline conveyor belt (Döös et
al., 2004). It is also possible to combine these two separately determined stream-functions to one which corresponds to the standard Eulerian stream-function, cf. Fig. 6C. From this diagram it is
recognised that the fluxes associated with the ”Baltic-proper circulation cell” are an order of magnitude larger than the transport
from the river Neva and thus dominate the vertical circulation in
the Gulf of Finland.
Instead of projecting the stream-function against depth, this
can be done versus density, salinity or temperature. These less intuitively clear visualisations have the advantage of revealing more
of the physical processes underlying the circulation. Figs. 7A–C
and 8A–C thus show the Lagrangian overturning stream-function
against density and salinity, respectively. These diagrams manifest
pronounced similarities, underlining the well-known fact that the
stratification and circulation of the Baltic are mainly determined
by the salinity, not the temperature. Consequently Fig. 9 A–C,
showing the stream-function versus temperature, differs significantly from those based on salinity and density.
A useful feature associated with using these density- or salinitybased projections (Figs. 7A–C and 8A–C) is that they facilitate
a qualitative as well as quantitative understanding of how the
salinity characteristics of water originating from the Baltic proper
gradually are diluted as it circulates in the Gulf of Finland. This
deep overturning cell is associated with a flux of around 5000
88
DM Report no. 93
Paper III
σ0 (kg-1m3
0
750
350
4
-350
-750
A) River Neva Meridional Stream-function
8
24°E
25°E
26°E
27°E
28°E
29°E
σ0 (kg-1m3
0
4500
1500
4
-1500
-4500
B) Baltic Proper Meridional Stream-function
8
24°E
25°E
26°E
27°E
28°E
29°E
σ0 (kg-1m3
0
4500
1500
4
-1500
-4500
C) Total Meridional Stream-function
8
24°E
25°E
26°E
27°E
28°E
29°E
Fig. 7. Lagrangian overturning stream-functions projected with density (σ0 ) as the
vertical coordinate in the diagrams. The upper panel (A) shows the behaviour of
the water debouching into the Gulf of Finland from the river Neva. The middle
panel (B) represents the motion of the Baltic water entering across the Hanko-Hiiuma transect. The bottom panel (C) shows a combination of the results from the
upper two panels, and corresponds to the standard Eulerian stream-function.
89
Using Lagrangian Trajectories to Study a Gradually Mixed Estuary.
0
Salinity
750
350
5
-350
-750
A) River Neva Meridional Stream-function
10
24°E
25°E
26°E
27°E
28°E
29°E
0
Salinity
4500
1500
5
-1500
-4500
B) Baltic Proper Meridional Stream-function
10
24°E
25°E
26°E
27°E
28°E
29°E
0
4500
Salinity
Paper III
1500
5
-1500
-4500
10
C) Total Meridional Stream-function
24°E
25°E
26°E
27°E
28°E
29°E
Fig. 8. Lagrangian overturning stream-functions projected with salinity as the vertical coordinate in the diagrams. The upper panel (A) shows the behaviour of the
water debouching into the Gulf of Finland from the river Neva. The middle panel
(B) represents the motion of the Baltic water entering across the Hanko-Hiiuma
transect. The bottom panel (C) shows a combination of the results from the upper
two panels, and corresponds to the standard Eulerian stream-function..
90
DM Report no. 93
Paper III
Temperature (C)
20
750
350
10
-350
-750
A) River Neva Meridional Stream-function
0
24°E
25°E
26°E
27°E
28°E
29°E
Temperature (C)
20
4500
1500
10
-1500
-4500
B) Baltic Proper Meridional Stream-function
0
24°E
25°E
26°E
27°E
28°E
29°E
Temperature (C)
20
4500
1500
10
-1500
-4500
0
C) Total Meridional Stream-function
24°E
25°E
26°E
27°E
28°E
29°E
Fig. 9. Lagrangian overturning stream-functions projected with temperature as the
vertical coordinate in the diagrams. The upper panel (A) shows the behaviour of
the water debouching into the Gulf of Finland from the river Neva. The middle
panel (B) represents the motion of the Baltic water entering across the Hanko-Hiiuma transect. The bottom panel (C) shows a combination of the results from the
upper two panels, and corresponds to the standard Eulerian stream-function.
91
Paper III
Using Lagrangian Trajectories to Study a Gradually Mixed Estuary.
m3 /s, hereby significantly exceeding the value of 4000 m3 /s deduced from the depth- projected stream-functions. This discrepancy between the results from the different projections indicates
that the meridional slope of the isopycnals/isohalines plays a very
subordinate role for the vertical circulation in the Gulf of Finland.
In the depth-projected results there are, however, indications
of a shallow circulation cell which is absent in the corresponding
salinity- and density-projected results. This feature is possibly
due to the meridional slope of the isopycnals/isohalines, similar
to what holds true for the Southern-Ocean Deacon Cell (Döös and
Webb, 1994; Döös, 1994), but disappears when lateral averaging
is carried out over isohalines and isopycnals.
