Masterarbeit SvenjaReith
Department of Physics and Astronomy
University of Heidelberg
Master thesis in Physics submitted by
Svenja Reith
born in Gelnhausen
2014
Spatiotemporal slope measurement of short wind waves under the influence of surface films at the Heidelberg Aeolotron
This Master thesis has been carried out by Svenja Reith at the
Institute of Environmental Physics in Heidelberg under the supervision of
Prof. Dr. Bernd Jähne
Abstract:
A novel, high resolution Imaging Slope Gauge (ISG) at the Heidelberg Aeolotron is presented. The instrument allows measurements of the twodimensional slope of short winddriven water waves at unprecedented temporal and spatial resolution. Wave number spectra are measured up to k = 13 000 rad/m with a temporal resolution of more than 1500 Hz. The high sampling frequency eliminates aliasing up to wave numbers of about k = 2660 rad/m. A new nonlinear calibration and data processing work flow has been implemented to convert raw camera images into wave slope in the range of ±0.96. The average statistical measurement error is estimated to be ∆s rms
=
0.018, a significant improvement compared to previous
Color Imaging Slope Gauge setups.
Spectrally resolved measurements of the influence of various surfaceactive materials (surfactants) on smallscale waves are reported. The wave damping effects of different substances are analyzed. It is shown that, except for low wind speeds, gas transfer velocities across the airsea boundary layer can be parametrized with the mean square slope of the waves, independent of the specific type of surfactant that is used.
Zusammenfassung:
Ein neues hochauflösendes bildgebendes Messinstrument (Imaging Slope Gauge) am Heidelberger Aeolotron wird beschrieben. Das Instrument ermöglicht Messungen der Neigung von kurzen winderzeugten Wasserwellen mit bisher unerreichter zeitlicher und räumlicher Auflösung. Wellenzahlspektren werden bis zu Wellenzahlen von k = 13 000 rad/m und mit einer zeitlichen Auflösung von mehr als
1500 Hz gemessen. Die hohe zeitliche Aufnahmefrequenz verhindert Aliasing bis hin zu Wellenzahlen von k = 2660 rad/m. Eine neue nichtlineare Kalibrierungsund Auswerteroutine wurde implementiert um KameraRohbilder in Wellenneigung im Bereich ±0.96 umzurechnen. Der mittlere statistische Messfehler beträgt
∆s rms
=
0.018, eine deutliche Verbesserung gegenüber älteren Color Imaging Slope
Gauge Aufbauten.
Spektral aufgelöste Messungen des Einflusses von unterschiedlichen oberflächenaktiven Substanzen (Surfactants) auf kleinskalige Wellen werden präsentiert. Die
Wellendämpfungseffekte verschiedener oberflächenaktiver Substanzen werden analysiert. Es zeigt sich, dass die GasaustauschTransfergeschwindigkeiten durch die Grenzschicht zwischen Luft und Wasser durch oberflächenaktive Substanzen
(außer für sehr niedrige Windgeschwindigkeiten) durch die mittlere quadratische Neigung parametrisiert werden können, unabhängig von der verwendeten oberflächenaktiven Substanz.
Contents
I Background
9
11
Gravity waves and Capillary waves
. . . . . . . . . . . . . . . . . . . . 11
. . . . . . . . . . . . . . . . . . . . . . . . . 13
. . . . . . . . . . . . . . . . . . . . . . . . . . 13
Theories of Wave Generation by Phillips and Miles
. . . . . 13
Spectral Description of the Wave Field
. . . . . . . . . . . . . . . . . 14
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Slope Probability distribution
. . . . . . . . . . . . . . . . . . 19
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Waves and AirSea Gas Exchange
. . . . . . . . . . . . . . . . . . . . 21
. . . . . . . . . . . . . . . . . . . . . . 21
Measuring and Modelling Gas Exchange
. . . . . . . . . . . 22
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
. . . . . . . . . . . . . . . . . . . 25
Surfactants used for this thesis
. . . . . . . . . . . . . . . . . 26
. . . . . . . . . . . . . . . . . . . . 28
Effects of surfactants on water waves and airsea gas transfer
29
. . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 Foundations in Signal Processing
31
3
I
III
IV
Discrete Fourier Transform (DFT)
. . . . . . . . . . . . . . . . . . . . 31
The Alias Effect, Digital Filtering and the NyquistShannon
. . . . . . . . . . . . . . . . . . . . . . . . 33
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
. . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Spectral leakage and energy loss due to windowing
. . . . . 35
II Methods
39
4 The Imaging Slope Gauge (ISG) as a technique to measure water wave surface slopes
41
Slope measurements vs. height measurements
. . . . . . . . . . . . 42
Methods for water wave surface slope measurements
. . . . . . . . 43
. . . . . . . . . . . . . . . . . . . . 43
. . . . . . . . . . . . . . . . . . . . 45
Concepts of the Imaging Slope Gauge (ISG)
. . . . . . . . . . . . . . 46
Advantages and Limitations of an ISG setup
. . . . . . . . . 49
. . . . . . . . . . . . . . . . 50
53
The Heidelberg Wind/Wave Facility “Aeolotron”
. . . . . . . . . . . 53
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Coupling of Light Source and Camera
. . . . . . . . . . . . . . . . . . 61
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
. . . . . . . . . . . . . . . . . . . . . . . . . . 62
. . . . . . . . . . . . . . . . . . . . . . . . . . . 65
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6 Measurement Campaigns in the Heidelberg Aeolotron
67
. . . . . . . . . . . . . . . . . . . . . . 67
. . . . . . . . . . . . . . . . . . . . . . 69
Wave Field Equilibrium Measurements
. . . . . . . . . . . . 69
Continuous Wind Speed Measurements
. . . . . . . . . . . . 70
. . . . . . . . . . . . . . . . . . . . . . . 73
. . . . . . . . . . . . . . . . . . . . . . . 73
III Data Analysis & Discussion
75
77
. . . . . . . . . . . . . . . . . . . . . . . . . . . 79
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
. . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Second Step: Slope Calculation
. . . . . . . . . . . . . . . . . . . . . 90
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
8 Characterization of the Setup
97
Determination of the Frame Rate
. . . . . . . . . . . . . . . . . . . . 97
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
. . . . . . . . . . . . . . . . . . . . . . . . . 101
. . . . . . . . . . . . . . . . . . . . . . . . . 101
Measurement of the Lens Float Target
. . . . . . . . . . . . 103
. . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Measurement of the Wavelet Target
. . . . . . . . . . . . . . 107
Spatial Distribution of Calibration Lenses
. . . . . . . . . . . . . . . 109
Influence of the nonideal imaging properties of the Fresnel Lens
. 111
9 The Influence of Surfactants on Water Waves and Gas Transfer 113
Surfactants and Water Wave Slope
. . . . . . . . . . . . . . . . . . . 113
Surfactants and Mean Square Slope
. . . . . . . . . . . . . . . . . . . 114
Surfactants, Mean Square Slope and Gas Transfer Velocities
. . . . 121
The Effects of Surfactants on Smallscale Waves
. . . . . . . . . . . 127
133
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
136
V
VI
IV Appendix
151
153
A.1 Wind speeds for the Aeolotron campaign 2013
. . . . . . . . . . . . 154
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
A.3 Uncorrelated Mean Square Slope Timeseries
. . . . . . . . . . . . . . 157
A.4 Correlated Mean Square Slope Timeseries
. . . . . . . . . . . . . . . 162
A.5 Omnidirectional Saturation Spectra
. . . . . . . . . . . . . . . . . . . 167
1
Introduction
The Ocean, AirSea Gas Exchange, and the Importance of Waves
Not only has the ocean exerted a strong fascination on people for ages but it is also a giant reservoir with major relevance as an ecosystem as well as for climatic processes.
It is important to examine the processes of heat, gas and momentum exchange between the ocean and the atmosphere in order to understand the climate system as well as climate change. Furthermore, at least heat and momentum exchange contribute to the driving forces of large scale oceanic circulation and influence temperature and humidity in the atmosphere and thus modify the atmospheric circulation and the
hydrological cycle [ IPCC , 2013 ].
Climate change is a topic of special concern for humanity. Regarding this, the ocean is of particular importance as a net sink for anthropogenic carbon dioxide
(CO
2
) emissions. The oceanic reservoir of DIC (dissolved inorganic carbon) is more
( than 50 times higher that the atmospheric one (see
At the same time, the oceans currently take up CO
∧
=
2 Gt C yr
−
1
;
2 at a rate of about 7 Gt CO
2 yr
−
1
Regional airsea fluxes of CO
2
are largely unknown [ Donelan and Wanninkhof
Wanninkhof and McGillis [ 1999 ] proposed parametrizations of the gas ex
change velocity with wind speed and, with that, found a yearly carbon intake of the ocean between 1.4 and 2.2 Gt C yr
−
1
. Many other parametrizations linking transfer velocities to wind speed have evolved over the past years which state different
functional relations, including piecewise linear [ Liss and Merlivat , 1986 ], quadratic
, 2000 ] relations. The model of
Wanninkhof and McGillis [ 1999 ]
states a cubic relation.
1
Chapter 1 INTRODUCTION
2
Figure 1.1.: The global carbon reservoirs and fluxes. Image taken from
Yet the wind does not influence gas transfer velocities directly. There is evidence from laboratory as well as field studies that (windinduced) waves, especially those with wavelengths between millimetres and decimetres, have a major impact on gas
transfer velocities [ Jähne , 1985 ;
, 2004 ]. The largest resistance to airsea
gas transfer of sparingly soluble gases like CO
2 lies in the aqueous mass boundary
layer at the airwater interface where turbulence processes are suppressed [
Jähne and Haußecker , 1998 ]. Semiempirical parametrizations linking the gas transfer
velocity with wind speed, such as the one proposed by
and
Wanninkhof and McGillis [ 1999 ], are widely used. The reason for this are the
difficulties of directly measuring nearsurface turbulence which is attributed to be
the quantity which controls gas transfer velocities [ Lamont and Scott , 1970 ;
1986 ]. The most important processes influencing nearsurface turbulence and thus
transfer velocities are microscale wave breaking 1
[ Banner and Phillips , 1974 ] and
1
The term microscale wave breaking describes the breaking of steep winddriven gravity waves with
decimeter wavelengths without entraining air [ Zappa et al.
was shown by
[ 2004 ] that at moderate wind speeds, microscale wave
breaking is the cause for up to 75 % of the overall gas transfer. Langmuir circulations appear to be important for gas exchange processes especially during initial wave
growth [ Veron and Melville , 2001a ]. Furthermore, the presence of surface active
substances (surfactants) leads to lower transfer velocities and gas exchange rates
, 2004 ] due to their damping effect
In general, a physics based parametrization of gas transfer velocities is desirable.
This has led to the search for alternative parametrizations of gas exchange processes.
The parameter mean square slope 3
of the water surface σ
2 has proven to be a promising
Alternative candidates which have been proposed are the turbulent kinetic energy
(TKE) dissipation [ Lamont and Scott , 1970 ;
and the divergence of the flow field at the surface [ McKenna and McGillis , 2004 ;
Banerjee , 2007 ; Asher et al.
This thesis focuses on mean square slope. It has been shown to correlate well with
microscale breaking effects [ Zappa et al.
, 2004 ] and gas transfer velocities. Several
measurements of mean square slope have been conducted in the field [ Cox and
] and in the laboratory [ Wu , 1971 ;
, 1999 ]. The common objective
of the studies presented here is to achieve a better understanding of the physical foundations of the link between gas transfer and water waves. As one step towards this goal, laboratory measurements of water wave surface slope will be presented in this thesis.
Wave Slope Imaging Techniques
A variety of imaging techniques for measuring water wave slopes has evolved over the past decades. Two basic groups of techniques can be distinguished. The first one is the group of height measurements which includes stereo methods with two cameras
[ Laas , 1905 ; Kohlschütter , 1906 ; Laas , 1906 , 1921 ;
Hilsenstein , 2004 ]. The second group comprises optical slope measurement
methods which are based on the reflection or the refraction of light at the inclined airwater interface.
The first successful application of reflectionbased methods was made by
Munk [ 1954b ] who measured water surface slope from photographs of sun glitter.
Stilwell [ 1969 ] made qualitative measurements of water wave surface slopes on the
open ocean using the diffuse light scattered in the sky as a light source with infinite
2
Smallscale Langmuir circulations occur due to nonlinear interactions between waves and the shear
Teixeira and Belcher , 2002 ] and
appear as helices whose axis is almost aligned with the wind direction [ Caulliez , 1998 ].
3
Mean square slope is a measure for the surface roughness of the wave field.
1.0
3
Chapter 1 INTRODUCTION
extent. Other methods include the Reflective Stereo Slope Gauge (RSSG) [ Schooley ,
Kiefhaber , 2010 ]. Recently, polarimetric slope imaging was developed by
[ 2008 ] which allows for the measurement of water wave surface slope from the
polarisation of the reflected light.
Refractionbased methods reach back to the measurements of
development is the Laser Slope Gauge (LSG) [ Hughes et al.
use CCD or CMOS cameras to measure slopes in two dimensions. These include the imaging slope gauge (ISG;
Jähne and Riemer [ 1990 ]) and its successor, the color imaging slope gauge (CISG;
Objectives of this Thesis
Although a lot of progress has been made concerning water wave slope measurements and imaging techniques some open questions remain. These include the influence of salinity and chemical and biological surfactants on the wave field as well as the interactions between the wind and the wave field, and the processes of wind input and energy dissipation. Recently, an increasing amount of studies sets the focus on the
connection between surfactants and airsea interaction processes [ Gade et al.
The present study makes an effort to shed light on the question which differences in the wave field are observable depending on surfactant type and concentration. For this purpose an imaging slope gauge (ISG) was used.
The objective of this study was to put the ISG in operation at a wind/wave facility, the Heidelberg Aeolotron and to conduct some first water wave slope measurements with different kinds of natural and synthetic surfactants. This includes the improvement of the ISG setup and evaluation method at the Heidelberg Aeolotron as well as its application for spatiotemporal measurements of the properties of water waves.
For calibration of the instrument, a method based on a lens float calibration target
Rocholz , 2008 ] has been adapted.
From the technical side of view it was possible to increase the brightness of the
ISG illumination source which allows for measurements with smaller aperture and hence with improved image quality in terms of depth of field. Using the new high speed camera which is installed as part of the ISG setup it is now possible to record both components of water surface slope with an effective frame rate of more than
1500 Hz. In contrast to the CISG previously used by
Rocholz [ 2008 ] the combination
of these technical improvements now allows for not only measuring the 2D wave number vector, but also for the measurement of the frequencies of waves with wave
without aliasing effects.
Within the scope of this thesis, experiments with different types of surfactants have been conducted and evaluated. ISG data was recorded during a measurement
4
Note that according to
Apel [ 1994 ] waves with wave numbers larger than 6000 rad/m do not occur
at all.
4
campaign in May 2013 where, among others, active thermography measurements were conducted and transfer velocities for N
2
O and friction velocities were determined. The evaluation of the ISG data comprises a spectral description of the wave field using omnidirectional saturation spectra B(k) as well as a description with statistical parameters. For that, the dependency of mean square slope on wind speed and surfactant concentration is analysed, especially for naturelike surfactants. Furthermore, it is examined whether mean square slope is a better parameter for gas transfer velocities in the presence of waves than friction velocity is.
For November 2014, a measurement campaign with sea water and natural surfactants is planned at the Aeolotron. This work serves as a preparatory study for the planned campaign.
1.0
5
Part I.
Background
7
2
Theory of water waves
A detailed description of the basic equations of water wave physics starting at the continuity equation and the NavierStokes equation is omitted here as it is given in many textbooks on fluid mechanics such as
Kundu [ 2008 ]. Instead, this chapter will
present a short classification of water waves, briefly describe the generation of waves by wind, and give a spectral description of the wave field. In the final section, the influence of surface films on surface water waves is summarized.
2.1 Gravity waves and Capillary waves
Water waves can be classified according to their restoring forces. Gravity waves are displacements of the water surface which are restored by gravity. The term gravity wave describes waves with wavelengths between centimeters and several
hundred meters and is applied to waves at the water surface or internal waves 1
alike.
In contrast, capillary waves are restored by surface tension and thus restricted to domains of high curvature, that is very short wavelengths. Gravitycapillary waves populate the transition region in between gravity waves and capillary waves.
For linear gravitycapillary waves with wave number k, the surface elevation
η(⃗x, t) is
η(⃗x, t) grav
= a cos θ(⃗x, t)
(2.1)
1
Internal water waves are waves within the water body which are for example observable from density fluctuations.
9
Chapter 2 THEORY OF WATER WAVES and the corresponding phase speed c is given by c grav
=
√ g k
+
σk
ρ
(2.2) with the surface tension coefficient σ and the density ρ of the fluid.
Linear capillary waves are described by
η(x) cap
= h sin(2π ⋅ x
λ
)
.
(2.3)
Their phase speed is c cap
=
√
2π ⋅ σ
ρ ⋅ λ
.
(2.4)
The instantaneous height profile η of pure nonlinear gravity waves of wavelength
λ and wave number k has been approximated by
η stokes
≈ − a cos(kx) +
1
2 ka
2 cos(2kx) −
3
8 k
2 a
2 cos(3kx) (2.5) with a =
πH kλ and wave height H. The corresponding phase speed is given by c stokes
=
√ g k
(
1 + a
2 k
2
)
.
(2.6)
The steepness of the wave is described by a
2 k
2
.
An exact solution for pure nonlinear capillary waves has been given by
[ 1957 ] and yields a surface profile similar to an inverted Stokes wave. Their phase
speed is given by c crapper
=
¿
Á
À kσ
ρ
(
1 + a
2 k
2
)
−
1
4
16
.
(2.7)
These waves are called Crapper waves and occur as parasitic capillaries on the leeward side of gravity waves. Crapper waves have sharp troughs and flat crests whereas
Stokes waves have sharp crests and flat troughs.
10
Wave Generation by Wind
2.2 Wave Generation by Wind
2.2.1 Drag Modelling
Ocean water waves propagating along the sea surface can transport momentum in horizontal direction. Any variation in the properties of these waves leads to a horizontal gradient of the momentum flux which can be modelled as a force per unit area acting on the boundary. This force is called radiation stress. A frequently used approach is to parametrize the radiation stress with the wind speed at a specific height above the water surface which is a quantity that is rather easy to measure.
This approach is referred to as the bulk aerodynamic method of estimating surface
stress [ Jones and Toba , 2001 ].
Wind blowing over a water surface creates a shear stress τ at the water surface by friction. Assuming water to be a Newtonian fluid the shear stress τ defined as the force of friction F per area A is given by Newton’s law of friction as
F
A
=∶
τ = η
∂u
∂z
(2.8) with the strongly temperaturedependent material constant η – the dynamic viscosity –, velocity u and depth z. The ratio
∂u
∂z is the local shear velocity. In the literature the shear stress is often related to the shear velocity or friction velocity u
⋆ via density
ρ: u
2
⋆
∶=
τ
ρ
(2.9)
The shear velocity can be seen as a measure of momentum transfer in the turbulent wind field over the boundary layer. The watersided shear velocity u
⋆
,water is related to the airsided shear velocity u
⋆
,air via density: u
⋆
,water
=
√ ρ air
ρ water
⋅ u
⋆
,air
(2.10)
2.2
2.2.2 Theories of Wave Generation by Phillips and Miles
According to
Kinsman [ 1965 ] energy can be transferred from the wind to the water
by tangential stresses or by pressure fluctuations. When the wind starts to blow (low wind speeds) momentum is transferred via friction at the water surface resulting in tangential shear stress which produces laminar flow. With increasing wind speed the laminar flow becomes unstable and waves begin to form.
In contrast to that description the models of
Phillips [ 1957 , 1958 , 1977 ];
interpret resonant interactions between pressure fluctuations on the surface due to turbulence in the wind on the one hand and water waves on the other hand as the source of energy transfer from the wind to the waves. Both models are well
11
Chapter 2 THEORY OF WATER WAVES explained in
Janssen [ 2004 ] gives an historical overview of the
rather complicated topic of wind driven wave generation.
The first work on wind driven wave generation was made by
who believed that a pressure difference occurring on the lee side of existing waves was doing work on the water (”sheltering hypothesis“). This hypothesis was discarded because laboratory measurements proved that the occurring pressure differences are too small to explain measured wave growth rates.
Miles [ 1957 ] developed two separate theories of wind driven
wave generation due to a resonance phenomenon at the same time.
Phillips’ theory on wave generation by wind is based on resonant interactions between the pressure field of the wind on the one hand and the waves on the other hand. Thus the model of Phillips includes turbulent pressure fluctuations as the source of resonant forcing of surface waves, resulting in a linearly growing wave spectrum in time. The problem of Phillips’ model is that the effect has proven ineffective because it is of the order of the square of the density ratio of air and water.
Also it does not include any feedback mechanism which makes it applicable for the first stage of wave growth only because there the wave growth is linear. Later,
[ 1959a , b ] included partial feedback into Phillips’ theory resulting in exponential
wave growth.
In the same year as Phillips,
Miles [ 1957 ] published the socalled ”quasilaminar
approach“ which is based on free surface waves resonantly interacting with pressure fluctuations due to the waves. In contrast to the model of Phillips, Miles’ theory results in an effect that exhibits exponential growth and is of the order of the airwater density ratio. However, Miles’ model was highly debated because it neglects the role of turbulence phenomena in the air and other nonlinear effects like the interaction between mean flow and waves. Also, Miles’ theory is not in good agreement with field experiments, for example of
2.3 Spectral Description of the Wave Field
The wave field of wind driven water waves consists of a multitude of waves, each having its own amplitude, phase velocity and wavelength. Thus it is common practice to use a spectral description of the wave field.
12
Spectral Description of the Wave Field
2.3.1 Spectra
Fourier Decomposition
The first conceptual work on spectra was made by Joseph Fourier (1768  1830), who demonstrated that any function ζ(t) ∈ L
2
(
can be rewritten as an infinite
of sine and cosine functions with harmonic wave frequencies:
2.3
with coefficients a n
=
2
T
∫
T/2
−
T/2
ζ(t) cos(2πn f t)dt, and
ζ(t) = a
0
2
+
∞
∑
( a n=1 n cos(2πn f t) + b n sin(2πn f t)) b n
=
2
T
∫
T/2
−
T/2
ζ(t) sin(2πn f t)dt,
( n = 0, 1, 2, ...)
( n = 0, 1, 2, ...)
(2.11)
Here f ∶=
1
T denotes the fundamental frequency and a
0 represents the mean value of ζ(t) over the interval −
T
2
< t <
T
2
. This decomposition of the function ζ nowadays is called a Fourier series.
Wave Energy Density Spectrum
Phillips [ 1977 ] defines the wave energy density spectrum X(
k, ω) as the Fourier transform of the autocorrelation of the water surface displacement:
X(⃗k, ω, ⃗x, t
0
) =
1
(
2π)
3
⋅
∫
+∞
−∞ −∞
∫
+∞
ρ(⃗x, ⃗r, t
0
, t) ⋅ exp(−i( k⃗r− ωt)) d⃗rdt
(2.12) k describes the wave vector, ω is the frequency, ⃗x and ⃗r denote twodimensional spatial vectors and t
0 and t are temporal coordinates. The displacement ζ(⃗ the water surface relative to the mean free surface level has an expectation value of
2
3
L
2
L
2
( R/2π) is the space of 2π periodic functions from
R to
C with
(
R/2π) ∶= {ζ ∶ R/2π → C ∶ ζ measurable, over the interval −
T
2
< t <
T
2
∫
−
π
π
∣
ζ(t)∣
2 dt < ∞}
13
Chapter 2 THEORY OF WATER WAVES zero and is included into
via its autocorrelation function
ρ(⃗x, ⃗r, t
0 x, t
= lim
T→∞
0
)
ζ(⃗x + ⃗r, t
0
+ t)
1
8T XY
−
T
∫
T
−
X
∫
X
−
Y
∫
Y
ζ(⃗x, t
0
)
ζ(⃗x + ⃗r, t
0
+ t) d⃗x dt
0
(2.13)
In reverse, the autocorrelation of the surface displacement can be rewritten in terms of the wave energy density spectrum via an inverse Fourier transform:
ρ(⃗x, ⃗r, t
0
, t) =
∫
+∞
−∞ −∞
∫
+∞
X(⃗k, ω) ⋅ exp(i(⃗k⃗r− ωt)) d⃗kdω
(2.14)
Assuming a homogenous and stationary wave field X( k, ω, ⃗x, t
0
) =
X(⃗k, ω) the integration of the wave energy density spectrum over all frequencies and wave numbers yields the mean squared wave height:
−∞
∫
+∞
−∞
∫
+∞
X(⃗k, ω) d⃗kdω = ζ
2
(2.15)
and
The wave energy density spectrum denotes the distribution of wave energy among
1 different wave frequencies (or wavelengths). Multiplication of
with
ρg yields the mean potential energy, leading to the interpretation of X(⃗k, ω) as the
2 mean energy of a gravity wave with wave vector
Reduced Spectra: Wave number Energy Spectrum, Frequency Energy
Spectrum
Reduced spectra are obtained from the wave energy density spectrum X( k, ω) by integration. Integration over the wave number vector yields the frequency energy spectrum
Φ( k) =
∫
+∞
−∞
X(⃗k, ω) d⃗k
(2.16) whereas integration over all frequencies yields the wave number energy spectrum
Ψ(
−∞
∫
+∞
X(⃗k, ω) dω
(2.17)
Integrating the wave number energy spectrum over the components of the wave number vector separately (i.e. a onedimensional projection on the respective wave
14
Spectral Description of the Wave Field 2.3
number axis) leads to onedimensional transverse wave number spectra:
Ψ(k x
) =
−∞
∫
+∞
Ψ(k x
, k y
) dk y
Ψ(k y
) =
−∞
∫
+∞
Ψ(k x , k y
) dk x
(2.18a)
(2.18b)
Rewriting Ψ( k) as Ψ(k, θ) with k = ∣⃗k∣ and θ = arctan k y k x allows for calculating the omnidirectional spectrum by integrating over all directions of propagation:
Ψ(k) =
−
π
∫
+
π
Ψ(k, θ) dθ (2.19)
Similarly, the unidirectional wave number spectrum is obtained with different integration limits and a scaling with
1
π
:
1
π
−
∫
+
π
2
π
2
Ψ(k, θ) dθ (2.20)
Similar definitions follow from the frequency spectrum. Typically, frequency spectra can be obtained from measurements with capacitive wave wires which determine the surface elevation at one point in space. With imaging techniques it is possible to determine wave number spectra even for low temporal resolution. With sufficiently high temporal resolution, the full energy density spectrum can be obtained.
