Evaluation and development of
wave-current interaction in SWAN
A. J. van der Westhuysen and G. Lesser
Report
November 2007
Contents
1 Introduction ... 1–1 2 Formulations for wave-current interaction ... 2–1
2.1 Default wave-current interaction ... 2–1 2.2 Frequency-dependent effective current ... 2–2 2.3 Dissipation due to current interaction ... 2–2
3 Simulations ... 3–1
3.1 Analytical tests... 3–1 3.2 Laboratory cases ... 3–2 3.2.1 Experiment of Lai et al. (1989) ... 3–3 3.2.2 Experiment of Suastika et al. (2000) ... 3–4 3.3 Field case of Port Phillip Heads... 3–7 3.3.1 Data available for model comparisons... 3–8 3.3.2 Method... 3–9 3.3.3 Results ... 3–12 4 Conclusions ... 4–1 5 Recommendations... 5–1 6 Acknowledgements ... 6–1 References Appendices A Appendix... A–1
A.1 Input of 3-D current fields ... A–1 A.1.1 Alterations to 'Waves' ... A–2 A.1.2 Alterations to SWAN ... A–3 A.2 Interpolation and transformation to effective currents ... A–6 A.3 Distribution of effective current information... A–10 A.3.1 Propagation in geographical space ... A–10 A.3.2 Propagation in spectral space ... A–11 A.3.3 Computation of source terms ... A–20 A.3.4 Determining the quadrants in the matrix solution ... A–21 A.3.5 Diverse modifications – e.g. ICUR settings... A–21
List of Tables
3.1 Experimental conditions of Suastika (2004) considered.
List of Figures
Section 3.1
3.1 Analytical case a31: Waves in following current. 3.2 Analytical case a32: Waves in opposing current. 3.3 Current fields for slanting current analytical cases. 3.4 Analytical case a33: Waves in opposing slanting current. 3.5 Analytical case a34: Waves in following slanting current.
Section 3.2
3.6 Laboratory experiment of Lai et al. (1989): Integral parameters. 3.7 Laboratory experiment of Lai et al. (1989): Wave spectra.
3.8 Flume experiment of Suastika et al. (2000). Integral wave parameters of observations, for cases S1 and S2.
3.9 Flume experiment of Suastika et al. (2000). Observed wave spectra, for cases S1 (top) and S2 (bottom).
3.10 Flume experiment of Suastika et al. (2000): Case S1 (Q=0.120 m3/s). Integral wave parameters.
3.11 Flume experiment of Suastika et al. (2000): Case S1 (Q=0.120 m3/s). Wave spectra 3.12 Flume experiment of Suastika et al. (2000): Case S2 (Q=0.078 m3/s). Integral wave
parameters.
3.13 Flume experiment of Suastika et al. (2000): Case S2 (Q=0.078 m3/s). Wave spectra.
Section 3.3
3.3-1 Location of Melbourne, Australia
3.3-2 Location of Port Phillip Heads showing location of offshore Triaxys Buoys and
AWAC instruments
3.3-3 Divers recovering an AWAC instrument from PP Heads (Photo by Professional Diving Services Pty Ltd.)
3.3-4 Typical wave height time series. Tidal current speed flood positive at location of AWAC instrument in PP Heads
3.3-5 Storm 1 Pt. Lonsdale wind record
3.3-6 Storm 1 Variation in offshore wave conditions 3.3-7 Storm 1 Scatter in measured offshore wave spectra 3.3-8 Storm 3 Pt. Lonsdale wind record
3.3-12 SWAN model bathymetry (relative to AHD) showing SWAN model output locations
3.3-13 SWAN model output locations. Enlarged view of PP Heads
3.3-14 Time series of measured and modelled current speed. Storm 1 at location of AWAC instruments
3.3-15 Time series of measured and modelled current speed. Storm 3 at location of AWAC instruments
3.3-16 Profiles of measured and modelled current speed. Storm 1 at location of RBO AWAC
3.3-17 Profiles of measured and modelled current speed. Storm 1 at location of RB AWAC
3.3-18 Profiles of measured and modelled current speed. Storm 3 at location of RBO AWAC
3.3-19 Profiles of measured and modelled current speed. Storm 3 at location of RB AWAC
3.3-20 Storm 1 Variation in depth and current along Curve01. 3.3-21 Storm 3 Variation in depth and current along Curve01.
3.3-22 Storm 1 Validation of SWAN model set up. Recovery of measured spectra at offshore buoy location
3.3-23 Storm 3 Validation of SWAN model set up. Recovery of measured spectra at offshore buoy location
3.3-24 Storm 1 Map of depth averaged tidal current. Ebb tide 3.3-25 Storm 1 Map of depth averaged tidal current. Slack tide 3.3-26 Storm 1 Map of depth averaged tidal current. Flood tide
3.3-27 Storm 1 Map of Dingemans depth weighted tidal current. Ebb tide, Wave frequency 0.049Hz (T = 20s)
3.3-28 Storm 1 Map of Dingemans depth weighted tidal current. Ebb tide, Wave frequency 0.099Hz (T = 10s)
3.3-29 Storm 1 Map of Dingemans depth weighted tidal current. Ebb tide, Wave frequency 0.199Hz (T = 5s)
3.3-30 Storm 1 Map of Dingemans depth weighted tidal current. Flood tide, Wave frequency 0.049Hz (T = 20s)
3.3-31 Storm 1 Map of Dingemans depth weighted tidal current. Flood tide, Wave frequency 0.099Hz (T = 10s)
3.3-32 Storm 1 Map of Dingemans depth weighted tidal current. Flood tide, Wave frequency 0.199Hz (T = 5s)
3.3-33 Storm 1 Map of ratio of current speed to wave group velocity. Ebb tide, Constant (peak) wave period of 16.7s
3.3-34 Storm 1 Map of ratio of current speed to wave group velocity. Ebb tide, Constant (peak/2) wave period of 8.4s
3.3-35 Storm 3 Map of ratio of current speed to wave group velocity. Ebb tide, Constant (peak) wave period of 11.8s
3.3-37 Storm 1 Map of significant wave height. Ebb tide, Komen whitecapping, Depth averaged current
3.3-38 Storm 1 Map of significant wave height. Slack tide, Komen whitecapping, Depth averaged current
3.3-39 Storm 1 Map of significant wave height. Flood tide, Komen whitecapping, Depth averaged current
3.3-40 Storm 3 Map of significant wave height. Ebb tide, Komen whitecapping, Depth averaged current
3.3-41 Storm 3 Map of significant wave height. Slack tide, Komen whitecapping, Depth averaged current
3.3-42 Storm 3 Map of significant wave height. Flood tide, Komen whitecapping, Depth averaged current
3.3-43 Storm 1 Map of significant wave height. Ebb tide, WBJ whitecapping, Depth averaged current
3.3-44 Storm 1 Map of significant wave height. Slack tide, WBJ whitecapping, Depth averaged current
3.3-45 Storm 1 Map of significant wave height. Flood tide, WBJ whitecapping, Depth averaged current
3.3-46 Storm 3 Map of significant wave height. Ebb tide, WBJ whitecapping, Depth averaged current
3.3-47 Storm 3 Map of significant wave height. Slack tide, WBJ whitecapping, Depth averaged current
3.3-48 Storm 3 Map of significant wave height. Flood tide, WBJ whitecapping, Depth averaged current
3.3-49 Storm 1 Sections of significant wave height along Curve01. Showing alternative whitecapping formulations, Depth averaged current
3.3-50 Storm 3 Sections of significant wave height along Curve01. Showing alternative whitecapping formulations, Depth averaged current
3.3-51 Storm 1 Sections of significant wave height along Curve01. Showing alternative current integration methods, Komen whitecapping
3.3-52 Storm 3 Sections of significant wave height along Curve01. Showing alternative current integration methods, Komen whitecapping
3.3-53 Storm 1 Sections of significant wave height along Curve01. Showing alternative current integration methods, WBJ whitecapping
3.3-54 Storm 3 Sections of significant wave height along Curve01. Showing alternative current integration methods, WBJ whitecapping
3.3-55 Storm 1 Map of energy dissipation. Ebb tide, Komen whitecapping, Depth averaged current
3.3-56 Storm 1 Map of energy dissipation. Flood tide, Komen whitecapping, Depth averaged current
3.3-57 Storm 3 Map of energy dissipation. Ebb tide, Komen whitecapping, Depth averaged current
3.3-59 Storm 1 Map of energy dissipation. Ebb tide, WBJ whitecapping, Depth averaged current
3.3-60 Storm 1 Map of energy dissipation. Flood tide, WBJ whitecapping, Depth averaged current
3.3-61 Storm 3 Map of energy dissipation. Ebb tide, WBJ whitecapping, Depth averaged current
3.3-62 Storm 3 Map of energy dissipation. Flood tide, WBJ whitecapping, Depth averaged current
3.3-63 Storm 1 Sections of energy dissipation along Curve01. Showing alternative current integration methods, Komen whitecapping
3.3-64 Storm 1 Sections of energy dissipation along Curve01. Showing alternative current integration methods, Westh whitecapping
3.3-65 Storm 1 Sections of energy dissipation along Curve01. Showing alternative current integration methods, WBJ whitecapping
3.3-66 Storm 3 Sections of energy dissipation along Curve01. Showing alternative current integration methods, Komen whitecapping
3.3-67 Storm 3 Sections of energy dissipation along Curve01. Showing alternative current integration methods, Westh whitecapping
3.3-68 Storm 3 Sections of energy dissipation along Curve01. Showing alternative current integration methods, WBJ whitecapping
3.3-69 Storm 1 Map of absolute mean wave period. Ebb tide, Komen whitecapping, Depth averaged current
3.3-70 Storm 1 Map of absolute mean wave period. Slack tide, Komen whitecapping, Depth averaged current
3.3-71 Storm 1 Map of absolute mean wave period. Flood tide, Komen whitecapping, Depth averaged current
3.3-72 Storm 1 Sections of absolute mean wave period. Showing alternative current integration methods, Komen whitecapping
3.3-73 Storm 1 Energy spectra. Showing alternative current integration methods, Komen whitecapping
3.3-74 Storm 1 Sections of absolute mean wave period. Showing alternative whitecapping formulations, Depth averaged current
3.3-75 Storm 1 Energy spectra. Showing alternative whitecapping formulations, Depth averaged current
3.3-76 Storm 3 Map of absolute mean wave period. Ebb tide, Komen whitecapping, Depth averaged current
3.3-77 Storm 3 Map of absolute mean wave period. Slack tide, Komen whitecapping, Depth averaged current
3.3-78 Storm 3 Map of absolute mean wave period. Flood tide, Komen whitecapping, Depth averaged current
3.3-79 Storm 3 Sections of absolute mean wave period. Showing alternative current integration methods, Komen whitecapping
3.3-81 Storm 3 Sections of absolute mean wave period. Showing alternative whitecapping formulations, Depth averaged current
3.3-82 Storm 3 Energy spectra. Showing alternative whitecapping formulations, Depth averaged current
3.3-83 Storm 1 Map of mean wave direction. Ebb tide, Komen whitecapping, Depth averaged current
3.3-84 Storm 1 Map of mean wave direction. Slack tide, Komen whitecapping, Depth averaged current
3.3-85 Storm 1 Map of mean wave direction. Flood tide, Komen whitecapping, Depth averaged current
3.3-86 Storm 3 Map of mean wave direction. Ebb tide, Komen whitecapping, Depth averaged current
3.3-87 Storm 3 Map of mean wave direction. Slack tide, Komen whitecapping, Depth averaged current
3.3-88 Storm 3 Map of mean wave direction. Flood tide, Komen whitecapping, Depth averaged current
3.3-89 Storm 1 Sections of mean wave direction. Showing alternative current integration methods, Komen whitecapping
3.3-90 Storm 3 Sections of mean wave direction. Showing alternative current integration methods, Komen whitecapping
3.3-91 Storm 1 Spectra of mean wave direction. Showing alternative current integration methods, Komen whitecapping
3.3-92 Storm 3 Spectra of mean wave direction. Showing alternative current integration methods, Komen whitecapping
3.3-93 Measured 2D Spectra at RipBank AWAC. Showing distribution of energy in direction and frequency
3.3-94 Storm 1 Map of mean directional spreading. Ebb tide, Komen whitecapping, Depth averaged current
3.3-95 Storm 1 Map of mean directional spreading. Slack tide, Komen whitecapping, Depth averaged current
3.3-96 Storm 1 Map of mean directional spreading. Flood tide, Komen whitecapping, Depth averaged current
3.3-97 Storm 3 Map of mean directional spreading. Ebb tide, Komen whitecapping, Depth averaged current
3.3-98 Storm 3 Map of mean directional spreading. Slack tide, Komen whitecapping, Depth averaged current
3.3-99 Storm 3 Map of mean directional spreading. Flood tide, Komen whitecapping, Depth averaged current
3.3-100 Storm 1 Sections of mean directional spreading. Showing alternative current integration methods, Komen whitecapping
3.3-101 Storm 3 Sections of mean directional spreading. Showing alternative current integration methods, Komen whitecapping
3.3-103 Storm 3 Spectra of directional spreading .Showing alternative current integration methods, Komen whitecapping
3.3-104 Influence of including diffraction in SWAN simulation. Storm 3, Ebb tide, Komen whitecapping, Depth averaged current
List of Symbols
Symbol Units Description
BJ - Proportionality coefficient for surf breaking
EB - Proportionality coefficient for triad interaction\
BJ - Breaker parameter for surf breaking
rad/s Absolute radian frequency
rad/s Instrinsic radian frequency
o
Wave direction
Cds - Proportionality coefficient for whitecapping dissipation
CJON m
2
s-3 Proportionality coefficient for bottom friction
c m/s Wave phase velocity
cg m/s Wave group velocity
d m Water depth
E(f) m2/Hz Variance density
f Hz Intrinsic wave frequency
g m/s2 Gravitational acceration
Hm m Maximum wave height
Hm0 m Significant wave height
k rad/m Wavenumber
MDir oTN Mean direction
Q m3/s Discharge
Qb - Fraction of breaking waves
smax - Maximum steepness
s - Spectral mean steepness
Spr o Directional spreading
Stot m2/Hz/o/s Total source term
Tm01 s Mean absolute wave period
Tp s Peak absolute wave period
t s Time
U
m/s Depth-integrated current velocityU
m/s Frequency-dependent effective current velocityU10 m/s Wind speed at a height of 10 m
V
m/s Three-dimensional current velocity fieldx,y m Horizontal spatial coordinates
1
Introduction
The spectral wind wave model SWAN (Booij et al. 1999) plays a key role in the estimation of the Hydraulic Boundary Conditions (HBC) for the primary sea defences of The Netherlands. Since some uncertainty remains with respect to the reliability of SWAN for application to the geographically complex area of the Wadden Sea, a number of activities have been initiated under project H4918 ‘Uitvoering Plan van Aanpak SBW-RVW Waddenzee’ (Plan of Action on the Boundary Conditions for the Wadden Sea) to devise a strategy for the improvement of the model. This activity is carried out in parallel with a measurement campaign that is being undertaken in the Wadden Sea to assist in the establishment of the boundary conditions (‘SBW-Veldmetingen’).
In this context, hindcast studies have recently been carried out with SWAN for the Wadden Sea (WL 2006, Royal Haskoning 2006 and WL 2007b), in which model results were compared with buoy observations taken in the Amelander Zeegat tidal inlet. The objective of these studies was to determine the predictive skill of SWAN for a number of severe storm conditions, including a range of wind and wave directions, high water levels and strong tidal currents through the Amelander Zeegat tidal inlet. These hindcasts indicated that, at the buoy locations positioned around the tidal inlet, the largest discrepancies between model results and observations are found at short fetches at the lee of the barrier islands, and in the main tidal channel ('Borndiep'). Wave-current interaction was shown to have a significant influence on wave conditions in the tidal channel. In particular, under conditions of opposing current, wave heights and mean absolute wave periods were found to be overestimated by the model (Alkyon 2007; WL 2007b). A subsequent sensitivity analysis (WL 2007a) has shown that effects of wave-current interaction in the tidal inlet persist up to the mainland coast of Friesland (in particular changes to the wave period and direction). It is therefore important to verify the performance of wave-current interaction in SWAN, and to investigate refinements to its implementation. Two aspects of wave-current interaction are considered here, namely the dispersion of waves travelling on a current, and the dissipation of waves that are steepened due to an opposing current.
Comprehensive reviews on the problem of waves propagating into a current are given by Peregrine (1976), Jonsson (1978), Peregrine and Jonsson (1983), Jonsson (1990) and Thomas and Klopman (1997). A specific aspect considered here is the influence of the current profile on the dispersion of the waves. Presently, the SWAN model assumes a depth-uniform current when computing wave-current interaction. This is a good approximation of nature in many applications, for example for tidal currents near a coast. However, due to stratification and/or shearing of the current, depth-uniformity is not always a good assumption, and in such situations one has to consider the depth-varying current V(z). High-frequency components, for example, may only be affected by the upper layers of the current field, whereas low-frequency components may experience influence of the current over the entire water column. Dingemans (1997) gives an analytical expression for the dispersion relation in the case where the current varies linearly with depth, in other words where
V''(z) = 0, where the double prime denotes the second derivative with respect to the depth.
follow the formulation of Kirby and Chen (1989), as presented by Dingemans (1997), who produce an expression for the first-order correction of the phase velocity of a given wave
component. This can also be interpreted as the weighted mean of the depth-varying
current V(z) over the water column, as experienced by this wave component. This weighted mean velocity is the velocity that enters the dispersion relation to first order. In contrast to these solutions to the 'gradually varying' problem of wave-current interaction, Swan and James (2001) present an analytical solution to the interaction between a regular wave train and a depth-varying current, appropriate to the equilibrium solution arising from an established wave-current interaction. In their model, the interacting current can vary arbitrarily with depth, and can have significant vorticity distribution. In particular, it considers the case where swell propagates on a wind-driven surface current. The model of Swan and James (2001) is not considered in this study, since such wind-driven surface currents were not present in the (observed and computed) flow fields considered here. Concerning the modelling of the wave energy, the ray description of water waves on an ambient current has been initiated by the introduction of the concept of radiation stress by Longuet-Higgins and Stewart (1960, 1961, 1964). Bretherton and Garret (1968) formulated this in terms of wave action. When a wave field meets an adverse current, the ray description predicts an increase in wave amplitude owing to the reduction in wave energy transport velocity in the direction of propagation. The resulting steepening of the waves may lead to wave breaking. However, in wave blocking situations, because of the vanishing value of the energy transport velocity at the blocking point, the ray theory predicts infinite wave amplitude at the blocking point (a singular solution). Observations of waves breaking on an adverse current have been reported by Lai et al. (1989), Chawla and Kirby (1998, 2002) and Suastika et al. (2000). Concerning modelling, Ris (1997) have indicated that the wave energy dissipation is underestimated in SWAN in (near) blocking situations, leading to a strong overestimation in the significant wave height. This may be due to insufficient modelling of the enhanced wave energy dissipation due to wave breaking on strong adverse currents, but also due to the inadequacy of the adopted ray approximation for the waves near the caustics. Dedicated models for wave energy dissipation of waves breaking on adverse currents have been presented by Ris (1997), Chawla and Kirby (1998, 2002) and Suastika (2004), all based on the bore-based model of Battjes and Janssen (1978). In his model, Suastika (2004) distinguishes between the dissipation in the region running up to the blocking point (the far field, where the ray approximation is still valid) and at the blocking point itself (near field, where a singularity is found). We note, however, that wave blocking is not the main focus of the present study.
The aim of this study is to investigate the general accuracy of wave-current modelling in SWAN. Initially, this study was directed at assessing whether the modelling of wave-current interaction can be improved by the use of depth-varying current fields, integrated using the method proposed by Dingemans (1997), as opposed to depth-averaged fields. However, since initial results have shown the critical role of wave dissipation in adverse currents, a number of options for the modelling of wave dissipation have also been considered. The objective was therefore broadened to investigate the influence on model accuracy of using depth-varying currents in combination with a number of available expressions for wave dissipation in ambient currents.
performance of SWAN is first checked for a collection of analytical situations of wave-current interaction. These feature interaction with following, opposing and slanting wave-currents, but no dissipation of wave energy. Subsequently, the modelling of dissipation as a result of the steepening of waves due to Doppler-shifting is investigated. For this, the laboratory flume experiments of Lai et al. (1989) and Suastika et al. (2000) are considered, which are designed to investigate wave blocking. These expressions feature strong opposing currents, which cause Doppler-shifting, dissipation, and a degree of wave blocking. Due to the experimental technique of Suastika et al. (2000), the current profiles are, however, uniform over the water depth. In the SWAN simulations of this experiment, wave-current interaction is considered in combination with a number of alternatives for whitecapping dissipation. Lastly, to verify the performance of wave-current interaction in SWAN in field applications, a field case recorded at Port Phillip Bay Heads, Australia, is investigated. This field case features swell-dominant wave fields, under the influence of strong following and opposing currents in the narrow Heads leading into Port Phillip Bay. The current fields at Port Phillip Bay Heads have a clear variation in current velocity with depth, so that the influence of including depth-varying current in SWAN can be assessed. For this investigation, an implementation in the Delft3D modelling suite was made that communicates a computed 3D current field to SWAN, which subsequently computes the frequency-dependent effective current using Dingemans (1997). In the simulation of this field case, the influence depth-varying currents are investigated together with the impact of alternative dissipation expressions.
This study was carried out by André van der Westhuysen and Giles Lesser. The internal quality assurance and review was carried out by Ap van Dongeren, and the external review was done by Ad Reniers (University of Miami/Delft University of Technology).
2
Formulations for wave-current interaction
This section presents the default and new implementations for wave-current in SWAN investigated in this study. The investigated implementations comprise, firstly, the introduction of a frequency-dependent effective current based on a depth-varying current velocity field and, secondly, the enhancement of dissipation in the presence of (counter) current. Full details of the implementation of these expressions in the SWAN source code are given in Appendixes A and B.
2.1
Default wave-current interaction
In the stationary mode of SWAN, phase-averaged quantities of the wave field are modelled using the action balance equation, given by (Booij et al. 1999):
tot x g
c N c N S
c U N . (2.1)
The first term of (2.1) denotes the propagation of wave energy in two-dimensional geographical x-space, with cg the group velocity and
U
the ambient current velocity. The second term represents the effect of shifting of the intrinsic radian frequency due to variations in depth and mean currents. The third term represents depth-induced and current-induced refraction. The quantitiesc
andc
are the propagation velocities in spectral space ( , ), in which is the wave propagation direction. The right-hand side contains the source termS
tot that represents all physical processes that generate, dissipate or redistribute wave energy.The rates of change in geographical and spectral space are obtained using linear wave theory and the conservation of wave crests (Whitham 1974, LeBlond and Mysak 1978 and Mei 1983): 2
1
2
1
2
sinh 2
gdx
kd
k
c
U
U
dt
kd
k
(2.2) gd
d
U
c
U
d
c k
dt
d
t
s
(2.3) 1 d d U c k dt k d m m , (2.4)0 1 ( , ) ( , , ) d U x y V x y z dz d . (2.5)
The use of depth-averaged current in (2.2) to (2.4) does not take into account the depth into the water column over which a wave component of a particular frequency is affected by the current field. High-frequency components, for example, may only be affected by the upper layers of the current field, whereas low-frequency components may experience influence of the current over the entire water column. Expression (2.5) may therefore only be appropriate for the latter wave components.
2.2
Frequency-dependent effective current
Dingemans (1997, p.86) presents an approximate method, originally proposed by Kirby and Chen (1989), for incorporating the influence of a depth-varying current field on the wave spectrum. In this method, the effective current experienced by each wave frequency is obtained by: 0 2 ( , ) ( , , ) cosh 2 ( ) sinh 2 d k U x y V x y z k z d dz kd , (2.6)
Expression (2.6) effectively weights the orbital velocity of the wave at a particular depth with the magnitude of the current at that same depth, and integrates this product over the entire water column. For a logarithmic current profile, higher frequency components, whose orbital velocities are contained to the upper layers of the water column, would therefore have a higher effective current when using (2.6) than when using the expression (2.5) applied in the default version of SWAN. For low-frequency components, on the other hand, (2.6) is expected to yield similar results to (2.5).
In this study, expression (2.6) was implemented in the pre-processing routines of SWAN (see Appendix A). When provided with a depth- and space varying (3D) current field, the effective current is computed per frequency using (2.6) at each geographical grid point. Effectively, the depth-dependent current field is transformed into a frequency-dependent current field. The latter is used in the expressions (2.2) to (2.4).
2.3
Dissipation due to current interaction
expression is more effective in the modelling of the rapid dissipation occurring near the blocking point. The bore-based expression used by (Ris 1997) is formulated in analogy to the Komen whitecapping expression, and reads:
2 max , , BJ ds b s k S C Q E s k (2.7)
where
k
is the mean wavenumber computed with Tm01, the mean steepness iss
k E
totand is the mean relative radian frequency. The proportionality coefficient Cds = 2 BJ/ ,
where BJ is set to 1. The variable Qb is the fraction of breaking waves, determined by
2
1
8
ln
b tot b mQ
E
Q
H
, (2.8)in which a maximum wave height Hm is defined based on a limiting steepness
max 2 m s H k . (2.9)
The limiting steepness smax is set to 0.14, based on Miche’s criterion for the limiting
steepness of an individual breaker. We note that Chawla and Kirby (1998) show that when propagating on a counter current, waves can break at a lower steepness than this. However, this finding is not incorporated in the present study. Ris (1997) demonstrates that expression (2.7) enhances the dissipation for waves exceeding a mean steepness of s = 0.08, such as can occur in strong adverse currents. On the other hand, Ris (1997) shows that under active wave growth situations, young, steep waves are excessively dissipated when (2.7) is used to supplement the Komen formulation. As a result, the roller-based expression has not been taken up in the default SWAN model. We note that in addition to the enhanced dissipation leading up to the blocking point, SWAN also includes dissipation at the blocking point itself: the caustic at the wave blocking point is solved by artificially removing wave energy from the model where the spectral density exceeds a threshold value due to the blocking. Since wave blocking is not the focus of this study, we did not alter this treatment of the blocking point in the model.
The present study considers, in addition to the Komen expression, the performance of the saturation-based whitecapping expression of Van der Westhuysen (2007), henceforth referred to as the Westh formulation. As will be shown in the next section, the Westh expression, like the Komen formulation, is provides too little dissipation in the presence of adverse currents. Therefore, the whitecapping expression Westh is supplimented by (2.7) in situations with ambient current. This is achieved, in analogy to the implementation of Ris (1997), by
,
max
,
wc wc WESTH BJ
S
S
S
, (2.10)young wind sea identified by Ris (1997), a switch has been implemented to activate (2.7) only where a significant following or opposing ambient current is present. This is done by increasing the maximum wave height, Hm, in the expression for Qb in conditions with weak
or no current: ,min
.max 1 ,
g m mc
H
H
U
(2.11)This altered maximum wave height H’m replaces Hm in (2.8). Therefore, where the current is
weak relative to the minimum computed group velocity (|cg,min/
U
| >> 1), the maximumwave height is increased (H’m/Hm >> 1), so that the fraction of breakers Qb and hence the
roller-based dissipation SBJ are suppressed. Through cg,min the switch is linked to the highest
frequency in the computed spectral range, since this is the first component to experience shoaling an adverse current. This combined expression is referred in the following as the
WBJ formulation. We note that this additional bore-based expression does not affect
3
Simulations
This section presents a number of simulations, of increasing complexity, in which the performance of the default and newly-proposed implementations for wave-current interaction are evaluated. These cases range from standard analytical tests (Section 3.1), to laboratory flume tests (Section 3.2), and finally to a field situation of a tidal inlet (Section 3.3).
3.1
Analytical tests
The first step in the evaluation is to compare the model results of SWAN against well-established analytical expressions for the effect of current on the wave field (current-induced refraction and shoaling), when neglecting dissipation. These tests are taken from the so-called ONR Testbed for wave models (Ris et al. 2002). The situation modelled is a monochromatic, long-crested wave field, with a wave height of 1.0 m and period of 10 s, in the presence of following, opposing and slanting currents. This situation is modelled in SWAN by setting Hm0 = 1.0 m and using a narrow Gauss distribution in frequency space
around Tp = 10 s, and a directional distribution of cos 500
( ). In all these simulations, SWAN is run over a propagation distance of 4000 m, with a spatial discretisation of 40 m in the propagation direction, and spectral discretisations of f/f = 0.1 and = 10 to 20.
The first case considered is that of a non-uniform following current that increases in the down-wave direction from 0 to 2 m/s, shown in Figure 3.1. This figure also presents the simulation results by SWAN, as well as the analytical solution for the significant wave height. The latter is given by (e.g. Phillips 1977; Jonsson 1990):
2 2 0 2 0
(
2 )
c
H
H
c c
U
(3.1) where 1 2 0 01
1
1 4
2
2
c
U
c
c
(3.2)in which the subscript 0 indicates reference values outside the current. Figure 3.1 shows that under the influence of the non-uniform following currents the analytical solution of the
significant wave height reduces from Hm0 = 1.0 m at the up-wave boundary to about
Hm0 = 0.8 m over a propagation distance of 4000 m. The theoretical value of the peak
absolute wave period remains constant over the domain. Figure 3.1 shows that the significant wave height is well reproduced by SWAN. Since in this case the dissipation of wave energy plays little or no role, the model results are not noticeably influenced by the choice of whitecapping expression. The simulated peak absolute period Tp remains constant,
in agreement with theory.
analytical solution to this case is given by (3.1) and (3.2) above, and reproduced in Figure 3.2.
Under opposing non-uniform current, the analytical solution of the significant wave height increases from Hm0 = 1.0 m at the up-wave boundary to about Hm0 = 1.4 m at the
down-wave end of the domain. The theoretical value of the peak absolute down-wave period remains constant over the domain. In Figure 3.2 it can be seen that SWAN overestimates the increase in significant wave height somewhat (with both the Komen and Westh whitecapping expressions), but reproduces the constant peak absolute period Tp well.
The third analytical case considered is that of waves propagating at an angle over a non-uniform opposing current field that increases in strength from 0 to 2 m/s in the positive y direction (Figure 3.3, top panel). The analytical solutions for significant wave height and wave direction of this case are given by, for example, Hedges (1987) and Jonsson (1990) and reproduced in Figure 3.4:
0 0 2 0 0 cos( ) arccos [ cos( )] gk Uk (3.3) and 0 0 sin(2 ) sin(2 ) H H (3.4)
Figure 3.4 shows that under the slanting, opposing current, analytical result of the mean wave direction is shifted by about 4 degrees over a propagation distance of 4000 m. Also, the significant wave height is somewhat increased (by about 0.04 m). The theoretical value of the peak absolute wave period again remains constant over the domain. The SWAN results of these parameters are in excellent agreement with these analytical results.
The final analytical case considered is that of waves propagating at an angle over a non-uniform following current field that increases in strength from 0 to 2 m/s in the positive y direction (Figure 3.3, bottom panel). The analytical solutions for the significant wave height and wave direction of this case are given by (3.3) and (3.4), and reproduced in Figure 3.5, together with the peak absolute period (constant). Figure 3.5 shows that under the slanting, following current the mean wave direction is shifted by approximately 5 degrees over a propagation distance of 4000 m. Over this distance, the significant wave height is reduced somewhat (about 0.03 m). Figure 3.5 shows that these alterations to the wave field are well reproduced by SWAN, with only a small (0.2 degree) difference between the simulated mean direction and the analytical solution.
3.2
Laboratory cases
3.2.1 Experiment of Lai et al. (1989)
Lai et al. (1989) investigated transformation of the wave spectrum under strong adverse currents in an 8 m-long, 0.75 m-deep flume. A current flow was induced along the flume, which was contracted by the presence of a shoal (bottom panel of Figure 3.6), resulting in a local increase in the current velocity from U = -0.13 m/s to -0.22 m/s over the shoal. Waves were mechanically generated at the downstream end of the flume. In the case considered here, the incident wave field had a significant wave height of Hm0 = 0.019 m and a mean
period of Tm01 = 0.5 s. The top panels of Figure 3.6 show that the observed significant wave
height reduces strongly over the shoal due to blocking by the counter current. Correspondingly, the observed absolute mean wave period increases going over the shoal. Figure 3.7 shows that this increase in the mean period is due to the blocking of wave components with absolute frequencies higher than approximately 2 Hz.
This experiment was simulated in the one-dimensional mode of SWAN (parallel depth contours), with discretisations in geographical and spectral spaces of x = 0.02 m, f/f = 0.1 and = 0.50, by which a long-crested wave field was approximated. The source terms for triad interaction, depth-induced breaking and whitecapping dissipation were activated. The first two source terms, which had small magnitudes, were applied with their default settings. For whitecapping, the Komen, Westh and WBJ expressions were applied in three separate simulations. A small amount of under-relaxation ( = 0.01) was applied to remove an instability in the model (oscillations in results over consecutive iterations), which occurred irrespective of the whitecapping expression used.
Figure 3.6 presents the simulation results using the three different expressions for whitecapping dissipation. It can be seen that the simulations with both the default (Komen) and the Westh whitecapping expressions produce a strong increase in the significant wave height on the upslope of the shoal, where the counter current strongly increases in the direction of the waves. Here no observations are available to verify the model results. The wave height in both the Komen and Westh simulations decreases upon reaching the top of the shoal, and again moving over the downslope. Over these regions the wave height is
overestimated by both model variants. The mean absolute wave period Tm01 is locally
with the observations. In particular, the wave height and period results do not display the strong respective increases in decreases at the upslope of the bar (where observations are unfortunately not available). These results agree with those presented by Ris (1997). Figure 3.7 shows the corresponding wave spectra produced using the WBJ expression. It can be seen that the spectral shape agrees well with the observations, and that energy levels are reduced by about 50% relative to the Komen and Westh simulations. However, down-wave of the bar the observed energy density is still overestimated.
3.2.2 Experiment of Suastika et al. (2000)
The second laboratory flume experiment considered is that of Suastika et al. (2000), who studied partial and complete wave blocking under adverse current. Suastika et al. (2000) used a 35 m long flume, with a 12 m measurement section at its centre. Waves were mechanically generated at the one end of the flume. A water head difference induced current flow along the flume, running in the up-wave direction. Within the measurement section, the discharge entering at the upstream (and down-wave) end was gradually withdrawn through the bottom of the flume by pumps, so that the current was zero at the downstream (up-wave) end of the measurement section. The result was a counter-current that reduced approximately linearly in the up-wave direction. At the respective ends of the flume, the total flume width and height (0.8 by 1.0 m) were available to the waves and current, but within the measurement section the flow was contracted to 0.4 m by 0.7 m by a false wall and bottom. This false perforated bottom allowed withdrawal of discharge into the adjacent dummy half of the flume, which acted as a sump for the suction pumps. However, the suction of discharge through this false bottom had two additional consequences: firstly, the current profile became depth-uniform due to the flow extraction and, secondly, the perforated false bottom introduced an additional source of dissipation (see below).
3.2.2.1 Experimental conditions
Suastika (2004) reports the measurement results of both periodic and random waves for both partially and fully blocking situations of this experiment. Of this test program, two cases are considered in the present study, namely random wave conditions featuring full and partial blocking, referred to here as conditions S1 and S2 respectively. The details of these conditions are given in Table 3.1 below.
Case Hm0 Tp Q Condition
[m] [s] [m3/s]
S1 0.05 1.1 0.12 Fully blocking
S2 0.05 1.1 0.078 Partially blocking
Table 3.1: Experimental conditions of Suastika (2004) considered.
Firstly, we consider the observed results of these two cases, as presented by Suastika (2004). Figure 3.8 presents the evolution of the significant wave height and mean period of cases S1 and S2, along with their respective current profiles. Concerning first the fully blocking condition S1, it is seen that the significant wave height increases steadily up to about
x = 22 m as the waves advance in the upstream direction. Down-wave from this point the
wave energy is dissipated almost entirely. The mean wave period of this condition is seen to first reduce as the increasing adverse current is met, due to the shoaling of the higher frequency components. However, as the blocking region is approached, the mean period increases, as progressively lower frequencies become blocked. The partial blocking condition S2, on the other hand, displays a more modest increase in the significant wave height moving in the upstream direction. After reaching a maximum at x = 23 m, the significant wave height decreases gradually. The mean wave period can be seen to first reduce upon meeting the adverse current, after which it increases. However, since the spectrum is only partially blocked, the increase in frequency is not as strong as for Case S1. Figure 3.9 presents the corresponding spectral development along the flume for the two cases. The top panel of Figure 3.9 shows the spectral evolution of the fully blocking condition S1. The initial transformation of the spectrum is an increase of the variance density of the high frequencies, due to shoaling on the current (e.g. x = 16.2 m). Subsequently, these frequencies become blocked, and frequencies at the spectral peak strongly increase in energy density. Proceeding towards the blocking region, frequencies at the spectral peak are first shoaled and then blocked, resulting in a down-shift and strong reduction of the spectral peak. The bottom panel of Figure 3.9 presents the spectral evolution of the partially blocking case S2. As above, it can be seen that the higher frequencies are the first to increase in energy density due to the adverse current. Subsequently, the energy densities at the spectral peak increases somewhat up to about
x = 23 m, where the maximum significant wave height was found (Fig 3.8). From this point,
it can be seen that energy at frequencies higher than about 1.2 Hz are zero, due to the wave blocking. At the spectral peak, the energy density is reduced somewhat, but no components are blocked, so that there is practically no shift of the initial spectral peak to lower frequencies as in Case S1.
3.2.2.2 Modelling
Suastika (2004) modelled the random wave conditions S1 and S2 with a spectral wave model featuring a range of dissipation mechanisms. These include energy dissipation in the side wall boundary layers, dissipation due to orbital motion through the perforated false bottom, and dissipation due to wave breaking (based on the bore-based model of Battjes and Janssen (1978)). These dissipation mechanisms were derived both for the portion of the flume leading up to the blocking point (far field dissipation) and at the blocking point itself (near field dissipation). The latter treatment is required for the caustic in the wave amplitude that forms at the blocking point. Here, a non-singular solution for the wave amplitude is obtained by the retention of extra partial derivative terms, called high-order dispersive terms (Whitham, 1974), see Suastika (2004).
3.2.2.3 Full blocking
Figure 3.10 presents the observed and simulated significant wave height and mean period for the full blocking case S1. It was seen above that the significant wave height increases up to the blocking region due to the adverse current. The results of the SWAN simulations show that both the Komen and Westh expressions strongly overestimate this observed increase wave height in the adverse current. Furthermore, the decrease in wave height leading up to the blocking point appears to start too early along the length of the flume, especially when using the Westh whitecapping expression. Similarly, the mean wave period undershoots the observations in the region of x = 15 m, but then increases more strongly than observed values further upstream. Figure 3.10 also presents the simulation results obtained with the WBJ breaking expression, to show the effect of enhanced dissipation. This dissipation term yields much lower values of significant wave height, which are in better agreement with the observations. However, the mean wave period still overestimates observed values, although the undershoot in the region of x = 15 m is reduced.
Figure 3.11 presents the corresponding simulated spectra. It can be seen that in the simulations with Komen and Westh the energy density at higher frequencies are strongly overestimated before they are blocked (e.g. x = 16.2 m), resulting in the overestimation of significant wave heights seen above. The observed energy levels are better reproduced when using the WBJ dissipation expression. Beyond x = 16.2 m, it can be seen that in the model frequencies are blocked that are not yet blocked in the observations, resulting in a stronger down-shift in the model than in the observed spectra. This overestimation of the down-shift corresponds with the overestimation of the mean period in this region, and to the inaccurate prediction of the location of the blocking region along the flume. Saustika (2004) argues that in the observations the blocking of waves is delayed due to the increase of the phase velocity through amplitude dispersion, as described for example by Stokes higher order wave theory. Previously, Chawla and Kirby (1998, 2002) have shown the importance of taking amplitude dispersion into account in the wave action propagation velocities of models for wave-current interaction. Not only can the location of the blocking point be affected, but wave heights and periods leading up to the blocking point can also be altered. Since in SWAN the linear dispersion relation is applied, the waves in the present simulation are blocked further downstream in the flume, and at a lower counter current velocity, than in the observations. Wave heights and period measures leading up to the blocking point are potentially also affected, but this cannot be verified from the present results. We note that in cases with smaller wave height, for which amplitude dispersion effects are less, the location of the blocking region is predicted better by SWAN (results not shown). We conclude that the absence of amplitude dispersion effects in SWAN could negatively affect the accuracy of model predictions, in particular for higher wave heights.
3.2.2.4 Partial blocking
the flume, but overestimated by about 0.1 s at the down-wave end of the flume (x = 24 m). Figure 3.12 shows that the significant wave heights produced using the WBJ dissipation expression are markedly lower than the other model variants, but that it is still significantly overestimated. We note that in this partially blocking case, the additional (artificial) dissipation afforded by the treatment of wave blocking in SWAN has a much smaller overall effect than in the fully blocking Case S1. The underprediction of the dissipation by the whitecapping terms is therefore more clearly seen here. By applying the WBJ expression, the prediction of the mean wave period is improved in the region x = 16-18 m, but gives a worse prediction than the other simulations towards the end of the flume. The location of the partial blocking region around x = 23 m is well predicted in this case, although the simulated decrease in wave height appears too abrupt. Figure 3.13 presents the corresponding simulated wave spectra for Case S2. Concerning the Komen and Westh results, it can be seen that along most of the flume the simulated spectral shape agrees well with the observations, but that they contain too much energy. This is especially the case for higher frequencies, which are strongly shoaled, and evidently not dissipated sufficiently (x = 18.2 m and 20 m). The simulation using WBJ strongly improves the agreement with the observations, in particular at the higher frequencies. However, at x = 22 m and beyond, all model variants predict that components higher than 1 Hz are blocked, causing a down-shift in the simulated spectra that is not found in the observations.
The results presented in this section show that when using the conventional whitecapping expressions in SWAN (Komen and Westh), greater wave steepnesses are allowed in the model than in reality, leading to the strong overestimation of significant wave heights. Applying a more energetic wave breaking expression based on the bore-based model of Battjes and Janssen (1978) alleviates this problem. However, in the partial-blocking Case S2, the absence of the additional (artificial) dissipation due to the treatment of blocking in SWAN highlights remaining inadequacies in the dissipation expressions. However, the remaining difference between model results and observations may in part be ascribed to the additional dissipation mechanisms particular to the experimental setup (side wall dissipation and dissipation due to the perforated false bottom). An overestimation of the mean wave period is found in strong adverse current, which is stronger in fully blocking conditions than for partial blocking. This appears to be directly related to the premature blocking of frequencies, which in turn can be related to the use of the linear dispersion relation in SWAN and hence the omission of amplitude dispersion effects.
3.3
Field case of Port Phillip Heads
The Heads of Port Phillip Bay, Melbourne, Australia (PP Heads) (Figures 3.3-1 and 3.3-2) are subject to strong tidal currents and an energetic wave climate. The tidal flows at PP Heads are predictable and are approximately aligned with the approach direction of the incoming swell waves. Typical depth-averaged ebb and flood current velocities reach 2 to 3 m/s and offshore significant wave height is typically in the range 1 to 4 m with a peak swell period in the range of 10 to 20 s. Shorter period wind-sea components may also be present.
sometimes limited by conditions in PP Heads. As such, considerable effort has been expended by the Port of Melbourne to understand the wave and tidal conditions in PP Heads and this has resulted in the collection of high-quality data with synchronous measurements of offshore wave conditions and inshore tidal current velocities and directional wave spectra at several locations within PP Heads. The bathymetry within PP Heads is also well known thanks to recent high-resolution surveys.
The extensive dataset of wave and current conditions collected in PP Heads is an extremely valuable resource for validating and/or improving models of wave-current interaction. This study uses this unique dataset to evaluate the performance of the wave-current interaction formulations implemented in the SWAN wave model.
3.3.1 Data available for model comparisons
Port of Melbourne Corporation has collected offshore wave data at PP Heads for many years using Triaxys wave buoys. Two wave buoys are permanently deployed, in close proximity to each other, approximately 10 km southeast of PP Heads (Figure 3.3-2, blue dots). These buoys measure waves that are unaffected by the tidal flows in and out of Port Phillip Bay and produce directional wave spectra based on 20 minute data bursts every half hour. Wave parameters and directional spectra are analysed on the buoy and are transmitted to a data collector at the Port of Melbourne Shipping Management Centre via VHF radio. Due to the limitations of the buoys’ power supplies, the full two-dimensional wave spectra are not transmitted to shore. Instead the “MeanDir” or “1.5D” spectrum consisting of the energy, mean direction and directional spreading width for each frequency band is transmitted and stored. “MeanDir” wave spectra are available for the “Offshore” Triaxys buoy location since July 2004.
Waves and currents in PP Heads are measured using Nortek AWAC instruments bolted to the rocky channel floor on the centreline of the shipping channel in approximately 18 m of water (Figure 3.3-2, red dots and Figure 3.3-3). The AWACs record velocity profiles in 1 m vertical bins every 10 minutes and perform a 17 minute wave burst every hour. A unique feature of the AWAC instrument is that it incorporates a vertical acoustic surface-tracking beam sampled at 2Hz which allows accurate measurement of the water surface elevation even under the challenging conditions experienced in PP Heads. Other instruments, either surface buoys or bottom-mounted instruments, tend to be strongly affected by the tidal current and produce questionable results during times of high current speed. Pairs of AWAC instruments have been deployed in PP Heads for 6-week deployments more or less continuously since February 2006. The locations of the AWAC deployments considered for this study are known of as Rip Bank Outer (RBO) and Rip Bank (RB) and are shown overlaid on the SWAN model bathymetry in Figure 3.3-13.
Wave conditions within PP Heads are strongly affected by the tidal streams through PP Heads and fluctuate much more rapidly than the offshore wave conditions. Figure 3.3-4 presents a typical time series of offshore and inshore wave heights. The offshore Triaxys buoy data shows no effect of tidal conditions whereas inshore, at the AWAC location, a strong tidal signature in the recorded wave height is clearly visible.
and Lorne (refer Figure 3.3-2). A recent high-resolution (2 m) digital bathymetry obtained by multi-beam hydrographical survey was also provided by the Port of Melbourne.
3.3.2 Method
3.3.2.1 Approach
This study aims to test the ability of the SWAN wave model to capture the influence of currents on the wave field measured at PP Heads. As well as testing the standard formulations available in SWAN, we also seek to test two new whitecapping formulations (Westh and WBJ, as described in Section 2.3) and an improved vertical integration of current (Dingemans (1997), as described in Section 2.2). In order to achieve these aims we followed the following procedure:
1. Inspect the available data and select times of interest (see Section 3.3.2.2 below). 2. Run a hydrodynamic (tidal) model for the times of interest and store 3D tidal
current fields for use in SWAN.
3. Set up a SWAN model covering the area of interest.
4. Back-transform wave spectra for the times of interest from the Triaxys buoy location to the offshore boundary of the SWAN model.
5. Simulate wave conditions for consecutive ebb, slack, and flood tides for each time of interest.
6. Check that the SWAN model was correctly set up by comparing computed and measured wave spectra at the Triaxys buoy location.
7. Repeat SWAN simulations for the various current integration and whitecapping formulations to be tested.
8. Extract SWAN output and compare simulations with each other and with the measured wave conditions to evaluate the performance of the SWAN model under the selected conditions and for the various formulations.
These steps are reported in more detail in the following sections.
3.3.2.2 Selection of test simulations
during the 7.5 hours of Storm 1 are overlaid. The mean values were extracted and used to produce model boundary conditions.
Storm 2: The second storm identified was later rejected from selection and forms no further part of this analysis or report.
Storm 3: The second storm selected, “Storm 3”, (27/10/2006 20:10 - 28/10/2006 03:20 AEST) occurs during a time of high distant swell coinciding with a strong local wind. Offshore significant wave height was approximately 4.4 m with a spectral peak period of approximately 12 seconds. Wind at Point Lonsdale was strong at 17m/s and the wind direction at Point Lonsdale gradually veering from 270 to 220 degrees during the course of the storm (Figure 3.3-8). Although the wind direction varies slowly, the measured offshore waves are again reasonably consistent (Figure 3.3-9) and, despite the gradual increase in wave energy observed during the storm, are again assumed constant for the duration of the storm in order to simplify comparisons of inshore wave conditions between ebb and flood tide. Figure 3.3-10 shows the scatter around the mean spectral parameters over the duration of the storm. The scatter in spectral wave energies recorded at the Triaxys buoy during this storm is high. This is predominantly noise in the spectral analysis performed by the buoy as the area under the spectrum (m0, and hence the significant wave height) doesn’t fluctuate nearly as widely during the storm (Figure 3.3-9).
During each storm the time of peak ebb and flood tides were identified, along with the time of near-slack. These six times were selected as times for model – measurement comparison. For each time multiple simulations are required however as each condition is simulated for three different current integration techniques (no current, depth averaged, and frequency-dependent depth weighted according to the method of Dingemans – as described in Section 2.2 of this report). Multiple simulations are also required to test the effect of the three different whitecapping formulations implemented (Komen, Westh, and WBJ – as described in Section 2.3).
All in all these factors result in a matrix of 54 separate SWAN simulations to test the various combinations of conditions and modelling approaches. The results of these simulations are presented in Section 3.3.3 below.
3.3.2.3 Model setup
Bathymetry data were obtained from a number of sources and adjusted to depths below Australian Height Datum (AHD, approximately equivalent to mean sea level). These were then interpolated onto the SWAN model grid using WL | Delft Hydraulics’ QUICKIN software using a combination of spatial averaging and interpolation. Raw bathymetry data resolution varied from around 2 m in PP Heads up to > 1 km near the outer edges of the model domain. The resulting model bathymetry is shown in Figure 3.3-12. The deep canyon cutting through PP Heads is clearly resolved in the model bathymetry, but falls landward of our AWAC locations of interest. Of more importance, the submerged ebb tidal delta is clearly visible offshore with a crest elevation of approximately -18m. The outer boundary of the SWAN model lies in approximately 80 – 90 m of water along most of its length.
Other SWAN model parameters were left at their “default” settings, as noted in Table 3.2 below
Parameter Setting
Directional resolution 10 degree bins over full circle
Frequency resolution 50 bins spread from 0.03 Hz to 1.0 Hz
Wind Spatially constant, per storm
Depth induced breaking Battjes and Janssen (1978), BJ = 0.73
Bottom friction JONSWAP, CJON = 0.067 m2s-3
Triads LTA, EB = 0.1, CUTFR = 2.2
Diffraction Deactivated
Wind growth Activated
Whitecapping Activated – Komen, Westh, or WBJ
Quadruplets Activated
Refraction Activated
Frequency shift Activated
Numerical convergence criterion 30 iterations required
Table 3.2: SWAN parameter and configuration settings.
Boundary conditions were obtained by trial and error modification of the wave spectra measured at the offshore Triaxys location. These are provided to SWAN as 1.5D spectral boundary condition files. Identical spectra are applied to offshore, northeast and southwest boundaries of the SWAN model.
For Storm 3 the observed wind direction of between 270 and 220 degrees at Point Lonsdale had to be modified to 215 degrees in order to get the direction of the higher frequency wave components to match the observations at the Triaxys buoy location. We therefore assume that the wind directions measured at Point Lonsdale are not representative of the bulk of the wind acting on the fetch length up-wave of the Triaxys buoy location. This is not felt to be a serious limitation as the measured wave spectra at the Triaxys buoy location are recovered well by the model (see Section 3.3.2.4 below).
on the model bathymetry. The depth profile along Curve01 is shown in Figure 3.3-20. The measurements at the locations of the AWAC instruments are indicated by red stars in the cross section figures.
3.3.2.4 Validation of model setup
Hydrodynamic (tidal) current fields were produced using an existing, calibrated, three-dimensional Delft3D model of Port Phillip Bay. The model uses 1 m thick constant z-layers in the vertical direction to prevent unrealistic vertical mixing occurring as the tidal flow crosses the 90 m deep canyon in PP Heads. The model is driven using measured tidal water levels from the Lorne tide gauge, spatially adjusted for the phase lag experienced by the tide along the outer model boundary. Wind forcing is not included in the model.
Figures 3.3-14 and 3.3-15 show time series of depth-averaged modelled and measured current speeds for each AWAC location for each storm period. The precise times to be used for the SWAN simulations of waves during ebb, slack, and flood are also identified in the figures. Vertical profiles of the measured and modelled current speed are shown in Figures 3.3-16 to 3.3-19. Figures 3.3-20 and 3.3-21 show cross sections of depth and long-and cross-channel current speed along Curve01 for both Storm 1 long-and Storm 3. Signed current speeds are designated as positive in the flood direction.
Generally the hydrodynamic model performs extremely well for the times and locations tested. The model performs better at RB than at RBO and does tend to slightly over-estimate ebb current speeds at the RBO AWAC location, however both the time series and profiles achieved by the model appear satisfactory for our purposes. The influence of the strong wind experienced during Storm 3 is just evident in the upper 2-3 m of the measured velocity profiles. This effect is not captured by the hydrodynamic model but is expected to be of little consequence for the relatively long-period swell waves that are the primary focus of this study.
The validation of the setup of the SWAN model is achieved by comparing measured and modelled wave spectra at the location of the offshore Triaxys wave buoy. Figures 3.3-22 and 3.3-23 contain these comparisons for Storm 1 and Storm 3 respectively. The figures also contain the spectra applied to the offshore model boundary; this indicates the degree of transformation of the waves occurring from the model boundary to the Triaxys buoy location. The model does an excellent job of recovering the measured spectra.
3.3.3 Results
3.3.3.1 Tidal currents
Maps of the depth average tidal currents computed by the Delft3D hydrodynamic model at the selected times of ebb, slack, and flood for Storm 1 are shown in Figures 24 to 3.3-26. The maps of currents for Storm 3 are not presented as they are of a similar nature. Maps of the frequency-dependent depth-weighted tidal currents created by the Dingemans approach to current integration are shown in Figures 3.3-27 to 3.3-32. The current fields are frequency dependent because the lower-frequency (longer period) waves “feel” the effect of the current to a greater depth than the high frequency (short period) waves – as described by Equation 2.6. Only the current fields for ebb and flood tide are presented. The variation in the current felt by the different frequencies of waves plotted (wave period from 5 to 20 s) only vary very subtly. The variations are logical, with the higher frequency (shorter period) waves feeling a somewhat higher current speed. Careful inspection also reveals that the longer period (low frequency) waves also feel a current that deflects more across the deep canyon in PP Heads. This is also to be expected as the long period waves will feel the deep transverse current flowing through the canyon to a much greater extent than the short period waves.
On the basis of these plots is seems unlikely that the modelled waves at locations RBO and RB will be altered very greatly by the choice between depth-averaged or Dingemans current applied in the SWAN model.
A good indication of the significance of the currents in terms of their effect on the action balance of the waves is the ratio of current speed to wave group (energy) velocity. This ratio is shown in Figures 3.3-33 to 3.3-36 for both the peak offshore wave period (Tp) and half
this period (Tp/2) for both Storm 1 and Storm 3 during ebb tide. For Storm 1 the peak
offshore wave period was 16.7 s for Storm 3 this reduces to 11.8s. The figures show that although the currents are strong, they do not completely block waves at either Tp or Tp/2. In
fact, for the worst case, current speeds only just reach half the group velocity of the Tp/2
(5.9 s) waves found in Storm 3. Clearly, waves at higher frequencies (shorter periods) will also be present and may be totally blocked; however energy levels at these frequencies are low in the offshore wave condition. These figures confirm that the conditions selected for modelling are those targeted by this study: the partial blocking of wave energy by opposing currents. The abrupt edge of the ebb tidal jet is clearly visible in the figures.
3.3.3.2 Wave height and energy dissipation
Cross-sections of significant wave height along Curve01 clearly contrast the effects of the different whitecapping formulations. Figures 3.3-49 and 3.3-50 compare the modelled wave heights for ebb, slack, and flood tides using depth-average current integration for each of the whitecapping formulations. Little difference is observed between formulations for both slack and flood tides, however on ebb tide the Komen and Westh formulations lead to a dramatic over-estimation of significant wave height at both AWAC locations. Use of WBJ whitecapping leads to a significant under-estimation of wave height on ebb tide. As the model somewhat over-estimates significant wave height during slack and flood tides when no whitecapping of these long swell waves is expected, these results probably also indicate that the model does not include sufficient dissipation due to bottom friction as there a few other processes that could dissipate the required energy during these conditions. This hypothesis is in line with previous (unpublished) experience using SWAN and the Jonswap bottom friction formulation in relatively shallow water depths.
Cross-sections of significant wave height are also useful to demonstrate the impact of changing the current integration method. Figures 3.3-51 and 3.3-52 compare the results of the different current integration methods for Storms 1 and 3 when using the Komen whitecapping formulation. As expected, changing current integration method makes virtually no difference near slack tide. During flood tide neglecting the effect of current decreases the model performance, however the difference between depth-average and Dingemans’ approaches is negligible. During ebb tide, neglecting the current altogether actually dramatically improves the model results as this removes the dramatic over-estimation of significant wave height caused by the lack of dissipation in the Komen whitecapping formulation. This (fortunate?) result should be regarded as two errors cancelling. Figures 3.3-53 and 3.3-54 compare the current integration methods for Storms 1 and 3 when using the WBJ whitecapping formulation. It is clear from these figures that the overall level of dissipation computed by the WBJ formulation is much closer to the measurements than when using the Komen formulation although in this case rather too much dissipation is computed during ebb tide.
To identify where significant energy dissipation is computed by SWAN maps of energy dissipation patterns are made. Figures 3.3-55 to 3.3-58 present dissipation patterns for Storm 1 and Storm 3 on ebb and flood tides, using the Komen whitecapping. Figures 3.3-59 to 3.3-62 present similar plots for the WBJ whitecapping formulation. Energy dissipation rates above 10 W/m2 are regarded as “high”. In some cases the formulations can result in extremely high rates of energy dissipation. These levels are more clearly seen in the cross section plots which follow.