Summary and Discussion
Even if the trajectory results reported here must be interpreted
within the framework of our a priori knowledge of the horizontal as well as vertical circulation in the Gulf of Finland, they can
provide some additional understanding of the estuarine mixing
dynamics. In the present study it has been shown that the main
part of this estuary is dominated by water from the Baltic proper
and that the most pronounced mixing with Neva water takes place
over a rather small area in the inner parts of the Gulf, the location
of this zone tending to fluctuate somewhat. This variation does
not appear to be directly linked to the freshwater influx from the
river Neva, as has earlier been suggested. Rather, there appears to
be a weak correlation between the mixing in the system and the
long-term variations of the wind-forcing, a result which is consonant with the outcome of recent investigations due to Meier and
Kauker (2003).
A closer examination of the vertical dynamics of the Gulf
of Finland using Lagrangian overturning stream-functions confirmed the picture of a system dominated by a haline-driven circulation from the Baltic proper with a magnitude of about 5000
m3 s−1 . When this circulation was projected on the depth, it was
also possible to detect a closed additional shallow circulation cell,
most likely caused by a meridional slope of the sea-surface height.
In a general sense, the present study has shown the feasibility of using Lagrangian trajectories for studying complex and/or
less well-defined estuaries. When investigating systems of this
type, a balance must be struck between choosing a simple and robust model approach as typified by mass-balance models (which,
92
DM Report no. 93
Paper III
however, require certain assumptions to be fulfilled) and more
advanced 3-D numerical models. This latter formalism requires
significant implementation efforts, but yields a much better understanding of the physical processes and furthermore minimizes
the prerequisites to be fulfilled by the system under consideration. 3-D modeling, however, generates an abundance of highly
resolved data in time and space. Even if physical processes are described in a satisfactory manner, it remains a challenge to specify
the ”representative state” of the system, a common goal in estuarine studies. In situations such as these, Lagrangian-trajectory
methods can serve a useful purpose, since these techniques are
capable of providing a coherent synthesis of the time-evolution
of large data-sets, while still including any intrinsic variability of
the system. It is also possible to define prognostic scalars characterizing the estuaries, which facilitates a systematic comparison
between different systems. The approach of using Lagrangian
trajectories for describing estuarine systems can thus be regarded
as a way to merge some of the advantages accruing to simple
mass-balance models with those associated with the use of more
sophisticated 3-D numerical models.
References
Andrejev, O., Myrberg, K., Lundberg, P., 2004. Age and renewal
time of water masses in a semi-enclosed basin- application to
the Gulf of Finland. Tellus 56A, 548–558.
Blanke, B., Arhan, M., Madec, G., Roche, S., 1999. Warm water paths in the equatorial Atlantic as diagnosed with a general circulation model. Journal of Physical Oceanography 29,
2753–2768.
Blanke, B., Raynaud, S., 1997. Kinematics of the Pacific Equatorial
Undercurrent: a Eulerian and Lagrangian approach from GCM
results. Journal of Physical Oceanography 27, 1038–1053.
Döös, K., 1994. Semi-analytical simulation of the Meridional
Cells in the Southern Ocean. Journal of Physical Research 24,
1281–1293.
Döös, K., 1995. Inter-ocean exchange of water masses. Journal of
Geophysical Research 100 (C7), 13499–13514.
Döös, K., Webb, D. J., 1994. The Deacon Cell and the other
meridional cells in the Southern Ocean. Journal of Physical Research 24, 429–442.
93
Paper III
Using Lagrangian Trajectories to Study a Gradually Mixed Estuary.
Jurva, R., 1951. Ympäröivät meret. Suomen maantieteellinen
käsikirja, Suomen maantieteellinen seura, Helsinki, 121–144.
Meier, H., Döscher, R., Faxén, T., 2003. A multiprocessor coupled ice-ocean model for the Baltic Sea: Application to salt inflow. Journal of Geophysical Research.
Meier, H. E. M., Kauker, F., 2003. Modeling decadal variability of the baltic sea: 2. role of freshwater inflow and
large-scale. Journal of Geophysical Research 108:C11, 3368,
doi:10.1029/2003JC001799.
Palmén, E., 1930. Untersuchungen über die Strömungen in
den Finnland umgebenden Meeren. Commentationes PhysicoMathematicae, Societas Scientarium Fennica 12.
Witting, R., 1912. Hydrographische Beobachtungen in den Finland umgebenden Meeren. Kaiserlicher verlag.
94
IV
Baltic Sub-Basin Turnover Times Examined Using
the Rossby Centre Ocean Model.