Power Spectra and Slope Spectra
The definitions given above can be made with surface displacement ζ or with surface slope s. In the following derivation, let ˆ
1
(
2π)
3
∫ ζ(⃗x, t) exp(−i⃗k⃗x) d⃗x be the
Fourier transform of the surface displacement and ˆs be the Fourier transform of the surface slope. The squared absolute value of the Fourier transform of surface displacement and surface slope define the power spectra of displacement F( and slope S(
⃗
2
, respectively.
⃗
2
The total slope wave number power spectrum S( k) can be decomposed into the alongwind component and the crosswind component:
S(⃗k) = S x
( k) + S y
( k)
(2.21)
Due to slope being the derivative of surface height 4
the inherent properties of the
4 s x
∂
∂x
ζ(⃗x), s y x) =
∂
∂ y
ζ(⃗x)
15
Chapter 2 THEORY OF WATER WAVES
Fourier transform yield the following relations: s
ˆ x s
ˆ y
= − ik x
ζ
ˆ
= − ik y
ζ
ˆ
∣ s
ˆ x
∣
2
+ ∣ s
ˆ y
∣
2
= k
2
∣ ˆ
2
(2.22a)
(2.22b)
(2.22c)
With a sufficient amount of statistically independent images of the water surface displacement at different points in time ζ i
( ⃗ i
) the wave number energy spectrum can be obtained from the mean of the power spectra of displacement F i
(
⃗ and vice versa:
Ψ( k) ≈ F(⃗k) =
1
N
N
∑ i=1
F i
( k) =
1
N
N
∑ i=1
∣ ˆ i
(
⃗ 2
=
1
N
N
∑ i=1 k
−
2
∣ s
ˆ x
∣
2
+ ∣ s
ˆ y
∣
2
= k
−
2
S(⃗k)
(2.23)
Saturation Spectrum / Wave Spectrum Models
For the analysis of small scale water waves, the wave number energy spectrum as defined in
is commonly rewritten as the dimensionless saturation spectrum which is defined as
B(⃗k) = k
4
Ψ(
⃗
2
S(⃗k)
(2.24)
The main advantage of this notation is the reduced range of the spectrum which is useful especially for plotting purposes. Other names for the saturation spectrum are the degree of saturation or the curvature spectrum. The latter term is comprehensible because the second derivative of the surface elevation (i.e. the curvature) is k
4
Ψ( k).
The term degree of saturation was invented by
Phillips [ 1958 ] who conducted research
on the fundamental form of wave spectra.
introduced the saturation range model based on a stationary wave field with the assumption that the energy input from the wind and the dissipation by breaking waves compensate each other. This leads to a saturation of the wave field and an upper limit for the wave number. This upper limit is assumed to be independent of the wind induced energy input and thus determined by the restoring capillary and gravity forces only. With the additional assumption that B depends on the direction of the wave number vector relative to the wind (given by the angle θ) only,
used dimensional arguments to deduce that for gravity waves the wave number energy spectrum is of the form
Ψ(
⃗
−
4
(2.25)
Here f (θ) is an angular spreading function which describes the directionality of the waves and is not specified in further detail. β is a proportionality constant which may differ for the gravity regime and the capillary regime.
16
Spectral Description of the Wave Field 2.4
Many authors developed models in order to predict the shape of the saturation spectrum for gravity waves.
a k
1
2 dependency of the saturation spectrum, but for different regimes of the wave number:
Ψ( k) ∝ u
⋆ k
−
7
2
B(⃗k) ∝ f (θ)u
⋆ k
1
2
(2.26)
The model of
is based on a local equilibrium of the spectral flux and includes nonlinear interactions between waves as well as different mechanisms of dissipation due to the breaking of waves, turbulent diffusion, and viscous dissipation.
The energy input is assumed to stem from the turbulent wind field.
In contrast,
Zakharov and Filonenko [ 1967 ];
[ 2004 ] developed a model which contains analogies to the
turbulence model of
Kolmogorov [ 1941 ]. This is why it is called the weak turbulence
model. The underlying assumption of this approach is that the energy input occurs at small wave numbers (i.e. large wavelengths) and after that the energy is transferred by nonlinear interactions until it is dissipated at large wave numbers.
2.3.2 Slope Probability distribution wave surface slope. It also allows for a statistical analysis of the wave field. One can
I(⃗s, ⃗δ) = {(s
′ x
, s
′y
)∣ s x
−
δ x
< s, ⃗δ) of measuring a value ⃗s = (s x s
′
≤ s x
+
δ x and s y
−
δ y
< s
′y
≤ s y
, s y
)
T
+
δ y inside the interval
} when performing a single measurement from the slope probability distribution:
P(⃗s, ⃗δ) = s
′ x s
′ x
∫
+
δ
−
δ x x s
′ y s
′ y
−
δ
∫
+
δ y y
ρ(⃗s
′
) ds
′ x ds
′ y
(2.27)
In order to meet the definition of a probability density function the slope probability distribution is normalised to 1:
∫ ∫
ρ(⃗s
′
) ds
′ x ds
′ y =
1 (2.28)
5 valid for large wave numbers in the gravity wave regime
17
Chapter 2 THEORY OF WATER WAVES
2.4 Mean Square Slope
Mean square slope is another parameter which is frequently used to describe the wave field.
[ 1987 ] showed that it correlates well with the transfer velocity
k for air sea gas interaction which is one of the main reasons this parameter is of interest for the scientific community. For a stationary wave field fulfilling the principle of superposition the central limit theorem of statistics leads to P(⃗ a Gaussian distribution:
P(⃗s) = ρ
Gauß
=
1
2πσ x
σ y
⋅ exp (− s
2
2σ x s
2y
−
2σ
)
(2.29) with zero mean:
⟨ x⟩ =
1
2πσ x
σ y
∫ ∫ exp (−
2σ s
2 x s
2y
−
2σ
) dx dy
Then the sum of the two variances σ x and σ y yields mean square slope s
2
:
⟨ s
2
⟩ = ⟨ s
2 x ⟩ + ⟨ s
2 y ⟩ =
σ
2
+
σ
2
=
σ
2
(2.30)
(2.31)
A rotation of the coordinate system such that one axis is aligned with the wind direction has been established. Then the components of slope and of the variance are called the upwind component σ
σ
2 or s c or s u along this axis and the crosswind component along the perpendicular axis. The larger the ratio
σ
2
σ
2 of the two components of the variance is, the more isotropic is the wave field.
Wind driven water waves are a little more complicated than this simple model.
First, nonlinear interactions of the water waves lead to a violation of the principle of superposition. Second, parasitic capillary waves exhibit a preference for the downwind side of gravity waves which results in an asymmetry of the probability distribution. Third, due to physical constraints, the steepness of waves is limited.
These limitations can be modelled by the inclusion of additional parameters.
Cox and Munk [ 1954b ] introduced the GramCharlier distribution to do this modelling
for the twodimensional case:
P gc
(
η, ξ) =
1
2πσ c
σ u
⋅ exp (−
η
2
+
ξ
2
2
)
⋅ [
1 −
+
1
C
21
(
ξ
2
−
2
1
4
C
22
(
ξ
2
−
1)η −
1)(η
2
−
1
6
C
03
(
η
3
−
1) +
1
24
3η) +
C
04
(
η
4
−
6ξ
2
1
24
C
40
(
ξ
4
−
+
6ξ
3) + . . . ]
2
+
3)
(2.32)
18
Waves and AirSea Gas Exchange which they found to be sufficient for slopes up to η = ξ = 2.5
21 and C
03 describe the skewness of the distribution which represents the asymmetry of the distribution and the coefficients C
40
, C
22 and C
04 describe the peakedness which is a measure of the deviation of the steepness of the distribution compared to the Gaussian distribution. All these coefficients have to be evaluated from the data.
In
Kiefhaber [ 2014 ], a brief overview of different methods for determining mean
square slope is given.
2.5 Waves and AirSea Gas Exchange
Water waves are an important parameter for the exchange processes of heat, trace gases and momentum between the ocean and the atmosphere. Over the last years, the search for a physically based model of airsea gas exchange which includes the effects of the wave field arose. This section serves as a brief summary of airsea gas exchange and the influence of water waves.
2.5.1 Transport Mechanisms
Turbulent and molecular motion are the two fundamental mechanisms of mass
transport in fluids 7 . Both can be described by Reynolds decomposition similar to
Fick’s law for diffusion, but with a turbulent diffusion coefficient K(z) in addition to the molecular diffusion constant D. Assuming homogeneity in the horizontal directions x and y, the flux j c of a trace gas is given by the concentration gradient j c
= −(
D + K(z)) ▽ c = −(D + K(z))
∂c
∂z
.
(2.33)
Molecular diffusion dominates in the boundary layers at the airwater interface which have a typical thickness of less than a millimetre whereas turbulent transport is more efficient on large scales. Close to the interface, the size of turbulent eddies shrinks as the eddies cannot penetrate the interface. The turbulent diffusion coefficient K(z) is significantly reduced close to the interface.
2.5
6
Note that η and ξ denote normalized slope components η =
7
Among others, the term fluid refers to air and water.
s
σ c c and ξ =
σ s u u
19
Chapter 2 THEORY OF WATER WAVES
Transfer Velocity
The rate of airsea gas transfer can be described macroscopically by integration 8
of
c(z
2
) − c(z
1
) = − j c
∫ z
1 z
2
(
D + K(z))
−
1 dz ∶= j c
R = j c k
.
(2.34)
This defines the transfer velocity k and its inverse, the transfer resistance R, which contain the turbulent structures in the boundary layer.
2.5.2 Measuring and Modelling Gas Exchange
Direct measurements of gas concentration profiles in the boundary layers are highly demanding, especially in the presence of waves, because of the small thickness of the layers in combination with the large amplitude of the motion at the water surface.
Various models have been proposed to describe the transfer velocity k [ Lamont and
Deacon , 1977 ; Coantic , 1986 ]. The model of
by the waterside friction velocity u
⋆
, the Schmidt number Sc =
ν
D with the kinematic viscosity ν, the dimensionless Schmidt number exponent n and a dimensionless scaling parameter β to describe the strength of nearsurface turbulence: k =
1
β u
⋆
Sc
− n
.
(2.35)
Both β and n depend on the wave field but the exact dependency is still unknown.
1
[ 1979 ] give a Schmidt number exponent of
for a rough, wavy water
2
2 surface and
Deacon [ 1977 ] proposed a value of
for a smooth water surface. A
3 gradual transition between these limiting cases was found by
[ 2013 ] for which the facet model
Jähne and Bock [ 2002 ] accounts. A similar model
was reported by
Evidence exists that wind speed alone is not sufficient to parametrize transfer
velocities [ Wanninkhof et al.
[ 1987 ] suggest the mean square
slope σ
2 of the waves as a parameter which has empirically proven to correlate well with gas transfer velocities
Kiefhaber [ 2014 ]. Yet a fundamental
link between gas exchange and mean square slope is missing whereas it exists for parameters such as the surface divergence or turbulent kinetic energy dissipation
(TKE). Nevertheless, mean square slope has the significant advantage that it can be measured on small and large scales with comparably low effort.
8
The integration is valid as presented when mass sources or sinks are absent because j c is constant due to mass conservation.
20
Surface Films
2.6 Surface Films
Surface films, also called surfactants 9
are a species of molecules which enrich at the surface of a liquid (e.g. for thermodynamic reasons). Surfactants influence surface tension: adding surfactants to a liquid lowers the liquids surface tension and modifies the free surface boundary condition in the tangential direction. Here the viscoelasticity plays an important role: the surface film is pushed together and pulled apart due to the orbital movement in a wave. This process dissipates energy transferred from the wind field. This in turn suppresses capillary waves and capillarygravity
Alpers [ 1989 ] explained that
not only capillary waves, but also short gravity waves are damped by surface films due to the Marangoni effect. In the following sections the most important properties of surface active agents will be described. Their classification into different subtypes will be explained and those used for experiments for this thesis will be explained in further detail.
2.6
2.6.1 Surfactants
Surfactants usually are amphiphilic substances. Thus they consist of at least two parts; the polar (hydrophilic) head determining the properties of the molecule and the hydrophobic tail consisting of a – possibly branched or aromatic – hydrocarbon chain. Therefore, surfactants consist of a component that is soluble in water and one that is insoluble in water but soluble in oil and other unpolar solvent. Also some oligomers and polymers act like surfactants.
Due to their hydrophobic parts surfactants tend to form clusters in the water body (called micelles, see
) as soon as a material dependent concentration,
the so called “critical micelle concentration (CMC)” is reached. Below the CMC surfactants form a monomolecular layer at the water surface in order to minimize the contact between the hydrophobic tails of the molecules and the surrounding water molecules.
The hydrocarbon tail of surfactants often contains an even number of carbon atoms and strongly influences the surface packing of the surfactant. An amide
bond improves surface packing ( van Os [ 1997 ]) and planar ring structures such as
]), thus increasing the CMC ( Myers
[ 2005 ]). Large, stiff tails (sterol ethoxylates) lead to a long time in order to reach an
equilibrium state of surface tension ( Holmberg [ 2001 ]).
Typically surfactants are enriched at the water surface and reduce the free energy at the surface and thus also the surface tension γ which can be described by the
9
surface active agents
21
Chapter 2 THEORY OF WATER WAVES
hydrophobic tail hydrophilic head
Micelle
Figure 2.1.: Surfactants sticking to the airwater interface. As soon as the critical micelle concentration (CMC) is reached, micelles are formed. Image taken from
Gibbs isotherm ( Equation 2.36
− dγ = ∑ i
Γ i dµ i (2.36) where Γ i is the surface excess of component i and µ i is the chemical potential of component i. Important effects of surfactants – apart from lowering the surface tension and wave damping – include foam stabilization, emulsification, dispersion
(immersion wetting) and the enhancement of wetting.
In the oceans surfactants are of major importance for biological organisms and perform many different tasks such as the transport across cellular membranes or building lipid bilayer membranes. Most naturally occurring oceanic surfactants are produced by phytoplankton during photosynthesis. They also occur as byproducts when dead organisms are degraded by microorganisms. Amongst others, the amount of surfactants found at the ocean surface varies due to hydrodynamics at the interface
as well as the availability of nutrients to surfactantproducing organisms ( Wurl et al.
22
Surface Films
Micelles
Micelles formed by surfactants are highly soluble, polar clusters which exhibit small surface activity. They may consist of different types of surfactants and can be regarded as a reservoir for dissolved surfactants. Their relaxation times are in the order of
ms ( Shah [ 1998 ]). The critical micelle concentration (CMC) at which micelles start
to form is determined by the charge of the polar head group, the length of the hydrocarbon chain, the number of double bonds in the hydrocarbon chain and is
influenced by cosolutes such as salts or alcohols, temperature and pressure ( Farn
[ 2008 ]). When the CMC is reached, surfactant solubility often increases enormously
because excess molecules are transported into the water body to form micelles.
2.6.2 Classification of Surfactants
Surfactants can be grouped into soluble and insoluble species. Soluble surfactants adsorbed to the surface have a relatively higher concentration than in the bulk of the liquid whereas insoluble surfactants adsorbed to the surface have rearranged such that the forces occurring between the molecules are minimal. Surfactants can be classified according to the charge of their polar head group into anionic, cationic, nonionic and zwitterionic surfactants. An overview over different classes of surfactants is given in
Nagel [ 2013 ]; here the most important properties are repeated. A more
detailed description of the classification of surfactants and their chemical properties is given in
Anionic surfactants
Anionic surfactants span the largest class of surfactants and are those which are most commonly used. Their polar head group is composed of a carboxyl, sulfate, sulfonate or phosphate group. In water, they dissociate into an anion and a cation, which often is an alkaline metal. Anionic surfactants are generally sensitive to water hardness. Sulfates are rapidly hydrolysed by acids, whereas the other types of anionic surfactants are stable unless under extreme conditions.
Cationic surfactants
Cationic surfactants are mostly composed of permanently charged quaternary ammonium cations (NR
4
+
) or amines (NH
2
R
1
, NHR
2
,NR
3
).
Amines only act as surfactants in protonised state, i.e. for low to neutral pH values whereas quaternium amonium is insensitive to pH. Other types of cationic surfactants include phosphonium, sulfonium & sulfoxonium, but they are rarely used due to their high costs. Cationic surfactants exhibit high water toxicity because they are hydrolytically stable. Since most materials are negatively charged when they are surrounded by an aquaeous medium the positively charged head group of cationic surfactants is attracted, leading to a strong adsorption of the cationic surfactant to most surfaces.
2.6
23
Chapter 2 THEORY OF WATER WAVES
Nonionic surfactants
Nonionic surfactants consist of polyether or polyhydroxyl head groups. They are insensitive to water hardness and their physiochemical properties are not affected by electrolytes; in return they exhibit temperaturedependent behaviour: with increasing temperature the solubility of nonionic surfactants in wa
ter decreases ( Atwood and Steed [ 2004 ]). A major drawback of nonionic surfactants
is their high skinirritating potential.
Zwitterionic surfactants
Finally, zwitterionic surfactants which are the smallest and most expansive group of surfactants contain two charged groups of different sign.
The positively charged group is composed of ammonium whereas the negatively charged group varies, but is commonly a carboxylate. Zwitterionic surfactants contain a subgroup named amphoteric surfactants which can react as an acid as well as a base. Zwitterionic surfactants are generally stable in acid and alkaline surroundings and are insensitive to hard water. Moreover they exhibit excellent dermatological properties.
2.6.3 Surfactants used for this thesis
Naturally occurring surfactants exhibit a large variation of weight and chemical composition. For this work the influence of surfactants on water wave surface slope was investigated. Therefore, five different kinds of surfactants were used. Their chemical structure is depicted in
Natural surface films
During the first type of experiments surfactants were chosen to mimic natural surface films. The chemical composition of the sea surface microlayer was analysed by
Ćosović and Vojvodić [ 1998 ] and others who
found polysaccharides, lipids and fatty acids to be the main components. A mixture of the following four surfactants was used to reproduce natural conditions:
Dextran
Dextran is a soluble surfactant which is naturally produced by synthesis of sucrose by various species of bacteria. Dextran is a branched glucan contain
ing a mixture of different polysaccharides 10
with a molecular weight of 10 000 u to
50 000 000 u.
[ 2009 ] demonstrate the usability of dextran as a model
substance for the influence of polysaccharides on airsea gas exchange.
24
10
Polysaccharides consist of glucose molecules with the molecular formula (C
6
H
10
O
5
) n
.
Surface Films 2.6
(b) Triton X100
(a) Dextran
(c) Palmitic Acid (PA)
(d) MonoGalactosylDiacylglycerol
(MGDG)
(e) Phosphatidylglycerol (PG)
Figure 2.2.: Chemical structure of the surfactants used for the experiments in this thesis.
R
1 and
R
2 denote organic radicals. Image taken from
MonoGalactosylDiacylglycerol (MGDG)
MonoGalactosylDiacylglycerol is an insoluble surfactant that is produced by algae and is contained in the membranes of chloroplasts. It belongs to the chemical class of glycolipids which are lipids with attached carbohydrates and has a neutral headgroup. MGDG is one of the most common glycolipids in plant cells.
Phosphatidylglycerol (PG)
Phosphatidylglycerol is an insoluble surfactant which occurs in the cell membranes of bacteria, algae and other marine organisms. Chemically it belongs to the class of glycerophospholipids and is used as a model substance for lipids.
25
Chapter 2 THEORY OF WATER WAVES
Palmitic Acid (PA)
Palmitic Acid (CH
3
(CH
2
)
14
COOH, hexadecanoic acid in
IUPAC nomenclature) is an insoluble surfactant which is naturally produced by marine organisms during the synthesis of fatty acids. It is used as a model substance for fatty acids. PA has a molar mass of 256.42 g/mol and a density of 0.8527 g/cm
3 at
62
○
C ( Lide [ 2005 ]). The solubility of PA in water is about 0.04 mg/l at 25
○
Synthetic surfactant
Because of its wellknown properties and comparability to older studies one part of the experiments were conducted with synthetic surface active substance Triton
X100.
Triton X100
Triton X100 (C
14
H
22
O(C
2
H
4
O)
11 , scientific name polyethylene
glycol p(1,1,3,3tetramethylbutyl)phenyl ether) belongs to the class of nonionic surfactants. It is soluble and synthetically produced. Its molecular mass is 647 g/mol and its density is 1.07 g/cm
3
. The critical micelle concentration is at 0.22 mol/l–
0.24 mol/l and its viscosity is 2.4 g/(cm s) at 25
○
2.6.4 The seasurface microlayer
The sea surface microlayer (SML) is the boundary layer interface at the oceanic water surface with a thickness of a few tens to hundreds of micrometers. The sea surface microlayer is characterized by its distinctive physicochemical properties which differ from those of the subsurface water below. The SML consists of a highly hydrated loose gel of tangled macromolecules and colloids at the airwater interface
[ 2013 ]) and includes surfactants. As such, it is remarkably stable:
even under conditions with high turbulence it covers most of the water surface and
it remains stable up to a wind speed of at least 10 m/s ( Wurl et al.
[ 2008 ]). After mixing the water
the SML is rapidly reformed at timescales of typically less than 1 min ( Cunliffe et al.
[ 2013 ]). An important effect influencing the SML is the socalled bubble scavenging:
the formation of the SML is enhanced by rising bubble plumes ( Zhou et al.
Wallace and Duce [ 1978 ]). It has been found that the SML plays a fundamental role
in airsea gas exchange processes and is linked to the production of aerosols. The bacterioneuston which contributes to the SML has been shown to contribute to the
airsea gas exchange for at least some types of gas ( Cunliffe et al.
26
11
On average, n=9.5.
Surface Films 2.6
Figure 2.3.: Interactions involving the Sea Surface Microlayer. Image taken from
2.6.5 Effects of surfactants on water waves and airsea gas transfer
Several authors ( Levich [ 1962 ];
Lucassen [ 1970 ]) gave an explanation of the damping of capillary ripples by surface
films. Today it is believed that during the propagation of the small waves local changes of the hydrodynamic boundary conditions occur, i.e. the water surface is locally compressed and dilated. This in turn leads to local changes of surface tension which can be described by a complexvalued viscoelastic modulus.
Mann [ 1989 ] gave a corrected dispersion relation for surface waves in the presence
of surfactants which was later verified by several laboratory experiments.
Today it is believed that winddriven water waves are damped by surfactants due to one or more of the following mechanisms:
• A finite viscoelastic modulus is created by the surfactant which leads to a change in the dispersion relation of the waves. This in turn increases energy dissipation.
• A change of the surface roughness of the water due to the surfactant leads to a decrease of the energyflux from the wind to the water waves.
• The modifications of the wave field, which occur because of the presence of a surfactant, change the effect of the nonlinear interactions of the waves leading to faster dissipation of energy.
The generation of water waves by wind is also changed in the presence of surfactants as the energy input by wind shear as well as energy transfer and energy
27
Chapter 2 THEORY OF WATER WAVES dissipation in the wave field are modified. When surfactants are present a critical wind speed has to be exceeded in order to generate waves.
that this is not true for a clean water surface whereas
show that highly concentrated surfactants suppress the generation of water waves by wind up to a critical wind speed but at very high wind speeds their effect vanishes.
In general, the effect of surfactants on the wave field and nearsurface turbulence is larger for low wind speeds which is explained by a rupture of the surface film at higher wind speeds leading to a mixing of the surfactants with the bulk water.
Also, the wind profile changes in the presence of surfactants due to the induced change of surface roughness.
Concerning airsea gas exchange,
Liss and Slinn [ 1983 ] showed that gas molecules
are not directly obstructed in passing the interface by the surfactant molecules.
Nevertheless surfactants reduce airsea gas transfer by up to 50 % ( Frew et al.
or even 60 % for low to medium wind speeds ( Krall [ 2013 ]).
2.6.6 Marangoni effect
Marangoni waves are predominantly longitudinal waves occurring in the boundary layer which have been experimentally detected by
Lucassen [ 1968a , b ]. They are
visible as oscillations of the airwater interface whose wavelengths depend on surface
, 1982 ]. Marangoni waves occur due to instabilities of the
airwater interface which are evoked by surface tension gradients (e.g. stemming from surface films). A tangential force generated by these gradients acts as restoring force.
Viscoelastic surface films influence the flow patterns in the boundary layer because they modify the boundary conditions of the surface. In the presence of surface films,
Marangoni waves are important because of their interactions with transversal gravitycapillary waves which cause local compression and dilation of the surface film. These in turn lead to surface tension gradients which excite Marangoni waves. Nonlinear wavewave interactions result in a transfer of energy from longer waves to an energy sink in the Marangoni region. Marangoni waves are strongly damped on the scale of
one wavelength [ Alpers , 1989 ]. This effect is associated with strong velocity gradients
in the boundary layer which leads to an enhancement of viscous dissipation. Thus, the Marangoni effect leads to strong damping of short gravity waves. Surface films with higher elasticity lead to stronger wave damping.