Published in Ambio, v. 33 (4-5), pp 257-260, 2004
Published in Ambio, v. 33 (4-5), pp 257-260, 2004
Baltic Sub-Basin Turnover Times Examined Using the Rossby
Centre Ocean Model
Bror Jönsson , Peter A. Lundberg , Kristofer Döös
Department of Meteorology/Physical Oceanography, Stockholm University,
SE-10691 Stockholm, Sweden
Abstract
Not least when judging the possible effects of climate change
it proves necessary to estimate the water-renewal rates of limited marine areas subject to pronounced external influences. In
connection with the SWECLIM programme this has been undertaken for two ecologically sensitive sub-basins of the Baltic, viz.
the Gulf of Riga and Gdansk Bay. For this purpose two methodologically different approaches have been employed, based on
mass-balance budgets and analysis of Lagrangian trajectories, respectively. When compared to the results obtained using the
Lagrangian technique, the box-model approach proved to be adequate for the Gulf of Riga representing a morphologically highly
constrained basin, whereas it demonstrated certain shortcomings
when applied to the more open topographic conditions characterizing Gdansk Bay.
INTRODUCTION
Present concerns about the effects of increasing CO2 levels in
the atmosphere are not least focused on the consequences global
change may have for our habitat and living conditions. These
questions have been dealt with extensively within the SWECLIM
programme, a Swedish effort on regional climate modeling for
studying the possible effects of climate change on the Nordic regionRäisänen et al. (2003). When attention is directed towards
the regional marine environment, it must, from an oceanographic
standpoint, be underlined that climate change can be expected to
Paper IV
Baltic Sub-Basin Turnover Times Examined Using the RCO Model.
affect our seas in two ways; directly via the atmospheric forcing, but also indirectly via altered runoff conditions. These latter
changes may be physical as well as chemical and biological, in
particular as regards trace elements and nutrients essential for a
well-balanced marine ecosystem. Particularly in semi-enclosed
basins such as the Baltic the river-borne influx of substances plays
a critical role for the nutrient dynamics of the sub-basins. A
highly relevant example is e.g. provided by a number of recent
studies Humborg et al. (2000, 2002); Ittekkot et al. (In Press)
emphasizing the altered role of riverine dissolved silica for the
plankton communities of the Baltic. These investigations have focused on how the silicate flux to the Baltic Sea has changed over
the last century as a result of the hydroelectric exploitation of
Finnish and Swedish rivers, which may have led to considerable
sedimentation of particulate matter taking place already in the
power-station reservoirs. Other important nutrient-balance studies have focused on well-defined sub-basins of the Baltic, such as
Himmerfjärden on the east coast of Sweden Elmgren and Larsson
(2001) and the Gulf of Riga Wassmann and Tamminen (1999).
Since good insights concerning the matters above is essential for management programmes and related activities, much research has been devoted to budget studies. The standard tool
when dealing with this class of problems has been mass-balance
modeling based on conservative quanties, an approach originating from ideas formulated at the beginning of last century Knudsen (1900). This concept has proved its worth in varied contexts, but it must be kept in mind that box-model approximations represent a lowest-order approach with concomitant errors
and uncertainties. This is, nevertheless, the technique favored by
LOICZ (Land Ocean Interactions in the Coastal Zone)Gordon et al.
(1996) for establishing global-scale nutrient budgets, a prime
reason being that it often is the only way to obtain first estimates
of these ecologically important fluxes. In the present study the
relevance of standard mass-balance models based on conservative
quantities will be investigated for two typical sub-basins of the
Baltic. The analysis of these results will be undertaken on the basis of a comparison with results obtained from a 3-dimensional
numerical circulation model. The ultimate purpose of the study is
to serve as an aid for the choice of adequate tools when estimating
the possible biological consequences of an altered climate, and in
particular how these changes might be expected to affect such an
ecologically sensitive semi-enclosed sea as the Baltic.
98
Ambio 33(4-5) pp 257-260
Paper IV
15˚
20˚
60˚
58˚
56˚
54˚
Fig. 1. Map of the Baltic proper. The Gulf of Riga and Gdansk Bay are heavily
shaded.
Next the two modeling concepts are introduced and discussed,
whereafter the subsequent sections deal with applications to our
two model areas, viz. the Gulf of Riga and Gdansk Bay, as well
as with a comparison between the results obtained using the two
approaches. (The horisontal delimitations of these two areas are
shown in the Baltic map of Figure 1.) The study is concluded
by a discussion of the advantages and drawbacks associated with
each of the methodologies as well as an outlook towards possible
future work.
Methods
Mass-balance budgets, as further described in a technical box, are
based on using averaged quantities, viz. the pertinent variables
have been integrated over space as well as time. These models
consequently have a low spatial resolution, and indeed tend to
yield better results the longer the time-scales are. In short, box
models may be characterized as comparatively robust. The drawback is that they normally do not lead to a deeper understanding
of the physical processes within the system.
Numerical 3-D circulation models, on the other hand, can
have a high resolution in time and space and hence permit pro-
99
Paper IV
Baltic Sub-Basin Turnover Times Examined Using the RCO Model.
cess studies in the interior of the basin under consideration. The
timescales are not subject to any a priori limitations (except those
associated with the numerical Courant-Friedrichs-Lewy criterion),
but these models may require considerable computer resources
and and are not always robust, partly because the results tend to
be rather dependent on the parameterizations implemented in the
numerical realization of the model. (Many physical processes are
on a sub-grid level or poorly understood and therefore not possible to model without approximations.) For the present investigation the Rossby Center Ocean model (RCO) Meier et al. (2003)
has been used. This finite-difference model, has 41 depth levels
and a horizontal resolution of 2 nautical miles in the runs used
here. Given standard meteorological forcing, the model yields
the evolution in time of the velocity, temperature and salinity
fields.
Due to the considerable difference between the ways in which
the two classes of models have been formulated, it is somewhat
of a challenge to devise means whereby the results can be directly
compared. (Note e.g. that the volume flows in box-models are
usually estimated using a conservative tracer such as salt, and thus
represent a combination of advective flow and diffusive fluxes, in
contrast to the 3-D model results.) A first approach to reducing
the large quantity of data arising from a numerical model could be
to integrate all fluxes across the delimitations between the basin
under consideration and the ambient water body, and to hereafter
compare with the corresponding fluxes determined using a massbalance model. It must, however, be underlined that budgets of
the latter type yield averaged mean fluxes to or from the system as
a whole, whereas the numerical model provides the local fluxes
at the boundaries of the modelled system. This may constitute a
problem if e.g. a more-or-less permanent vortex is located at the
basin boundary. A typical example hereof was found in a recent
study Andrejev et al. (2004) of the exchange between the Gulf
of Finland and the Baltic proper, where the integrated fluxes in
each direction across the Hanko-Ossmusaar transect proved to be
on the order of 3000 km3 /a, values to be compared to a massbalance budget estimate of around 600 km3 /a for the water flux
from the Gulf of Finland Andrejev et al. (2004) . In cases like
these a more appropriate diagnostic variable for characterizing
the system is the global turnover time Bolin and Rodhe (1973).
This quantity is, by definition, integrated over time as well as
space and can furthermore be calculated for both of the mod-
100
Ambio 33(4-5) pp 257-260
Paper IV
eling approaches. Turnover time additionally serves as a useful
indicator of the physical processes taking place in the system, and
this quantity is thus frequently used in biogeochemical studies,
hereby being of considerable importance in itself when evaluating the geochemical and ecological characteristics of the basin
under consideration.
Calculating the turnover time of a reservoir from a massbalance budget is easily accomplished on the basis of the estimated fluxes and the volume of the system. It is a less straightforward matter for 3-dimensional numerical models, where an indirect approach is required. In the present study Lagrangian trajectories Döös (1995); Blanke and Raynaud (1997) have been employed for this purpose. This methodology is most conveniently
applied off-line to velocity fields originating from a numerical
ocean model. It is based on following the individual tracks of
numerous, initially prescribed, water parcels.
The Lagrangian trajectories are based on an algorithm that
computes particle paths in a three-dimensional velocity field. This
algorithm calculates true trajectories for a given stationary velocity field. In the numerical ocean model, the divergence is
discretized and allows the exact calculation of three-dimensional
streamlines within each box of the three-dimensional mesh. For a
stationary field, such streamlines define exact trajectories of particles in the model. These streamlines indirectly comprise the
subgrid-scale parameterisations of the numerical ocean model by
using its velocity field. This method has been applied in many
different studies of the world ocean circulation Döös (1995);
Blanke and Raynaud (1997); Speich et al. (2002); Drijfhout et al.
(2003); Döös et al. (2004)
Convection has not been taken into account in the present
study, but has been tested in previous investigations by assigning
a water parcel a random depth whenever it enters a convectively
unstable water column Döös (1995). Like the velocities used to
calculate the trajectories, these convection events also follow from
the ocean general circulation model. These studies showed, however, that the effects of convection did not contribute significantly
to the integrated transports.
Results
As already touched upon, the alternative modeling approaches
will be applied to two test areas of different character, where
101
Baltic Sub-Basin Turnover Times Examined Using the RCO Model.
1
0.9
0.8
0.7
0.6
Nt/N0
Paper IV
0.5
0.4
0.3
0.2
0.1
0
0
500
1000
1500
2000
Time (days)
Fig. 2. Normalized temporal decay of the number of trajectories remaining in the
Gulf of Riga. The four differently dashed lines represent the results from trajectory
experiments commencing in January, April, July and October 1985 and the solid,
somewhat irregular, curve shows their average. The smooth line represents a theoretical decay curve calculated on the basis of the estimated box-model turnover
time 825 days (indicated by the arrow on the time axis).
consequently the dynamics governing the water exchange can be
expected to demonstrate different properties.