28
3
Foundations in Signal Processing
The process of wave imaging transforms the continuous wave signal into a discrete
(digital) dataset represented by intensity values at given pixel positions. Discrete signals have to be treated slightly differently than continuous signals. The most important concepts of signal processing of discrete data which are necessary for the evaluation of ISG raw data are presented in the following sections. These include the discrete Fourier transform (DFT), the theory of aliasing, spectral analysis and digital filtering using window functions.
3.1 Discrete Fourier Transform (DFT)
The discrete Fourier transform (DFT) is a transform which converts a finite and equally spaced signal into a finite combination of complex sinusoids which represent a discrete and periodic frequency spectrum. Thus, the DFT converts the sampled function from its original domain (spatial or temporal) to the frequency domain.
For application in image processing, the 2DDFT is of special interest. For a signal
S = (s m,n
) of size (M, N) it is defined as
Definition 3.1 (2D Discrete Fourier Transform (2DDFT))
s
ˆ k,l
=
M−1 N−1
∑ m=0
∑ n=0 s m,n
⋅ e
−
2πi⋅ mk
M e
−
2πi⋅ nl
N where k = 0, . . . , M − 1 and l = 0, . . . , N − 1.
The inverse transform is given by
(3.1)
29
Chapter 3 FOUNDATIONS IN SIGNAL PROCESSING
Definition 3.2 (2D Inverse Discrete Fourier Transform (2DIDFT))
s m,n
=
1
MN
M−1 N−1
∑ k=0
∑ l=0 s
ˆ k,l
⋅ e
2πi⋅ mk
M e
2πi⋅ nl
N where m = 0, . . . , M − 1 and n = 0, . . . , N − 1.
(3.2)
For twodimensional images, the indices k and l are often called wave numbers. The extension of the DFT to three dimensions is obvious and is not given here explicitly.
Often, the third dimension is time which becomes an (angular) frequency ω in
Fourier space.
Important properties of the DFT are completeness, orthogonality and periodicity:
Completeness
The DFT is an invertible and linear transformation F ∶ C
M
→ C
M
. Here C denotes the set of complex numbers. This means that the DFT maps any Mdimensional complex vector onto another Mdimensional complex vector for any M > 0. Concurrently, the inverse mapping IDFT coexists for every Mdimensional complex vector.
Orthogonality
The vectors e k
= [ exp(
2πi
M km) ∣ m = 0, 1, . . . , M − 1] the set of Mdimensional complex vectors:
T form an orthogonal basis over e
T k e
⋆ k
′ =
M−1
∑ m=0
( exp(
2πi
M km))⋅(exp(
2πi
M
(− k
′
) m)) =
M−1
∑ m=0 exp(
2πi
M
( k−k
′
) m) = M δ kk
′ where ○
T denotes transposition, ○
⋆ denotes complex conjugation and δ kk
′ denotes the Kronecker delta.
Periodicity
The periodicity can be shown directly from the definition: s
ˆ k+M,l
∶=
=
M−1 N−1
∑ m=0
M−1
∑ m=0
∑ n=0
N−1
∑ n=0 s m,n s m,n
⋅ e
−
2πi⋅ m(k+M)
M e
−
2πi⋅ nl
N
⋅ e
−
2πi⋅ mk
M e
−
2πiM
´¹¹¹¹¸¹¹¹¹¶ e
−
2πi⋅ nl
N
1
=
M−1 N−1
∑ m=0
∑ n=0 s m,n
⋅ e
−
2πi⋅ mk
M e
−
2πi⋅ nl
N
= s
ˆ k,l .
30
Discrete Fourier Transform (DFT)
Plancherel Theorem and Parseval’s Theorem
The Plancherel theorem and Parseval’s theorem are two results from harmonic analysis which relate a function in its natural domain to its Fourier spectrum. Let ˆ k k be the (1D)DFTs of f n and g n , respectively. The Plancherel theorem states:
Definition 3.3 (Plancherel Theorem)
N−1
∑ n=0 f n g
∗ n =
1
N
N−1
∑ k=0
F
ˆ k
ˆ
∗ k where the star denotes complex conjugation.
(3.3)
A special case of the Plancherel theorem is Parseval’s theorem. It reads:
Definition 3.4 (Parseval’s Theorem)
N−1
∑ n=0
∣ f n
∣
2
=
1
N
N−1
∑ k=0
∣ ˆ k
∣
2
(3.4)
3.1.1 The Alias Effect, Digital Filtering and the NyquistShannon
Sampling Theorem
Usually the discrete signal is created by discretizing a continuous signal. The term sampling describes the process of converting a function of continuous time or space into a numeric sequence such as a function of discrete time or space. The samples of a function f (t) are commonly denoted by f [n] ∶= f (nT) for integer values of n.
The spectra obtained from the DFT will be the correct discrete representation of the spectra of the underlying signal when the Nyquist–Shannon sampling theorem is not violated. The Nyquist–Shannon sampling theorem provides a connection between continuous and discrete signals:
Definition 3.5 (NyquistShannon Sampling Theorem)
If a function f contains no frequencies higher than B, it is completely determined by giving its ordinates at a series of points spaced
1
2⋅B
seconds apart ( Shannon [ 1949 ]).
This means that the sampling frequency f s has to be more than twice the maximum
frequency B of the original bandlimited signal 1
in order to avoid a loss of information during sampling. The frequency f
Nyquist
∶=
1
2 f sampling is then called the Nyquist frequency. The term Nyquist rate describes the minimum sampling rate f sampling
∶=
2B that satifies the Nyquist–Shannon sampling theorem for a given signal. Thus, the
Nyquist rate can be interpreted as a property of the continuous signal, whereas the
Nyquist frequency is seen as a property of the discrete system.
1
A bandlimited signal is a signal which exhibits a zero power spectrum for frequencies ν > B.
3.1
31
Chapter 3 FOUNDATIONS IN SIGNAL PROCESSING
Sampling with a higher frequency than the Nyquist rate does not lead to any gain of information. When the sampling frequency is too low or the original signal is not bandlimited then the reconstruction of the signal leads to nonlinear imperfections which are called aliasing and cannot be reconstructed. When the sampling frequency is predetermined, for example by the sampling method, the continuous function f is usually filtered before sampling in order to to eliminate high frequencies and avoid aliasing effects. An appropriate filter type is a lowpass filter which is referred to as an antialiasing filter in this context.
If the Nyquist–Shannon sampling theorem is not satisfied, the frequency components above f
Nyquist of the sampled signal cannot be distinguished from lowerfrequency components mirrored at the sampling frequency which are then called aliases.
The Nyquist–Shannon sampling theorem is applicable to functions of time, where the term “frequency” has its original meaning, but also to functions of other domains, such as space (e.g. for a digital image). In the latter case, the units of measure of the frequencies has to be adapted accordingly.
3.2 Spectral analysis
The term spectral analysis refers to the process of identifying frequency components in data. Most often, spectral analysis is used to determine the frequency content of an analog (continuous) time signal f (t). A frequently used strategy to perform spectral analysis is to sample the analog signal in a first step, then truncate the discretized
with a window function (see
), and finally compute the DFT of
the filtered data. The DFT ˆ spectrum is represented by the absolute value of the DFT.
3.3 Windowing
The main assumption of the discrete Fourier transform is that the input signal is finite and periodic. For many physical signals, such as images, the periodicity condition is not fulfilled. For such signals, a method to overcome this limitation is to assume a periodic extension of the signal. This leads to yet another problem: the spectrum is modified artificially, which is undesirable when one is interested in the “real”, physical spectrum. Applying a window function to the signal before performing the
DFT reduces this effect, although it cannot be avoided completely.
A window function is defined as follows:
2
The discretized dataset usually consists of a finite set of uniformly spaced timesamples of the signal f (t).
32
Windowing
Definition 3.6 (Window Function)
In a narrow sense, a window function is a function which is exactly zero at the beginning and at the end of a data block (e.g. at the borders of an image) and outside the data blocks’ domain. A broader definition allows the function to not assume identically zero in the data blocks’ domain. In that case it is required that the product of the window multiplied by its argument is square integrable, and that the function approaches zero
sufficiently fast ( Cattani and Rushchitsky [ 2007 ]).
Depending on the exact shape of the window function, it affects the spectrum in a slightly different way. In general it is desired that the frequency response, i.e. the
Fourier transform of the window function exhibits low amplitudes at side lobes. The window function is chosen with respect to the width of the central lobe as a tradeoff
).
3.3.1 Hann Window
The Hann window is a window function named after Julius von Hann. It is also known as the Hanning window, von Hann window or raised cosine window and it is defined by w(n) =
1
2
[
1 − cos (
2πn
N
)]
, (3.5) with n = 0, . . . , N − 1. One of the most important applications of the Hann window is the filtering of a signal prior to conversion to the frequency domain by performing a Fourier transform. The advantage of the Hann window compared to other window functions is the good frequency resolution and low spectral leakage. The drawback is the slightly reduced amplitude accuracy.
3.3.2 Spectral leakage and energy loss due to windowing
Due to the limited size and the nonperiodicity of a typical signal a phenomenon called spectral leakage occurs after the DFT. All frequencies and wave numbers which do not occur in the basis of the DFT are not periodic in the image window. This results in discontinuities when the signal is periodically extended which in turn lead to the occurence of frequencies in the spectrum, which are not part of the original signal. Metaphorically speaking the “energy” of these frequencies is distributed onto the adjacent frequencies during the DFT, resulting in a leakage of energy to “wrong” frequencies. This effect is reducing by windowing.
In general, applying a window function to a signal leads to a reduction of the total power carried by the signal. This can be corrected by a normalization factor f which is multiplied with the spectrum, resulting in a correction of the amplitudes in
3 small width of the main lobe leading to better frequency selectivity
4 large width of the main lobe leading to the desired small side lobe amplitudes.
3.3
33
Chapter 3 FOUNDATIONS IN SIGNAL PROCESSING
Fourier space. In this section, a general form of f is derived as well as the specific f for a Hann window in one and two dimensions.
According to Parseval’s theorem (see
discrete signal S of length N in the spatial domain equals its energy density ˜
E of a
E in the
Fourier domain
E ∶=
N−1
∑ n=0
∣
S[n]∣
2
=
1
N
N−1
∑ k=0
∣ ˆ
2
E.
(3.6)
This means that it is possible to apply a window to the signal in the spatial domain and to correct the amplitudes in the Fourier domain afterwards.
Assume a signal S with finite extent which is (pixelwise, denoted by ⋆ 6 ) multiplied
by a window function W. Let ⟨○⟩ denote the ensemble average and ¯ denote the time average (or the spatial average for spatial signals).
Assuming the ensemble average of the signal to be homogeneous, the ensemble average of the windowed signal is given by
⟨
W ⋆ S⟩ =
1
L
∫
W dx ⋅ ⟨S⟩ with ¯
1
L
∫ W dx.
Since the squared signal is of interest for the energy density:
⟨
W ⋆ S
2
⟩ =
1
L
∫
W dx ⋅ ⟨S
2
⟩
(3.7)
(3.8)
Now substitute W = ˜
2
.
⟨ ˜
2
⋆
S
2
⟩ =
1
L
∫
˜ 2 dx ⋅ ⟨S
2
⟩
(3.9)
⟨( ˜
2
⟩ =
1
L
∫
˜
2 dx ⋅ ⟨S
2
⟩
(3.10)
For a 1DHann window ˜
2π⋅x
L
)) the term
1
L
∫
˜
2 dx in
5
Technically, the terms energy, energy density and power have to be taken with care. In the language
6 of signal processing, the signal S usually is a function of time which results in ⟨S
2 a power spectrum which is a function of frequency ω and has the dimension of
⟩ describing energy
. For spectrum as a function of wave number energy area
.
time application on images, the signal S is a spatial signal. Thus ⟨S
⃗
2
⟩ describes the energy density k. The corresponding dimension is
For two matrices A, B of the same dimension m × n the Hadamard product or pixelwise product
A ⋆ B is the matrix of the same dimension as the operands with elements given by (A ⋆ B) i, j
=
(
A) i, j
⋅ (
B) i, j .
34
Windowing 3.3
can be evaluated as f
Hann1D ∶=
1
L
0
∫
L
˜ 2 dx =
1
L
1
=
4L
0
∫
L
0
∫
L
(
1
4
(
1 − cos(
2π
L
(
1 − 2 ⋅ cos(
2π
L
⋅ x))
2
) dx x) + cos
2
(
2π
L x)) dx
1
=
4L
= [ x −
2L
2π sin(
2π
L x) + x
2
+
L
8π
⋅ sin(2 ⋅
=
3
8
2π
L x)]
L
0
(3.11)
For a 2DHann window ˜
2π⋅x
L x
))⋅
0.5 (1 − cos (
2π⋅y
L y
)) a similar calculation results in f
Hann2D
∶=
L x
1
⋅
L y
(
L x
,L y
)
∫
(
0,0)
˜ 2 dx dy = (
3
8
)
2
=
9
64
(3.12)
Then the correctly normalized signal is given by ⟨S
3DHann window and 3D signal it is ⟨S
2
⟩ =
2
⟩ =
512
⟨( ˜
2
⟩
.
27
64
⟨( ˜
2
⟩ and for a
9
35
Part II.
Methods
37
4
The Imaging Slope Gauge (ISG) as a technique to measure water wave surface slopes
This chapter outlines the most important techniques which are used to measure water wave surface slopes. An overview of the existing techniques available for measuring the geometrical properties of water waves is given in
chapter explains the underlying foundations as well as the limitations of the different methods. Finally, the characteristic features of the Imaging Slope Gauge are described in more detail.
39
Chapter 4 THE IMAGING SLOPE GAUGE (ISG) AS A TECHNIQUE TO . . .
Pointbased Methods
Optical Methods
Stereo Methods
Imaging Methods
Laser Slope
Gauge
Scanning Laser
Slope Gauge
Statistical
Methods
Imaging
Polarimeter
Imaging Slope
Gauge
Color Imaging
Slope Gauge
Sunglitter Method
(Cox & Munk)
Reflective/Refractive
Slope Gauge
Nonoptical Methods
Acoustical Methods Wire Probes
Accelerometer Buoys Pressure Sensors
Radar
Resistive Wire Probes Capacitive Wire Probes
Figure 4.1.: Overview of a selection of the existing techniques for measuring geometrical properties of the sea surface. Adapted from
4.1 Slope measurements vs. height measurements
When measuring geometrical properties of water waves two main approaches can be distinguished: slope measurements and height measurements. Both techniques are in principal equivalent because it is possible to obtain height information from slope measurements via integration at the cost of the mean surface elevation. This is feasible because water wave surface slopes are the gradient of surface elevation. A short description of this equivalence is given in
One important difference between the two approaches is due to the fact that water wave surface slope measurements require the acquisition of two slope components at the same time whereas height measurements  as surface elevation is a scalar quantity  do not. This makes water wave surface slope measurements technically more demanding. On the other hand the wave height displays a rather large variation depending on wave length whereas the variability of wave slopes is almost constant
for a broad range of wave lengths ( Jähne and Schultz [ 1992 ]). This is the main reason
why this thesis is focused on water wave surface slope measurements.
In principle, two different types of techniques for measuring water wave surface slopes and heights are available, optical methods and nonoptical methods. Nonoptical techniques include measurements of the wave amplitude using capacitive or resistive wire probes, pressure sensors or accelerometer buoys. Conventional
stereophotogrammetric methods (
Laas [ 1905 , 1906 ]; Kohlschütter [ 1906 ]; Laas [ 1921 ];
Schuhmacher [ 1939 ]) are an example for optical techniques used for height measure
40
Methods for water wave surface slope measurements ments. It has been shown that they exhibit insufficient height resolution for small waves (see
4.2 Methods for water wave surface slope measurements
In this chapter, the focus lies on optical techniques for measuring water wave surface slopes as most nonoptical methods interact with the wave field and are not suitable for measuring capillary waves due to this limitation. Optical methods for measuring
water wave surface slopes are based on the principle of either refraction ( Cox [ 1958 ];
Zhang and Cox [ 1994 ]) or reflection
[ 1994 ] gives a detailed review of the advantages and disadvantages of
the different optical techniques for measuring water wave surface slopes. Their areas of application can be summarized as follows:
Reflectionbased methods are particularly useful for field measurements. Methods
such as the Stilwell technique ( Stilwell [ 1969 ]) require a homogeneous illumination
of the sky, suffer from a large degree of nonlinearity between intensity and water wave surface slope.
Refractionbased methods can resolve small structures on the water surface and are applicable over a broad range of wave lengths with a rather small degree of nonlinearity.
4.2.1 Reflectionbased Methods
Reflectionbased methods including Stilwell photography ( Stilwell [ 1969 ]) are partic
ularly suitable for field measurements since no equipment has to be submerged into the water. Thus, the wave field remains undisturbed. However, as mentioned before,
[ 1994 ] have demonstrated that these techniques perform satisfactorily
for a rather narrow slope range only.
Reflectionbased methods found on the fact that the water surface reflects light according to the reflection condition: the viewing angle α relative to the surface normal equals the angle of incidence β of a light ray. This is illustrated in
From the positions of all reflexes in a series of images the slope distribution of the surface can then be determined.
4.2
41
Chapter 4 THE IMAGING SLOPE GAUGE (ISG) AS A TECHNIQUE TO . . .
Sun Glitter Method by Cox and Munk
Cox and Munk [ 1954a , b ] are known for
the first successful application of this principle in order to acquire statistical water wave surface slope data in field measurements. The glitter of the sun on the sea surface is photographed from a plane. When the reflection condition is fulfilled a reflex is visible on the photo. Since the reflex condition varies depending on the position in the image a slope can be assigned to each position. From this, a distribution of the sun glitter can be obtained which allows for the computation of the water wave surface slope distribution. A theoretical study of the patterns of the specular reflexes which are observed on a randomly moving surface was performed by
Stilwell Method
The Stilwell method which is named after
extension of the method of
Cox and Munk [ 1954a , b ]. It makes use of the diffuse
scattered light of the sky which acts as a light source with infinite extent. With the assumptions of a quasiuniform radiance of the sky and an optimal geometry of the imaging system
Stilwell [ 1969 ] showed that the optical spectrum
of an image of the water surface is linearly related to the water wave surface slope
spectrum 1 . This model has been extended to second order by
allows to describe the quality of the linearity.
Limitations
Measurement techniques based on reflection are theoretically limited to slopes between ±1 due to the doubling of angles by reflection (reflex condi
( Jähne and Schultz [ 1992 ]).
Another important limitation of reflectionbased techniques  apart from the restriction to a narrow range of slopes  is the low reflectivity of the water surface
for small angles of incidence 3
as well as the disturbances introduced by light that
penetrates the surface and experiences backscattering 4 .
[ 1994 ] also showed that the nonlinearities between measured inten
sities and water wave surface slopes are significantly larger for reflectionbased techniques than for refractionbased methods. Reflectionbased techniques are highly demanding in terms of the required size of the light source. Thus they are not easily usable with artificial light sources which is another reason why techniques based on refraction are preferred for laboratory measurements.
42
1 for small wave slopes.
2
Practically, this would require a light source of infinite extent.
3
The reflectivity of a smooth water surface is less than 5 % for angles of incidence smaller than 50°
4 socalled upwelling light
light ray
Methods for water wave surface slope measurements
to observer
α
β air water
α air water to observer
(a) Reflection of light at the water surface
d
Δx
β
(b) Refraction of light at the water surface
Figure 4.2.: Comparison of light reflection ( 4.2a
The reflection condition for
is
α = β whereas the refraction condition for
given by Snell’s law reads n a
⋅ sin α = n w
⋅ sin β with refractive indices for air (n a
) and water ( n w
), respectively. In the case of refraction it is possible to determine the slope of the surface from the measured deflection ∆x = d ⋅ tan(α − β).
4.2.2 Refractionbased Methods
Refractionbased methods are particularly useful for water wave surface slope imaging because they are capable of measuring large slopes and exhibit a large degree of linearity over a large range of slopes.
The fundamental principle of refraction based techniques is the deflection of light at the airwater interface according to Snell’s law: A light ray which enters an interface (here: airwater interface) with an angle of incidence α relative to the surface normal is refracted at the interface such that n a
⋅ sin α = n w
⋅ sin β (4.1) with an angle of refraction β and refractive indices for air (n a
) and water (n w
), respectively (see
). If the variations of water height are negligible (h ≈
const.), the deflection ∆x = d ⋅ tan(α − β) of the light ray at the bottom of the water body can be used to determine water wave surface slope.
4.2
43
Chapter 4 THE IMAGING SLOPE GAUGE (ISG) AS A TECHNIQUE TO . . .
Method of Cox
The first application of refraction for water wave surface slope measurements was described by
Cox [ 1958 ] who placed a glass wedge filled with ink
below a wave tank and illuminated it uniformly from below. Seen from above this results in a logarithmic decrease of brightness in one direction. Using a photometer, the brightness of a point on the water surface can be measured and be related to the corresponding component of water wave surface slope.
For this technique the main source of error is due to the neglect of large height variations for higher wind speeds which result in an error in the slope data. Apart from that the measured intensity depends on curvature and slope as well. Nevertheless the method served as a starting point for further development.
this technique such that it is capable of measuring the curvature distribution of water waves.
Wright and Keller [ 1971 ] extended the method of Cox to an imaging
technique that is capable of measuring one component of water wave surface slope in a confined area.
Laser Slope Gauge
Another frequently used technique that can be used for water wave surface slope measurements at a single point is the Laser Slope Gauge (LSG).
It consists of a laser beam which is refracted at the airwater interface according to
Snell’s law depending on the surface slope. It is detected using a positionsensitive optical receiver. This method can only yield a slope value for one point of the surface.
Among others,
Prettyman and Cermak [ 1969 ];
Long and Huang [ 1976 ] have applied
this technique.
Later the method has been extended by scanning the laser beam across the surface in one or two dimensions – the so called Scanning Laser Slope Gauge (SLSG)
Bock and Hara [ 1995 , 1992b , a ];
4.3 Concepts of the Imaging Slope Gauge (ISG)
The Imaging Slope Gauge (ISG) is a refractionbased method for measuring water wave surface slopes. It was used for the measurements which were conducted for this thesis. The specific setup that was used for the measurements is depicted in
and explained in
. Here, only the general concept of the ISG is
illustrated.
Historically, the Imaging Slope Gauge was first mentioned by
and described by
Color Imaging Slope Gauge (CISG) to measure water wave surface slope from the deflection of refracted light. A 2dcolor scheme was installed as light source which encodes both slope components as a specific color value.
similar method to determine various statistical and geometrical properties of water waves. His experimental setup consists of a color CCD camera which is focused
5
Although the system presented was not referred to as Imaging Slope Gauge in the original article.
44
Concepts of the Imaging Slope Gauge (ISG) on the mean water height and an illuminated color gradient below a fresnel lens which itself is installed below the water tank which ensures a unique relationship
between color and water wave surface slope. For field measurements (e.g. [ Klinke ,
[ 1994 ] demonstrate that it is preferrable to submerge the light
source and install the camera on the air side because this setup ensures smaller nonlinearities.
Jähne and Schultz [ 1992 ] describe the calibration and explore the
accuracy of different ISG setups.
camera
4.3
telecentric lens top window air water surface water air bottom window
Fresnel lens air f illumination screen
Figure 4.3.: 1D simplified view of an ISG setup. Image modified after
The ISG at the Heidelberg Aeolotron is based on a socalled telecentric illumination in combination with objectspace telecentric optics.
depicts the basic components of an ISG setup. A camera is placed on top of the water surface such that the aperture is positioned in the focal point of a large “telecentric lens” which itself is placed in between the camera and the water surface. This guarantees that only those light rays which leave the water surface parallel to the optical axis are refracted onto the image sensor of the camera and that all rays which enter the telecentric lens parallel to the optical axis are correctly focused onto the sensor of the camera.
The optical axis is oriented perpendicular to the flat water surface. This way the large lens in combination with the camera aperture yield objectspace telecentricity which implies a constant magnification factor independent of object distance. The setup is aligned such that the mean water surface is positioned in the second focal plane of the large lens. Underneath the bottom window of the wind/wave facility,
45
Chapter 4 THE IMAGING SLOPE GAUGE (ISG) AS A TECHNIQUE TO . . .
a large Fresnel lens is placed in the air space between the bottom window and the illumination source installed at the very bottom. For an ideal setup, the distance between the Fresnel lens and the illumination source equals the focal length of the
Fresnel lens. The Fresnel lens guarantees telecentric illumination, i.e. that all rays that leave the lens under an angle δ relative to the optical axis have the same point of origin on the illumination screen.
Thus, the setup provides a unique relation between the point of origin of the light rays which enter the camera for each pixel and the corresponding water surface slope tan α. This relation is independent of the position on the water surface and of the height of the waves. As shown in
[ 1994 ], the relation is almost linear. Yet
it remains to identify the point of origin of the light that enters the camera aperture.
When measuring one slope component only it is sufficient to have an intensity gradient in the direction in which slope is supposed to be measured. For two dimensions the idea of the color imaging slope gauge (CISG) described in the previous section is to use an RGB camera and a colorcoded light scheme. Then two of the color gradients can be used for position coding and the third color channel can be applied for normalization as well as the correction of disturbing effects such as those induced by lens curvature. A significant disadvantage of this method is the chromatic aberration of the telecentric lens which leads to misalignments of the
different color components (see [ Kiefhaber , 2014 ]).
Another approach, which is used for this thesis, is applicable with a high speed monochrome camera and a programmable light source which is capable of switching between different states in short time. The underlying idea of this approach is to implement four different brightness gradients A, B, C, D in the alongwind directions x and x as well as the crosswind directions y and y (see
intensities I x and I y can be computed from two intensity gradients in opposite directions in x and ydirection, respectively:
I x
=
A − B
A + B and I y
=
C − D
C + D
(4.2)
The combination of both components (I x , I y
) makes it possible to identify the origin of the light rays on the illumination source from the camera images uniquely. The usage of normalized intensities is referred to as ratio imaging and it has the significant advantage that small imperfections, for example in the bottom window and the light source, are automatically compensated.
Between 2009 and 2011 Roland Rocholz constructed a CISG setup at the Heidelberg Aeolotron. It was used during several experiments to conduct research on
airwater gas exchange ( Krall [ 2013 ], Kräuter [ 2011 ]). Based on this setup, the new
ISG setup was constructed in 2013.
46
Concepts of the Imaging Slope Gauge (ISG)
A
B
4.3
C D
Figure 4.4.: Light source intensity gradients (A,B,C,D) for an ISG setup.
4.3.1 Advantages and Limitations of an ISG setup
For the approach using the brightness gradients A, B, C, D as described it is impossible to determine both slope components from one image which is the reason why a high speed camera in combination with fast electronics for the light source are necessary in order to reduce the time between two consecutive images to a minimum.
The main advantage of this approach compared to a CISG setup is the possibility to avoid chromatic aberration. At the same time, the demand for homogeneity of the light source is reduced due to ratio imaging.
Since the ISG is a refractionbased technique either the camera or the light source
have to be submerged into the water 6 . Although this has been tried in the field
([ Klinke , 1996 ]) the ISG is mainly used in laboratory applications.
The footprint on the water surface is restricted by the size of the large telecentric lens because of the geometry of the light rays for the telecentric setup. The measurable
slope range depends on the size of the light source and the optical components 7 .
Wave height cannot be determined from ISG data directly, since slopes are measured.
In simple terms, as slopes are the gradient of height, the reconstruction of wave height is possible by integration up to an additive factor. This leads to the requirement of additional instruments to measure the water height at one point at least. Yet the ISG is ideally suited to study smallscale waves and their dynamics in a laboratory setup.
6
This is true at least for field measurements; it is avoidable in the lab depending on the construction of the wave channel.
7
For the current data evaluation method at the Heidelberg Aeolotron it is in fact as well limited by the range of slope values available in the calibration target, see
47
Chapter 4 THE IMAGING SLOPE GAUGE (ISG) AS A TECHNIQUE TO . . .
4.3.2 Ray Geometry of the ISG Setup
In order to understand the ISG setup and to evaluate the data the relation between water surface slope and the position on the light source is described in this section.
Consider the trace of a light ray from the camera on top of the setup to the illumination source underneath the bottom of the wind/wave facility as it is shown in
. The light rays which are focused on the image sensor have been
refracted at the water surface according to Snell’s law ( equation (4.1) ). In order to
leave the water surface under an angle
α relative to the water surface normal (i.e.
parallel to the optical axis) they have to pass the water under an angle γ relative to the optical axis. At the faces of the (thick) bottom window the ray is refracted twice, leading to a small displacement and a change of the angle.
As described in
Rocholz [ 2008 ] a light ray that leaves the light source with the
angle δ enters the camera if and only if s = (s s x y
) = ( cos Φ sin Φ
) ⋅ √ n
2
+ ( n
2 w
∣ tan δ∣
−
1) tan
2
δ −
√
1 + tan
2
δ
, (4.3) s, n w water and Φ denotes the polar angle of the light ray.
≈
1.33 is the refractive index of
Denoting the deviation of the examined light ray from the origin on the light source which is given by the principal axis of the fresnel lens as
(
∆x
∆y
) =
√
(
∆x)
2
+ (
∆y)
2
⋅ ( cos Φ sin Φ
)
, (4.4) and using the definition of
∣ tan δ∣ =
√
(
∆x)
2 f
+ (
∆y)
2
, (4.5) with the focal length of the telecentric Fresnel lens f, the relation between water surface slope ⃗ s x y
) = (
∆x
∆y
) ⋅ √ f
2 n
2 w
+ ( n
2 w
1
−
1)((∆x)
2
+ (
∆y)
2
) −
√ f
2
+ ((
∆x)
2
+ (
∆y)
2
)
.
(4.6)
48
Concepts of the Imaging Slope Gauge (ISG) 4.3
camera telecentric lens air water
γ
α
β
γ
90°
α water surface bottom window air
δ
Fresnel lens
δ air
ε
f illumination screen principal axis of Fresnel lens
Figure 4.5.: 1D simplified view of the light ray geometry of the ISG setup. Image modified after
49
5
Experimental Setup
All experiments described in this thesis were conducted in the Heidelberg wind/wave facility “Aeolotron” which is characterized in
the measurement setup of the Imaging Slope Gauge at the Aeolotron is specified. Then
the technical details of the high speed camera ( section 5.3
) and the custommade programmable and highpower LED light source ( section 5.4
setup are described. The doublesided telecentric imaging setup at the Aeolotron allows to sample waves up to highfrequency capillary waves. Due to the highspeed camera this can be done without noticeable aliasing. Finally, in
the calibration targets used for the experiments are specified in detail.
5.1 The Heidelberg Wind/Wave Facility “Aeolotron”
The Aeolotron in Heidelberg is an annular wind/wave facility which was constructed for the analysis of airwater interaction processes (exchange of mass, momentum, and heat). The measurements performed here include measurements of the exchange of volatile substances as well as the application of imaging techniques for the analysis of the wave field or heat exchange. The Aeolotron has an inner diameter of ≈8.7 m and a mean circumference of 29.2 m at the inside wall. A photographic view into the flume of the facility is provided in
. It consists of a ring shaped water flume
of approximately 61 cm width and a height of 2.41 m and is parted into 16 segments.
Since the first description in
[ 1999 ] it has been modified during several
periods of construction work which includes a new design for wind generation. Two
axial fans mounted onto the ceiling of the tank in sections 4 and 12 ( Figure 5.3
used to generate wind. Typical wind speeds during past experiments reach up to approximately 10 m/s (at the usual tank filling of 1.0 m, see
51
Chapter 5 EXPERIMENTAL SETUP
Figure 5.1.: The Aeolotron: Photographic View into the facility. The wind is blowing from right to left.
Photo: AEON Verlag & Studio, Hanau 2010.
52
Figure 5.2.: Rendered view of the Aeolotron with new system for wind generation. Wind is generated such that the air is flowing counterclockwise. The ISG is mounted at segment 13 which can be identified in the picture by the framework for the installation of optical setups in light grey. Image taken from
The Heidelberg Wind/Wave Facility “Aeolotron” the facility is equipped with a counter current pump.
shows a rendered view of the facility with the new system for wind generation.
Experiments are typically conducted with deionized water at a water depth of about
1.0 m, which corresponds to a water volume of about 18.0 m
3
. The air space then comprises about 24.4 m
3
. The Aeolotron is thermally isolated and mostly gastight.
Ambient parameters such as wind speed, temperatures in the water and in the air space and humidity can be measured at reference positions in segments 15 (wind speed measurements with a fananemometer; temperature measurements with two
Pt100 temperature sensors) and 2 and 13 (humidity sensors). In addition, the water height can be determined with a ruler.
In
Krall [ 2013 ], the gas concentration measurement system is described. It contains
two spectrometers one of which is used for direct air side concentration measurement. Furthermore, it consists of a pump installed in segment 6 which is connected to a membrane oxygenator for water side gas concentration measurement. The oxygenator is used to equilibrate the gas concentration in the water with an air parcel which can then be pumped into a FTIR spectrometer for analysis via infrared spectroscopy. The sampling location for air side concentration measurements is in segment 2. From the gas concentration data, transfer velocities can be calculated.
13
15 16
14
1 wind direction
ISGwindow air sampling 1 air sampling 2
12 wind generator 1
2 window
3
11 air outlet wind generator 2
4
10 fresh air inlet
9
8 water sampling
5
7
6
Figure 5.3.: Schematic view of the Aeolotron explaining the segment numbering scheme. The
ISG window is installed in segment 13, segments 16 and 14 contain a large window which allows for direct visual access. The positions of the two wind generator fans, the sampling locations as well as the fresh air in and outlets are shown as well. Image taken from
A disadvantage of annularly shaped flumes are the inertial forces which occur when the water body is moving. They give rise to the formation of secondary flow,
5.1
53
Chapter 5 EXPERIMENTAL SETUP i.e. flow which is less strong than the primary flow in wind direction but which is oriented perpendicular to it. The secondary flow is superimposed on the primary flow and leads to changes of the properties of the total flow due to the geometry of
the facility only ( Bopp [ 2014 ];
Ilmberger [ 1981 ]). Another restriction which occurs
due to the geometry of the flume is the reflection of waves at the walls which leads to conditions different to those on the open ocean.
However, another fact makes the conditions in the Aeolotron more similar to an open water situation like the ocean than the conditions in a linear windwave
channel are. Due to the annular shape of the flume the fetch 1
is quasi unlimited and a stationary wave field can evolve. In contrast to linear facilities, the physical conditions are independent from the position of the measurement device due to the circular geometry of the flume. Furthermore, in contrast to linear facilities, there is no need for a wave absorber.
5.2 ISG
mirror f = 2.0 m telecentric lens window
1.4 m
1.0 m window
Fresnel lens f = 0.76 m light source
Figure 5.4.: Wave imaging setup at the Aeolotron. Image modified after
The imaging setup at the Heidelberg Aeolotron is depicted in
body is illuminated from below with a programmable light source (see
through a window made from 4 layers of Lexan®polycarbonate sheets and high
with a total thickness of 3.2 cm. In between the bottom window and
1
The term fetch describes the length on which the wind acts on the water surface.
2
Lexgard®RS1250 laminate [ General Electric Plastics ]
54
ISG the light source a large Fresnel lens with a focal length of f f
=
0.762 m and a diameter of d f
=
0.89 m is placed in order to ensure “telecentric illumination”. The latter is defined by the following property: as the light source is positioned in the focal plane of the Fresnel lens all rays that leave the light source at a common position will be parallel to each other after passing the Fresnel lens. A tilted mirror is fixed above the top window of the facility. The mirror directs the light rays into a planoconvex lens made from BK7 with a diameter of d = 0.32 m and a focal length of f = 2 m (see
). Both the mean water surface and the aperture of the lens of a high speed
camera (decribed in
) are placed in the focal planes of the planoconvex
lens. This way an object space telecentricity is achieved which ensures that all light rays which are focused on the image sensor of the camera have been parallel to the optical axis before entering the “telecentric” lens. The setup is constructed such that the light rays which enter the camera leave the (flat) water surface normal to it. This is achieved by placing the aperture of the camera lens in the focal plane of the telecentric lens. By this means changes in the size of imaged structures due to varying distances to the image sensor because of the waves can be avoided. This is
important because the height 3
of the waves at the Aeolotron can reach more than
60 cm at high wind speeds. The accuracy of the imaging system is explored in detail in
5.3
Figure 5.5.: Wave imaging setup at the Heidelberg Aeolotron: telecentric lens and tilted mirror.
3
Height is meant here as the vertical distance between wave crest and wave trough.
55
Chapter 5 EXPERIMENTAL SETUP
5.3 Camera and Lenses
In our setup a monochrome pco.dimax high speed
CMOS camera (PCO AG) was used ( figure 5.6
Data was acquired with a spatial resolution of
960 x 768 pixel at a frame rate of 6030 fps. The resulting effective frame rate for slope measurements is one fourth of this value, 1507.5 fps, because four raw images are required to compute a complete
2D wave slope image (see
surements of 2014 were performed with fnumber
8 and a Zeiss lens with f = 100 mm. This equals a footprint on the water surface of 227 × 182 mm corresponding to 0.24 mm per pixel.
Figure 5.6.: pco.dimax
camera used in the ISG setup.
For the 2013 measurements a Nikon lens with f = 105 mm was used with fnumber
5.6. For the measurements with the Nikon lens the footprint on the water surface is
203 × 166mm which corresponds to a pixel size of 0.22 mm in object space. For each raw image, integration time is 140 µs. A summary of the most important technical data of the pco.dimax camera is given in
Table 5.1.: Technical data of the pco.dimax high speed camera; taken from
sensor type full resolution (hor x ver) pixel size (hor x ver) shutter mode spectral range quantum efficiency maximum frame rate (full resolution) frame rate (@ 960 x 768 pixel) exposure/shutter time dynamic range internal RAM data interface
CMOS
2016 x 2016 pixel
11 µm x 11 µm global
290 nm . . . 1100 nm
50 % @ peak
1279 fps
6030
1.5 µs . . . 40 ms
12 bit
36 GB
GigE, USB2.0, CameraLink
56
Light Source
5.4 Light Source
The custommade programmable light source which is used for the experiments consists of a total of 1704 red highpower LEDs (Cree XLamp XPE red) which have a peak wavelength at 630 nm and a maximum continuous current of 700 mA.
For ISG application, they are operated in Flash mode as described below. They are positioned in 35 columns with 24 LEDs each in alongwind direction and 24 rows with
36 LEDs each in crosswind direction. For each row or column, the corresponding
LEDs are placed at a distance of 2.1 cm and are wired in series.
shows a photographic view of the light source with the LEDs. Custom electronics is used in order to control the LED currents between 0 mA and 1000 mA with a resolution of 12 bit separately for each row and column. Fast multiplexer units allow for a fast swapping between preprogrammed brightness gradients (A,B,C,D) as depicted in
. Details of the electronics are given in
In order to create a homogeneous luminance gradient an acrylic diffusing screen is placed on top of the LEDs at a distance of 4 cm.
shows the light source with the diffusing screen on top.
For the 2013 measurements the maximum LED current was 100 mA for both directions. For the 2014 measurements, the maximum LED current has been raised to 300 mA. This leads to higher light intensities which allows for measurements with smaller fnumber.
For ISG measurements, the light source is programmed as follows: The four preprogrammed brightness gradients (A,B,C,D) are activated one after another with a frequency of f ISG
=
6000 Hz
∧
=
δt = f
1
ISG
=
0.167 ms (see
is referred to as a subsequence for the rest of this thesis. A variable number Q of these subsequences is activated with a temporal distance of δT = 4 ⋅ δt = 0.667 ms, leading to Q ⋅ 4 (temporally) equally spaced illumination gradients being activated after another and forming one ISG illumination sequence. P of these sequences are recorded with a frequency of f
Seq
=
1
∆T
. The time during which each individual illumination gradient is active is given by ∆t = 0.140 ms. Thus the duty cycle D
′ is calculated as follows: D
′
=
∆t
δT
=
0.140 ms
0.667 ms
=
21%.
5.4
57
Chapter 5 EXPERIMENTAL SETUP
(a) Photograph of the light source without acrylic diffusion screen showing the
LED rows and columns.
(b) Light source with acrylic diffusing screen.
(c) Schematic view of the light source. Image taken from
A
B
C
1 cm
24 LEDs per column
D
Figure 5.7.: The ISG light source which is used for illumination.
58
p: 1
δT q: 1 2 3 4 5
A B C D A B C D A B C D A B C D A B C D
Coupling of Light Source and Camera
2
1
A B C D
2
A B C D
3
A B C D
4
A B C D
5
A B C D
5.5
∆t
δt
∆T
Figure 5.8.: Example for two consecutive ISG illumination sequences consisting of five subsequences each. With the nomenclature introduced in
P=2 and
Q=5 for this example. Each subsequence consists of each of the illumination brightness gradients A,B,C,D as depicted in
for a time ∆t, the time between the beginnings of two consecutive gradients is δt and the time between the beginnings of two consecutive sequences is ∆T.
5.5 Coupling of Light Source and Camera
The light source and the camera have to be coupled in order to obtain one camera image per illumination gradient. This is done in two different ways depending on the experiment. The first one is internal triggering. Here the camera is set to record images with a frequency of f ISG . The camera trigger output signal is then used to trigger the light source electronics which is programmed to switch between the four illumination gradients consecutively. For the experiments conducted during this study, internal triggering was used to measure long image sequences (20000 raw images). The second method is external triggering where an external function generator is used to trigger camera, light source and possibly other measurement devices. External triggering was used for statistical measurements, where each sequence contains 20 raw images recorded with a frequency of f
ISG . The time interval between individual sequences is controlled by the external trigger signal.
59
Chapter 5 EXPERIMENTAL SETUP
5.6 Calibration targets
Several calibration targets are used for the experiments performed for this thesis.
They are constructed such that they can float on the water surface and can be imaged with the ISG setup. A graphical overview of the targets is given in
with schematic drawings of each target on the left hand side.
5.6.1 Lens float target
The lens float target consists of 24 planoconvex lenses made from BK7 glass 4
with a diameter of d = 50 mm and a radius of curvature of R = 51.68 mm. The lenses are glued into the bottom of a float, leading to a small portion of the lenses being hidden.
The “real” visible radius is analyzed in
shows a schematic drawing of the lens float target; a photograph is given in the same figure.
depicts a single lens floating on the water surface together with the corresponding ray geometry for a light ray entering from the water side. Because the lenses consist of a material with a different refractive index than that of water the light ray is refracted at the waterlens interface and at the lensair interface. This leads to a change of the angle under which the light ray enters the air space compared to the case if the lens consisted of water with the same surface slope. Thus a function is required which translates lens surface slope – which is known from the lens’ geometry – into the corresponding water surface slope. In the following chapters this function will be referred to as “water equivalent slope function” and its inverse as “inverse water equivalent slope function”. It is nonlinear and can be described after
follows: s = (s s y x
) ≈ n g n w
−
−
1
1
⋅ √
1 − ( x
R x
R
)
2
− ( y
R
)
2
+
[ n g
⋅ ( n g
−
1)
2
2 ⋅ n g
+ n w
⋅ n w
⋅ ( n w
⋅ ( n w
−
1)
2
] ⋅ ( n g
−
1)
2
−
1)
⋅
⎛
⎝
√
1 − ( x
R x
R
)
2
⎞
− ( y
R
)
2
⎠
3
(5.1)
Here n g
≈
1.515 and n w
≈
4
3 denote the refractive indices of BK7 glass and water at position on the lens relative to the centre of the lens surface.
is only an approximation of ⃗ expansion of the exact function up to third order. It shows that the xcomponent of water equivalent slope is not independent of the ycomponent of the position on the lens for higher slope values and vice versa. Therefore it has to be considered during the calibration process.
visualizes the equivalent slope of the water surface
4
BK7 glass has a refractive index of n g
≈
1.515 at the peak wavelength of the red LEDs at 625 nm
60
Calibration targets 5.6
(a) Lens float target with known slopes and sizes. Slopes are explained in
5 cm
36 cm
(b) Wavelet Target with known slopes. Schematic drawing modified after
[ 2008 ]. The numbers indicate the corresponding slope values.
0.00
0.25
0.50
0.75
1.25
1.00
1.25
1.00
0.75
0.50
0.00
0.25
side view top view
(c) Schematic drawing of a unit cell of the MTF target and photograph. Schematic drawing modified after
Rocholz [ 2008 ]. The numbers indicate the hole diameter and
the grid size and are given in mm.
10
10
3.9
1.0
3.3
10
8.2
0.82
0.56
10
2.7
0.68
2.2
Figure 5.9.: Calibration Targets for accuracy testing. Left hand side: Schematic drawing.
Right hand side: Photograph.
61
Chapter 5 EXPERIMENTAL SETUP
r
R air glass water
62
Figure 5.10.: Single lens of the lens float target with ray geometry. Image modified after
as a function of the position on a lens float lens as calculated from
As the lenses can be considered spherical the relation between the slope at the surface of the lens ⃗ the lens surface ⃗
′
= ( s x r = (x, y)
, s y
)
T
T is and the position on the lens relative to the centre of s
⃗
′
= ( tan(α
′ x
) tan(α
′y
)
) =
√
R
2
− ∣⃗
2
(5.2)
This can be plugged into
to obtain the water equivalent slope function: n g n w
−
1
−
1
⋅ ⃗
′
+
[ n g
⋅ ( n g
−
1)
2
2 ⋅ n g
+ n w
⋅ n w
⋅ ( n w
⋅ ( n w
−
1)
2
] ⋅ ( n g
−
1)
2
−
1)
⋅ ⃗
′
3
(5.3)
The lens float target is useful for calibration purposes because slope varies continuously in alongwind and crosswind direction. This way a single set of calibration images is sufficient to yield both components of the slope calibration function f (x, y).
A fundamental assumption made during the calibration process with the lens float target is that the calibration function f is sufficiently homogeneous, i.e. within
Calibration targets 5.6
0.41
0.33
0.25
0.17
0.08
0.00
0.17 0.33 0.50
x/R
0.8
0.6
0.4
0.2
0
0.41
0.33
0.25
0.17
0.08
0.00
0.17 0.33 0.50
x/R
0.8
0.6
0.4
0.2
Figure 5.11.: Equivalent slope of the water surface in x direction (left) and in y direction
(right) as a function of the position on a lens float lens (x,y) normalized with the radius of curvature R.
the spatial extent of a lens the position dependency of f is weak. This is true for a telecentric illumination which is set up correctly. A qualitative experimental analysis for the given setup is presented in
Experimental difficulties arise from the fact that the lens float target has to be perfectly balanced without any air bubbles trapped below. Secondly, the water body has to be completely at rest which requires some time of waiting. Furthermore, the centre of the lenses has to be determined precisely in order to ensure a correct calibration result from
. This is because the theoretical model for lens
surface slope and an intensity ratio image of a lens are matched in order to obtain the calibration function (see
5.6.2 Wavelet target
The wavelet target is made from acrylic and consists of a total of 11 planes with different slope values in one horizontal direction. In the perpendicular horizontal direction its slope is constantly zero. The wavelet target is depicted in
and a schematic drawing indicating the slope values of each plane is provided as well. The plexiglass planes have a finite thickness which causes a displacement of the light rays but no change of their inclination because the two bounding surfaces are parallel to each other. Because slope is changing in one direction only the wavelet target is harder to use for calibration purposes. For this work it is used for testing the calibration result only.
63
Chapter 5 EXPERIMENTAL SETUP
5.6.3 MTF target
The MTF target is a thin metal stencil containing a 10 mm ± 10 µm grid of holes with varying diameter. It was designed to measure the Modulation Transfer Function
(MTF) of the camera system. For this thesis it is solely used for calculating the footprint of the camera on the water surface and for determining the scaling factor between image coordinates in pixels and real world coordinates in millimetres. An illustration of a unit cell of the MTF target with the individual hole diameters is given in
as well as a closeup photograph.
64
6
Measurement Campaigns in the
Heidelberg Aeolotron
Two major measurement campaigns have been carried out at the Heidelberg Aeolotron for this thesis, one in 2013 with naturelike surface films and one in 2014 with clean water and Triton X100. In addition, some separate small experiments have been conducted in order to characterize the ISG setup.
6.1 2013 Aeolotron Measurements
The experiments in spring 2013 have been conducted in cooperation with Klaus
SchneiderZapp, School of Marine Science and Technology, University of Newcastle,
United Kingdom. This campaign was carried out in order to investigate the influence of naturally occurring surfactants on airsea gas exchange.
The ISG was used to determine mean square slope values as well as to investigate wave number spectra.
Simultaneously, gas exchange measurements for N2O and C2HF5 were conducted similar to
Krall [ 2013 ]. Using FourierTransformationInfrared (FTIR) spectroscopy
the gas concentration was measured in the air space and in the water space. Using a mass balance method, transfer velocities were determined in evasion measurements.
Schmidt number exponents were determined from both gases. Active thermography measurements with scanned infrared laser lines were performed by
order to determine heat transfer. Friction velocities were measured similar to
The measurements have been conducted at seven different wind speeds. In
the different conditions are summarized. Therein u re f is the reference wind speed as
65
Chapter 6 MEASUREMENT CAMPAIGNS IN THE HEIDELBERG AEOLOTRON it is measured in the Aeolotron (see
wind describes the frequency which is set at the frequency converter which drives the wind generators.
The time of measurement varies between the conditions and ranges from about 30 minutes for the highest wind speed up to more than two hours for the lowest wind speed. On each day, measurements started with the lowest wind speed. After each condition, the wind was turned off for a short while for thermography measurements of the drift velocity of the water surface. This time is assumed to be short enough in order to keep the water velocity at an (almost) constant level. Then the next wind speed was set and after 15 minutes of waiting time for the water body to reach its new equilibrium velocity all data acquisition was started again. More information on the wind speed during the experiment is given in
in the appendix.
The water surface was not skimmed before the experiment 1 . Further details of the
experimental conditions are provided in
The first two sets of measurements were conducted with the soluble surfactant dextran (see
). The first set was conducted with a concentration of
1 mg/l
∧
=
and the second set with 2 mg/l
∧
=
36 g total. For the third and fourth set of measurements, 7.2 mg of the soluble surfactant palmitic acid, 95 mg of the unsoluble glycolipid monogalactosylacylglycerol (MGDG) and 36 mg of the insoluble phospholipid phosphatidylglycerol (PG) were added to the water with the higher dextran concentration of 2 mg/l. This mixture was chosen to mimic naturally occurring surfactants and is referred to as full mix hereafter. The individual constituents are described in
ISG settings:
The camera lens which was used for the experiments is a Nikon
Micro Nikkor 105 mm, f =1:2.8 lens. The fstop was set to 5.6. For the first day and the first wind speed condition on the second day, ISG image sequences of 4 raw images each were taken at f = 6030 Hz with a delay of 1.496 s between the first images of two consecutive sequences. For all other wind speed conditions, sequences of 20 raw images each were taken with the same settings. At the end of each condition
except for the first condition of the first day 3 , a long sequence of 20000 images was
taken at f = 6030 Hz (see
for details).