The Gulf of Riga represents a morphologically almost autonomous system (only connected to the Baltic via the narrow
and shallow Irbe and Moon straits), with a comparatively large
freshwater supply from the river Daugava Otsmann et al. (1997).
These two features acting in conjunction provide a text-book example of a situation where box-model considerations are highly
appropriate, i.e. well-defined exchanges, inputs and salinity differences. Calculations of this type have previously been undertaken for the period from 1977 to 1995 Savchuk and Swaney
(1999), resulting in an estimated turnover time of 825 days.
For the Lagrangian investigation of the turnover time characterizing the Gulf of Riga the RCO model results for the years
1980-1993 were used. The off-line experiments were initiated
by letting the model spin up for five years, whereafter ”release”
of trajectories took place in January, April, July and October 1985
so as to examine possible seasonal variability. 9000 ”particles”,
each corresponding to a water volume of 108 m3 , were evenly
distributed throughout the Gulf of Riga. (As determined from
a set of auxiliary experiments, this initial ”particle density” was
well above that required for the results to ”converge”.) Their
trajectories were hereafter followed for a time-span of approxi-
102
Ambio 33(4-5) pp 257-260
Paper IV
mately 6 years. Figure 2 shows the normalized temporal evolution of the number of trajectories remaining in the gulf, a quantity intimately related to the residence time of water in this volume. From the diagram it is immediately evident that the renewal
process is only insignificantly affected by seasonal variations. The
figure also includes a theoretical decay curve, calculated on the
basis of the box-model estimate of the turnover time, viz.
t
Nt = N0 · e(− τ ) ,
where Nt is the current number of trajectories remaining in the
control volume, N0 is the number of initially marked particles
and τ is the box-model turnover time. The exponential decay
is seen to correspond rather well to the average of the four ”experimental” curves (which in turn do not differ markedly from
one another). It has thus been concluded that the two approaches
to determine the ventilation of the Gulf of Riga appear to work
equally well, most likely since this aquatic system to a high degree satisfies the a priori requirements for successful box-modeling
of non-reactive tracers such as salt.
When attention is directed towards Gdansk Bay it is recognized that this water body, with its open boundary towards the
Baltic proper, does not conform particularly well to the classical
prerequisites for establishing mass-balance budgets. A significant
salinity difference between the bay and the ambient water masses
of the southern Baltic is, however, present. Since the fresh-water
flux from the river Vistula moreover is known from hydrological
records, a box-model approach has previously been used to provide a lowest-order estimate of the exchange Witek et al. (2003).
In this study the northerly delimitation of Gdansk Bay was taken
to coincide with latitude 54◦ 50’ N, the examined period spanning between 1993 and 1998, and the resulting turnover time
proved to be 15 days. A Lagrangian approach, based on the
RCO-model fields from the Baltic used above, has also been implemented for the same region. This analysis pertained to 3000
marked water parcels, i.e. the same initial particle density as in
the Gulf of Riga case. The experiments were conducted along
analogous lines, although for a period of 3 years, and the resulting decay patterns are shown in Figure 3, which also includes an
exponential decay curve established on the basis of the 15-day
box-model turnover time. This latter decay is seen to be more
rapid than the average of those calculated using the Lagrangian
formalism (where least-squares curve fit and use of the theoret-
103
Baltic Sub-Basin Turnover Times Examined Using the RCO Model.
1
0.9
0.8
0.7
0.6
Nt/N0
Paper IV
0.5
0.4
0.3
0.2
0.1
0
0
200
400
600
Time (days)
800
1000
1200
Fig. 3. Normalized temporal decay of the number of trajectories remaining in
Gdansk Bay. The four differently dashed lines represent the results from trajectory
experiments commencing in January, April, July and October 1985 and the solid,
somewhat irregular, curve shows their average. The smooth line represents a theoretical decay curve calculated on the basis of the estimated box-model turnover
time 15 days (indicated by the arrow on the time axis). Note the different asymptotic properties of the two independently established sets of results.
ical decay equation introduced above yields a turnover time of
around 115 days). This discrepancy between the 3-Dimensional
model and the mass-balance results is most likely due to the box
model being highly sensitive to the prescribed salinity difference.
This quantity, which is very small in the Gdansk-Bay case, may
have thus been prescribed with insufficient accuracy Savchuk and
Swaney (1999). Also, the problems concerning small-scale fluxes
over the basin boundary mentioned earlier probably apply here
Andrejev et al. (2004).
The trajectory results furthermore show considerable variance
between the outcome for the four different months at which the
experiments were initiated. This variability is consistent with the
only weakly constricted character of Gdansk Bay, with its long
open boundary towards the southern Baltic, since variations in
the meteorological forcing can be expected to exert considerable
influence on the exchange processes across this somewhat arbitrary delimitation of the control volume.
Seasonality thus probably exerts a fundamental impact on the
dynamics of Gdansk Bay. (As an attempt to avoid biased results
due to seasonal variations, each of the trajectory experiments was
initiated at a different time of the year.) In particular Figure 3 also
shows a considerable degree of variance between the different
runs. Although it is beyond the scope of the present study to
104
Ambio 33(4-5) pp 257-260
Paper IV
examine these variations in detail, a reflection that can be made is
that events over short time-scales, such as storms, appear to have
at least as strong an impact on the variations as do the seasonal
influences.
The asymptotic properties of the Lagrangian decay curves in
Figures 2 and 3 also merit some comments. As particularly evident from the Gdansk-Bay results, the number of remaining trajectories in this basin does not conform especially well to the
exponental-decay ”predictions”. Commonly a systematic discrepancy of this type (which frequently is encountered when applying the Lagrangian methodology to examine water-renewal problems) is ascribed to individual trajectories being trapped at the
boundaries of the basin under consideration. The presently employed trajectory scheme Döös (1995) is, however, especially designed so as to minimize the number of such occurrences, why
the explanation has to be sought in another direction. To judge
from a detailed analysis of the global behaviour of the trajectories
originating from Gdansk Bay once they have crossed the boundary of this control volume, a considerable number of these ”water parcels” enter either the main Baltic circulation gyre around
Gotland or a well localized bottom-water gyre encompassing the
Gdansk Basin as well as Gdansk Bay. The investigation revealed
that a significant number of these ”fugitives” eventually reentered
the control volume, a phenomenon which is clearly seen from the
trajectories shown in Figures 4 and 5. This recirculation feature is
completely absent from the box-model framework which is based
on assuming an infinite and thus unchanging external reservoir.
Hence the not totally compatible results obtained using the two
modelling approaches.
On the basis of these results it has been concluded that Gdansk
Bay is less suited for the application of box modeling than the Gulf
of Riga. The causes and consequences of this state of affairs will
next be examined
Discussion and Conclusions
In the present study the water-renewal problem has primarily
been dealt with in order to elucidate possible consequences of
climatic change as, within the SWECLIM programme, envisaged
over a regional Nordic scale comprising the Baltic Sea. Since,
however, the box-model concept is also widely used for more
direct purposes such as environmental management, the present
105
Paper IV
Baltic Sub-Basin Turnover Times Examined Using the RCO Model.
66 oN
Time(monts)
50
40
63 oN
30
60 oN
20
10
57 oN
54 oN
10 oE
15oE
o
20 E
o
25 E
o
30 E
Fig. 4. The figure shows every hundredth of the in toto 3000 trajectories used for
the Gdansk Bay investigation, where the colouring indicates trajectory age (blue
young, red old).
66 oN
63 oN
60 oN
57 oN
54 oN
10 oE
15oE
o
20 E
o
25 E
o
30 E
Fig. 5. The figure shows a selection of trajectories recirculating to Gdansk Bay,
where colour is used to characterize the individual trajectories.
106
Ambio 33(4-5) pp 257-260
Paper IV
attempt at validation based on a direct comparison with the results
from a numerical ocean model is also of a more general interest.
It was found that the mass-balance budget approach worked
well for systems fulfilling the classical prerequisites that the control volume is subject to pronounced topographical constraints
and that the fluxes to and from the system are well defined, cf.
the results pertaining to the Gulf of Riga. Gdansk Bay, on the
other hand, proved to be less well adapted to this approach, since
it is a comparatively open system where, furthermore, the salinity
difference between the bay and the open Baltic is small.
The estimates of turnover-times based on Lagrangian trajectories were not associated with this manifest drawback, but it must
be kept in mind that the application of this technique is rather
cumbersome and requires access to highly resolved velocity fields
from a numerical ocean model.
An important point is to which extent the respective results
are generalizible to reactive species such as nutrients, not least
since the LOICZ guidelines Gordon et al. (1996) emphasize that
budget considerations of this type have an important role to play
for estimates of the global bio-geochemical cycle. Even if the
question is not explicitly addressed in this study, any modelling
of active tracers heavily rely on a correct estimation of waterfluxes to and from the studied system, and hence benefit from a
proper choice of method. The two different model approaches,
however, use completely different methodology to incorporate
active tracers, making the comparison to extensive for this study.
The box-model approach outlined by LOICZ critically need a correct hydrodynamical budget for estimating sources and sinks in
the system. The 3-D numerical models, on the other hand, use
a separate process model to describe the dynamics of reactive
species. Here, the accuracy of the process model is more critical
for the outcome than the exact behaviour of the hydrodynamical 3-Dimensional model. One could therefore expect the boxmodel approach to be more susceptible to errors in the water flux
estimation, than the 3-D numerical model approach.