66
1
The term skimming describes the process of cleaning the water surface to remove surface films. For that, a small barrier is mounted in the flume perpendicular to the direction of the main current and the wind such that its lower part touches the water surface. Then the wind is turned on at very low wind speed which leads to the water surface including any remaining surfactants and particles being pushed into a channel inside the barrier; leaving a clean water surface. The channel is continuously emptied by a pump.
2 per 18 000 l, which is one filling of the Aeolotron at the water height of 1 m as used for these experiments.
3
The long sequence for the first day of measurements is missing due to PC memory issues
2014 Aeolotron Measurements 6.2
Table 6.1.: Wind speeds and conditions for the 2013 Aeolotron campaign. Conditions where complete ISG data is available are marked with an x. For the conditions marked by
(x), the length of each image sequence was accidentally set to 4 raw images instead of the desired 20. For the conditions marked with a star
⋆
, the long sequence of 20000 raw images is missing.
Date f wind u re f
[
Hz]
[ m s
] surfactant water height
5
1.48
7
2.20
9
2.89
12
3.88
16
5.11
22
6.77
29
8.42
mg
30/04/2013 1 l
Dextran mg
03/05/2013 2 l
Dextran
08/05/2013 Full mix
10/05/2013 Full mix
100.4 cm
100.7 cm
99.7 cm
100.0 cm
( x)
(x) x x
⋆
(x) x x x
⋆
(x) x x x
(x) x x x
(x) x x x
(x) x x x
(x) x x x
6.2 2014 Aeolotron Measurements
The measurements in 2014 have not been evaluated during the scope of this thesis, but the experiments are described in detail for future reference.
6.2.1 Wave Field Equilibrium Measurements
The measurements in early 2014 have been conducted together with
determined friction velocities for a clean and for a filmcovered water surface (Triton
X100). ISG data is available for a wide range of wind speeds and was recorded in order to calculate mean square slope values for several different wind speeds at the given surfactant conditions. In
, the conditions are summarized. Therein,
the values for u re f are given for clean water according to
On 21/02/2014, the first set of measurements was conducted with clean water without surfactants. The remaining six sets of measurements were conducted with the soluble surfactant Triton X100 (see
) in two different concentrations.
For the first three days of measurements with surfactant (25/02/2014,26/02/2014 and 27/02/2014), a concentration of
0.6 g
18 m
3
∧
=
0.033 g/m
3 was used. The last three sets (28/02/2014, 02/03/2014 and 03/03/2014) were conducted with a concentration
3.0 g of
18 m
3
∧
=
0.167 g/m
3
. In order to conduct water height measurements, 5 g of the hydrophilic, pHsensitive fluorescent dye pyranine have been added to the water before starting the measurements on 25/02/2014 and 26/02/2014 each.
The time of measurement varies between the conditions and ranges from about
67
Chapter 6 MEASUREMENT CAMPAIGNS IN THE HEIDELBERG AEOLOTRON
10 minutes for the conditions measured on 26/02/2014 and 02/03/2014 up to more
than two and a half hours for the lowest wind speeds on the other days 4 . On the two
days just mentioned, a dense range of wind speeds was covered in order to obtain a coarse understanding of mean square slope values for many different wind speeds.
As a result, the waiting time between two ISG measurements had to be drastically reduced, thus sacrificing the fulfilment of wave equilibrium conditions. For the other days, the waiting time between two measurements have been sufficient to guarantee equilibrium conditions concerning the wave field, even for low wind speeds. In between the individual conditions on 26/02/2014 and 02/03/2014, the wind was shut
down for at least 15 min in order to allow for a decline of the wave field 5 . On these
days, the ISG measurement was started approximately 5 min to 10 min after the wind was started for each condition. On the other days, the wind was not shut down between the individual conditions and ISG measurements were started at about
after the new wind speed was set.
The water surface was skimmed before the first day of measurement with clean water in order to avoid surface contaminations.
ISG Settings:
For the experiments a Zeiss 100 mm, f =1:2 lens was used at an fstop of 8. Sequences of 20000 raw images were taken at f = 6000 Hz for each condition (see
for details).
6.2.2 Continuous Wind Speed Measurements
In between 26/02/2014 and 28/02/2014 additional ISG measurements have been performed where the frequency f wind at the frequency converter which drives the wind generation was changed continuously from 6 Hz to 22 Hz and back. This corresponds to wind speeds between u clean re f
=
1.86 m/s to 6.72 m/s. The frequency was changed with a rate of ∆ f wind
=
0.005 Hz/s. An overview is given in
One such measurement took approximately one hour. Sequences of 4 raw images were taken at f = 6000 Hz with a spacing of 500 ms between two sequences (see
and
for details).
These measurements were conducted with the same water, surfactant concentrations and camera lens as the equilibrium measurements described in
68
4
Note, that only one ISG measurement is conducted for each sequence due to the rather long time needed for data transfer and storage (approximately 30 min for a sequence of 20000 raw images).
5
Note that the mean water velocity will not reach zero during that time.
6
The exact time depends on the wind speed with more time to wait for an equilibrium state at low wind speeds.
2014 Aeolotron Measurements 6.2
Table 6.2.: Wind speeds and conditions for the 2014 Aeolotron campaign I. Conditions which were measured on the respective day are marked with an x.
f wind u re f
[
Hz]
[ s m
]
5
1.52
6
1.86
7
2.16
8
2.51
9
2.79
10
3.29
11

12
3.88
Date surfactant water height
21/02/2014 clean
25/02/2014 water
0.6
g
18 m
3
Triton
26/02/2014
27/02/2014
X100
0.6
g
18 m
3
Triton
X100
0.6
g
18 m
3
Triton
99.5 cm
99.8 cm
100.0 cm
99.8 cm x x x x x x x x x x x x
28/02/2014
02/03/2014
X100
3.0
g
18 m
3
Triton
X100
3.0
g
18 m
3
Triton
100.0 cm
100.0 cm x x x x x x x
03/03/2014
X100
3.0
g
18 m
3
Triton
100.0 cm x x
X100 f wind u re f
[
Hz] 13 14 15 16 17 18 19 20 21 22 29 36
[ m s
]
4.58
5.08
5.69
6.29
6.72
8.47
9.74
Date
21/02/2014
25/02/2014 x x
26/02/2014 x x x x x x x x x x
27/02/2014 x
28/02/2014 x
02/03/2014 x x x x x x x x x x
03/03/2014 x x x x x
69
Chapter 6 MEASUREMENT CAMPAIGNS IN THE HEIDELBERG AEOLOTRON
Table 6.3.: Wind speeds and conditions for the 2014 Aeolotron campaign II
Date
26/02/2014
27/02/2014
28/02/2014
28/02/2014
Condition
0.6
g/m
3
18
0.6
g/m
3
18
3.0
g/m
3
18
3.0
g/m
3
18
Triton X100
Triton X100
Triton X100
Triton X100
Wind Speeds
6 Hz → 22 Hz
1.86 m/s → 6.72 m/s
22 Hz → 6 Hz
6.72 m/s → 1.86 m/s
22 Hz → 6 Hz
6.72 m/s → 1.86 m/s
6 Hz → 22 Hz
1.86 m/s → 6.72 m/s
Wind Speed
Change
0.005 Hz/s
0.005 Hz/s
0.005 Hz/s
0.005 Hz/s
70
Wind Speed in the Aeolotron
6.3 Wind Speed in the Aeolotron
Due to the annular shape of the Aeolotron the wind profile does not exhibit the typical logarithmic shape. As shown in
Jähne [ 1980 ] this has almost no influence on
transfer processes but it makes the comparison with field data more complicated.
In the Aeolotron, the wind is varied by setting the frequency f wind at the frequency converter which drives the wind generators. Wind speed u re f is measured in the centre of the air side of the flume right below the ceiling using a hydrometric vane and a pitot tube as described in
Bopp [ 2014 ]. The wind speed u
re f at a given frequency f wind at the frequency converter varies depending on the amount of water, of surfactants, and on the amount and position of measurement devices in the flume.
In order to compare the wind speed with measurements in the field and in other facilities, the facilityspecific reference wind speed u re f is of little use. Instead, the friction velocity u
⋆ which is separately determined and converted into the value u
10 as described in
10 is the wind speed which were measured in ten meters height on the open ocean if the same friction velocity u
⋆ as in the
Aeolotron is assumed.
6.3.1 Other measurements
In addition to the large measurement campaigns mentioned before, several test measurements with different targets have been performed with the ISG. The images of the wavelet target evaluated in
were recorded on 12/06/2014 with a
Zeiss 100 mm, f =1:2 lens at an fstop of 8.
6.3
71
Part III.
Data Analysis & Discussion
73
7
Processing Routine
The raw images acquired with the ISG setup do not directly express any useful physical information. Several processing steps are necessary to extract this information from the raw images (see
During the first step the images of the lens float calibration target are used to obtain a lookup table (LUT) which allows for the conversion of measured intensity ratios in alongwind and crosswind directions into the corresponding surface slope values. This step will be called the calibration step.
In a second step, the slope calculation step, the raw images of the water surface are preprocessed. After that the LUT from the previous step is applied to the preprocessed data. This way images containing slope information for both directions are obtained.
In the third and last step, the analysis step, mean square slope, wave number energy spectra and other statistical information is extracted from the slope images of the previous step.
In the following chapter, the three steps are explained in detail. All calculations were done in MATLAB® R 2013b. The raw data is stored in raw format and the results of intermediate steps are stored in raw format or in HDF5 format.
75
Chapter 7 PROCESSING ROUTINE
Step 1  Calibration Step:
Calculate Averaged Lookup Tables from Lens Float Raw Images
Step 2  Slope Calculation Step:
Calculate Slope Images from Raw Data
Step 3  Analysis Step:
Calculate Mean Square Slope and Wave Number Power Spectra and Saturation Spectra from Slope Images
Figure 7.1.: Overview of the ISG data processing routine.
76
First Step: Calibration 7.1
7.1 First Step: Calibration
1 (Dark)
Camera Dark Image
Step 1: Calibration Step
P f
· Q f
· (A,B,C,D)
Lens Float Raw Images
Q
0
· (A
0
,B
0
,C
0
,D
0
)
Zero Slope Raw Images
(A
0,mean
,B
0,mean
,C
0,mean
,D
0,mean
,)
Averaged Zero Slope Images
(Nx mean
, Ny ) mean
Normalized Averaged Zero Slope Images
P f
· Q f
· (X,Y)
Normalized Lens Float Images
R f
· (X lens
,Y lens
)
Single Lens Images
R f
· (XLUT, YLUT)
Lookup Tables
(XLUT mean
, YLUT mean
)
Averaged Lookup Tables
Figure 7.2.: Processing routine step 1: Calibration step. (A,B,C,D) refers to the illumination wedges as described in
. The indices 0 and f indicate zero slope and
lens float, respectively. Q indicates the number of subsequences (A,B,C,D) per image sequence and P indicates the number of image sequences.
The calibration data consists of one dark image of the camera taken on 30/04/2014 as depicted in
, one sequence of zero slope images (A
0
, B
0
, C
0
, D
0
) of length Q
0 for each day of measurement as shown in
and P f sequences of lens float images (A, B, C, D) of length Q f each as given in
(A,B,C,D) refer to the respective illumination wedges as described in
and the indices 0 and f indicate zero slope and lens float, respectively.
77
Chapter 7 PROCESSING ROUTINE
depicts all steps which are necessary in order to prepare the calibration data in a preprocessing step and to produce an empirical calibration “function” in the form of averaged lookup tables from the calibration data. The lookup tables are explained in
78
Figure 7.3.: Camera Dark Image
Dark.
7.1.1 Preprocessing
The camera installed at the ISG setup provides intensity images. In order to calculate slope images from which spectra can be deduced the raw data of the experiment as well as the calibration data has to undergo some preprocessing steps first.
Dark Image and Zero Images
The zero slope images (A
0
, B
0
, C
0
, D
0
) are averaged over sequences of length Q
0 right after acquisition. The raw data is not saved, but the averaged zero slope images
(
A
0,mean
, B
0,mean
, C
0,mean
, D
0,mean
) are saved in tif format. One averaged image is saved for each illumination wedge. The lengths Q
0 are given in
for each day of measurement.
shows the averaged zero slope images. Next, these are normalized as follows:
Nx mean
Ny mean
=
=
A
0,mean
A
0,mean
C
0,mean
−
B
0,mean
+
B
0,mean
−
D
−
2 ⋅ Dark
0,mean
C
0,mean
+
D
0,mean
−
2 ⋅ Dark
(7.1)
(7.2)
The resulting images Nx mean and N y mean represent the intensity ratio for zero slope in alongwind and crosswind direction, respectively. As can be seen in
the normalization process suppresses disturbing features in the raw images such as
First Step: Calibration 7.1
defects in the window or dirt on the water surface, the light source or the optical components which are visible as darker areas in the raw images.
Table 7.1.: Number of images used to average the zero slope images for each day of measurement. When two values are given, data with both lengths exists for that day.
Date # of images Q
0
⋅
4
30/04/2013 5 ⋅ 4 = 20
03/05/2013 250 ⋅ 4 = 1000
08/05/2013 1508 ⋅ 4 = 6032
10/05/2013 5 ⋅ 4 = 20
21/02/2014 5 ⋅ 4 = 20
25/02/2014 5 ⋅ 4 = 20 or 50 ⋅ 4 = 200
26/02/2014 5 ⋅ 4 = 20
27/02/2014 5 ⋅ 4 = 20 or 50 ⋅ 4 = 200
28/02/2014 5 ⋅ 4 = 20
02/03/2014 50 ⋅ 4 = 200
03/03/2014 5 ⋅ 4 = 20
12/06/2014 50 ⋅ 4 = 200
Lens Float Target Images
Each day a total of P f
image sequences of the lens float target are recorded. Each image sequence consists of a multiple of subsequences of 4 images with different illumination wedges (A,B,C,D) each. Example images are depicted in
For each of these Q f subsequences one intensity ratio image X of the component in alongwind direction x is calculated from A and B and one intensity ratio image Y of the component in crosswind direction y is calculated from C and D. This is done similarly as for the zero slope images according to the following equations:
X =
Y =
A − B
A + B − 2 ⋅ Dark
C − D
C + D − 2 ⋅ Dark
−
Nx mean
−
Ny mean
(7.3)
(7.4)
The result after normalization is shown in
and will be referred to as normalized lens float (intensity) images (X,Y). The main advantages of this so called ratio imaging are that it corrects for lens effects at the water surface as well as that it reduces the demands to the homogeneity of the light source. This is demonstrated
1
Remember that the index f labels the image sequence as lens float target data.
79
Chapter 7 PROCESSING ROUTINE when comparing
and
. The individual LEDs of the light source
are visible in the raw images although a diffusion screen is used to homogenize the light source. In the normalized images, they are not visible any more.
80
First Step: Calibration
(a) Averaged Zero Slope Data. Left hand side: illumination wedges
A
0,mean upwind direction x. Right hand side: illumination wedges
C
0,mean
, D
, B
0,mean
0,mean in crosswind direction y.
in
200
400
600
200
400
600
7.1
200 400 600 800 position [px]
200 400 600 800 position [px]
1,000 1,100 1,200 intensity [gray value]
1,200 1,300 1,400 intensity [gray value]
200
400
600
200
400
600
200 400 600 800 position [px]
200 400 600 800 position [px]
1,000 1,100 1,200 intensity [gray value]
1,000 1,100 1,200 1,300 intensity [gray value]
(b) Normalized Averaged Zero Slope Data. Left hand side: alongwind component
Nx mean
. Right hand side: crosswind component
Ny mean
.
200
400
600
200
400
600
200 400 600 800 position [px]
200 400 600 800 position [px]
−
1 0 1 2 norm. intensity []
⋅
10
−
2
4 6 norm. intensity []
8
⋅
10
−
2
Figure 7.4.: Zero slope images. Averaged raw data and normalized images. (data of
30/04/2013)
81
Chapter 7 PROCESSING ROUTINE
(a) Raw data. Left hand side: illumination wedges in upwind direction x. Right hand side: illumination wedges in crosswind direction y.
200
400
600
200
400
600
200 400 600 800 position [px]
200 400 600 800 position [px]
0 500 1,000 1,500 intensity [gray value]
0 500 1,000 1,500 intensity [gray value]
200
400
600
200
400
600
200 400 600 800 position [px]
200 400 600 800 position [px]
0 500 1,000 1,500 intensity [gray value]
0 500 1,000 1,500 intensity [gray value]
(b) Normalized Images. Left hand side: upwind direction x. Right hand side: crosswind direction y.
200
400
600
200
400
600
200 400 600 800 position [px]
200 400 600 800 position [px]
82
−
0.5
0 0.5
normalized intensity []
−
0.5
0 0.5
normalized intensity []
Figure 7.5.: Lens detection process: Raw data and normalized images (data of 30/04/2013).
First Step: Calibration
7.1.2 Lookup Tables
After data preprocessing, a lookup table is created from the normalized lens float target calibration images which links calculated slope data ⃗ x with measured intensity ratios x
, I y
]
T
.
, s y
]
T
7.1
Lens Detection
As a first step, the separate lenses have to be detected in the lens float calibration images. This is implemented with the MATLAB® function imfindcircles using a twostage circular Hough transform with a sensitivity factor of 0.97 and a fixed edge gradient threshold of 0.01.
Exemplary images of several stages of this process are shown in
The circle detection is performed on a binary representation of a gradient image of the first illumina
tion wedge A of a 4image sequence 2
Therefore the intensity threshold to obtain the binary representation is set to 200. The result after circle detection is shown in
An additional step is the automatic validation of the detected lenses where only those lenses are kept which lie entirely inside the image domain. The result of this step is depicted in
The circle detection is rather errorprone which is why a manual selection of correctly detected lenses is performed afterwards. Examples of misdetected lenses are given in
tracted lens images are discarded because they depict a lens of the lens float target which is glued such that it exhibits an irregular shape at the outside (see
and
ages do not have to be discarded because the center of the lens is determined correctly but it is done here for additional accuracy at the outside of the lenses.
Secondly, water droplets on top of the lenses or air bubbles below the lenses due to experimental inac
fails for some images (c.p.
2 after correction for the camera dark image
200
400
600
200
400
600
200
400
600
(a) Gradient Image
200
200
400
400
600
(b) Detected Lenses
600
(c) Validated Lenses
800
800
200 400 600 800
Figure 7.6.: Lens Detection
Process (data of
30/04/2013)
83
Chapter 7 PROCESSING ROUTINE occurs very rarely; for the data of 30/04/2013 only two such cases were detected out of a total of 400 individual lens images as the output of the automatic validation step.
After correction for zero slope images the detected lenses are stored in HDF5 format together with a flag indicating the result of the manual selection process. The position of the lens in the calibration image is stored as well.
For the 2013 Aeolotron campaign data and 12/06/2014,
gives the number of lenses which were detected and validated by the algorithm and the number of lenses which were selected by hand for LUT computation.
Table 7.2.: Number
R f of lens float target lens image pairs (
X mean lookup table pair (
X − LUT mean
, Y − LU T mean l ens
, Y l ens
) used to create a
) for each day of the 2013
Aeolotron campaign
Date
30/04/2013
03/05/2013
08/05/2013
10/05/2013
12/06/2014
# of lenses detected # of valid lenses
400
2916
638
606
1659
271
2013
256
255
802
84
First Step: Calibration 7.1
(a) Wrong position
100
200
100 position [px]
200
−
0.5
0 normalized intensity []
(b) Air bubble or water droplets
100
200
100 position [px]
200
−
0.5
0 0.5
normalized intensity []
(c) Badly shaped lens
100 100
200 200
100 position [px]
200 100 position [px]
200
−
0.5
0 0.5
normalized intensity []
−
0.5
0 normalized intensity []
Figure 7.7.: Examples of misdetected lenses (data of 30/04/2013).
0.5
85
Chapter 7 PROCESSING ROUTINE
LUT calculation
In a first step, a lookup table pair (XLUT, YLUT) is calculated for each of the
R f valid lens image pairs (X lens
, Y lens
) from the previous step separately. In a second step, these are averaged in order to obtain a single mean lookup table pair
(
XLUT mean
, YLUT mean
) for each day of measurement.
(
X
For the first step, all pixels outside the lens are set to NaN in the lens image pairs lens , Y lens
)
. Also, the theoretical slope value expected at each pixel according to
is calculated and translated into water equivalent slope via
Then an integervalued LUT index pair (LUTindex x
, LUTindex y
) in the interval
(0,2000) is calculated for each pixel of the lens image pairs according to
(
LUTindex x
, LUTindex y
) = ( floor((X lens
+
1) ⋅ 1000), floor((Y lens
+
1) ⋅ 1000)) .
(7.5)
Now each pixel (m, n) is associated with a slope value pair (sx theor
, s y theor
)
, a normalized intensity value pair (I x
, I y
) as well as a LUT index pair (LUTindex x
, LUTindex y
)
.
Next, a LUT pair (
XLUT, YLUT) is calculated by looping through all pixels (m, n) of a LUT index image LUTindex x/y and assigning the corresponding slope component sx theor
( m, n) to the respective LUT image pixel (i, j) which is determined by the LUT indices at position (m,n):
XLUT(i, j) =XLUT (LUTindex x
( m, n), LUTindex y
( m, n)) = sx theor
( m, n)
YLUT(i, j) =YLUT (LUTindex x
( m, n), LUTindex y
( m, n)) = sy theor
( m, n) (7.6)
When a value (
LUTindex x
( m, n), LUTindex y
( m, n)) occurs more than once the respective values are averaged.
As mentioned before, the second step comprises of averaging the LUT pairs to obtain one mean LUT pair (
XLUT mean
, YLUT mean
)
. After that missing values
(
XLUT mean
, YLUT mean
)
.
shows the LUT pair right after averaging and
gives the final LUT pair after interpolation and averaging.
86
3 using the Matlab® function scatteredInterpolant
4
Matlab® function medfilt2
5
Matlab® function imfilter
First Step: Calibration 7.1
500
1,000
(a) Averaged LUT pair XLUT mean
(left) and YLUT mean
(right)
0 0
500
1,000
1,500
2,000
0 500 1,000 1,500 2,000
LUTindex x
1,500
2,000
0 500 1,000 1,500 2,000
LUTindex x
500 1,000 1,500 unscaled slope []
500 1,000 1,500 unscaled slope []
(b) Smoothed and interpolated LUT pair XLUT mean
(left) and YLUT mean
(right).
0 0
500
1,000
500
1,000
1,500
2,000
0 500 1,000 1,500 2,000
LUTindex x
1,500
2,000
0 500 1,000 1,500 2,000
LUTindex x
−
0.5
0 0.5
slope []
Figure 7.8.: LUT of 30/04/2013.
−
0.5
0 0.5
slope []
87
Chapter 7 PROCESSING ROUTINE
7.2 Second Step: Slope Calculation
1 (Dark)
Camera Dark Image
Step 2: Slope Calculation Step
P· Q · (A,B,C,D)
Raw Images
(Nx mean
, Ny mean
)
Averaged Normalized Zero Slope Images
P · Q · (X,Y)
Normalized Images
(XLUT mean
, YLUT mean
)
Averaged Lookup Tables
P · Q · (S x
, S y
)
Slope Images
Figure 7.9.: Processing routine step 2: calculating slope images from raw data.
In order to calculate slope images from raw data (A, B, C, D) the LUT pair
(
XLUT mean
, YLUT mean
) created in the previous step is applied to the normalized image pairs (X, Y ). The normalization process is described in
malized images are scaled onto the interval (0,2000) similar to
to obtain two LUT indices. Next, a linear LUT index image is calculated according to index = round(Y) ⋅ 2000 + round(X).
(7.7)
Then the two slope images are calculated from
S x
S y
=
XLUT mean
( index)
=
YLUT mean
( index).
Finally, these images are saved in raw format with single precision.
(7.8)
88
Third Step: Analysis
7.3 Third Step: Analysis
In
the steps necessary to calculate the desired mean square slope values and the saturation spectra from the slope images are visualized.
P · Q · (S x
, S y
)
Slope Images
Step 3: Analysis Step
or
P · (Mean x
, Mean y
)
Sequencewise Mean Slope Components
(Total_Mean x
, Total_Mean y
)
Total Mean Slope Components
P · Q · (iMSS x
, iMSS y
)
Imagewise Mean Square Slope Components
P · Q · (iMSS)
Imagewise Total Mean Square Slope
7.3
(MSS x
, MSS y
)
Averaged Mean Square Slope Components
P · Q · (S x
, S y
)
Slope Images
(MSS)
Averaged Total Mean Square Slope
P · Q · (S(k x
, k y
))
2D Wave Number Power Spectra
(S mean
(k x
, k y
))
Averaged 2D
Wave Number Power Spectrum
P · Q · B(k x
,k y
)
2D Saturation Spectra
(B mean
(k x
,k y
))
Averaged 2D Saturation Spectra
Figure 7.10.: Processing routine step 3: calculating mean square slope and spectra from slope images.
Mean Square Slope Calculation
In order to calculate mean square slope (mss) from the P sequences of slope image pairs (S x
, S y
) p,q of length Q each the following procedure is followed:
• A mean slope pair (mean x
, mean y
) p is calculated from each sequence by averaging (S x
, S y
) p,q along the temporal and spatial dimensions:
( mean x , mean y
) p
=
1 m ⋅ n ⋅ q
∑ m,n,q
(
S x , S y
) m,n,p,q .
(7.9)
Recall that m ∈ [1, M] and n ∈ [1, N] are indices which indicate the pixel position (rows and columns).
89
Chapter 7 PROCESSING ROUTINE
• Next, from each image pair (S x , S y
) p,q an imagewise mean square slope pair
( iMSS x
, iMSS y
) p,q is calculated according to the following equation: iMSS x/y,p,q
=
1
M ⋅ N
M
∑ m=1
N
∑ n=1
(
S m,n x/y,p,q
− mean x/y,p
⋅
1)
2
.
(7.10)
Here, the square is to be understood as a pixelwise matrix multiplication in the
sense of the Hadamard product 6
and 1 represents a matrix of ones the same size as S x and S y .
• Next, one way of continuing the calculations is to average the mean square slope components over all images of all sequences:
MSS x/y
=
Q
2
P
∑ q=1
∑ p=1
(
2
Q
⋅
1
P
⋅ iMSS x/y,p,q
)
.
(7.11)
• Finally, the total mean square slope value is calculated as the sum of both components:
MSS = MSS x
+
MSS y .
(7.12)
• Alternatively, the last two steps can be calculated in reverse order. Then a total mean square slope value iMSS p,q
= iMSS x,p,q
+ iMSS y,p,q (7.13) is calculated from each mean square slope pair (iMSS x
, iMSS y
) p,q first with the averaging being conducted in the last step:
(
MSS) =
Q
2
∑ q=1
P
∑ p=1
(
2
Q
⋅
1
P
⋅ iMSS p,q
)
.
(7.14)
Saturation Spectra
The basic idea behind calculating saturation spectra is fairly simple and consists of applying the discrete Fourier Transform (DFT) to the slope image data. Special care has to be taken of the correct normalization of the spectra. For this work, saturation spectra are calculated from the N sequences of slope image pairs (S x
, S y
) p,q of length q max
=
Q
2 each according to the following routine which is based on
6
For two matrices A, B of the same dimension m × n the Hadamard product A ⋆ B is the matrix of the same dimension as the operands with elements given by (A ⋆ B) i, j
= (
A) i, j
⋅ (
B) i, j .
90
Third Step: Analysis
1
7.3
0.8
200
400
600
0.6
0.4
0.2
0
100 200 300 400 500 600 700 800 900 position [px]
Figure 7.11.: 2D Hann window.
Wave number power spectra
• First, wave number power spectra S
x
, S
y
are calculated by applying a raised
to the slope image pairs (S x , S y
) p,q , then applying the 2DDFT and finally taking the square of the absolute value of the transformed images.
Next the resulting spectra are averaged over all k images of a sequence and multiplied with a factor of f which occurs because of the energy loss due to windowing (see
). The whole step can be written as follows:
S x
[
W ⋆ S x
]( k x
, k y
) ∶=
∶= f ⋅
Q
2
∑ q=1
M
∣
∑ m=1
N
∑ n=1
(
W ⋆ S f ⋅ ˜S
x
[
W ⋆ S x
]( k x
, k y
)
) ⋅ exp (−2πi ⋅ [
( k x
−
1)(m − 1)
M
+
( k y −
1)(n − 1)
N
])∣
2
(7.15) where ⋆ denotes the Hamard product of two matrices, W is the raised cosine window as depicted in
W = w(M) ⋅ w(N) T
=
0.5 (1 − cos ( and 1 ≤ k x
≤
M, 1 ≤ k y
≤
N.
2π⋅(0∶M−1)
(
M−1)
)) ⋅
0.5 (1 − cos (
2π⋅(0∶(N−1))
T
N−1
))
• The factor f equals
64
. This is derived in
9
• Finally, the total wave number power spectrum components are the sum of
7 also called Hann window, Hanning window or von Hann window; see
91
Chapter 7 PROCESSING ROUTINE the two components: S(k x , k y
) ∶=
S
tot
( k x , k y
) =
S x
( k x , k y
) +
S y
( k x , k y
)
• Special care has to be taken concerning the correct normalization of the spectra. The total wave number power spectrum S(k x , k y
) obtained in the last step is normalized with a factor g =
1
M
2
N
2
∆k x
∆k y which is described in
[ Rocholz , 2008 , Chapter 6.7] leading to the correctly normalized spectrum
S
norm
( k x , k y
) = g ⋅ S(k x , k y
)
.
Logpolar spectra
• From the wave number power spectra on a Cartesian grid logpolar spectra are calculated because of their constant relative wavenumber resolution. For this, the wave number power spectrum S norm
( k x
, k y
)
onto a logpolar grid using an approach based on
in a logpolar wave number power spectrum S(log(k), θ). The two grids are depicted in
Kiefhaber [ 2014 ] the drawback of this
representation is that energy conservation is not guaranteed for small wave numbers.
k y k x
92
Figure 7.12.: Conversion from a Cartesian grid (black) to a logpolar (red) grid: Close to the origin, the resolution of the logpolar grid is much higher than that of the
Cartesian grid. Far away from the origin, the reverse is true. Image taken from
8 using the Matlab® function scatteredInterpolant
Third Step: Analysis
Omnidirectional spectra
• Next, the logpolar wave number power spectrum S(log(k), θ) is integrated over all directions θ in order to obtain the omnidirectional wave number power spectrum S(log(k)).
Omnidirectional saturation spectra
• Finally, the omnidirectional wave number power spectrum
S(log(k)) is multiplied with k
2 which leads to the omnidirectional saturation spectrum
B(log(k)) = k
2
⋅
S(log(k)).
(7.16)
7.3
93
8
Characterization of the Setup
Before the accuracy of the setup is explored it is of interest to examine its limitations.
Therefore the temporal and spatial resolution is analysed in
explores the detection limits concerning mean square slope values. After that, the accuracy of the setup is explored in
8.1 Determination of the Frame Rate
In order to resolve all wave numbers and frequencies (which fit into the image section) in the 3D spectrum B( certain frame rate (or sampling frequency) is necessary. The system is capable of measuring at f system−limit
=
6030 Hz. As 4 raw images have to be taken and offset against each other, the effective temporal resolution is lowered by a factor of 4 and amounts to f e f f system−limit
=
1507.5 Hz. The smallest structures that can be resolved by the measurement system are limited by the spatial resolution of the camera. Let
(
∆x, ∆y) be the size of the corresponding pixel spacing on the water surface. Then the smallest wavelengths which can be resolved are
λ min,x
=
2∆x
λ min,y
=
2∆y.
(
∆x, ∆y) can be calculated from the footprint (X, Y ) of the camera image on the water surface by dividing it by the amount of pixels in the corresponding direction. For the 2013 measurements, (X, Y ) = (203 mm, 166 mm) and for the 2014 measurements (X, Y ) = (227 mm, 182 mm).
For gravitycapillary waves it is known that the square of the phase velocity c phase
95
Chapter 8 CHARACTERIZATION OF THE SETUP is given by c
2 phase
=
ω
2 k
2
= g k
+ k ⋅ σ
ρ
(8.1) where in the case of waves at the interface of air and (pure) water σ ≈ 0.0727 N/m at 20
○
C and ρ ≈ 1 × 10
3 kg/m
3
.
Resolving this for ω yields
ω(k) =
√ g ⋅ k + k
3
⋅
σ
ρ
.
(8.2)
From this the minimum frequency which is necessary to follow the fastest waves is calculated according to f limit
=
2 ⋅
ω max
2 ⋅ π
=
4
π
¿
⋅
Á
À g ⋅
2 ⋅ π
λ min
+
(
λ
2⋅π min
ρ
)
3
⋅
σ
.
(8.3)
Furthermore, the maximum wave number which is still resolvable by the given system is obtained from k max,x
=
2π
λ min,x
π
=
∆x k max,y
=
λ
2π min,y
=
π
∆y
.
This yields:
Exp.
∆x ∆y k max,x k max,y f limit,x f limit,y
2013
0.203 m
960 px
0.166 m
768 px
14 857 rad/m 14 535 rad/m 19 706 Hz 19 068 Hz
2014
0.227 m
960 px
0.182 m
768 px
13 286 rad/m 13 257 rad/m 16 666 Hz 16 611 Hz
The limiting frequencies are larger than the effective frequency by a factor of 11 to
13. Nevertheless, this is of no concern for the desired measurements because waves with wave numbers which correspond to these frequencies do not exist at all.
[ 1994 ] found that the highest wave numbers which can occur are about 6000 rad/m
because above that limit, the effect of viscous damping, which scales with k
2
, becomes too large. Experiments by
suggest a maximum wave number in the vicinity of 1000 rad/m.
argues that this might be an artefact arising from data recording and evaluation and the limit found by
Apel [ 1994 ] is the correct physical quantity.
k f
Plugging k limit,exp
=
1000 rad/m into
yields a frequency limit of limit,exp limit,theo
=
=
366.4 Hz which is much lower than the system limit of the ISG. With
6000 rad/m the frequency limit is located at f limit,exp
=
5065.3 Hz. With the given effective frame rate of the ISG, waves with a maximum wave number of
96
Detection Limits
2660 rad/m can be resolved in the spectra without aliasing effects. Since the spectral energy close to 6000 rad/m is smaller than that at small wave numbers by several orders of magnitude the effects of the resulting aliasing are rather small.
Thus, in contrast to the one used by
Rocholz [ 2008 ], the current ISG setup allows
for measurements of wave spectra without or at least with small spatial and temporal aliasing effects.
8.2 Detection Limits
In order to analyse the detection limit for slope and mean square slope of the ISG
setup, 10 zero image sequences of length 20 each 1
were taken on 30/04/2013, normalized, converted to slope and then averaged pixelwise. Afterwards the variance was calculated for each pixel and averaged over the whole image. This yields the minimum value of mean square slope which can be measured with the given setup.
The reason for this is that the noise level acts as an additive offset on the physical contribution to mean square slope from the waves because variances are additive quantities. The value resulting from the calculations mentioned is σ
2 min
=
3.2189⋅10
−
(corresponding to a statistical slope measurement error of ∆s rms
=
0.018) which is a
major improvement compared to the detection limit of the CISG setup of 2011 which is in the order of σ
2 min,CISG
=
0.0025 (see
Kiefhaber [ 2014 ]). Yet one should bear in
mind that, in a strict sense, only values much larger than the given limit represent the “real” physical value of mean square slope.
The maximum slope value that can be detected with the current setup and calibration method is given by the value at the border of a lens float lens (see
for a description of the calibration process and
for a description of the lens float target). It is s max
= ±
0.9648 after conversion to water equivalent slope. Absolute values of slope being larger than this limit are treated as NaN values and ignored for further evaluation leading to an underestimation of mean square slope when high slopes are present. In order to extend the range of slopes to be covered an extrapolation of the LUT is pursued. Using the MATLAB® function scatteredInterpolant two different extrapolation methods are examined on the alongwind component of the averaged LUT X − LU T mean of 30/04/2013. The results are presented in
can clearly be seen that these extrapolation methods do not yield sufficient results as linearity is not preserved outside the known slope range. Thus extrapolation is not used for data evaluation for this thesis. If a target with higher slope ranges were used during the calibration process, the slope range which can be covered would ultimately be limited by the size of the light source.
8.2
1
As described in
this corresponds to a sequence length of Q=5 normalized intensity ratio images.
2
The individual components are given by σ
2 min,x
=
2.7685 ⋅ 10
−
4 and σ
2 min,y
=
4.5038 ⋅ 10
−
5
97
Chapter 8 CHARACTERIZATION OF THE SETUP
(a) Linear method.
0
500
1,000
1,500
2,000
0 500 1,000 1,500
LUT index in x direction
(b) Nearest Neighbour method.
2,000
0
500
0.5
1
2
0
−
1
−
2
−
3
1,000
0
1,500
−
0.5
2,000
0 500 1,000 1,500
LUT index in x direction
2,000
Figure 8.1.: Extrapolated LUT of 30/04/2013. Comparison of linear extrapolation and nearest neighbour extrapolation.
98
Accuracy of the ISG setup
8.3 Accuracy of the ISG setup
The accuracy of the ISG setup is tested with specified calibration targets with known properties. The targets are described in
8.3.1 Lens Float Target
shows an image of the xcomponent of the reconstructed slope of the lens float target with slope values converted to water equivalent slope. Recalling the
calibration process ( section 7.1.2
and
) the LUT applied to the normalized
lens float data is obtained from averaging the LUTs of many 3
single lenses detected in all lens float images of the same day. The individual LUTs are calculated from linking the normalized intensity ratios of a lens float target lens with the corresponding slope values which are known from the geometry.
In
a profile through the centre of
in vertical direction is depicted and
shows a profile in horizontal direction.
The profile in vertical direction is expected to be constant because the same horizontal slope component is located at the same horizontal distance from the lens centre in each lens. Deviations arise because
• the lens float target is not orientated perfectly parallel to the image axes. This implies different slope values because the profile is taken at changing horizontal distance from the lens centres.
• the position of the light source is not orientated perfectly parallel to the image axes. This way the coordinate system of the image is rotated compared to that of the light source which results in isolines of constant slope being rotated, too.
• a possible positiondependency of the calibration function due to nonideal behaviour of the optical components which has not been considered. This will be addressed in
(
The “jumps” in between the lenses which are visible in
are about 0.06
∧
=
3 % of the total slope range) which can be explained as follows: although the slope component in x direction remains constant the slope component in y direction is switching signs at a rather large absolute value of slope. The approximation made in
leads to larger errors in the reconstructed slope then because the two slope components cannot be considered independently.
The profile in horizontal direction ( Figure 8.2c
) is expected to be the same for all
lenses, but it is not expected to be constant but to follow the green line in
which is the theoretical expectation calculated from the lens geometry according
3
As described in
typically more than 250 individual lens images are detected and used.
8.3
99
Chapter 8 CHARACTERIZATION OF THE SETUP
200
0.5
0
200
400
400
0
600
200 400 600 800 position [px]
(a) Slope component in x direction. The red lines indicate the locations of the profiles.
1
0.5
−
0.5
600
800
−
0.2 0 0.2 0.4
water equivalent slope []
(b) Profile in vertical direction.
0
−
0.5
Profile maximum value for r=25mm maximum value for r=24mm theoretical expectation
−
1
0 200 400 600 800 1,000 position [px]
(c) Profile in horizontal direction.
Figure 8.2.: Resconstructed lens float target slope in units of water equivalent slope. LUT of
12/06/2014 applied to data of 12/06/2014.
to
and
. From these equations the maximum value to be
expected for both components of water equivalent slope is calculated as s = 0.96 for a lens radius of r = 25 mm (red line in
) and as s = 0.90 if we assume a lens
radius of r = 24 mm (black line). The two reference values are given here because of the uncertainties concerning the correct value of the visible lens radius which arise from the construction of the lens float target (see
and
values obtained by applying the LUT calculated from all lens float target images of the same day are in good agreement with that.
shows the deviation of the horizontal profile of the central lens from the theoretical expectation (green line in
100
Accuracy of the ISG setup 8.3
the deviation along the profile is σ = 0.0182 which corresponds to 0.91 % of the total slope range [−1, 1]. Furthermore, a trend is clearly visible in the figure. The absolute value of the deviation from the theoretical expectation becomes higher with growing distance to the lens centre. This might be explained with the distancedependent inaccuracy of the approximation made in
when translating lens slope in water equivalent slope.
0.040
0.020
0.000
−
0.020
−
0.040
400
Deviation from theoretical expectation
Standard deviation σ = 0.0182
450 500 position [px]
550
Figure 8.3.: Resconstructed lens float target slope. LUT of 12/06/2014 applied to data of
12/06/2014. Deviation from theoretical expectation.
8.3.2 Measurement of the Lens Float Target
The calibration process (see
) is based on matching the theoretically ex
pected slope of the lens float target with an intensity ratio image of the target. Also the ISG footprint on the water surface is determined by determining the size of a lens in the image and comparing this to the size of a lens in reality. Thus, the accuracy of the calibration process depends on a correct determination of the lens radius. The lenses of the lens float target are glued into the float. This way, the visible diameter of the lenses might differ slightly from the 50 mm ± 0.1 mm which are expected as given by the lens specifications. Thus, a calliper is used in order to validate the radius of the visible part of each of the lenses. The measured diameters are given in
One of the lenses has been repaired leading to an irregular shape of the visible part of it due to excessive glue. It was excluded from all calculations. Thus, no diameter is given for this lens.
101
Chapter 8 CHARACTERIZATION OF THE SETUP
48.9
mm
48.1
mm
48.5
mm
49.1
mm
49.0
mm
48.6
mm
48.2
mm
48.8
mm
48.8
mm
49.0
mm
49.0
mm
47.8
mm
49.0
mm
49.0
mm
48.5
mm
48.5
mm
49.0
mm
48.8
mm
48.7
mm
48.9
mm
47.9
mm
49.1
mm
47.7
mm
Figure 8.4.: Results of measuring the lens float target with a calliper. View from the bottom.
The numbers indicate the visible diameter of the lenses in mm. Lenses marked in red are affected by shadowing effects due to their mounting.
102
From the measured diameters, a mean radius is calculated to be r = 24.33 mm ± (0.05 mm) stat
± (
0.2 mm) sys
This is a significant deviation from the expected radius of r ref
=
25 mm. Yet it might still be possible to actually depict the entire lens due to light being refracted at the lens surface: All light rays which reach the camera sensor leave the lens in vertical direction (see
). Consider the trace of a light ray which leaves the lens
right at its border (marked in green in
). If it has been refracted such that
it is not obstructed by the lens mounting and the glue then the camera image will still depict the entire lens without shadowing effects. In the following the minimum open diameter d min of mounting and glue is calculated which does not lead to an obstruction of the light rays at the border of the lens. The calculations were proposed by
Balschbach [ 2014 ]. For simplicity a 2D case is considered. The lens surface slope
at the border of the lens (i.e. at a distance r max=25 mm from the centre) is given by
as tan α = r max
√
R
2
− r
2 max
.
Furthermore Snell’s law yields: sin(β) = sin(α) ⋅ n air n
BK7
.
(8.4)
Accuracy of the ISG setup 8.3
Simple geometric considerations lead to the following calculation: z = tan(γ) ⋅ ET
= tan(α − β) ⋅ ET
⎛
= tan
⎜
⎜
⎜ arctan
⎛
⎝ r max
√
R
2
− r
2 max
⎝
⎞
⎠
− arcsin
⎛ sin (arctan (
⎜
⎜
⎜ n
BK7
⎝
√
R r max
2
− r
2
)) ⋅ n air
⎞ ⎞
⎟
⎟
⎟
⎟
⎟
⎟
⋅
ET
⎠ ⎠
(8.5) where ET = 3.55 mm is the edge thickness of the lenses as given by the manufacturer.
Thus an open diameter of the mounting and glue of at least d min
=
2 ⋅ r max
−
2 ⋅ z =
48.35 mm would not have any effect on the visible lens diameter. This corresponds to a radius of r min
= d min
2
=
24.175 mm. Five of the lenses of the lens float target have a smaller visible diameter and are thus affected by shadowing effects. This leads to the conclusion that calibration accuracy could be increased by choosing a different target for determining the optical resolution of the ISG. The MTF target is a good candidate its dimensions are wellknown and there is no additional refraction occuring at the water surface. Another alternative is the use of a chessboard pattern.
r
max
R
γ
α n
BK7
n
air
β
α
ET z
Figure 8.5.: Ray geometry at the border of a lens float lens.
ET = 3.55 mm is the edge thickness of the lenses as given by the manufacturer.
8.3.3 Wavelet Target
shows the reconstructed slope of the wavelet target (see
gives the slope component in vertical direction which is expected to be zero if the target is aligned parallel to the image axes.
depicts a profile along the red line which shows that the reconstructed slope values in y direction are close to zero indeed with a standard deviation of 0.008. The standard deviation is calculated for the left half of the profile only (pixels 7 to 508) because the black areas in the right half of the target correspond to areas where the horizontal slope
103
Chapter 8 CHARACTERIZATION OF THE SETUP component is too large to be reconstructed with the LUT obtained from the lens float target or even too large to be reached by a light ray coming from the light source and moving on towards the camera sensor. Because the horizontal slope component is too large to be included in the LUT and the LUT is calculated from both slope components the application of the LUT for slope reconstruction leads to NaN values in the vertical component, too (see
depicts the slope component in horizontal direction which is expected to vary according to the specification of the wavelet target (see
the red line a profile is drawn which is given in
those slope components which are reconstructible with the LUT obtained from the lens float target images are shown as red lines. The expected slope values as well as the measured (reconstructed) slope values are summarized in
visible that the deviation from the expected value is increasing for higher absolute values of slope. A possible explanation for that is the (spherical) aberration of the large Fresnel lens which becomes larger with increasing distance to the centre of the lens. The position where the light ray passes the Fresnel lens depends on the position in the image as well as on surface slope. Nevertheless, in general the lens is passed further away from the centre (where aberration is larger) for larger slope values.
Table 8.1.: Wavelet Target Slope: Reconstructed values, expected values and deviation; all given in units of slope. The slope segments are visible in
and are numbered from left to right.
Slope
Segment
1 2 3 4 5 6 7 8
Measured
Slope
0.017
0.258
0.485
0.791
0.955
NaN NaN NaN
Expected
Slope
0.00
0.25
0.50
0.75
1.00
1.25
Deviation
0.017
0.008
0.015
0.041
0.045

1.25

1.00

104
Accuracy of the ISG setup 8.3
(a) Reconstructed slope component in y direction.
0.5
200
400
0
(b) Reconstructed slope component in y direction. Profile along red line.
0.10
0.05
600
200
400
600
200 400 600 800 position [px]
−
0.5
(c) Reconstructed slope component in x direction.
0.5
0
−
0.5
0.00
−
0.05
−
0.10
0 200 400 600 800 position [px]
(d) Reconstructed slope component in x direction. Profile along red line.
1
0.5
0
−
0.5
200 400 600 800 position [px]
−
1
0 200 400 600 800 position [px]
Figure 8.6.: Resconstructed wavelet target slope. LUT of 12/06/2014 applied to data of
12/06/2014.
8.3.4 Measurement of the Wavelet Target
The slope of the wavelet target was manually measured in order to investigate whether the observed deviations of the reconstructed slope values from the expectations stem from the reconstruction process or the construction of the target itself. For this purpose, a (red) laser is installed pointing vertically downwards onto the target.
∆y
Then the target is tilted by an angle φ = arcsin (
) until the laser beam is reflected
∆x into its point of origin (see
). This angle equals the slope of the target at
the segment where the laser point is reflected.
The length of the target was measured to be ∆x = 37.2 cm. The values obtained are summarized in
. They are in good agreement with the expected values
which are also given in
105
Chapter 8 CHARACTERIZATION OF THE SETUP
Table 8.2.: Measuring the wavelet target. When two values for ∆y are given, the measurement was repeated to obtain an estimate for the experimental error. ∆x = 37.2 cm.
segment
distance ∆y [cm] angle φ [rad] slope tan φ []
reference value []
k j l g h i f e c d a b
0
8.9; 9.2
16.4
22.4; 22.0
26.1
29.2; 29.2
28.9
26.4; 26.3
22.0
16.5; 16.5
8.9
0
0
0.24; 0.25
0.46
0.65; 0.63
0.78
0.90; 0.90
0.89
0.79; 0.79
0.63
0.46; 0.46
0.24
0
0
0.25; 0.26
0.49
0.75; 0.73
0.98
1.27; 1.27
1.23
1.01; 1.00
0.73
0.49; 0.49
0.25
0
1.25
1.00
0.75
0.50
0.25
0.00
0.00
0.25
0.50
0.75
1.00
1.25
Laser
a b c
φ
d e f
∆x
g h i
∆y
j k l
.
φ
Figure 8.7.: Measuring the wavelet target using laser reflection.
106
Spatial Distribution of Calibration Lenses
8.4 Spatial Distribution of Calibration Lenses
The calibration process does not consider any spatial inhomogeneities of the setup except for those corrected by the ratio imaging. Nevertheless, spatial inhomogeneities due to the optical setup might appear as positiondependent intensity ratios of the lens float calibration target lenses.
depicts the spatial distribution of the
centres of the lens float calibration target lenses 4
for different days of measurement of the 2013 campaign and of 12/06/2014. One can see that the positions of the lenses relative to the whole image section are scattered rather widely, providing a spatially well averaged LUT.
The curve and lineshaped patterns in the position images occur because the lens float calibration target is moving slowly across the field of view of the camera when a longer image sequence is taken.
Nonetheless, the dependence of the final LUT on the position of the lens float lens used for its calculation is evaluated in order to justify the use of a spatially averaged
LUT. For this purpose the position dependency of the measured intensity ratio at a given slope value is evaluated in the next section.
8.4
4 after manual selection, see
107
Chapter 8 CHARACTERIZATION OF THE SETUP
(a) Position of Lens Float Lens Centers, data of 30/04/2013
(b) Position of Lens Float Lens Centers, data of 03/05/2013
600
400
200
600
400
200
0
0 200 400 600 800 position [px]
(c) Position of Lens Float Lens Centers, data of 08/05/2013
0
0 200 400 600 800 position [px]
(d) Position of Lens Float Lens Centers, data of 10/05/2013
600
400
200
600
400
200
0
0 200 400 600 800 position [px]
0
0 200 400 600 800
(e) Position of Lens Float Lens Centers, data of 12/06/2014 position [px]
600
400
200
0
0 200 400 600 800 position [px]
Figure 8.8.: Positions of the lens float lens centers.
108
Influence of the nonideal imaging properties of the Fresnel Lens
8.5 Influence of the nonideal imaging properties of the
Fresnel Lens
The position dependency of the measured intensity at a given slope value is demonstrated using the lens float calibration target and a gridded foil which is placed right on top of the diffusing screen of the light source.
depicts an image of the first illumination wedge A with setup as described. The position of the gridded foil compared to the lens centres varies slightly from lens to lens. This variation gives a qualitative impression of the distortion. As suggested by
8.5
100
200
300
400
500
600
700
800
600
400
200
2,200
2,000
1,800
1,600
1,400
1,200
1,000
100 200 300 400 500 600 700 800 900 position [px]
Figure 8.9.: Raw image of first illumination wedge
A with transparent gridded foil placed on top of the light source and lens float target on the water surface. Data of
30/10/2013.
quantitative analysis could be performed by determining the centre of the light
first and covering the whole light source with an opaque material except for a small area around this point. Then the position of the uncovered point are varied and the corresponding positions seen through the lens float lenses are evaluated depending on the position of the lens in the image.
5 more specifically, the point on the light source where zero slope light rays stem from
109
9
The Influence of Surfactants on Water
Waves and Gas Transfer
In this chapter, the 2013 Aeolotron data is evaluated with a particular focus on mean square slope. Furthermore, wave slope power spectra obtained from the reconstructed slope data are presented. Since one row of LEDs of the ISG light source failed from the second day of the measurement campaign on and could not be repaired during the campaign, the LUT computed from the lens float calibration target images of the first day was used to reconstruct slope data for all conditions
(see
for a description of the reconstruction process). Since the optical setup was not modified and the mean water height was approximately constant for all days of measurement, this should have a small impact only on the results.
9.1 Surfactants and Water Wave Slope
It is expected that the presence of surfactants on the water surface influences water wave slopes (see
example wave images are presented. The alongwind component of slope is depicted for two different amounts of surfactant and three different wind speeds each. In each column, approximately the same wind speed was present and in each row, the same surfactant was used. The top row depicts the smaller amount (and only one type) of surfactant (1 mg/l Dextran) versus the full mix surfactant in the second row, and wind speed increases from left to right.
It is clearly visible that for lower wind speeds, small capillary waves are strongly suppressed by the full mix surfactants (see
). This is not the case for the
Dextran condition, where small waves are visible. For the highest wind speeds, the wave fields look very similar for both conditions. For intermediate wind speeds,
111
Chapter 9 THE INFLUENCE OF SURFACTANTS ON WATER WAVES AND . . .
Data of 30/04/2013, 1 mg/l Dextran. Wind Speeds of 1.48, 3.88 and 8.42 m/s.
Data of 10/05/2013, full mix. Wind Speeds of 1.48, 3.88 and 8.42 m/s.
Figure 9.1.: Wave slope images measured with the ISG. Displayed is the alongwind slope component; the wind is blowing from the left.
the wave field for the Dextran condition is growing whereas the full mix condition still has a flat water surface. For the highest wind speeds, slope values outside the measuring range of the ISG occur. They are visible in black in the images. For the calculation of mean square slope, these values are ignored and do not contribute.
Thus, mean square slope is underestimated for high wind speeds.
9.2 Surfactants and Mean Square Slope
Among other things, the Imaging Slope Gauge at the Heidelberg Aeolotron can be used to provide mean square slope data for the analysis of the effect of wavesuppression by surfactants on gas transfer velocities. Mean square slope has proven
to yield a better parametrization of gas transfer velocities than wind speed ( Frew et al.
[ 1987 ]). For the 2013 Aeolotron measurements as described in
mean square slope data has been calculated with the procedure detailed in
. Transfer velocities were provided by
velocities by
Bopp and Jähne [ 2014 ]. For comparison, the results of the 2011 Aeolotron
measurements as summarized in
Kiefhaber [ 2014 ] are given as well.
112
Surfactants and Mean Square Slope
Uncorrelated Mean Square Slope over Time
In order to obtain mean square slope values for a longer period of time, sequences of 20 raw images each were used to calculate one averaged value for mean square slope (see
chapter 7 ). This yields approximately one value for mean square slope per
1.5 s. Mean square slope varies on smaller timescales, thus the correlation between the individual values of mean square slope calculated here is very low.
presents the resulting mean square slope values plotted over time 1
for the different wind speed conditions for the data of 08/05/2013 (full mix). Therein, the black curve is mean square slope as calculated. The red curve represents a running mean with a window size of 100 datapoints (which equals 150 s) and the orange curves gives the standard deviation with respect to the running mean. Note the different scaling of the vertical axis. The data for the other days is presented in
The current measuring technique provides a good means to describe the wave field, although it is not suitable to make a general statement concerning the physical source of the observed patterns. In the following, an exemplary description of some noticeable patterns will be given.
The data as depicted in
exhibits various patterns. The mean square slope signal remains remarkably constant for wind speeds up to 2.89 m/s. Fluctuations occur on rather small scales compared to those at higher wind speeds. This is consistent with the observation described in
that waves with short wavelengths appear to be suppressed by the full mix surfactant. For a wind speed of 2.20 m/s some sequences were corrupted during data storage which leads to runaway values in the reconstructed mean square slope data and drastically increases the standard deviation.
For a wind speed of 3.88 m/s the running average of mean square slope remains relatively constant (in the given example, at a value of about 1.7 × 10
−
3
), but the standard deviation is increasing over time. This higher variability in mean square slope corresponds to a more inhomogeneous wave field. In contrast, for a wind speed of 5.11 m/s the running average of mean square slope increases over time and the standard deviation as well. For higher wind speeds, both remain relatively constant again. In combination with the increase of gas transfer velocities for this wind speed
(which will be presented in
) this leads to the conclusion that the surface
film was ruptured during this condition. Apparently, in the transition zone between large and small suppression of the wave field by surfactants, the variability of the wave field increases at lower wind speeds compared to mean square slope. All in all, mean square slope is higher for higher wind speeds as one expects because of the larger energy input from the wind field into the wave field.
9.2
1
Note that the zero point of the time axis is the beginning of the respective wind speed condition and thus differs for each plot.
113
Chapter 9 THE INFLUENCE OF SURFACTANTS ON WATER WAVES AND . . .
1.72
1.71
1.7
1.69
1.68
1.67
1.66
1.65
1.64
1.63
0
⋅
10
−
3 u re f
=
1.48 m/s
3300
6600
9900 time [s]
1.9
1.85
1.8
1.75
1.7
1.65
1.6
1.55
1.5
1.45
0
⋅
10
−
3 u re f
=
2.20 m/s
600
1200
1800
2400 time [s]
⋅
10
−
3 u re f
=
2.89 m/s
⋅
10
−
3 u re f
=
3.88 m/s
1.8
1.75
1.7
1.65
1.6
0 1350 2700 4050 time [s] u re f
=
5.11 m/s
60
50
40
30
20
10
0
0
⋅
10
−
3
900
1800
2700 time [s]
5
4
3
2
1
0
0
⋅
10
−
3
900
1800
2700 time [s] u re f
=
6.77 m/s
150
100
50
0
0
600
1200
1800 time [s]
220
200
180
160
140
120
100
80
60
40
20
0
⋅
10
−
3 u re f
=
8.42 m/s
600
1200
1800 time [s]
Figure 9.2.: Mean Square Slope plotted over time for different values of u re f
. Data of
08/05/2013 (full mix). Note the different scaling of the vertical axis. The black line gives mean square slope data, the read line gives a running mean with a window size of 150 s and the orange lines give the standard deviation.
114
Surfactants and Mean Square Slope
Choice of Running Mean Window Size
In order to obtain a reliable estimate for the variability of mean square slope on longer time scales, many mean square slope data points have to be averaged over time for each dataset. The choice of the averaging period is crucial. In general, a larger averaging time yields a more constant mean value. Shorter averaging times result in higher temporal resolution. The best choice of averaging time depends on the time scale of interest. As an example, the dataset of 30/04/2013 with the lower
Dextran concentration at the highest wind speed is analysed with different window sizes for the running mean. The results for two extreme cases for the window size are presented in
. Naturally, the running mean exhibits more variation for
the smaller window size and gives almost constant values for the large window size.
Running Mean window size of 15 s.
⋅
10
−
3
250
200
150
100
50
0
0 450 900 1350 time [s]
Running Mean window size of 675 s.
⋅
10
−
3
250
200
150
100
50
0
0 450 900 1350 time [s]
Figure 9.3.: Mean Square Slope plotted over time with different window sizes of the running mean. Data of 30/04/2013 (1 mg/l Dextran) for a wind speed of 8.42 m/s. The black line gives mean square slope data, the red line gives the running mean and the orange lines give the standard deviation.
Experimentally, a value between 100 s and 250 s has found to yield best results for the given datasets. To determine this value, the running standard deviation for mean square slope was calculated with different window sizes between 15 and 300 s for each wind speed condition of 30/04/2013. For each dataset and window size, the resulting vector was averaged. Finally, the averaged standard deviation was plotted over the window size. To facilitate the comparison between the different wind speed conditions, the value was normalized to a normalized mean standard deviation of 1.
The results are given in
. When the normalized mean standard deviation
approaches a constant value, then mean square slope varies on smaller timescales and the minimum window size to obtain a stable mean square slope value is found.
For the given dataset, the normalized mean standard deviation becomes sufficiently constant in the range between 100 and 250 s.
As no temporal resolution of mean square slope is necessary for the analysis performed in the following sections, one value for mean square slope has been calculated for each wind speed and surfactant condition with the exception of the
9.2
115
Chapter 9 THE INFLUENCE OF SURFACTANTS ON WATER WAVES AND . . .
1.03
1.02
1.01
1
0.99
0.98
0.97
1.48 m/s
2.20 m/s
2.89 m/s
3.88 m/s
5.11 m/s
6.77 m/s
8.42 m/s
0.96
0 30
60
90 120 150
180
210 240 270 300 window size [s]
Figure 9.4.: Normalized standard deviation plotted over running mean window size for the dataset of 30/04/2013 (1 mg/l Dextran).
fifth wind speed (5.11 m/s) on 08/05/2013 and 10/05/2013 which have been parted into four parts each in order to be consistent with gas transfer velocity data. The results are summarized in
Correlated Mean Square Slope over Time
In addition to short image sequences of 20 raw images each, long time series of surface slope were measured during the 2013 experiment. The current measuring technique allows for mean square slope measurements with a temporal resolution of
4 up to ∆t = 1/s ≈ 0.66 ms. For the measurements presented here the full temporal
6030 resolution of the ISG was exploited. One value for mean square slope is available per
0.66 ms. Thus, it is expected that the resulting time series of mean square slope are more correlated than those described in the previous section. Due to the limited amount of memory available in the camera (see
3.32 s was taken.
116
Surfactants and Mean Square Slope
??–?? in the appendix present the resulting mean square slope values plotted over time for the different experimental conditions. Here, only the dataset of 10/05/2013
is briefly discussed ( Figure 9.5
). Again, note the different scaling of the vertical axis.
Remarkable patterns are visible in the data of 08/05/2013 and 10/05/2013, which are the full mix surfactant cases, for wind speeds of 2.89 m/s and 3.88 m/s. In contrast to the other datasets, mean square slope varies almost periodically. Provided that small capillary waves are suppressed by the surface film and the wave field thus consists of mainly gravity waves with longer amplitudes, the frequencies occurring in the time series of mean square slope corresponds to the frequency of these waves. For
the datasets mentioned above, a beatlike pattern 2
occurs. This may indicate that two frequencies close to each other make up the main part of the mean square slope time series. For all other datasets, the signal is not that regular. In
the two slope components of the dataset at a wind speed of 3.88 m/s on 10/05/2013 are shown.
Wave structures are clearly visible but appear to be moving in crosswind direction as well. Interestingly, this effects occurs only for the two full mix conditions which were measured on two different days at the same intermediate wind speeds. As explained later in this section, the nexthighest wind speed 5.11 m/s covered with the given experiment is the wind speed where the surface film appears to rupture.
Other noticeable patterns occur for 2.20 m/s on 03/05/2013, and for 8.42 m/s on
10/05/2013 towards the end of the sequences. Here, mean square slope appears to rapidly drop to zero. This is not a physical phenomenon, but occurs due to technical difficulties during the saving process of the raw data.
9.2
2
German: Schwebung
117
Chapter 9 THE INFLUENCE OF SURFACTANTS ON WATER WAVES AND . . .
1.46
1.45
1.45
1.44
1.44
1.43
1.43
1.42
1.42
1.41
1.41
0
⋅
10
−
3 u re f
=
1.48 m/s
1.47
1.46
1.45
1.44
1.43
1.42
1.41
0
⋅
10
−
3 u re f
=
2.20 m/s
1 2 time [s] u re f
=
2.89 m/s
3 1 2 time [s] u re f
=
3.88 m/s
1.52
1.5
1.48
1.46
1.44
1.42
1.4
1.38
1.36
0
⋅
10
−
3
1 2 3
3.5
3
2.5
2
1.5
1
0
⋅
10
−
3
1 2 3 time [s] u re f
=
5.11 m/s time [s] u re f
=
6.77 m/s
60
55
50
45
40
35
30
25
20
0
⋅
10
−
3
1 2 time [s]
240
220
200
180
160
140
120
100
80
60
40
0
⋅
10
−
3
3 1 2 time [s]
180
160
140
120
100
80
60
40
20
0
⋅
10
−
3 u re f
=
8.42 m/s
1 2 3 time [s]
Figure 9.5.: Mean Square Slope plotted over time for different values of u re f
. Data of
10/05/2013 (full mix). Note the different scaling of the vertical axis.
3
3
118
Surfactants, Mean Square Slope and Gas Transfer Velocities 9.3
(a) Alongwind component of slope.
(b) Crosswind component of slope.
Figure 9.6.: Example for the slope components at a wind speed of 3.88 m/s on 10/05/2013
(full mix). Wave patterns are clearly visible. Alongwind direction is leftright and crosswind direction is topbottom.
9.3 Surfactants, Mean Square Slope and Gas Transfer Velocities
In order to parametrize gas transfer velocities with mean square slope, one average mean square slope value is calculated for each wind speed condition on the four days of measurement in 2013 by averaging the uncorrelated time series. Mean square slope is plotted over the friction velocity in
shows the gas transfer velocities for nitrous oxide (N
2
O) scaled to that of CO
2 at 20
○
C in fresh water, k
600
, as calculated by
plotted over mean square slope. For comparison, k
600 plotted over friction velocities as calculated by
Bopp and Jähne [ 2014 ] is presented in
Friction velocities, gas transfer velocities for and mean square slope values as measured with the CISG and evaluated by
Kiefhaber [ 2014 ] are given for the 2011
Aeolotron campaign, which is described in detail in
Kiefhaber [ 2014 ], for comparison in
to
The error bars given for the mean square slope values for the 2013 measurements reflect the standard deviation of the mean square slope time series calculated for a window size of one (see
). This “error” arises from a physical origin
rather than from the measuring device because mean square slope is a property of the wave field which varies on the timescales resolved by this measurement. The real measurement error is smaller.
shows a plot of mean square slope over the friction velocity u
⋆
,water
. As expected, mean square slope grows with growing friction velocity. The data is in good agreement with the measurements from 2011. The full mix conditions exhibit a behaviour similar to the measurements of 2011 with the higher concentration of
119
Chapter 9 THE INFLUENCE OF SURFACTANTS ON WATER WAVES AND . . .
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
30/04/2013 1mg/l Dextran
03/05/2013 2mg/l Dextran
08/05/2013 Full mix
10/05/2013 Full mix
01/03/2011 clean water
03/03/2011 0.6g Triton X100
08/03/2011 3g Triton X100
10/03/2011 3g Triton X100
0.01
0.009
0.008
0.007
0.006
0.005
0.004
0.003
0.002
0.1
0.2
0.3
0.4
0.5 0.6 0.7
0.9 1 u
⋆
,water
[ cm s
]
Figure 9.7.: Mean square slope plotted over friction velocity. The values for the friction velocity are preliminary results.
2
120
Surfactants, Mean Square Slope and Gas Transfer Velocities
Triton X100. For these conditions, mean square slope remains constantly small until a critical wind speed is reached where the surface film ruptures and a sudden increase of mean square slope occurs. For higher wind speeds, mean square slope grows in both cases. For the two Dextran conditions and low wind speeds, mean square slope is closer to the value at the clean water condition.
Surprisingly, mean square slope for the 2013 measurements is higher than the value of the 2011 clean water condition at the highest friction velocities, yet still compatible within the error tolerances. This might be due to improper calibration of the CISG used for the 2011 measurements. Reference measurements with clean water with the new ISG setup are advisable to quantify this effect.
illustrates the effect of wave suppression by surfactants (represented by mean square slope) on the gas transfer velocities k
600
. The red line indicates a curve fit of the form k = A ⋅ (mss)
B with fit parameters A = 479 and B = 1.15 to the data of 2013. The curve was fitted to the data points with mean square slope larger than 0.015 only (red dots). Data points with blue dots were excluded from the fit.
For comparison,
Krall [ 2013 ] found fit parameters of A
2011
=
478 and B
2011
=
1.11 for the 2011 data.
Surfactants are known to modify the hydrodynamic boundary conditions at the water surface and thus to suppress waves (see
transfer velocities k are reduced compared to clean water conditions. For mean square slope values larger than ≈ 0.02, the transfer velocity correlates well with mean square slope for all surfactants shown in
slightly larger transfer velocities at a given value of mean square slope, but is still close to the surfactant cases and is compatible within the error bars. For low values of mean square slope, there is no correlation visible. For these conditions, almost no waves are present, thus energy is transferred from the wind field to the water body by the generation of shear currents only. Then waves cannot be responsible for nearsurface turbulence which influences gas transfer, but the shear current is.
It is not surprising that mean square slope as a parameter related to waves cannot describe the dependency of gas transfer on this process.
depicts the transfer velocity k
600 over the friction velocity u
⋆
,water
. As
found in earlier studies [ Krall , 2013 ] the curves connecting the individual datasets of
each day of measurement differs for different surfactant concentrations. In comparison with
, this shows that mean square slope is better suitable as a quantity
to parametrize the transfer velocity k
600 than the friction velocity is.
It is noticeable that in contrast to theoretical expectations, the mean square slope values for a Dextran concentration of 1 mg/L at a given wind speed are lower than those for a Dextran concentration of 2 mg/L (see
seen in the gas transfer velocities ( Figure 9.9
). A possible explanation for that is the
presence of additional contaminations in the water 3
which act like surfactants and
3
The water surface was not skimmed before the experiments and additional contaminations might have entered the water body during sampling procedures.
9.3
121
Chapter 9 THE INFLUENCE OF SURFACTANTS ON WATER WAVES AND . . .
cm h
10
9
8
7
6
5
4
3
2
60
50
40
30
20
30/04/2013 1mg/l Dextran
03/05/2013 2mg/l Dextran
08/05/2013 Full mix
10/05/2013 Full mix
01/03/2011 clean water
03/03/2011 0.6g Triton X100
08/03/2011 3g Triton X100
10/03/2011 3g Triton X100
480 * (mss)
1.15
0.9
1
0.8
0.7
0.6
0.5
0.4
0.3
0.002
0.004 0.006
0.01
0.03
mean square slope []
0.05 0.07 0.1
Figure 9.8.: Transfer velocity plotted over mean square slope. The values for the transfer velocity are preliminary results. The red line indicates a curve fit of the form k = A ⋅ (mss)
B with fit parameters
A = 480 and B = 1.15 to the data of 2013. The curve was fitted to the data points with mean square slope larger than 0.015 only
(red dots). Data points with blue dots were excluded from the fit. For comparison,
Krall [ 2013 ] found fit parameters of
A
2011
=
478 and B
2011
=
1.11 for the 2011 data.
122
Surfactants, Mean Square Slope and Gas Transfer Velocities 9.3
cm h
10
9
8
7
6
5
4
3
2
60
50
40
30
20
30/04/2013 1mg/l Dextran
03/05/2013 2mg/l Dextran
08/05/2013 Full mix
10/05/2013 Full mix
01/03/2011 clean water
03/03/2011 0.6g Triton X100
08/03/2011 3g Triton X100
10/03/2011 3g Triton X100
0.9
1
0.8
0.7
0.6
0.5
0.4
0.3
0.1
0.2
0.3
0.4
0.5 0.6 0.7
0.9 1 u
⋆
,water
[ cm s
]
Figure 9.9.: Transfer velocity plotted over friction velocity. Preliminary results.
2
123
Chapter 9 THE INFLUENCE OF SURFACTANTS ON WATER WAVES AND . . .
inhibit gas exchange and the generation of high slope waves.
All in all, the results confirm that mean square slope is a parameter which is suitable to describe gas exchange for different kinds of surfactant concentrations, especially for the higher wind speed conditions covered (u ref
>
6 m/s). Mean square slope is more strongly correlated with the gas transfer velocity than friction velocity is. For lower wind speeds, the correlation is low, because gas transfer is enhanced by shearinduced turbulence which cannot be described by a wave field parameter like mean square slope.
124
The Effects of Surfactants on Smallscale Waves
9.4 The Effects of Surfactants on Smallscale Waves
Directional Slope Wave Number Power Spectra
At this point, slope wave number power spectra are calculated from the slope data as described in
(see
for an example). As these are hard to interpret, omnidirectional power spectra S(k) are calculated from them by integrating over all directions (see
for a description of this process). An example is provided in
. Due to the large range covered and to allow for
comparison with other studies ( Rocholz [ 2008 ];
Kiefhaber [ 2014 ]), dimensionless
omnidirectional saturation spectra B(k) = k
2
⋅
S(k) (see
in the following section to compare different spectra.
9.4
Omnidirectional Saturation Spectra
Omnidirectional saturation spectra are a good tool to study the suppression of the wave field by surfactants.
depicts omnidirectional saturation spectra grouped by the surfactant used for seven different wind speeds each. Therein, for all surfactants, the dashed line indicates the noise level. Due to the multiplication with k
2
, white noise in the slope power spectra S(k) is amplified at high wave numbers
(B(k) = k
2
⋅
S(k)).
In general, the spectral energy content of the wave field as represented by the omnidirectional saturation spectra shown here increases with wind speed as expected.
The overall shape of the spectra is similar to that observed in previous studies (see
Kiefhaber [ 2014 ] for examples).
For the two full mix conditions ( Figures 9.11c
to
) it is clearly visible that the
surface film is intact up to the fourth wind speed condition (about 3.88 m/s, green curves) and is ruptured above 5.11 m/s (dark blue). The spectral energy content increases rapidly between these two conditions. The transition appears to be smoother
for the two Dextran cases ( Figures 9.11a
to
). The same observation was made in
the mean square slope data ( Figure 9.8
). Additional measurements for several wind
speeds in the transition zone might help to understand the influence of rupturing surface films on water waves.
For lower wind speeds, the full mix surfactant effectively suppresses small and mediumscale waves. Compared to the two Dextran cases, wave suppression is much higher for the full mix cases. For higher wind speeds (dark blue, violet and black curves, 5.11 m/s, 6.77 m/s, 8.42 m/s), the spectra are very similar for all surfactant cases, indicating that the type of surfactant present becomes insignificant for the wave field at sufficiently high wind speeds. An explanation for that is that the surface film is mixed into the bulk water (e.g. by breaking waves) at higher wind speeds. For higher wind speeds, the spectra look very similar to the clean water case studied by
125
Chapter 9 THE INFLUENCE OF SURFACTANTS ON WATER WAVES AND . . .
(a) Logarithm of the 2D Power Spectrum S(⃗k). The numbers on the axes indicate the wave number vector components k x and k y
.
14661
8934
3207
2520
8247
13974
14661 8858
3054 2749 kx [px]
8552
14355
8 10 12 14 16 18 20 22
(b) Omnidirectional Power Spectrum S(k)
(c) Omnidirectional Saturation Spectrum
B(k)
10
−
4
10
−
1
10
−
7
10
−
2
10
−
10
10
1
10
2
10
3
10
4 10
−
3
10
1
10
2
10
3
10
4
Figure 9.10.: Comparison of some different types of spectra. Data of 30/04/2013 (1 mg/l
Dextran) for a wind speed of 8.42 m/s.
Compared to the measurements with Triton X100 as evaluated by
[ 2014 ], the wave suppression at low wave numbers (below 100 rad/m) is less promi
nent. Apparently, wave suppression is less effective for the surfactants studied here compared to Triton X100.
It is noticeable that the red curve (lowest wind speed 1.48 m/s) displays characteristic “bumps” at a k of about 2 × 10
2 and 5 × 10
2 for the first three surfactant cases. It is not visible for the last surfactant condition, which is essentially the same as the third one. It is likely to be an artifact from data evaluation without physical origin in the wave field.
For comparison, the spectra at a wind speed of 3.88 m/s and 5.11 m/s are given for all surfactant cases in
(see
for the other conditions). As
126
The Effects of Surfactants on Smallscale Waves 9.4
(a) Data of 30/04/2013, 1 mg/L Dextran.
(b) Data of 03/05/2013, 2 mg/L Dextran.
10
−
2
10
−
3
10
−
4
10
−
5
10
−
6
10
1
10
2
10 k [rad/m]
3
(c) Data of 08/05/2013, full mix.
10
4
10
10
−
2
−
3
10
−
4
10
−
5
10
−
6
10
1
10
2
10 k [rad/m]
3
(d) Data of 10/05/2013, full mix.
10
4
10
−
2
10
−
3
10
−
2
10
−
3
10
−
4
10
−
5
10
10
−
4
−
5
10
−
6
10
1
10
2
10 k [rad/m]
3
10
4
10
−
6
10
1
10
2
10 k [rad/m]
3
10
Figure 9.11.: Omnidirectional saturation spectra
B(k) = S(k) ⋅ k
2 plotted over k for the different surfactant conditions. Wind speed increases from condition 1 to condition 7 in the order red ( 1.48 m/s), blue( 2.20 m/s), yellow(
2.89 m/s), green( 3.88 m/s), dark blue( 5.11 m/s), violet( 6.77 m/s), black( 8.42 m/s). The dashed line gives the noise level (white noise in S(k)).
4
127
Chapter 9 THE INFLUENCE OF SURFACTANTS ON WATER WAVES AND . . .
mentioned in the previous section, the curves of the two Dextran cases appear to be the wrong way round because the spectral energy content is higher for the case with more surfactant. As before, an explanation for this phenomenon could be the pollution of the water with additional substances due to not skimming the surface.
The curves for the two full mix cases were recorded on different days, but with the same surfactant concentration. Thus, they are almost identical as expected.
Furthermore,
illustrates that for a wind speed of 3.88 m/s, the spectral energy content for the two Dextran cases (30/04/2013 & 03/05/2013) is much higher than that for the two full mix cases (08/05/2013 & 10/05/2013). When the surface film has ruptured for the full mix cases at a wind speed of 5.11 m/s the spectral energy content of the Dextran cases and of the full mix cases becomes almost the same, indicating that the influence of the surfactant becomes independent of the type of
All in all, the analysis of the omnidirectional saturation spectra shows that the two types of surfactant have different influences on the water waves for the concentrations used. In the full mix cases, waves are suppressed more effectively (especially in the high wave number range) than in the Dextran cases. This might be due to the insoluble components of the full mix surfactant which are not as easily mixed into the bulk water.
128
4
Probably due to mixing into the bulk water.
The Effects of Surfactants on Smallscale Waves 9.4
(a) Condition 4. Wind speed of 3.88 m/s
10
−
2
10
−
3
10
−
4
30/04/2013
03/05/2013
08/05/2013
10/05/2013
10
−
5
10
−
6
10
1
10
2 k [rad/m]
10
3
10
4
(b) Condition 5. Wind speed of 5.11 m/s
10
−
2
10
−
3
10
−
4
10
−
5
30/04/2013
03/05/2013
08/05/2013
10/05/2013
10
−
6
10
1
10
2 k [rad/m]
10
3
10
4
Figure 9.12.: Omnidirectional saturation spectra
B(k) = S(k) ⋅ k
2 plotted over k for conditions
4 and 5. The color code for the wind speeds is the same as in
129
10
Conclusion and Outlook
10.1 Conclusion
The ISG at the Heidelberg Aeolotron
In this work, the new high speed Imaging Slope Gauge at the Heidelberg Aeolotron
and
) has successfully been put into operation for the first time.
Moreover, a data evaluation routine ( chapter 7 ) similar to that described by
[ 2008 ] has been implemented for the new setup.
Regarding the ISG setup, the brightness of the illumination source ( section 5.4
has been increased compared to the first tests described in
for measurements with smaller aperture 1
and hence with improved image quality in terms of depth of field.
For the calibration procedure, a method based on a lens float calibration target (see
) has been adapted ( chapter 7
) and investigated ( section 8.3
The setup has been characterized regarding the range of wave numbers which can
be resolved in saturation spectra ( section 8.1
). Thanks to the new high speed camera
) which is installed as part of the ISG setup it is now possible to record
both components of water surface slope with an unprecedented frame rate of more than 1500 Hz. With the current setup, waves with wave numbers up to 2660 rad/m can be included into saturation spectra without aliasing effects. Detection limits for
slope and mean square slope ( section 2.4
) have been explored for the current setup
). Compared to the old CISG setup of 2011, the detection limit for mean
square slope is lowered by a factor of ≈ 10 and assumes a value of 3.2 × 10
−
4
. Slope
1
At the moment, measurements at an fstop of 8 are possible.
131
Chapter 10 CONCLUSION AND OUTLOOK values between ±0.965 in alongwind and crosswind direction are covered by the new setup.
A spatial dependency of the calibration method remains due to the nonideal optical imaging characteristics of the Fresnel lens, yet the resulting error appears
to
Waves and Gas Exchange in the Laboratory
The influence of different types of natural and synthetic surface films ( section 2.6
on water waves has been analysed using mean square slope time series and wave number saturation spectra. Data was recorded for seven wind speed conditions with reference wind speeds between 1.48 m/s to 8.42 m/s for each surfactant.
The effect of a mixture of naturelike surfactants on water wave surface slopes has
than that of Dextran ( section 9.1
5.11 m/s. Up to this wind speed, naturelike surfactants have been found to suppress waves, especially short and mediumscale waves, very effectively.
The averaging time necessary to obtain a stable estimate for mean square slope
was found to lie in between 100 s and 250 s ( section 9.2
slope data has been compared with data from a previous experiment at the Aeolotron with a different surfactant and clean water. The results give further evidence that, when waves are present, mean square slope is better suited for parametrizing gas
transfer velocities than friction velocity is ( section 9.3
). In the literature, evidence is
given which demands for a replacement of existing wind speed parametrizations for gas transfer velocities with other models. The present work supports with mean square slope.
For the higher wind speeds covered, the relation between mean square slope and transfer velocities has been shown to be similar for several types of soluble and insoluble surfactants.
The damping effect of surfactants on water waves has been studied using a spectral
description of the wave field through wave slope saturation spectra ( section 2.3
). All in all, the results were found to be consistent with findings reported
in
132
2 with the chosen reference concentrations
Outlook
10.2 Outlook
From the technical side of view, new calibration targets which cover a higher range of slope values are desirable in order to take advantage of the full capacity of the
ISG. Furthermore, the effective frame rate of the system may be increased by using a more complex illumination scheme which allows for the determination of both slope components from three raw images.
In the future, an MTF correction as examined by
setup should be further investigated in order to obtain corrected saturation spectra for large wave numbers. With this improvement, light may be shed on the question of spectral cutoff. Furthermore, it should be analysed whether the Fourier Decomposition Method (FDM), which includes phase information as described in
Fahle [ 2013 ], can be applied to the ISG data in order to include waves
with smaller wave numbers into the spectra.
The possible future range of application for the ISG at the Aeolotron will include providing wave parameters such as mean square slope for gas exchange measurements, yet it is not limited to this. Specific experiments targeted at the understanding of wind driven water waves, especially their suppression by surfactants, can be conducted. A question of special interest, which arose during the interpretation of the results of the 2013 Aeolotron campaign, is the specification of the processes which occur in the transition zone between wind speeds where waves are largely suppressed by surfactants and wind speeds where the influence of surfactants on the wave field is diminished.
Future experiments will include combined measurements of wave parameters, gas and heat exchange, sea spray and bubble bursting with sea water at the Aeolotron which are planned for November 2014.
10.2
133
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Part IV.
Appendix
149
A
Appendix
151
Anhang A APPENDIX
A.1 Wind speeds for the Aeolotron campaign 2013
In this section, the reference wind speed curves for the 2013 Aeolotron campaign are depicted.
Wind Speed on 30/04/2013.
8
6
4
2
0
0 2 time [s]
4
⋅
10
4
Wind Speed on 08/05/2013.
Wind Speed on 03/05/2013.
8
6
4
2
0
0 2 time [s]
4
⋅
10
4
Wind Speed on 10/05/2013.
8
6
4
2
8
6
4
2
0
0 2 time [s]
4
⋅
10
4
0
0 1 2 time [s]
3
⋅
10
4
4
Figure A.1.: Wind speed on the different days of the Aeolotron campaign 2013.
A.2 Datasets
In this section, the entire dataset for mean square slope, transfer velocities and
friction velocities for the 2013 Aeolotron campaign ( section 6.1
Aeolotron campaign is presented. f denotes the frequency at the wind generator, σ
2 is mean square slope. k
600 is the gas transfer velocity for nitrous oxide (N
2
O) scaled to that of CO
2 at 20
○
C in fresh water as calculated by
Friction velocities were provided by
Bopp and Jähne [ 2014 ]. Mean square slope data
for the 2011 dataset is taken from
Kiefhaber [ 2014 ]. Values marked by NaN are not
available. Quantities preceded by ∆ give the measurement error.
152
Datasets A
16
16
16
16
5
7
9
12
22
29
16
22
29
5
7
9
12
5
7
9
12
16
16
16
16
22
29 f
[Hz]
16
22
29
5
7
9
12 date
30/04
03/05
08/05
10/05
Table A.1.: Dataset for the 2013 Aeolotron campaign.
Surfactant u ref u
⋆
,water
∆u
⋆
,water k
600
∆k
600
1 mg/l
Dextran
[m/s] [cm/s] [cm/s]
1,48 0,196
2,20 0,195
2,89
3,88
0,410
0,547
5,11
6,77
8,42
0,754
1,197
1,751
0,009
0,009
0,023
0,034
0,053
0,094
0,166
[cm/h] [cm/h]
σ
2
[]
∆σ
[]
2
1,269 0,06345 0,0053 0,0015
2,066 0,1033 0,0102 0,0035
3,065
5,134
0,15325
0,2567
0,0122
0,0147
0,0043
0,0059
11,149 0,55745 0,0406 0,0126
25,607 1,28035 0,0815 0,0197
40,914 2,0457 0,1211 0,0263
1 mg/l
Dextran
Full mix
Full mix
1,48 0,214
2,20 0,325
2,89
3,88
0,429
0,557
5,11
6,77
8,42
0,759
1,213
1,777
1,48 NaN
2,20 NaN
2,89
3,88
0,313
0,424
5,11
5,11
5,11
5,11
6,77
8,42
0,719
0,719
0,719
0,719
1,177
1,742
1,48 0,158
2,20 0,233
2,89
3,88
0,327
0,439
5,11
5,11
5,11
0,706
0,706
0,706
5,11
6,77
8,42
0,706
1,189
1,745
0,0112
0,016
0,024
0,035
0,053
0,095
0,168
NaN
NaN
0,016
0,024
0,049
0,049
0,049
0,049
0,091
0,164
0,010
0,011
0,017
0,026
0,048
0,048
0,048
0,048
0,093
0,165
1,557
2,598
4,558
7,945
5,709
7,094
7,98
8,637
0,71
1,063
1,569
2,171
5,921
6,973
7,398
0,07785 0,0059 0,0007
0,1299 0,012 0,0037
0,2279 0,0153 0,0052
0,39725 0,023 0,0087
15,528 0,7764 0,0524 0,0136
30,666 1,5333 0,0869 0,0216
47,024 2,3512 0,1262 0,0277
0,657 0,03285 0,0017 0,0002
0,949 0,04745 0,0017 0,0001
1,515
2,2
0,07575
0,11
0,0017
0,0021
0
0,0008
0,28545
0,3547
0,399
0,43185 NaN
25,408 1,2704
42,128 2,1064
0,0355
0,05315
0,07845
0,10855
0,29605
0,0166
0,023
0,0276 0,01
0,0782
0,117
0,0018
0,0018
0,0018
0,0021
0,0183
0,01
0,01
NaN
0,0202
0,0268
0
0
0
0,0007
0,01
0,34865 0,0233 0,01
0,3699 0,0265 0,01
7,746 0,3873 0,029 0,0141
26,059 1,30295 0,0832 0,0204
42,089 2,10445 0,1211 0,0274
153
Anhang A APPENDIX f
[Hz]
7
9
3
5
12
16
22
29
12
16
22
29
7
9
3
5
16
22
29
5
5
7
9
12
7
9
12
16
22
29 date
Table A.2.: Dataset for the 2011 Aeolotron campaign.
Surfactant u ref u
⋆
,water
∆u
⋆
,water k
600
∆k
600
[m/s] [cm/s] [cm/s]
σ
2
[cm/h] [cm/h] []
01/03
03/03 clean water
0.6 g
Triton
X100
0,7333 0,11
1,414 0,15
2,056 0,24
2,691 0,34
3,619 0,52
4,795 0,82
6,466 1,38
8,251 2,14
0,8
1,46
0,11
0,17
2,091 0,25
2,717 0,35
3,65 0,53
4,851 0,84
6,502 1,39
8,288 2,15
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
0,598
2,648
3,922
6,153
0,028
0,124
0,186
0,298
0,500
0,731
1,096
2,110
3,205
7,070
24,042 1,274
44,061 2,713
0,024
0,035
0,053
0,103
0,157
0,352
∆σ
[]
2
0,0024 NaN
0,0073 NaN
0,0127 NaN
0,0163 NaN
9,419 0,466 0,0246 NaN
15,805 0,816 0,046 NaN
33,850 1,927
53,262 3,871
0,0782
0,1136
NaN
NaN
0,002
0,0019
0,0019
0,0076
0,0101
0,0205
0,0738
0,1144
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
08/03
10/03
3.0 g
Triton
X100
3.0 g
Triton
X100
1,463 NaN
2,075 0,15
2,695 0,28
3,675 0,45
4,927 0,66
6,625 1,06
8,372 1,72
1,438 NaN
2,074 0,15
2,718 0,28
3,659 0,45
4,898 0,66
6,604 1,06
8,369 1,72
NaN
0,02
0,03
0,03
0,02
0,09
0,24
NaN
0,02
0,03
0,03
0,02
0,09
0,24
0,738
1,100
1,618
2,207
0,037
0,055
0,082
0,113
2,995
11,275
0,156
0,618
31,019 1,962
0,756 0,036
0,0019
0,0019
0,0018
0,0018
0,0018
0,0351
0,1013
0,0017
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
1,139
1,655
0,055 0,0017 NaN
0,080 0,0017 NaN
2,582 0,126 0,0045 NaN
3,829 0,189 0,0065 NaN
11,734 0,606 0,0387 NaN
33,266 1,958 0,0997 NaN
154
Uncorrelated Mean Square Slope Timeseries
A.3 Uncorrelated Mean Square Slope Timeseries
In this section, the uncorrelated mean square slope time series as described in
are presented. Note the different scaling of the vertical axis. The black line gives mean square slope data, the red line gives a running mean with a window size of 150 s and the orange lines give the standard deviation.
A
155
Anhang A APPENDIX
90
80
70
60
50
40
30
20
10
0
Figure A.2.: Mean Square Slope plotted over time for different values of u re f
30/04/2013 (1 mg/l Dextran).
. Data of
⋅
10
−
3 u re f
=
1.48 m/s
10
9
8
7
6
5
4
3
2
1
0 1500 3000 4500
20
15
10
5
0
0
⋅
10
−
3 u re f
=
2.20 m/s
1050 2100 3150
⋅
10
−
3 time [s] u re f
=
2.89 m/s
25
20
15
10
5
0
0
⋅
0
10
−
3
750 time [s] u re f
=
5.11 m/s
450
1500
900
2250
1350
140
120
100
80
60
40
20
0
⋅
10
−
3 time [s] u re f
=
3.88 m/s
35
30
25
20
15
10
5
0
0
⋅
10
−
3
600
1200
1800 time [s] u re f
=
6.77 m/s
300
600 time [s]
900 time [s] u re f
=
8.42 m/s
220
200
180
160
140
120
100
80
60
40
20
0
⋅
10
−
3
450 900 time [s]
1350
156
Uncorrelated Mean Square Slope Timeseries A
Figure A.4.: Mean Square Slope plotted over time for different values of u re f
03/05/2013 (2 mg/l Dextran).
. Data of
⋅
10
−
3 u re f
=
1.48 m/s
8.5
8
7.5
7
6.5
6
5.5
5
4.5
4
3.5
0 300
600
900 1200 time [s]
⋅
10
−
3 u re f
=
2.20 m/s
25
20
15
10
5
0
0 1050 2100 3150 time [s]
35
30
25
20
15
10
5
0
0
⋅
10
−
3 u re f
=
2.89 m/s
750 1500 2250
50
40
30
20
10
0
0
⋅
10
−
3 u re f
=
3.88 m/s
450 900 1350 time [s] u re f
=
6.77 m/s time [s]
⋅
10
−
3 u re f
=
5.11 m/s
100
90
80
70
60
50
40
30
20
10
0
600
1200
1800
2400 time [s] u re f
=
8.42 m/s
220
200
180
160
140
120
100
80
60
40
20
0
⋅
10
−
3
450 900 1350 time [s]
160
140
120
100
80
60
40
20
0
0
⋅
10
−
3
300 time [s]
600
157
Anhang A APPENDIX
Figure A.6.: Mean Square Slope plotted over time for different values of u re f
08/05/2013 (full mix).
. Data of
⋅
10
−
3 u re f
=
1.48 m/s
1.72
1.71
1.7
1.69
1.68
1.67
1.66
1.65
1.64
1.63
0 3300
6600
9900 time [s]
⋅
10
−
3 u re f
=
2.20 m/s
1.9
1.85
1.8
1.75
1.7
1.65
1.6
1.55
1.5
1.45
0
600
1200
1800
2400 time [s]
⋅
10
−
3 u re f
=
2.89 m/s
1.8
1.75
1.7
1.65
1.6
0 1350 2700 4050
5
4
3
2
1
0
0
⋅
10
−
3 u re f
=
3.88 m/s
900
1800
2700 time [s] time [s]
60
50
40
30
20
10
0
0
⋅
10
−
3 u re f
=
5.11 m/s
900
1800
2700 time [s]
220
200
180
160
140
120
100
80
60
40
20
0
⋅
10
−
3 u re f
=
8.42 m/s
600
1200
1800 time [s]
150
100
50
0
0
⋅
10
−
3 u re f
=
6.77 m/s
600
1200 time [s]
1800
158
Uncorrelated Mean Square Slope Timeseries A
Figure A.8.: Mean Square Slope plotted over time for different values of u re f
10/05/2013 (full mix).
. Data of
⋅
10
−
3 u re f
=
1.48 m/s
1.82
1.81
1.8
1.79
1.78
1.77
1.76
1.75
1.74
1.73
1.72
0 2550 5100
7650 10200
⋅
10
−
3 u re f
=
2.20 m/s
1.86
1.84
1.82
1.8
1.78
1.76
1.74
1.72
1.7
1.68
0
1800 3600
5400 time [s]
⋅
10
−
3 u re f
=
2.89 m/s
1.9
1.85
1.8
1.75
1.7
1.65
0 1500 3000 4500 time [s] time [s]
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
⋅
10
−
3 u re f
=
3.88 m/s
900
1800
2700 time [s]
60
50
40
30
20
10
0
0
⋅
10
−
3 u re f
=
5.11 m/s
750 1500 2250 3000 time [s]
220
200
180
160
140
120
100
80
60
40
20
0
⋅
10
−
3 u re f
=
8.42 m/s
600
1200
1800
2400 time [s]
160
140
120
100
80
60
40
20
0
0
⋅
10
−
3 u re f
=
6.77 m/s
600
1200
1800 time [s]
159
Anhang A APPENDIX
A.4 Correlated Mean Square Slope Timeseries
In this section, the correlated mean square slope time series as described in
are presented. Note the different scaling of the vertical axis.
160
Correlated Mean Square Slope Timeseries
Figure A.10.: Mean Square Slope plotted over time for different values of u re f
30/04/2013 (1 mg/l Dextran).
. Data of u re f
=
2.20 m/s u re f
=
1.48 m/s
18
16
14
12
10
8
6
4
0
⋅
10
−
3
1 2 3
⋅
10
−
3 u re f
=
2.89 m/s time [s] u re f
=
3.88 m/s
20
15
10
35
30
25
20
15
10
⋅
10
−
3
5
0 1 2 3 0 1 2 3 time [s] u re f
=
5.11 m/s time [s] u re f
=
6.77 m/s
65
60
55
50
45
40
35
30
25
20
0
⋅
10
−
3
1 2 time [s] u re f
=
8.42 m/s
3
140
130
120
110
100
90
80
70
60
50
0
⋅
10
−
3
1 2 time [s]
3
250
200
150
100
50
⋅
10
−
3
0 1 2 time [s]
3
A
161
Anhang A APPENDIX
Figure A.12.: Mean Square Slope plotted over time for different values of u re f
03/05/2013 (2 mg/l Dextran).
. Data of u re f
=
1.48 m/s u re f
=
2.20 m/s
8.5
8
7.5
7
6.5
6
5.5
5
4.5
4
0
⋅
10
−
3
1 2 3
16
14
12
10
4
2
8
6
0
⋅
10
−
3
1 2 3 time [s] time [s] u re f
=
2.89 m/s u re f
=
3.88 m/s
20
18
16
14
12
10
8
6
0
⋅
10
−
3
3
28
26
24
22
20
18
16
14
12
0
⋅
10
−
3
3 1 2 time [s] u re f
=
5.11 m/s
⋅
10
−
3
1 2 time [s] u re f
=
6.77 m/s
80
75
70
65
60
55
50
45
40
35
30
0
⋅
10
−
3
1 2 time [s] u re f
=
8.42 m/s
3
250
200
150
100
50
0 1 2 time [s]
3
220
200
180
160
140
120
100
80
0
⋅
10
−
3
1 2 time [s]
3
162
Correlated Mean Square Slope Timeseries A
Figure A.14.: Mean Square Slope plotted over time for different values of u re f
08/05/2013 (full mix).
. Data of u re f
=
1.48 m/s
1.34
1.33
1.33
1.32
1.32
1.31
1.31
1.3
1.3
0
⋅
10
−
3
1 2 3 u re f
=
2.20 m/s time [s] u re f
=
2.89 m/s u re f
=
3.88 m/s
1.4
1.38
1.36
1.34
1.32
1.3
1.28
1.26
1.24
1.22
0
⋅
10
−
3
3
5
4.5
4
3.5
3
2.5
2
1.5
1
0
⋅
10
−
3
3 1 2 time [s] u re f
=
5.11 m/s
1 2 time [s] u re f
=
6.77 m/s
80
70
60
50
40
30
20
0
⋅
10
−
3
1 2 time [s] u re f
=
8.42 m/s
3
130
120
110
100
90
80
70
60
50
40
0
⋅
10
−
3
1 2 time [s]
3
240
220
200
180
160
140
120
100
80
60
0
⋅
10
−
3
1 2 time [s]
3
163
Anhang A APPENDIX
Figure A.16.: Mean Square Slope plotted over time for different values of u re f
10/05/2013 (full mix).
. Data of u re f
=
1.48 m/s u re f
=
2.20 m/s
1.46
1.45
1.45
1.44
1.44
1.43
1.43
1.42
1.42
1.41
1.41
0
⋅
10
−
3
1 2 3
1.47
1.46
1.45
1.44
1.43
1.42
1.41
0
⋅
10
−
3
1 2 3 time [s] time [s] u re f
=
2.89 m/s u re f
=
3.88 m/s
1.52
1.5
1.48
1.46
1.44
1.42
1.4
1.38
1.36
0
⋅
10
−
3
3
3.5
3
2.5
2
1.5
1
0
⋅
10
−
3
3 1 2 time [s] u re f
=
5.11 m/s
1 2 time [s] u re f
=
6.77 m/s
60
55
50
45
40
35
30
25
20
0
⋅
10
−
3
1 2 time [s] u re f
=
8.42 m/s
3
240
220
200
180
160
140
120
100
80
60
40
0
⋅
10
−
3
1 2 time [s]
3
180
160
140
120
100
80
60
40
20
0
⋅
10
−
3
1 2 time [s]
3
164
Omnidirectional Saturation Spectra
A.5 Omnidirectional Saturation Spectra
In this section, omnidirectional saturation spectra as described in
are presented. The color code for the wind speeds is the same as in
A
165
Anhang A APPENDIX
10
−
2
10
−
3
10
−
4
10
−
5
30/04/2013
03/05/2013
08/05/2013
10/05/2013
10
−
6
10
1
10
2 k [rad/m]
10
3
10
4
Figure A.18.: Omnidirectional saturation spectra
B(k) = S(k) ⋅ k
2 plotted over k for u re f
=
1.48 m/s
10
−
2
10
−
3
10
−
4
10
−
5
30/04/2013
03/05/2013
08/05/2013
10/05/2013
10
−
6
10
1
10
2 k [rad/m]
10
3
10
4
Figure A.19.: Omnidirectional saturation spectra
B(k) = S(k) ⋅ k
2 plotted over k for u re f
=
2.20 m/s
166
Omnidirectional Saturation Spectra A
10
−
2
10
−
3
10
−
4
10
−
5
30/04/2013
03/05/2013
08/05/2013
10/05/2013
10
−
6
10
1
10
2 k [rad/m]
10
3
10
4
Figure A.20.: Omnidirectional saturation spectra
B(k) = S(k) ⋅ k
2 plotted over k for u re f
=
2.89 m/s
10
−
2
10
−
3
10
−
4
10
−
5
30/04/2013
03/05/2013
08/05/2013
10/05/2013
10
−
6
10
1
10
2 k [rad/m]
10
3
10
4
Figure A.21.: Omnidirectional saturation spectra
B(k) = S(k) ⋅ k
2 plotted over k for u re f
=
3.88 m/s
167
Anhang A APPENDIX
10
−
2
10
−
3
10
−
4
10
−
5
30/04/2013
03/05/2013
08/05/2013
10/05/2013
10
−
6
10
1
10
2 k [rad/m]
10
3
10
4
Figure A.22.: Omnidirectional saturation spectra
B(k) = S(k) ⋅ k
2 plotted over k for u re f
=
5.11 m/s
10
−
2
10
−
3
10
−
4
10
−
5
30/04/2013
03/05/2013
08/05/2013
10/05/2013
10
−
6
10
1
10
2 k [rad/m]
10
3
10
4
Figure A.23.: Omnidirectional saturation spectra
B(k) = S(k) ⋅ k
2 plotted over k for u re f
=
6.77 m/s
168
Omnidirectional Saturation Spectra A
10
−
2
10
−
3
10
−
4
10
−
5
30/04/2013
03/05/2013
08/05/2013
10/05/2013
10
−
6
10
1
10
2 k [rad/m]
10
3
10
4
Figure A.24.: Omnidirectional saturation spectra
B(k) = S(k) ⋅ k
2 plotted over k for u re f
=
8.42 m/s
169
Erklärung:
Ich versichere, dass ich diese Arbeit selbstständig verfasst habe und keine anderen als die angegebenen Quellen und Hilfsmittel benutzt habe.
Heidelberg, den (Datum) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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