Both methodologies have drawbacks as well as advantages. It
should, however, be kept in mind that when applying an ensemble of box models to describe a larger aquatic system (such as the
Baltic proper), care should be taken that the various sub-basins
not be demarcated in too arbitrary a fashion.
Finally, it is well worth underlining the versatility of Lagrangian
methodology in the field of ocean circulation. As an example of
107
Paper IV
Baltic Sub-Basin Turnover Times Examined Using the RCO Model.
a possible application, it can be mentioned that a modification of
the trajectory analysis used in the present study will be employed
to determine whether it is the river Neva discharge or or the inflow from the Baltic proper that plays the dominant role for the
renewal of the Gulf of Finland water masses. The nature of this
task makes it highly unlikely that the problem could be resolved
solely on the basis of standard box-models.
Acknowledgments
This work was carried out within the SWECLIM programme. Financial support from the Foundation for Strategic Environmental
Research (Mistra) and the Swedish Meteorological and Hydrological Institute (SMHI) is gratefully acknowledged. We thank
M. Meier of SMHI for making the RCO model runs available
to us, and furthermore wish to acknowledge constructive comments by two unknown referees on a preliminary version of this
manuscript.
Authors
Bror Jönsson is a PhD student at the Department of Meteorology,
Stockholm University. He is mainly interested in coastal oceanography, particularly estuarine processes. His research areas include
the Baltic and coastal estuaries in Colombia. Peter Lundberg is
Professor in oceanography at Stockholm University. His main research focus is on physical processes in the Baltic and the North
Atlantic. His address: Department of Meteorology, Stockholm
University, SE-10691 Stockholm, Sweden Kristofer Döös is an associate professor in physical oceanography at the Department of
Meteorology at Stockholm University, where he has worked on
the Sweclim project and coordinated the EU project TRACMASS.
His address: MISU, SE-10691 Stockholm, Email: [email protected]
108
Ambio 33(4-5) pp 257-260
Paper IV
Technical Box: The Mass-Balance Budget
Box models as applied to exchange processes use a conservative
tracer, e.g. salt, together with the net transports in and out of the
control volume, to estimate the fluxes between the system and
the adjacent water body. The most basic type of mass-balance
budget assumes a constant volume V of the system. (Even if
this not normally is the case, a reasonable approximation can
be achieved by averaging data over long time-scales.) The system is furthermore taken to be well mixed, implying that the
conservative tracer is evenly distributed in space. The method
is generally based on the salinity difference between the system
and the ambient water body, and hence such a discrepancy is
required. Frequently the box-model formalism is applied to estuaries, viz. semi-enclosed areas of an intermediate salinity, which
implies some type of physical limitations on the freshwater input
as well as on the fluxes to and from the system. Such constraints
can be exerted by sills, sounds, or other morphological features.
The internal and external salinities Si and So as well as a possible riverine influx Qf are used to formulate a budget based on
conservation of volume as well as salt:
dV
= Qf + Qi + Qo ,
dt
dSi
V
= Sf Qf + So Qi − Si Qo ,
dt
where the salinity Sf of the river discharge is equal to zero for
the cases examined in the present study. Assuming stationary
conditions, viz. d/dt = 0, the unknown fluxes to (Qi ) and
from (Qo ) the system can be calculated from the resulting purely
algebraic equations. The outflow can hereafter be used to estimate
the turnover time τ = V /Qo .
References
Andrejev, O., Myrberg, K., Lundberg, P., 2004. Mean circulation
and water exchange in the gulf of finland — a study based on
three-dimensional modelling. Boreal Environmental Research
9, 1–16.
Blanke, B., Raynaud, S., 1997. Kinematics of the Pacific Equatorial
109
Paper IV
Baltic Sub-Basin Turnover Times Examined Using the RCO Model.
Undercurrent: a Eulerian and Lagrangian approach from GCM
results. Journal of Physical Oceanography 27, 1038–1053.
Bolin, B., Rodhe, H., 1973. A note on the concepts of age distribution and transit time in natural reservoirs. Tellus 25, 58–62.
Döös, K., 1995. Inter-ocean exchange of water masses. Journal of
Geophysical Research 100 (C7), 13499–13514.
Döös, K., Meier, K. M., Döscher, R., 2004. The Baltic Haline Conveyor Belt or The Overturning Circulation and Mixing in the
Baltic. Ambio 33 (4-5), 261–266.
Drijfhout, S. P., Vries, K. D., Döss, K., Coward, A., 2003. Eddyinduced transport of the Lagrangian structure of the upper
branch of the thermohaline circulation. Journal of Physical
Oceaography 33, 2141–2155.
Elmgren, R., Larsson, U., 2001. Science and Integrated Coastal
Management. Dahlem University Press Berlin, Ch. Eutrophication in the Baltic Sea Area, pp. 15–35.
Gordon, D. C., Boudreau, Mann, K. H., Ong, J. E., Silvert, W. L.,
Smith, S. V., Wattayakom, G., Wulff, F., Yanagi, T., 1996.
LOICZ Biogeochemical Modelling Guidelines. Vol. 5 of LOICZ
Reports & Studies. LOICZ, Texel, The Netherlands.
Humborg, C., Blomqvist, S., Avsan, E., Bergensund, Y., Smedberg, E., Brink, J., Mörth, C.-M., 2002. Hydrological alterations with river damming in northern Sweden: Implications for weathering and river biogeochemistry. Global Biogeochemical Cycles 16 (3), art. no.1039.
Humborg, C., Conley, D. J., Rahm, L., Wulff, F., Cociasu, A., Ittekkot, V., 2000. Silicon retention in river basins: Far-reaching
effects on biogeochemistry and aquatic food webs in coastal
marine environments. Ambio 29, 45–50.
Ittekkot, V., Humborg, C., Rahm, L., TacAn, N., In Press. Carbon
Silicon Interactions. SCOPE publications. Scope.
Knudsen, M., July 1900. Ein hydrographischer Lehrsatz. Annalen
der Hydrographie und Maritimen Meteorologie, 316-320.
Meier, H., Döscher, R., Faxén, T., 2003. A multiprocessor coupled ice-ocean model for the Baltic Sea: Application to salt inflow. Journal of Geophysical Research.
Otsmann, M., Astok, V., Suursaar, Y., 1997. A model for water
exchange between the Baltic Sea and the Gulf of Riga. Nordic
Hydrology 28 (4-5), 351–364.
Räisänen, J., Hansson, U., Ullerstig, A., Döscher, R., Graham,
L. P., Jones, C., Meier, H. E. M., Samuelsson, P., Willén, U.,
2003. European climate in the late 21st century: regional sim-
110
Ambio 33(4-5) pp 257-260
Paper IV
ulations with two driving global models and two forcing scenarios. Climate Dynamics DOI: 10.1007 / s00382-003-0365x.
Savchuk, O. P., Swaney, D. P., March 1999. Water and nutrient
budgets of the gulf of riga (mass-balance budget developed in
the scope of loicz).
URL
http://data.ecology.su.se/ mnode/
Europe/Gulf%20of%20Riga/ rigabud.htm
Speich, S., Blanke, B., de Vries, P., Döös, K., Drijfhout, S.,
Ganachaud, A., Marsh, R., 2002. Tasman leakage: a new route
in the global ocean conveyor belt. Geophysical Research Letters
29 (10.1029/2001GL014586, 55-1 - 55-4).
Wassmann, P., Tamminen, T., 1999. Pelagic eutrophication and
sedimentation in the Gulf of Riga: a synthesis. Journal of Marine Systems 23 (1-3), 269–283.
Witek, Z., Humborg, C., Savchuk, O. P., Grelowski, A., LysiakPastuszak, E., 2003. Nitrogen and Phosphorus Budgets of the
Gulf of Gdansk (Baltic Sea). Estuarine Coastal and Shelf Science
57, 239–248.
111
This thesis was prepared using LaTeX togheter with TexShop on Mac OS X. Figures were produced with the help of Matlab,
m-map, and Illustrator. Photos on the cover and this page were taken by the author in Cispata, Northern Colombia 2001 The
text font is Joanna 10pt. The cover fonts are Impact and Gill Sans. Insert paper is Gallerie One Silk, cover paper is Invercote G.
Printed 2005 at Duvbo Tryckeri AB, Stockholm
There are two contrasting methods for investigating estuarine systems
with oceanographic models. Mass-balance models are simple, robust
and easily applied, although limited by fundamental constraints. More
advanced 3-D models are more challenging to use but generate an
abundance of high resolution data. However, this wealth of data may
also lead to problems when attempting to specify the "representative
state" of the system, a common goal in estuarine studies.
In this thesis, different types of models suitable for investigating estuarine systems have been utilized in various settings. A mass-balance
model was used to investigate potential changes of water fluxes and
salinities due to the restoration of a mangrove estuary in northern
Colombia. Another study combined velocity-fields from a 3-D numerical model and Lagrangian-trajectory analyses (comment: this last bit
isn’t exactly popular science! Whether you change it or not depends
how popular you want to make it…) to investigate the mixing dynamics in the Gulf of Finland. Lagrangian-trajectory analysis was also used
as a tool to compare mass-balance and 3-D model results from the
Gulf of Riga and the Bay of Gdansk, highlighting when and where each
method was applicable.
The conclusion is that estuarine models not only require different
levels of effort for their implementation, but also yield results of
varying quality. When oceanographic aspects are to be taken into
account within Integrated Coastal Zone Management it is therefore
important to decide which model to use as early as possible in the
planning process, since this choice ultimately determines how much
information the model can be expected to provide about the physical
processes characterizing the system
Department of Meteorology
Stockholm University, 2005
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement