numerical modelling of wind wave energy dissipation at the bottom ...

KATHOLIEKE UNIVERSITEIT LEUVEN FACULTEIT WETENSCHAPPEN DEPARTEMENT GEOLOGIE–GEOGRAPHIE AFDELING HISTORISCHE GEOLOGIE FACULTEIT TOEGEPAST WETENSCHAPPEN DEPARTEMENT BURGERLIJKE BOUWKUNDE LABORATORIUM voor HYDRAULICA

NUMERICAL MODELLING OF WIND WAVE ENERGY DISSIPATION AT THE BOTTOM INCLUDING AMBIENT CURRENTS

Promotoren:

Proefschrift voorgedragen tot

Prof. J. Monbaliu

het behalen van het doctoraat

Prof. N. Vandenberghe

in the wetenschapen door ´ Roberto PADILLA–HERNANDEZ

Mei 2002

KATHOLIEKE UNIVERSITEIT LEUVEN FACULTEIT WETENSCHAPPEN DEPARTEMENT GEOLOGIE–GEOGRAPHIE AFDELING HISTORISCHE GEOLOGIE Redingenstraat 16bis, B-3000 Leuven FACULTEIT TOEGEPAST WETENSCHAPPEN DEPARTEMENT BURGERLIJKE BOUWKUNDE LABORATORIUM voor HYDRAULICA Kasteelpark Arenberg 40, B-3001 Heverlee

NUMERICAL MODELLING OF WIND WAVE ENERGY DISSIPATION AT THE BOTTOM INCLUDING AMBIENT CURRENTS

Jury: Prof. J. Berlamont, Voorzitter

Proefschrift voorgedragen tot

Prof. J. Monbaliu, Promotor

het behalen van het doctoraat

Prof. N. Vandenberghe, Promotor

in the wetenschapen

Prof. J. Battjes (TUDelft)

door

Dr. F.J. Ocampo–Torres (CICESE, México)

´ Roberto PADILLA–HERNANDEZ

U.D.C. 551.466 Mei 2002

c °2002 Faculteit Toegepast Wetenschappen, Katholieke Universiteit Leuven All rights reserved. No part of this book may be reproduced, store in a database or retrieval system or published in any form or in any way – electronically, mechanically by print, photoprint, microfilm or by any other means – without the prior written permission of the publisher Department of Civil Engineering, Catholic University of Leuven, Kasteelpark Arenberg 40, 3001 Heverlee, Belgium.

Wettelijk depot D/7515/2002/19 ISBN 90-5682-362-0

A Lilia Adrián, Andrea y Alejandro A mis Padres

Yo, Nezahualcóyotl, lo pregunto Acaso de veras se vive con ra´ız en la tierra? Nada es para siempre en la tierra: Sólo un poco aqu´ı. Aunque sea de jade se quiebra, Aunque sea de oro se rompe, Aunque sea plumaje de quetzal se desgarra. No para siempre en la tierra: Sólo un poco aqu´ı.

Nezahualc´ oyotl (1431-1472) Emperador Azteca

Abstract The exploitation and preservation of coastal zones has put some pressure to understand the complex dynamical interactions between all physical processes that take part in that zone. Wind waves are the most eminent form of energy in the oceans and they play a key role in coastal hydrodynamics. The wind waves link most of the physical (and perhaps biological) processes in the coastal zone, by affecting the vertical profile of the ambient current at the bottom, by taking an active role in sediment transport, by oxygenating the water column due to air–sea gas exchange through the turbulence generated at the surface and at the bottom, by helping to determine sites for productive reef development and shaping reef morphology as well as community structure, among other processes. For the simulation, forecasting and understanding of the evolution of the waves one of the most useful tools is numerical modelling. In this thesis the research (carried out partially under the EU–PROMISE project) about wind waves begun with the adaptation of the WAM wave model making it able to simulate the evolution of the wind waves in shallow water areas with high spatial resolution. Two aspects were considered for its improvement: the computational efficiency and the representation of the physical processes involved in the evolution of waves in shallow water areas. The resultant wave model was named WAM–PRO. A number of simple or idealized example applications were done to illustrate some of the enhancements included in WAM–PRO. As a result of all the changes and additions, it has become feasible and economical to explore the wave spectra evolution in coastal areas with the WAMC4–P model. When the waves propagate from deep water to shallow water areas, they are mixed up in several mechanisms which modify the environment and in turn the wave properties are affected. One of these interactions takes place with the ambient currents (by tides and surges). A module was developed to i

study wave–current interactions. This module enabled combined modelling of tides, surges and waves in shallow water at the North Sea scale. Another important interaction in which the waves take part is the wave– bottom interaction. This interaction is responsible for drastic changes in the wave field and, in a way, it determines how the interactions between waves and other phenomena, including those between waves and current. In this study an attempt is made to search for evidence to determine which friction formulation, amongst four, performs best or is more consistent in shallow water regions. Some models for wave energy dissipation by bottom friction are very elaborate and have some very fine qualities. Most of them are function of a roughness height, which most of the time, at least in wave forecasting, remains unknown for a given wave, current and sediment characteristics. In order to remove the roughness height to be constant in the bottom friction formulations, a new ripple geometry predictor (MN81) has been proposed. It is based on the model of Nielsen but including findings from other published investigations. The proposed model was coupled with the model of Christofferson and Jonsson (CJ85) for the wave–current bottom boundary layer. A set of measurements (waves, currents and ripples dimensions) in the field and in laboratory data (apparent roughness) were used to verify the ripple predictor MN81 and the WBBL model (CJ85). Without making specific assumptions for the different stages, the proposed MN81 model was able to fit the measurements in all ripple regimes. In turn the WBBL model (CJ85), using the ripple geometry from the MN81 model, reproduces the different flow regimes observed in the field and some parameters measured in laboratories by other researchers.

ii

Samenvatting De exploitatie en duurzaam beheer van kustzones heeft geleid tot enige druk om de complexe interacties die zich afspelen in deze zone beter te begrijpen. Windgolven zijn de meest zichtbare vorm van energie in de oceanen en ze spelen een belangrijke rol in de dynamiek van de kustzone. Windgolven zijn verbonden met de meeste fysische (en misschien ook biologische) processen in de kustzone. Ze benvloeden het stromingsprofiel aan de bodem. Zij spelen een aktieve rol bij sedimenttransport. Zij brengen zuurstof in de waterkolom door uitwisseling van gassen tussen lucht en water. En zij bepalen de plaatsen van productieve rifontwikkeling, om maar enkele processen te noemen. Numerieke modellering is een handig instrument om de evolutie van windgolven niet alleen te simuleren en te voorspellen, maar ook om ze beter te begrijpen. In dit onderzoekswerk over windgolven (gedeeltelijk uitgevoerd tijdens het EU–project PROMISE) werd eerst het WAM–golfmodel aangepast zodat het mogelijk werd om ook het golfveld met hoge spatiale resolutie in ondiep water te simuleren. Het resulterende model werd WAM–PRO genoemd. Een aantal eenvoudige en/of gedealiseerde toepassingen illustreren de in WAM– PRO aangebrachte verbeteringen. Als resultaat van al deze veranderingen en toevoegingen, is het nu mogelijk en vrij economisch om de evolutie van golfspectra in kustgebieden met het WAM-PRO model te berekenen. Wanneer golven zich van diep naar ondiep water voortplanten, ondergaan ze interacties met andere processen die aanwezig zijn in de kustzone. Twee belangrijke interacties zijn die tussen golven en stromingen en die tussen golven en de bodem. Er werd een module ontwikkeld om een golfmodel te koppelen aan een model voor stormopzet zodat de interactie tussen golven en stromingen kon worden bestudeerd. Met deze module is het mogelijk om gelijktijdig getij, stormopzet en golven op de schaal van de Noordzee te modelleren. De andere belangrijke interactie is de interactie aan de bodem. Deze interactie is verantwoordelijk voor veranderingen in het golfveld en bepaald tot op zekere hoogte iii

de interactie tussen golven en andere processen, o.a. tussen golven en stromingen. In dit onderzoek werd een poging gedaan om na te gaan welke van vier formuleringen voor bodemwrijving het meest consistent was in ondiep water. Sommige modellen voor golfenergiedissipatie zijn vrij uitgebreid en zijn kwalitatief sterk. De meeste onder hen zijn een functie van de ruwheidslengte. Deze is echter, tenminste bij golfvoorspellingstoepassingen, onbekend. Teneinde niet te moeten rekenen met een constante ruwheidslengte in de bodemwrijvingsformulering werd een nieuwe ribbelgeometriepredictor voorgesteld (MN81). Het is gebaseerd op het werk van Nielsen maar een aantal bevindingen beschreven in de literatuur werden erin verwerkt. Het voorgestelde model werd gekoppeld aan het grenslaagmodel voor golven en stromingen van Christoffersen en Jonsson (1985, CJ85). Datasets van veld– en labometingen (golven, stromingen en ribbeldimensies) werden gebruikt om zowel de ribbelgeometriepredictor (MN81) en het grenslaagmodel CJ85 te verifiren. De voorgestelde ribbelgeometriepredictor kon, zonder specifieke aannames voor de verschillende stadia, de metingen in alle ribbelregimes reproduceren. Anderzijds kon ook het grenslaagmodel, gebruik makende van de ribbelgeometrie verkregen met het MN81–model, de verschillende stromingregimes in de veldmetingen en een aantal parameters door andere onderzoekers gemeten in het laboratorium reproduceren.

iv

Table of Contents Abstract

i

Samenvatting

iii

Table of Contents

v

1 Introduction 1.1 The relevance of wind waves 1.2 History . . . . . . . . . . . 1.3 Aims of this study . . . . . 1.4 Overview of the thesis . . . 1.5 References . . . . . . . . . .

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1 1 2 3 4 5

2 Numerical modelling of wind waves 2.1 The beginning of the wave forecast . . . . . . . 2.2 Phase–resolving and phase–averaged models . . 2.3 The action balance equation . . . . . . . . . . . 2.4 Wind input . . . . . . . . . . . . . . . . . . . . 2.5 Quadruplets nonlinear wave–wave interactions . 2.6 Whitecapping . . . . . . . . . . . . . . . . . . . 2.7 Bottom friction . . . . . . . . . . . . . . . . . . 2.8 Triad nonlinear wave–wave interactions . . . . 2.9 Depth–induced wave breaking . . . . . . . . . . 2.10 References . . . . . . . . . . . . . . . . . . . . .

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3 WAM for fine-scale applications 31 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 Wave modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2.1 The standard WAMC4 model . . . . . . . . . . . . . . . 33 3.2.2 Difficulties in using WAMC4 in high–resolution applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 v

3.3 3.4

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35 36 36 37 50 53 53 55

4 Tide, surge and waves interactions 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Module for Combined Tide, Surge and Waves . . . 4.2.1 Overview of the models . . . . . . . . . . . . . . 4.2.2 Coupling procedure . . . . . . . . . . . . . . . . 4.3 North Sea Applications . . . . . . . . . . . . . . . . . . 4.3.1 Implementation . . . . . . . . . . . . . . . . . . . 4.3.2 Results from an uncoupled mode run . . . . . . . 4.4 North Sea Sensitivity Analysis . . . . . . . . . . . . . . 4.4.1 Description of the experiments . . . . . . . . . . 4.4.2 Sensitivity of waves, tides and surges to coupling 4.4.3 Sensitivity of waves to coupling during storms . . 4.4.4 Sensitivity of surges to waves . . . . . . . . . . . 4.5 Summary and Conclusions . . . . . . . . . . . . . . . . . 4.6 References . . . . . . . . . . . . . . . . . . . . . . . . . .

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59 59 61 61 66 67 67 68 72 72 73 76 83 88 90

5 The wave bottom friction formulations 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 5.2 The SWAN wave model . . . . . . . . . . . . . . . . 5.2.1 Expressions for the dissipation coefficient Cf 5.3 Numerical experiments . . . . . . . . . . . . . . . . . 5.3.1 Introduction . . . . . . . . . . . . . . . . . . 5.3.2 Statistical analysis . . . . . . . . . . . . . . . 5.3.3 Lake George . . . . . . . . . . . . . . . . . . 5.4 Summary and Conclusions . . . . . . . . . . . . . . . 5.5 References . . . . . . . . . . . . . . . . . . . . . . . .

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95 . 95 . 96 . 97 . 99 . 99 . 99 . 100 . 114 . 118

3.5 3.6 3.7

Model implementations . . . . . . . . . . . . WAM–PRO for fine–scale coastal applications 3.4.1 Introduction . . . . . . . . . . . . . . 3.4.2 Code modification . . . . . . . . . . . 3.4.3 I/O and some other peculiarities . . . Further work . . . . . . . . . . . . . . . . . . Summary and conclusions . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .

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6 Ripples in wave–current–bedform interactions 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 The Ripple Predictor of N81 . . . . . . . . . . . . . . . 6.2.2 The Model of CJ85 for the Wave Bottom Boundary Layer in Combined Wave and Current Flows . . . . . .

121 122 123 123 124

6.3

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6.5 6.6

Proposed Additions and Modifications to N81 Model . . . . 6.3.1 Relative Ripple Length . . . . . . . . . . . . . . . . 6.3.2 The Effective Shields Parameter . . . . . . . . . . . 6.3.3 Critical Shear Velocities . . . . . . . . . . . . . . . . 6.3.4 The Total roughness height . . . . . . . . . . . . . . 6.3.5 Proposed Model for Ripple Geometry: Summary. . . 6.3.6 Coupling Procedure . . . . . . . . . . . . . . . . . . Verification of MN81 . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Site Description and Data Set . . . . . . . . . . . . . 6.4.2 Numerical Results and Comparisons Between MN81 Model and the Data Set . . . . . . . . . . . . . . . . Summary and Conclusions . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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125 125 126 127 128 129 129 131 131

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7 The wave bottom boundary layer 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 The Model of CJ85 for the WBBL in Combined Wave– Current Flow . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 The Modified Model of Nielsen for Ripple Geometry. . . 7.3 Coupling Procedure . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Verification of CJ85 Model . . . . . . . . . . . . . . . . . . . . 7.4.1 Data sets . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 CJ85 Numerical Results and Comparisons with the Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . 7.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

147 147 148 148 151 152 152 152 153 162 165

8 Summary and conclusions 167 8.1 Recapitulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 8.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . 170

Chapter 1

Introduction “Se miente m´ as de la cuenta por falta de fantas´ıa: también la verdad se inventa.”

Antonio Machado

1.1

The relevance of wind waves

The importance of wind waves in nature starts from the moment they start growing. In their initial state the waves1 play a key role in air–sea gas exchange (Bock et al., 1999; Gemmrich and Farmer, 1999). Oceanic measurements of air-sea gas transfer velocities, wind speed, and whitecapping have shown that gas transfer rates are a function of both whitecap coverage and wind speed (Asher et al., 1998). Once the waves are generated by the wind they can travel thousands of kilometers with insignificant loss of energy (Snodgrass et al., 1966). By changing the surface roughness the waves enhance the frictional effects associated with wind–driven flows, modifying the surge elevation (Mastenbroek et al., 1993; Ozer et al., 2000; Osuna, 2002) and with their associated turbulence provoke changes of the tidal amplitude and phase (Davies and Lawrence, 1994). When the waves enter shallow water areas, they are mixed up in several mechanisms which modify the environment and in turn the wave properties are affected. In the complex conditions of coastal waters, the waves are involved in dynamic interactions with wind, bottom and ambient currents, and the dynamics are dominated by wave–bottom interactions. The growth by wind, propagation, non-linear interactions, energy decay and possibly the enhancement of whitecapping, are all linked to how the waves interact with the bottom. One of the most important processes is how the waves dissipate their energy. Among the different mechanisms for wave energy dissipation at the bottom, such as percolation, friction, motion of a soft muddy bottom and bottom scattering it appears 1

The term waves in this thesis is used exclusively for wind–waves.

1

2

CHAPTER 1. INTRODUCTION

that the bottom friction is the most important mechanism for energy decay in sandy coastal regions (Shemdin et al., 1978). The role of waves in sediment suspension and transport in sandy coastal regions has been investigated extensively (Nielsen, 1992; Soulsby, 1997). Dissipation of wave energy, through bottom friction, on reef flats has been found to be one of the main responsible mechanisms of the high productivity of coral reefs in low–nutrient tropical water (Hearn et al., 2001). Also, wave direction and wave power help to determine sites for productive reef development and shape reef morphology as well as community structure (Roberts, 1992). Depth–induced wave breaking is another process through which the wave energy is dissipated. This process becomes dominant over all other processes in the surf zone, giving rise to wave–driven flow. Over reefs, wave–driven flow is a critical factor in determining community distribution and production rates in coral reef ecosystem both by controlling the supply of nutrients and the level of turbulence on the reef and in some places wave breaking is the dominant flushing mechanism (Hearn, 1999). The exploitation and conservation of the coastal zone causes a growing need to increase our capability to model the coastal dynamics. For ship routing, gas and oil exploitation, transport and dispersion of dissolved and suspended matter, building of structures, fisheries and all other activities in the coastal zone the knowledge of the sea state is relevant. Reliable wave forecasts are required to estimate beach erosion, to design and operate ports and harbors and are also essential to dredging companies and for the safety of human settlement along the coast.

1.2

History

The first operational wave prediction scheme was developed by Sverdrup and Munk (1947). Since then several milestones in the research of wind waves can be recognized. The introduction of wave energy spectrum concept by Pierson (1952) and its use in the work of Gelci et al. (1956, 1957) about wave forecasting using the spectral energy transport equation, represent a turning point in the study and forecast of wind waves. At that time the processes of how the waves gain and lose their energy were poorly understood. It was in 1957 when Phillips and Miles, working independently, described theoretically the wind energy input to the wave field. The nonlinear wave–wave interactions were described by Hasselmann (1962). The importance of the nonlinear interactions for the wave spectrum development and evolution was revealed by the Joint North Sea Wave Project (JONSWAP) (Hasselmann et al., 1973). After the reliable description of two of the most important wave processes in deep water (wind input and whitecapping) a fast development of the numerical wave models took place, based on the energy transfer equation. Those models were classified according to the exclusion (first generation) or inclusion (second generation) of the nonlinear interactions source term (SWAMP, 1985). The Sea WAve Modelling Project (SWAMP,

1.3. AIMS OF THIS STUDY

3

1985) revealed shortcomings of those wave models. Neither the first generation nor the second generation wave models were able to reproduce the development of the wave spectrum. Although these models can be tuned to provide useful results for certain classes of wind fields, they are not reliable in extreme situations where the wind field changes rapidly. This can be explained in terms of the absence (first generation) or the inadequate parameterization of the nonlinear energy transfer. However, the inclusion of the nonlinear transfer was prohibitive in terms of computing cost. A breakthrough came with the simplification of the nonlinear transfer source term by Hasselmann et al. (1985). This work opened the way to develop third generation wave models. It was the effort of many scientists (Komen et al., 1994) which made possible the development of the WAM model, which is considered as the prototype third generation wave model.

1.3

Aims of this study

In shallow water the wave dynamics become more complicated by the presence of the bottom and intensified ambient currents. Besides, more physical processes show up, viz, refraction, bottom friction, percolation, depth limitation, wave energy scattering, wave–current interactions2 , triad–wave interaction and depth–induced wave breaking. All the physical processes mentioned above are linked to how the waves interact with the bottom. Each of those processes will be addressed in Chapter 2. Much progress has been made regarding the spectral modelling of wind waves, specially on the oceans and on shelf seas, and recently in shallow water areas. It is in the shallow water zone where the waves have the strongest interaction with several processes, and there it is precisely where is an increasing need for both better understanding and to increase our capability to model the coastal dynamics. Most of the numerical wind–wave models were developed to simulate the evolution of the wave field in deep waters, but being imperative the simulation of the wave field in shallow waters those models should be modified (if possible) in order to simulate the wave field in the coastal zone. To this respect the SWAN (Simulating WAves Nearshore) model is the proto–type of a model made specifically for simulation of the wave spectrum in shallow water (Ris et al., 1999) . The main objective of this study is to achieve a beter knowledge on dynamics of the waves in shallow water areas, particularly the interactions between waves, currents and bottom. To achieve this objective, the research was divided in specific objectives. The first objective is to improve the computational efficiency of the WAM model for coastal applications and, because of the intensive wave–bottom interactions, adding different bottom friction formulations and an expression for the energy dissipation due to depth–induced wave breaking. The second objective is to investigate the wave– current interactions through dynamic coupling between the improved WAM model 2

These wave–current interactions can be characterized as weak in oceans and shelf seas, and stronger in shallow water areas, Ozer et al., 2000; Osuna, 2002.

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(hereafter referred as WAM–PRO model) and a surge model. Due to the fact that the bottom friction plays a major role in the spectral energy balance the third objective is to investigate how the bottom friction formulations work in very shallow water areas and in almost ideal conditions. Finally the fourth objective is to investigate the wave–bedform interactions in a two–way coupled system. After all most of the bottom friction models are a function of the bottom roughness height.

1.4

Overview of the thesis

This thesis is a compilation of published or submitted articles with some additions and slight modifications. Conceptually the WAM model was conceived as a model for deep and intermediate waters. In WAM all the source functions that are important for deep water are represented. Its numerical scheme is appropriate to describe wave evolution on an oceanic scale. Simulating the waves in deep and intermediate waters, WAM had a great success in terms of scientific as well as operational purposes, but in order to run it in shallow water it is necessary to modify it and add some features. Chapter 2 gives an overview of wave processes that are relevant for wave simulation, including some relevant literature.3 In chapter 3 those new features that have been added and several necessary changes to implement the standard WAM–cycle4 (WAMC4) model code for shallow water applications are presented. This was done under the EU MAST III project, PROMISE (Pre–Operational Modelling in the Seas of Europe). Particular emphasis is placed on computer central processing unit (CPU) efficiency and improved input/output (I/O). Two processes that become important in shallow regions are the enhanced wave–bottom and wave–current interactions. In chapter 4 the development, testing and preparation for dissemination of a generic module in which tides, surges and waves are dynamically coupled is described. The main purpose of this chapter is to describe how the module has been developed and to report on a series of experiments dealing with the sensitivities of both models to coupling. Chapter 5 is devoted to search for evidence to determine which friction formulation (see Section 2.7) performs best in shallow water wave modelling. Despite the fact that some models for wave energy dissipation by bottom friction are very elaborate and have some very fine qualities, they are a function of the roughness height, which most of the time, at least in wave forecasting, remains unknown for a given wave, current and sediment characteristics. Chapter 6 is devoted to model and to analyze the ripple geometry field in wave–current flow situations. Chapter 7 deals with the description of the wave bottom boundary layer (WBBL) dynamics, through the numerical coupling of two models, a WBBL model and a ripple geometry predictor. A summary and conclusions are given in Chapter 8. 3

After all we should not remain ignorant about the work of the people on whose shoulders we stand.

1.5. REFERENCES

1.5

5

References

Asher W., Wang Q., Monahan E.C. and Smith P.M., 1998. Estimation of air-sea gas transfer velocities from apparent microwave brightness temperature. Mar. Tech. Soc. J. 32 (2), 32–40. Barnett T.P. and Wilkerson J.C., 1967. On the generation of wind waves as inferred from airborne measurements of fetch–limited spectra. J. Mar. Res., 25, 292– 328. Bock E.J., Hara T., Frew N.M. and McGillis W.R., 1999. Relationship between air-sea gas transfer and short wind waves. J. Geophys. Res., 104, (C11), 25821–25831. Booij N., Ris, R.C. and Holthuijsen L.H., 1999. A third-generation wave model for coastal regions: 1. Model description and validation. J. Geophys. Res., 104 (C4), 7649–7666. Davies A.M., and Lawrence J., 1994. Examining the influence of wind and wave turbulence on tidal currents, using a three–dimensional hydrodynamic model including wave–current interaction. J. Phys. Oceanogr., 24, 2441–2460. Gelci R., Cazalé H. and Vassal J., 1956. Utilization des diagrammes de propagation à la prévision énergétique de la houle. Bullletin d’information du Comité Central d’océanographie et d’études des cb otes, 8, No. 4, 170–187. Gelci R., Cazalé H. and J. Vassal, 1957. Prévision de la houle. La méthode des densités spectroangulaires. Bullletin d’information du Comité Central d’océanographie et d’études des cb otes, 9, 416–435. Gemmrich J.R. and Farmer D.M., 1999. Observations of the scale and occurrence of breaking surface waves J. Phys. Oceanogr., 29 (10), 2595–2606. Hasselmann K., 1962. On the non–linear energy transfer ina gravity wave spectrum. Part 1: General theory. J. Fluid Mech., 12, 481–500. Hasselmann K., Barnett T.P., Bouws E., Carlson H., Cartwright D.E., Enke K., Ewing J.I., Gienapp H., Hasselmann D.E., Kruseman P., Meerbrug A., M¨ uller P., Olbers D.J., Richter K., Sell W. and Walden H., 1973. Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Dtsch. Hydrogr. Z., A8(12), 95 p. Hasselmann, S., Hasselmann K., Allender J.H. and Barnett T.P., 1985. Computations and parameterizations of the nonlinear energy transfer in a gravity–wave spectrum. Part. II: Parameterizations of the nonlinear energy transfer for application in wave models. J. Phys. Oceanogr., 15, 1378-1391. Hearn, C., 1999. Wave–breaking hydrodynamics within coral reef system and the effect of changing relative sea level. J. Geophys. Res., 104 (C12), 30007–30019.

6


Hearn C.J., Atkinson M.J. and Falter J.L., 2001. A physical derivation of nutrientuptake rates in coral reefs: effects of roughness and waves. Coral Reefs, 20 (4), 347–356. Komen G.J., Cavaleri L., Donelan M., Hasselmann K., Hasselmann S., Janssen P.A.E.M., 1994. Dynamics and Modelling of Ocean Waves. Cambridge Univ. Press, Cambridge, 532 pp. Nielsen P., Coastal Bottom Boundary Layers and Sediment Transport, 1992. Advanced Series on Ocean Engineering. Vol. 4, World Scientific. Singapore, 324 pp. Osuna Ca˜ nedo P., 2002. On the high resolution simulation of the dynamic interaction between current and waves in coastal waters: an application to the North Sea. Ph.D. Thesis, Katholieke Universiteit Leuven, België, 185 pp. Ozer J., Padilla–Hernández R., Monbaliu J., Alvarez–Fanjul E., Carretero–Albiach J.C., Osuna P., Yu J.C.S. and Wolf J., 2000. A coupling module for tides, surges and waves. Coastal Eng., 41, 41–62. Phillips O.M., 1957. On the generation of waves by turbulent wind. J. Fluid Mech. 2, 417–445. Pierson W.J., 1952. A unified mathematical theory for the analysis, propagation and refraction of storm generated ocean waves, Parts I and II. N.Y.U., Coll. of Eng. Res. Div., Depto. of Meteorol. and Oceanogr., Prepared for the beach erosion Board Dept. of the army, and Office of Naval Res., Depto. of the Navy, 461 pp. Ris R.C., Holthuijsen L.H. and Booij N., 1999. A third-generation wave model for coastal regions: 2. Verification. J. Geophys. Res., 104 (C4), 7667–7681. Roberts H.H., Wilson P.A. and Lugo–Fernandez A., 1992. Biologic and geologic responses to physical processes – examples from modern reef system of the Caribean–Atlantic Region. Cont. Shelf Res., 12 (7–8), 809–834. Shemdin P., Hasselmann K., Hsiao S.V. and Herterich K., 1978. Non–linear and linear bottom interaction effects in shallow water. In: Turbulent Fluxes Through the Sea Surface, Wave Dynamics and Prediction, NATO Conf. Ser. V, Vol 1, 347–372 Snodgrass F.E., Groves G.W., Hasselmann K.F., Miller G.R., Munk W.H., and Powers W.H., 1966: Propagation of ocean swell across the Pacific, Philos. Trans. R. Soc. London, A: 259, 431–497. Soulsby R., 1997. Dynamics of marine sand. Tomas Telford Pub., London, UK, 249 pp.

1.5. REFERENCES

7

Sverdrup H.U. and Munk W.H., 1947. Wind sea and swell: Theory of relations for forecasting. H.O. Pub. 601, US Navy Hydrographic Office, Washington, DC, 44 pp. SWAMP group, 1985. Sea wave modeling project (SWAMP). An intercomparison study of wind wave predictions models, part 1: Principal results and conclusions. In: Ocean wave modelling, Plenum Press, New York, 256 pp.

8


Chapter 2

Numerical modelling of wind waves “Si alguna vez quieres hacer reir a Dios, cuentale tus planes” Abuela de Susana en “Amores Perros”

2.1

The beginning of the wave forecast

Three distinct approaches had been utilized for wind wave prediction: the empirical approach, based on dimensional analysis, and the discrete spectral and the parametric approach, both based on the action balance equation. The pioneering work of Sverdrup and Munk (1947) (SM), who developed the first operational wave prediction scheme,1 was based on dimensional analysis. Since then numerical wave modelling has made an impressive advance. SM represented the wave field in terms of the significant wave height Hs (x, t) and phase speed C(x, t), being x = (x, y) the horizontal position and t the time. They assumed that Hs and C were functions of the constant wind speed (U10 ), at the standard anemometer height of 10 m in the atmospheric boundary layer, the constant water depth h, the distance F (the so–called fetch) of the sea surface over which the constant wind is blowing up to the position x, the duration, tD of the constant wind up to the time t of prediction, and the gravitational acceleration g, viz. Hs , C = f (F, tD , U10 , g, h)

(2.1)

Classical dimensional analysis reduces the number of independent variables from 1 During the Second World War it was clear that any amphibious operation must have some way to forecast wave conditions. This problem was entrusted to Drs. Sverdrup and Munk.

9

10

CHAPTER 2. NUMERICAL MODELLING OF WIND WAVES

five to three:

µ ¶ gHs C gF gtD gh =f 2 , U 2 , U , U2 U10 U10 10 10 10

(2.2)

The function f must be determined from field and laboratory experiment. An alternative representation, and almost identical to that one of SM, was introduced by Kitaigorodskii (1962), who chose to characterize the wave field in terms of the directional–frequency spectrum at position and time (x, t) as a function of the constant wind speed (U10 ), and also, as in SM, h, F, tD and g. The popularity of the Kitaigorodskii representation follows from its adoption in the presentation of the results of the Joint North Sea Wave Project (JONSWAP) field experiment (Hasselmann et al., 1973). Although it was Pierson (1952) who introduced the spectrum approach to the ocean waves2 , Gelci et al., (1957) introduced the wave forcasting based on the energy density transport equation. Since then this method has become the most popular one.

2.2

Phase–resolving and phase–averaged models

Battjes (1994) considers two families of shallow water wave models. They are phase– resolving and phase–averaged models. Phase–resolving models describe the sea surface as a function of space and time and are used for rapidly varying wave conditions. These models can simulate the sea surface elevation with high accuracy having the advantage that diffraction, refraction effects and nonlinear wave–wave interactions (quadruplets and triads) are taken implicity into account in Hamiltonian– and Boussuneq–type models. However, in phase–resolving models the incorporation of wind effect is rather difficult. They are computationally very demanding and should be used only when strictly required. Only the mathematical description of strong diffraction and possibly of triad interaction requires these type of models. In the phase–averaged models the irregular sea surface is described by a spectral energy density function (or action density). In the Lagrangian approach the wave energy contained in each spectral component propagates independently along wave rays which makes the computation of nonlinear effects numerically inefficient and very often results in a chaotic waves rays patterns making their interpretation difficult. In the Eulerian approach the wave propagation is formulated on a grid. Every grid point has the information of the whole wave spectrum. With this the problem of chaotic wave patterns is avoided and the inclusion of generation, dissipation and nonlinear wave–wave interactions can be done efficiently. 2

According to Kinsman (1984) there were fragmentary use, not well justified, of the stochastic process and the power spectrum applied to wind waves before Pierson (1952). A group of four related theoretical unpublished notes (Longuet–Higgins, 1946; Barber, 1945, 1946, and Barber, Collins and Tucker, 1946), are sometimes identified as the beginning of the use of the stochastic process–spectrum in wind waves.

2.3. THE ACTION BALANCE EQUATION

11

The phase–resolving models are usually based on Hamiltonian approach (Radder, 1992), on a Boussinesq approach (Peregrine, 1966; Madsen and Sørensen, 1992; Agnon et al., 1999) or on the mild-slope equation (Mei, 1999), while phase–averaged models are based on a Lagrangian approach (i.e. wave rays) or Eulerian approach (i.e. grid model) or a combination of both.

2.3

The action balance equation

Nowadays most of the numerical models for wave forecasting are based on the action balance equation. The development of the phase–averaged models based on the wave spectrum started with the work of Gelci et al. (1957), but at that time very little was known about the source functions. The action balance equation in its general form in Cartesian coordinates reads: ∂N ∂ ∂ ∂ ∂ Stot + (cx N ) + (cy N ) + (cσ N ) + (cθ N ) = ∂t ∂x ∂y ∂σ ∂θ σ

(2.3)

Where N (t, x, y, σ, θ) is the action density spectrum, t is the time, σ is the intrisic frequency, θ is the wave direction measured clockwise from the true north, c x , cy are the propagation velocities in the geographical space, cσ , and cθ are the propagation velocities in the spectral space (frequency and directional space). The first term in the left–hand–side of this equation represents the rate of change of action density in time. The second and third term represent rectilinear propagation of action (with propagation velocities cx , cy respectively) in geographical space. The fourth term represents the effects of shifting of relative frequency due to variation in depths and currents (with propagation velocity cσ ). The fifth term represents propagation in directional space (with propagation velocity cθ ) due to current and/or depth-induced refraction. In the right hand side of the Equation 2.3, Stot represents all effects of generation and dissipation of the waves, generally written as the sum of a number of separate source terms, each representing a different type of process: S = Sin + Sds + Snl

(2.4)

Sin represents the generation of wave energy by wind, Sds the dissipation of wave energy due to whitecapping (Swc ), wave–bottom interaction (Sbf ) and, in very shallow water, depth-induced wave breaking (Sbk ), Snl is the wave energy transfer due to conservative nonlinear wave–wave interactions (both quadruplet Snl4 , and triad Snl3 , interactions). Each of these processes will be addressed in the following section, emphasizing the expressions that are used in the SWAN and WAM wave models The numerical wave models used in this thesis are phase–average models that use the Eulerian approach.

12

2.4


Wind input

Transfer of wind energy to the waves is generally described by the Phillips’ (1957), resonance mechanism and the Miles’ (1957) feed–back mechanism (or Jeffreys’, 1924, shelter mechanism) . The source term for these mechanisms is commonly described as the sum of linear and exponential growth: Sin = Ψinl + Ψine N (σ, θ)

(2.5)

in which Ψinl and Ψine depend on wave frequency and direction, and wind speed and direction. In the early stages of the wave–growth the linear term Ψinl is dominant but is quickly overridden by the exponential growth term Ψine N . There are several expressions for the linear term. The one that is used in the SWAN model is due to Cavaleri and Rizzoli (1981) (with a filter to avoid growth at frequencies lower than the Pierson–Moskowitz frequency, Tolman, 1992) Ψinl

H

=

· ¸4 1.5 × 10−3 H u max[0, cos(θ − θ )] ∗ w g 2 2π

=

· µ ¶−4 ¸ σ exp − ∗ σP M

with

σP∗ M =

(2.6) 0.13g 2π 28u∗

(2.7)

where u∗ is the wind friction velocity, θw is the mean wind direction, H is a filter and σP∗ M is the normalized equilibrium peak frequency for a fully developed sea state according to Pierson and Moskowitz (1964), reformulated as a function of the friction velocity, which is given by 2 (2.8) u2∗ = CSD U10 where CSD is the surface drag coefficient. According to Wu (1982) the relation of CSD to U10 is given by

CSD (U10 ) =

½

1.2875 × 10−3 (0.8 + 0.065 sm−1 × U10 ) × 10−3

for U10 < 7.5 ms−1 for U10 ≥ 7.5 ms−1

The exponential term is the so–called Miles’s feedback mechanism. Here two expressions are presented, the first one due to Komen et al. (1984) and the second one due to Janssen (1989, 1991). The expression of Komen et al. is a function of u∗ /c: µ ¶¸ · u∗ ρa 28 cos(θ − θw ) − 1 σ (2.9) Ψine = max 0, 0.25 ρw c in which ρa and ρw are the air and water density, respectively . It has been found experimentally that the drag coefficient is not a constant for a certain wind speed but depends on the sea state (Maat et al., 1991).

13

2.4. WIND INPUT

The expression for the exponential growth rate of gravity waves given by Janssen (1989, 1991) explicitly accounts for the interactions between the wind and the waves including the influence of the waves and the effect of the air–turbulence on the mean wind velocity profile. As Shown by Miles (1957), the growth rate of gravity waves due to wind depends on two parameters, x=

u∗ cos(θ − θw ) c

and

Ωm =

gz0 u2∗

(2.10)

where u∗ is the friction velocity, c the phase speed of the waves, θw the wind direction and θ the direction of waves propagation. The so–called profile parameter Ω m characterizes the state of the mean air flow through its dependence on the roughness height z0 . Through Ωm the growth rate depends on the roughness of the air flow, which, in turn depends on the sea state. The expression of Janssen (1989, 1991) for the exponential growth rate of gravity waves given by Ψine = ²βx2 σ

(2.11)

where ² is the air–water density ratio (ρa /ρw ), β is the so–called Miles parameter and is given by βm µ ln4 µ, µ≤1 (2.12) κ2 βm is a constant equal to 1.2 and κ is the von Kármán constant equal to 0.41. In terms of dimensionless critical height (zc ), which is defined as the height above the mean sea level where the wind speed U equals the wave phase speed, µ is expressed as µ = kzc β=

with k being the wavenumber. In terms of wave and wind quantities µ is given as µ=

µ

u∗ κc

¶2

Ωm eκ/x

(2.14)

Note that x and Ωm are given by Equations 2.10. The stress of the air flow over sea waves depends on the sea state and from a considerations of the momentum balance of air it is found that the kinematic stress is given by (Janssen, 1991) ·

κUzobs τ= ln(zobs /z0 ) where

ατ z0 = √ , g 1−y

¸2

y=

(2.15) τw τ

(2.16)

14


zobs is the mean height above the water, α is a constant equal to 0.01 and τw is the stress induced by the waves, given by τw = ²

−1

g

Z Z

Ψine N kdσdθ

(2.17)

Now with this set of equations, from 2.10 to 2.17, Ψine is determined for a given wind speed and sea state. Young (1999) remarks that the theories (and measurements) for wind input are essentially restricted to the deep water situation, and little is known on the effects of shallow water on wind input.

2.5

Quadruplets nonlinear wave–wave interactions

The first numerical wave models (the so–called first generation, see SWAMP, 1985) were applied successfully, however in order to reproduce the observed wave growth, the Phillips’ term (Ψinl in Equation 2.5) had to be chosen several orders of magnitude greater than estimates based on turbulent pressure measurements. The second factor Ψine similarly had to be increased almost an order of magnitude beyond the theoretical estimates of Miles or Jeffreys. Those wave models were unable to explain the pronounced overshoot phenomenon of a growing windsea observed by Barnett and Wilkerson (1967) and confirmed later by other researchers. Those issues were clarified through extensive field measurements (Mitsuyasu et al., 1971; Hasselmann et al., 1973) and then corroborated by wind–wave tank experiments (Mitsuyasu, 1968a, 1968b, 1969). The analysis of those data led to a completely changed view of the spectral energy balance of a growing windsea. According to the revised picture the principal source of energy, during the main growth phase of the waves on the low frequency face of the spectrum, was the nonlinear energy transfer from higher–frequency components to the peak and to the lower–frequency components, rather than direct wind forcing. Besides being responsible for the high growth rates on the forward face of the peak, the nonlinear energy transfer was also found to control the shape of the spectrum, including the development of the peak and playing an important role in determining the directional spread of the energy density function (Young and van Vledder, 1993). The overshoot phenomenon was explained by the existence of a very sharp peak, significantly more pronounced than for a fully developed spectrum. This spectral peak migrates towards lower frequencies as the wave spectrum develops. This migration of the peak is faster while the wind continues pumping energy to the wave field, and slower once the wind is not forcing the waves any more (the nonlinear interactions are weaker). The quadruplet wave–wave interaction mechanism can be expressed by the Boltzmann integral (Hasselmann, 1962, 1963a, 1963b). This describes the redistribution of energy over the spectrum.

2.5. QUADRUPLETS NONLINEAR WAVE–WAVE INTERACTIONS

15

Its numerical evaluation is extremely time consuming and not yet convenient in any operational wave model3 . In the SWAN and WAM wave models, the nonlinear interactions are computed using the Discrete Interaction Approximation (DIA) as proposed by Hasselmann et al. (1985). In the DIA the following quadruplet is considered in terms of frequency σ1 =

σ2

σ3 =

σ(1 + λ)

= σ+

σ(1 − λ)

−

σ4 =

=σ =σ

(2.18)

where λ is a constant coefficient set equal to 0.25. The two frequencies σ1 and σ2 are identical and so are the two wave numbers vectors corresponding to these frequencies. The other two frequencies σ3 and σ4 and wave numbers are of a different magnitude and direction. To satisfy the resonant condition for the first quadruplet, the wave number vectors with frequencies σ3 and σ4 lie at an angle of θ1 = −11.5o and θ2 = 33.6o to the two wave number vectors with frequencies σ1 and σ2 . The second quadruplet is the mirror of the first quadruplet. Here the wave number vectors with frequency σ3 and σ4 lie at the mirror angles of θ3 = 11.5o and θ4 = −33.6o . With this discrete interaction approximation, the source term Snl4 is given by: 00 0 + Snl4 Snl4 = Snl4

(2.19)

0 00 where Snl4 refers to the first quadruple (with θ1 = −11.5o and θ2 = 33.6o ) and Snl4 o o to the second quadruple (with θ1 = 11.5 and θ2 = −33.6 ) and 0 Snl4 = 2∆Ssnl4 (α1 ) − ∆Snl4 (α2 ) − ∆Snl4 (α3 )

(2.20)

in which α1 = 1, α2 = (1 + λ), α3 = (1 − λ). Each of the contributors (i = 1, 2, 3) is: ∆Snl4 (αi )

=

µ

¸ F (αi σ + , θ) F (αi σ − , θ) Cnl4 (2π) g + F (αi ) (1 + λ)4 (1 − λ)4 ¾ F (αi σ, θ)F (αi σ + , θ)F (αi σ − , θ) −2 (2.21) (1 − λ2 )4 2 −4

σ 2π

¶½

2

·

00 The constant Cnl4 is equal to 3 × 107 . The expressions for Snl4 are identical for the mirror directions. The above analysis is made for deep water. Numerical computations by Hasselmann and Hasselmann (1981) of the full Boltzman integral for water of arbitrary depth have shown that there is an approximate relation between transfer rates in 3

The understanding of small–scale marine processes rather than machine power starts appearing as a bottleneck for operational oceanography, but some processes that are already mathematically described, as the nonlinear wave–wave interactions, should wait some time before their full description can be implemented in an operational modelling.

16


deep water and water of finite depth. For a given frequency–direction spectrum, the transfer for finite depth can be computed as the transfer for infinite depth using a scaling factor R: Snl4 (finte depth) = R(kh)Snl4 (infinite depth)

(2.22)

where k is the mean wavenumber. This scaling relation holds in the range kh > 1, where the exact computation could be closely reproduced with the scaling factor µ ¶ µ ¶ 5.5 5x 5x R(x) = 1 + 1− exp − (2.23) x 6 4 with x = (3/4)kh. This approximation is used in the SWAN and WAM model. In the case the wind continues pumping energy into the wave field, the wave– growth cannot continue indefinitely. The wave–wave interactions are not, generally, able to transfer energy from a given frequency band to another as rapidly as it is supplied by the wind. The energy level contained by the wave field must be limited by another process. This process is the deep–water wave breaking, better known as the whitecapping.

2.6

Whitecapping

The mathematical development of the whitecapping formulation is due to Hasselmann (1974). The assumptions are that the whitecaps can be treated as a random distribution of perturbation forces, which are formally equivalent to pressure pulses, and that the scale of the whitecaps in space and time are small compared to the wavelength and period of the associated wave. The existence of an attenuation factor in Hasselmann’s model implies that the whitecaps are preferentially situated on the forward faces of the waves, thereby exerting a downward pressure on the upward moving water and hence doing negative work on the wave. The model yields a dissipation source function that is linear in both the spectral density and frequency. There is, however, a second attenuation mechanism associated with the passage of a whitecapping wave through a field of smaller waves. The breaking of large waves causes rapid attenuation of short waves in its wake (Banner et al., 1989). Both mechanisms are sensitive to the extent of the breaking, i.e. whitecap coverage, which itself depends on the overall steepness of the wave field. Combining those two processes, the dissipation function is expressed as Swc = Ψwc F (k) = −Cwc where

µ

¶m µ ¶n σ σF (k) α ˆP M σ ¯ α ˆ

(2.24)

m0 σ ¯4 (2.25) 2 g and Cwc , m and n are fitting parameters, σ ¯ is the mean radian frequency and α ˆ /ˆ αP M is a measure of the overall steepness of the wave field, m0 is the zero–th moment of α ˆ=

2.7. BOTTOM FRICTION

17

the wave spectrum. Expression 2.24 corresponds to the result of Hasselmann (1974) for n = 1. Noting that the whitecapping dissipation (Equation 2.24) scales with wavenumber as Ψwc ∼ k (2.26) and the wave growth (Equation 2.11) scales as Ψine ∼ k 3/2

(2.27)

an imbalance in the high–frequency range of the spectrum may be anticipated. Eventually wind input will dominate wave–breaking resulting in energy levels which are too high when compared with observations. Janssen et al. (1989) realized that the wave dissipation source function had to be adjusted in order to obtain a proper balance at the high frequencies. The dissipation source term (Equation 2.24) is thus extended as follows 2

Swc = Ψwc N = −Cwc hσi (hki m0 )

2

·

¶2 ¸ µ k (1 − δ)k N +δ hki hki

(2.28)

where Cwc and δ are constants, m0 is the total wave variance per square metre, k the wavenumber and hσi and hki are the mean angular frequency and mean wave number, respectively. There is hardly any experimental evidence regarding the dissipation of random sea due to whitecapping. As a consequence the whitecapping is perhaps the least understood mechanism in deep water.

2.7

Bottom friction

Komen et al. (1994) start from the linearized momentum equation for the bottom boundary layer flow reads 1 1 ∂τ ∂u + ∇p = (2.29) ∂t ρw ρw ∂z where t is time, z is the vertical coordinate, ρw is the water density, u and p the Reynolds–average horizontal velocity and pressure, respectively, and τ is the turbulent stress in the wave boundary layer. They obtain an expression for the wave energy dissipation due to bottom friction: À ¿ 1 τ (2.30) Sbf (k) = − · Uk g ρw where the bottom friction depends on the known free orbital velocity (Uk ) of the waves at the bottom and on the unknown turbulent bottom stress (τ ); the subscript k denotes a given wave number and h i denotes the ensemble average. An exact solution for τ in Equation 2.29 (and hence for Sbf (k) in Equation 2.30) does not exist, not even for a simple flow. To overcome this problem, several approaches have

18


been proposed. Most of the approaches result in a turbulent shear stress expressed as a function of a friction coefficient and of a free–stream orbital velocity (orbital velocity at the top of the boundary layer). There are two distinct formulations for τ ; the first is to retain a spectral description. The second is to represent the range of frequencies by a single frequency, for example the peak frequency, resulting in an integral form. Usually, τ is expressed in a drag law as τ = 1/2ρw CD (Uw )rms U, where CD is a drag coefficient, Uw is the wave orbital velocity at the bottom and Urms is the root mean square of the orbital velocity. Taking Cf = 1/2CD (U )rms results in τ = ρ w C f Uw

(2.31)

Substitution of Equation 2.31 in the dissipation Equation 2.30 yields for every wave component with wavenumber k: 1 Sbf (k) = − Cf h(Uk )2 i g

(2.32)

The mean square of the bottom velocity, h(Uk )2 i, can be associated with the wave component having the wavenumber k. Rewriting the expression 2.32 in terms of the wave spectrum, one obtains: Sbf (k) = −2Cf

k F (k) sinh 2kh

(2.33)

or equivalently (as expressed in SWAN model) Sbf (σ, θ) = −

Cf σ2 F (σ, θ) g sinh2 kh

(2.34)

where Cf is a dissipation coefficient with the dimension of ms−1 , F (k) and F (σ, θ) are the energy–density spectra in wavenumber–space and in frequency–direction space respectively. The vector k = (k1 , k2 )= (k cos θ, k sin θ) is the wavenumber vector with modulus k and direction θ, and σ is the relative frequency. The different formulations for bottom friction dissipation differ mainly in the expression given to the dissipation coefficient Cf . The expressions for bottom friction dissipation that are currently implemented in the SWAN model (version 40.01, Holthuijsen et al. 1999) and WAM– PRO are briefly explained below.

Expressions for the energy dissipation coefficient Cf The JONSWAP model This is the simplest expression for bottom dissipation. It was proposed by Hasselmann et al. (1973). Cf J (the extra character in the subscript stands for the different formulations) is assumed to be constant and is given by Cf J =

Γ g

(2.35)

19


where g is the acceleration due to gravity. From the results of the JONSWAP experiment they found a value for Γ of 0.038 m2 s−3 . As long as a suitable value for Γ is chosen, this expression performs well in many different conditions. The value for Γ can be different for swell and for wind sea. Bouws and Komen (1983) found that the JONSWAP expression with a value of 0.038 m2 s−3 for Γ yielded too low dissipation rates for depth–limited wind sea conditions in the North Sea. They selected a value of 0.067 m2 s−3 in order to obtain a correct equilibrium solution for a steady state. The JONSWAP formulation is also implemented as the default friction formulation in the WAM model (Komen et al., 1994).

The Madsen model Madsen et al. (1988) derived a bottom friction formulation based on the eddy– viscosity concept, fw (2.36) Cf M = √ hU 2 i1/2 2 where hU 2 i1/2 =

·Z Z

2gk F (σ, θ)dσdθ sinh 2kh

¸1/2

(2.37)

and fw is a non–dimensional friction factor. In the SWAN model the following formulation for fw is used (Jonsson, 1966):

fw ¸

· 1 1 √ + log10 √ 4 fw 4 fw

=

0.3

=

mf + log10

ab ≤ 1.57 KN ab for > 1.57 KN

for ·

ab KN

¸

(2.38)

Where mf = −0.08, ab is a representative near–bottom excursion amplitude: · Z Z ab = 2

1 F (σ, θ)dσdθ sinh2 kh

¸1/2

(2.39)

and KN is the bottom roughness height. Graber and Madsen (1988) implemented the expression of Madsen in a parametric wind sea model for finite water depths.

The Collins model Hasselmann and Collins (1968) derived a formulation for the bottom friction dissipation. They related the turbulent bottom stress to the external flow by means of a quadratic friction law. The dissipation coefficient they derived reads: ½ ¿ À¾ Ui Uj Cf = 2c δij hU i + (2.40) U where δij is the Kroneker delta function;h i denotes the ensemble average, U is equal to (U12 + U22 )1/2 , U1 and U2 are the near bottom orbital velocity components, and c is

20


a drag coefficient determined experimentally as a function of the bottom roughness. Hasselmann and Collins proposed a value for c equal to 0.015. Collins (1972) simplified the Expression 2.40 for the dissipation coefficient by leaving out the dependence on the direction of the wave component and by using the total wave induced bottom velocity: Cf C = 2chU 2 i1/2

(2.41)

where hU 2 i can be computed from Equation 2.37. Expression 2.41 is the one implemented in SWAN and WAM–PRO wave models. The value of the drag coefficient c was set to 0.015. Cavaleri and Rizzoli (1981) implemented this friction model in a parametric wave model.

The Weber eddy–viscosity model Weber’s model for the spectral energy dissipation due to friction in the turbulent wave boundary layer is based on the eddy–viscosity concept, as Madsen’s model. In this model the turbulent shear stress is parameterized in analogy with the viscous stress, with the coefficient of molecular viscosity replaced by a turbulent eddy–viscosity coefficient. By solving the Navier–Stokes equations in the turbulent boundary layer and using perturbation theory, Weber derived the following dissipation coefficient. Cf W = u∗ {Tk (ξ0 ) + Tk∗ (ξ0 )}

(2.42)

Cf W depends on the wave spectrum F (k), the water depth h, and the bottom roughness KN through the friction velocity u∗ , and on the radian frequency (σ = 2πf ) through ξ0 . µ ¶1/2 µ ¶1/2 4KN σ 4(y0 + h)σ = (2.43) ξ0 = κu∗ 30κu∗ The variable ξ0 reflects the ratio between the roughness height and the boundary layer thickness, which scales with u∗ /σ; κ is the von Kármán constant set equal to 0.41; (y0 + h) is the theoretical bottom level and Tk is defined as 1 Ker0 (ξ0 ) + iKei0 (ξ0 ) Tk (ξ0 ) = − κξ0 2 Ker(ξ0 ) + iKei(ξ0 )

(2.44)

Tk is a dimensionless complex function and depends on the radian frequency and thus on the wave number through the argument ξ0 . Ker + iKei is the zero order Kelvin function (Abramowitz and Stegun, 1965). The prime denotes the derivative with respect to the argument ξ0 . Details of the derivation of the dissipation coefficient in the eddy–viscosity model are given in Weber (1989, 1991).

The Christoffersen and Jonsson model The model of Christoffersen and Jonsson (1985) (CJ85 from now on) describes the velocity field and its associated shear stresses in a combined wave-current motion by

21


the means of two two-layer eddy viscosity models. The two models comprises the socalled current bottom boundary layer (CBBL) and the wave bottom boundary layer (WBBL). Both models have the same eddy-viscosity in the CBBL but different eddy viscosities in the WBBL. One of the two models (Model I) is valid for small values of uwbm /(kb ωa ) (“large roughness”), where uwbm is the amplitude of the wave particle velocity at the bed, ωa is the absolute angular frequency and kb is the roughness height. The other model (Model II) deals with large values of uwbm /(kb ωa ) (“small roughness”). The models facilitate analytical solutions. Here a brief description is presented. For a detailed description see CJ85. The assumptions are that the current is steady and that the bed is locally horizontal. By neglecting lateral shear stresses in vertical sections, the Coriolis force and tidal forces, the describing equations for horizontal equilibrium are ∂u 1 ∂ ∂u + (u · ∇)u + w + ∇p = ∂t ∂z ρ ∂z

µ ¶ τ ρ

(2.45)

Here u is the total horizontal particle velocity, t is time, z is the vertical Cartesian coordinate measured upwards from the bed, ∇ = (∂/∂x1 , ∂/∂x2 ) is the horizontal gradient operator, ρ is the water density, x1 and x2 are horizontal Cartesian coordinates, τ is the total shear stress in a horizontal section and w is the vertical particle wave velocity. The quantity p is the total pressure consisting of a steady (current) part and oscillating (wave), i.e. p = pc + pw

(2.46)

For the total horizontal velocity u we similarly have u = u c + uw

(2.47)

The subscripts c and w indicate the components due to the current (steady component) and the wave (periodic component). By taking the mean value of Equations 2.46 and 2.47 (time averaging over an absolute wave period, from now on symbolized by brackets h i) we have hpi = pc and hui = uc , since hpw i = 0 and huw i = 0. Using the eddy viscosity concept, the shear stress is expressed as ∂u ∂uc ∂uw τ =² =² +² ρ ∂z ∂z ∂z

(2.48)

where ² is the eddy viscosity. Since ² is assumed to be time-independent, then hτ c i τc = = ρ ρ

¿ À ∂uc ∂uw ∂uc ² +² =² ∂z ∂z ∂z

(2.49)

because huw i = 0. From equations (2.48) and (2.49) we therefore find with τ separated as τ = τc + τw

(2.50)

22


that τw ∂uw =² ρ ∂z

(2.51)

Equation (2.50) is also valid at the bed giving τ b = τ cb + τ wb

(2.52)

where the subscript b indicates a quantity at the bed. The current friction factor f c and the wave friction factor fw are defined in the following way τcb ≡

1 fc ρU 2 2

(2.53)

1 (2.54) fw ρu2wbm 2 where U is the depth average current speed and τwbm and uwbm are the amplitudes of τwb and uw at the bottom respectively. Using the following expression τwbm ≡

k k τwb = τwbm ei(ϕ+φb ) (2.55) k k where τwb is the magnitude of τ wb , and τwbm is the amplitude of τwb and φb is the phase lead of τ wb relative to the wave particle velocity just outside the WBBL, the quantity ϕ is the phase function given by ϕ = ωa t − k · x, the total bed shear stress is by eq (2.52) and (2.55) given by τ wb ≡

τb =

U k τcb + τwbm ei(ϕ+φb ) U k

(2.56)

in components τ b = τwbm (σ cos δ + cos(ϕ + φb ) cos α, σsinδ + cos(ϕ + φb )sinα) where σ≡

τcb τwbm

=

fc fw

µ

U uwbm

¶2

(2.57)

(2.58)

α and δ are the angles between the wave direction and the current direction with the axis of reference, say x1 , respectively. From equation (2.54) and (2.57) the maximum (total) bed shear stress is found τbm = τwbm m = with m≡

1 mfw ρu2wbm 2

p 1 + σ 2 + 2σ| cos(δ − α)|

(2.59)

(2.60)

The current friction factor fc is thus defined as for a pure current (no waves), and likewise the wave friction factor fw is defined as for a pure wave motion (no current).

2.8. TRIAD NONLINEAR WAVE–WAVE INTERACTIONS

23

In a combined wave-current motion, fc and fw naturally become dependent on each other. Is through the parameter m that the interaction between the waves and the current takes place. For pure wave motion σ = 0 and m = 1, and in the presence of a current σ > 0 and m > 1, depending on the strength of the interactions.

General considerations The JONSWAP friction model does not interpret bottom dissipation in terms of a physical mechanism such as percolation, friction or bottom motion. Madsen’s formulations differ from the expressions of Weber and CJ85 in the fact that Madsen’s model approximates the random wave field by an “equivalent” monochromatic wave. This approximation is applied at an early stage of their calculations. Therefore Madsen’s expression is only valid for a narrow, singled–peaked spectrum, in fact Collins’ drag–law dissipation expression is rederived. However in the Madsen formulation the friction coefficient fw depends explicitly on the wave field and on the roughness height. The expressions of Weber and CJ85 are able to compute the dissipation rate directly from the bottom roughness height through the stress parameterization. That offers the possibility to adapt the dissipation rate according to the changing roughness under different wave–current regimes. This could be important in some coastal areas. Moreover those expressions maintains the spectral description and can be applied to complex situations, where the wave field can not easily be represented by one wave component (Weber, 1991).

2.8

Triad nonlinear wave–wave interactions

Triad–wave interactions have been described as important source term in very shallow water. They are responsible for distributing energy from the central part of the spectrum to higher and lower harmonics. A first attempt to describe triad wave– wave interactions as a spectral energy source term was made by Abreu et al. (1992). However, their expression is restricted to non–dispersive shallow water waves and is therefore not suitable in many practical applications of wind waves. Further development was made by Eldeberky and Battjes (1995). They transformed the amplitude part of the Boussinesq model of Madsen and Sørensen (1993) into an energy density formulation and parameterized the biphase of the waves on the basis of laboratory observations (Battjes and Beji, 1992; Arcilla et al., 1994). The model appeared to be fairly successful in describing the essential features of the energy transfer from the primary peak of the spectrum to the super harmonics. A slightly different version, the Lumped Triad Approximation (LTA), was later derived by Eldeberky (1996). This LTA is used in SWAN. Ris (1997) extended the expression of Eldeberky and Battjes (1995) to include the spectral directions. The expression is + − Snl3 = (σ, θ) = Snl3 + Snl3

(2.61)

24


with ½ · ¸¾ + Snl3 = max 0, αEB 2πccg J 2 | sin(β)| F 2 (σ/2, θ) − 2F (σ/2, θ)F (σ, θ) and + − (2σ, θ) = −2Snl3 Snl3

(2.63)

where αEB is a tunable coefficient that controls the magnitude of the interactions, kσ/2 is the value of the wave number at frequency σ/2, β is the biphase parameterized as ¶ µ π π 0.2 β = − + tanh (2.64) 2 2 Ur with Ursell number (U r) given by g Hs T Ur = √ 8 2π 2 h2

2

(2.65)

with T = 2π/σ, and σ ¯ is given by −1 σ ¯ = Etot

Z

2π 0

Z

2π

σE(σ, θ)dσdθ

(2.66)

0

The interaction coefficient J is taken from Madsen and Sørensen (1993): J=

2 kσ/2 (gh + 2c2σ/2 )

kσ h(gh +

2 3 2 15 gh kσ

− 25 σ 2 h2 )

(2.67)

Although triad–wave interactions are considered dominant in very shallow water, in the SWAN model only the energy transfer to higher harmonics is considered (Ris, 1997). In WAM the triads interaction are not included yet.

2.9

Depth–induced wave breaking

In extreme shallow water depth–induced wave breaking dominates over all other process and it takes place when the wave height reaches its limiting value relative to the water depth. Battjes and Janssen (1978) proposed a bore–based model to compute the total rate of energy dissipation of random breaking waves. Analysis of laboratory observations made by Battjes and Beji, (1992), Vincent et al., (1994); Arcilla (1994) and by Eldeberky and Battjes (1996) had shown that the wave energy dissipation is frequency–independent. This lead Eldeberky and Battjes (1996) to formulate a simple spectral version of the model of Battjes and Janssen (1978) in which the spectral dissipation rate is proportional to the energy density. However Chen et al. (1997) inferred from observations and numerical simulations with a Boussinesq model that the spectral evolution across the surf zone is modelled more accurately by a dissipation

2.10. REFERENCES

25

that increases at high frequencies than by a frequency–independent dissipation. They found that, apparently, the increased dissipation at high frequencies is compensated by increased nonlinear energy transfers to those frequencies.4 Ris (1997) extended the expression of Eldeberky and Battjes (1995) to include the spectral directions: E(σ, θ) (2.68) Sbrk = −Ψbrk Etot where

1 σ ¯ 2 αQb Hm (2.69) 4 2π α is a free parameter of order one, Qb is the fraction of breaking waves determined by: 1 − Qb Etot = −8 2 (2.70) ln Qb Hm Ψbrk =

Hm is the maximum individual wave height that can exist at a given depth and σ ¯ is the mean frequency defined by Equation 2.66. The maximum wave height can be calculated from Hm = γh, in which γ is the breaker parameter and h is the total water depth. Battjes and Stive (1985) re-analyzed wave data of a number of laboratory and field experiments and found values for the breaker parameter (γ) varying between 0.6 and 0.83 for different types of bathymetry, with an average of 0.73. This default–value is used in SWAN. In the WAM-PRO model (see Chapter 3) the default value is set equal to 0.8. The expected relative importance of each of the typical process mentioned above for different regions is given in Battjes (1994) who argues that depth–induced refraction/shoaling, depth–induced wave breaking and triad–wave interactions are dominant in shallow waters.

2.10

References

Abramowitz M. and I.A. Stegun, 1965. Handbook of mathematical functions. National Bureau of Standards, Washington, DC. Abreu, M., A. Larraza and E. Thornton, 1992. Nonlinear transformation of directional wave spectra in shallow water. J. Geophys. Res., 97, 15579–15589. Agnon Y, Madsen P.A., Schaffer H.A., 1999. A new approach to high–order Boussinesq models J. Fluid Mech., 399, 319–333. Arcilla, A.S., J.A. Roelvink, B.A. O’Connor, A.J.H.M. Reniers and J.A. Jimenez, 1994. The Delta flume ’93 experiment. Proc. Coastal Dynamics Conf. ’94, 488–502. 4

Although the final results from the wave models using depth–induced wave breaking frequency–independent or frequency–dependent could not differ much, it is necessary that the physical processes themselves be represented, not the trends of some limited data set, in order to make reasonable predictions under a wide variety of conditions.

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Banner, M.L., I.S.F. Jones and J.C. Trinder, 1989. Wavenumber spectra of short gravity waves. J. Fluid Mech., 198,321–344. Barnett T.P. and J.C. Wilkerson, 1967. On the generation of wind waves as inferred from airborne measurements of fetch–limited spectra. J. Mar. Res., 25, 292–328. Battjes, J.A., 1994. Shallow water wave modelling. Proc. Int. Symp. Waves – Phys. Numer. Modell., University of British Columbia, Vancouver I, 1–23. Battjes J.A. and S. Beji, 1992. Breaking waves propagation over a shoal. Proc. 23 rd Int. Conf. Coastal Eng., 42–50. Battjes, J.A. and Janssen, J.P.F.M., 1978. Energy loss and set-up due to breaking of random waves. Proc. 16th Int. Conf. Coastal Eng., 569–587. Bouws E. and G.J. Komen, 1983. On the balance between growth and dissipation in an extreme, depth limited wind–sea in the Southern North Sea. J. Phys. Oceanogr., 13, 1653–1658. Cavaleri L. and Rizzoli P.M., 1981. Wind wave prediction in shallow water: theory and application. J. Geophys. Res., 86 (C11), 10961–10973. Chen Y., R.T. Guza and S. Elgar, 1997. Modeling spectra of breaking surface waves in shallow water. J. Geophys. Res., 102 (C11), 25035–25046. Collins, J.I., 1972. Prediction of shallow water spectra. J. of Geophys. Res., 93 (C1), 491–508. Eldeberky, Y., 1996. Nonlinear transformation of wave spectra in the nearshore zone. Ph.D. thesis, Delft University of Technology, Department of Civil Engineering, The Netherlands. 203 pp. Eldeberky Y, and J.A. Battjes, 1995. Parameterization of triad interactions in wave energy models, Proc. Coastal Dynamic Conf. ’95, Gdansk, Poland, 140–148. Eldeberky Y, and J.A. Battjes, 1996. Spectral modelling of wave breaking: Application to boussinesq equations. J. Geophys. Res., 101 (C1), 1253–1264. Gelci, R., H. Cazalé and J. Vassal (1957): Prévision de la houle. La méthode des densités spectroangulaires. Bull Inform. Comité Central Oceanogr. D’Etude Cˆ otes, 9, 416–435. Graber, H.C. and O.S. Madsen, 1988. A finite–depth wind wave model, 1, Model description. J. Phys. Oceanogr., 18, 1465–1483. Hasselmann, K., 1962. On the non–linear energy transfer ina gravity wave spectrum. Part 1: General theory. J. Fluid Mech., 12, 481–500. Hasselmann, K., 1963a. On the non–linear energy transfer in a gravity wave spectrum. Part 2: Conservation theorems; wave particle analogy; irreversibilty. J. Fluid Mech., 15, 273–281.

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27

Hasselmann, K., 1963b. On the non–linear energy transfer in a gravity wave spectrum. Part 3: Evaluation of energy flux and swell–sea interaction for a Newman spectrum. J. Fluid Mech., 15, 385–398. Hasselmann, K., 1974. On the spectral dissipation of ocean waves due to whitecapping. Boundary Layer Meteorology, 6, 107–127. Hasselmann, K., T.P. Barnett, E. Bouws, H. Carlson, D.E. Cartwright, K. Enke, J.I. Ewing, H. Gienapp, D.E. Hasselmann, P. Kruseman, A. Meerbrug, P. M¨ uller, D.J. Olbers, K. Richter, W. Sell and H. Walden, 1973. Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Dtsch. Hydrogr. Z., A8(12), 95 pp. Hasselmann, K. and J.I. Collins, 1968. Spectral dissipation of finite–depth gravity waves due to turbulent bottom friction. J. Mar. Res., Vol. 26, 1, 1–12. Hasselmann, S. and K. Hasselmann, 1981. A symmetrical method of computing the nonlinear transfer in a gravity–wave spectrum. Hamburger Geophys. Einzelschr., Serie A, 52, 8. Hasselmann, S., K. Hasselmann, J.H. Allender and T.P. Barnett, 1985. Computations and parameterizations of the nonlinear energy transfer in a gravity–wave spectrum. Part. II: Parameterizations of the nonlinear energy transfer for application in wave models. J. Phys. Oceanogr., 15, 1378–1391. Holthuijsen L.H., N. Booij, R.C. Ris, IJ. G. Haagsma, A.T.M.M. Kieftenburg and R. Padilla–Hernndez, 1999. SWAN Cycle 2 version 40.01. User Manual. Delft Univ. of Tech., the Netherlands. Janssen, P.A.E.M., 1989. Wave-induced stress and the drag of the air flow over sea waves. J. Phys. Oceanogr., 19, 745–754. Janssen, P.A.E.M., 1991. Qausi–linear theory of wind–wave generation applied to wave forecasting. J. Phys. Oceanogr., 21, 1631–1642. Janssen, P.A.E.M., P. Lionello, M. Reistad and A. Hollingsworth, 1989. Hindcast and data assimilation studies with the WAM model during the Seasat period. J. Geophys. Res., 94 (C1), 973–993. Jeffreys J., 1924. On the formation of waves by wind. Proc. R. Soc. London Ser. A107, 189–206. Jonsson, I.G. 1966. Wave boundary layers and friction factors. Proc. 10th Int. Conf. Coastal Eng., ASCE, 127–148. Kinsman B., 1984. Wind Waves. Their Generation and Propagation on the Ocean Surface. Dover Publications, Inc., New York. 676 pp. Kitaigorodskii, S.A., 1962. Applications of the theory of similarity to the analysis of wind generated wave motions as a stochastic process. Bull Acad. Sci. USSR Geophys. Sr. No. 1, 37 pp.

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Komen, S. Hasselmann, S. and K. Hasselmann, 1984. On the existence of a fully developed wind–sea spectrum. J. Phys. Oceanogr., 14, 1271–1285. Maat, N., C. Kraan and W.A. Oost, 1991. The roughness of wind waves. Boundary Layer Meteorology, 54, 89–103. Madsen O.S., Y.–K. Poon and H.C. Graber, 1988. Spectral wave attenuation by bottom friction: theory. Proc. 21st Int. Conf. on Coastal Eng., ASCE, 492– 504. Madsen, O.S. and O.R. Sørensen, 1992. A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly–varying bathimetry. Coastal Eng,. 18, 183–205. Mei C.C. 1999. Mild–slope approximation for long waves generated by short waves. J. Eng. Math., 35 (1-2), 43–57. Miles, J.W., 1957. On the generation of surface waves by shear flows. J. Fluid Mech., 3, 185–204. Mitsuyasu H., 1968a. A note on the nonlinear energy transfer in the spectrum of wind–generated waves. Rep. Res. Inst. Appl. Mech., 16, Kyushu Univ., 251–264. Mitsuyasu H., 1968b. On the growth of the spectrum of wind–generated waves. 1. Rep. Res. Inst. Appl. Mech., Kyushu Univ., 16, 459–465. Mitsuyasu H., 1969. On the growth of the spectrum of wind–generated waves. 2. Rep. Res. Inst. Appl. Mech., Kyushu Univ., 17, 235–243. Mitsuyasu H., R. Nakayama and T. Komori, 1971. Observations of the windwaves in Hakata Bay. Rep. Res. Inst. Appl. Mech., Kyushu Univ., 19, 37–74. Peregrine D.H., 1966. Long waves on a beach. J. Fluid Mech., 27 No.4, 815–827. Pierson W.J., 1952. A unified mathematical theory for the analysis, propagation and refraction of storm generated ocean waves, Parts I and II. N.Y.U., Coll. of Eng. Res. Div., Depto. of Meteorol. and Oceanogr. Prepared for the beach erosion Board Dept. of the army, and Office of Naval Res., Depto. of the Navy, 461 pp. Pierson W.J., L. Moskowitz, 1964. A proposed spectral form for fully developed wind seas based on the similarity theory of S.A. Kitaigorodskii. J. Geophys. Res., 69 (C24), 5181–5190. Radder, A.C., 1992. An explicit Hamiltonian formulation of surface waves in water of finite depth. J. Fluid Mech., 237, 435–455. Resio, D.T., J.H. Pihl, B.A. Tracy and C.L. Vincent, 2001. Nonlinear energy fluxes and the finite depth equilibrium range in wave spectra. J. Geophys. Res., 106 (C4), 6985–7000.

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29

Ris, R.C., 1997. Spectral modelling of wind waves in coastal areas. Ph.D. thesis. Delft University of Technology, the Netherlands, 160 pp. Sverdrup H.U. and W.H. Munk, 1947. Wind sea and swell: Theory of relations for forecasting. H.O. Pub. 601, US Navy Hydrographic Office, Washington, DC, 44 pp. SWAMP group, 1985. Sea wave modeling project (SWAMP). An intercomparison study of wind wave predictions models, part 1: Principal results and conclusions. In: Ocean wave modelling, Plenum Press, New York, 256 pp. Tolman, H.J., 1992. Effects of numerics on the physics in a third–generation wind– wave model. J. Phys. Oceanogr., 22 (10), 1095–1111. Vincent, C.L., J.M. Smith and J. Davis, 1994. Parameterization of wave breaking in models. Proc. of Int. Symp.: Waves – Physical and Numerical Modelling, Vol. II, Univ. of British Columbia, Vancouver, Canada, M. Isaacson and M. Quick (Eds.), 753–762 Weber, S.L., 1989. Surface gravity waves and turbulent bottom friction. Ph.D. thesis, Univ. of Utrecht, the Netherlands, 128 pp. Weber, N., 1991. Bottom friction for wind sea and swell in extreme depth–limited situations. J. Phys. Oceanogr., 21, 149–172. Young, I.R. and Ph. van Vledder, 1993. A review of the central role of the nonlinear interactions in the wind–wave evolution. Phil. Trans. R. Soc. London. A., 342, 505–524.

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Chapter 3

WAM for fine-scale applications “Pensar es dif´ıcil, actuar es mas dif´ıcil todav´ıa, pero actuar como se piensa es lo m´ as dif´ıcil del mundo”

Veto, base de apoyo del EZLN 1

Abstract

New features have been added and several necessary changes have been made to the standard WAM–cycle4 (WAMC4) model code to run it efficiently when applied to shallow water regions. The restriction of having a source term integration time step smaller than or equal to the propagation time step was relaxed, considerably reducing the computational time needed. An additional reduction in computing time and an increase in accuracy were obtained by introducing a split–frequency time step. There is now a choice of using the octant or quadrant coordinate system for the propagation of the wave energy. The source term integration and, more particularly, the use of a limiter on the energy growth, were studied. For the evaluation of the energy decay in shallow water areas, different bottom friction formulations and an expression for the energy dissipation due to depth-induced wave breaking were added to the code. The procedure to make nested runs was changed in order to save data storage space and time spent on input/output (I/O) operations. Several other changes were done in order to improve the accuracy in high–resolution applications. A number of simple or idealized example applications are included to illustrate some of these enhancements. 1

This chapter was adapted from the article published as “The spectral wave model WAM adapted for applications with high spatial resolution”, by Monbaliu, J., R. Padilla–Hern´ andez, J.C. Hargreaves, J.–C. Carretero, W. Luo, M. Sclavo and H. G¨ unther. Coastal Eng., 41, 41–62, 1999.

31

32

CHAPTER 3. WAM FOR FINE-SCALE APPLICATIONS

3.1

Introduction

The development of a high–resolution spectral wave model capable of dealing with shallow water2 conditions and incorporating the interaction due to tide and surge was an important aspect of the EU MAST III project, PROMISE (Pre–Operational Modelling in the Seas of Europe) (also see Prandle, 2000). Waves form an integral part of the envisaged pre–operational framework of mathematical modelling tools that were thought to be necessary to quantify the rates and scales of the sediment exchange between the coast and the nearshore zone. At the beginning of the project, the option of improving the capabilities of the WAM–cycle4 (WAMC4) model (G¨ unther et al., 1992) was decided upon. One objective in the PROMISE wave modelling effort was an intercomparison exercise on the North Sea scale for a 1–month period. The main purpose of this exercise was to align the efforts of the participating institutions. Details of this intercomparison exercise can be found in Monbaliu et al. (1997, 1999). Although this could not be considered as an exhaustive test, the different model implementations provided similar results and the agreement with buoy and satellite measurements was acceptable and comparable to other values published in the literature. Another objective was the use of a spectral wave model for applications in coastal areas, i.e. shallow water areas where high–spatial resolution (order of 1 km) is needed. However, the standard WAMC4 code presented a number of limitations. The coding developments made for these coastal scale applications will be described in more detail in this chapter, with particular emphasis on computer central processing unit (CPU) efficiency and improved input/output (I/O). Some developments are illustrated with simple or idealized example applications. A practical example of the use of this enhanced model can be found in Prandle et al. 2000. In the remainder of the text, we will refer to the PROMISE version of WAMC4 as WAM–PRO. In Section 3.2 reasons are given for choosing WAMC4 for this study. Section 3.3 shows the areas where WAM–PRO was implemented to test the additions and modification to the code. All the necessary changes an additions made in WAMC4, including a model implementations are described in Section 3.4. Some other changes and additions dealing with Input/Output and some other enhancements done to WAM–PRO are explained in Section 3.4.3. The work that has to be done in order to extend the capabilities of WAM–PRO for fine–scale resolution is discussed in Section 3.5. A list of symbols is given at the end.

2

The terms shallow, intermediate and deep water for wind waves are in function of the relation between the depth (h) and the wave number (k) viz., kh < π/10, π/10 < kh < π and kh < π for shallow, intermediate and deep water, respectively. It may be justified to modify the limits of these regions for a particular applications.

3.2. WAVE MODELLING

3.2

33

Wave modelling

The wave modelling effort in the PROMISE project envisaged different applications. In terms of spatial scale, they ranged from large–scale applications such as the North Sea where a resolution of the order of 20–50 km seemed adequate, to small–scale applications such as the Holderness area where a spatial resolution in the order of 1 km is needed. None of the applications had to deal with rapid variations. A phase–averaged model was therefore the logical choice for the PROMISE project. It contains the necessary physics to describe the spectrum adequately, covers the space and time scales needed, and is reasonable in terms of computational overheads. The model, WAMC4 (G¨ unther et al., 1992), was chosen because it was, at the start of the project, the only state–of–the–art model readily available in the public domain. The WAMC4 model is used, e.g., for global operational wave forecasting at the European Centre for Medium–Range Weather Forecasts (ECMWF), and regionally in many other meteorological centres around the world. In deep and intermediate water, wave hindcasts are fairly reliable and efficient. There was also considerable experience and familiarity with this model at a number of the institutes participating in PROMISE. During the time that the PROMISE project was underway, other researchers had devoted much effort to developing the SWAN (Simulating WAves Nearshore) model. This model is now also in the public domain (see, e.g., Ris, 1997; Ris et al., 1999; Booij et al., 1999). SWAN is very similar to WAM, but was conceived as a shallow water model. It uses an implicit scheme for wave propagation and includes relevant shallow water source terms such as depth–induced breaking and triad wave–wave interactions. The most recent version of SWAN–40.11, can be used on any scale relevant for wind generated surface gravity waves, it has a high–quality propagation with 3 rd -order diffusion and can be run in Cartesian or spherical coordinates (Holthuijsen, et al., 2000). Also worth mentioning is the development of the so–called K–model (G¨ unther and Rosenthal, 1997; Schneggenburger, 1998). It is a phase–averaged model specially designed for application to coastal tidal environments. A new feature of the K-model is the consideration of wave dissipation by interaction with turbulence. The dynamics is formulated to allow for a convenient treatment of non–stationary water level and current fields. As part of the PROMISE project, the K–model was validated and applied for investigations in the Sylt–Rømø tidal basin (Schneggenburger et al., 2000).

3.2.1

The standard WAMC4 model

WAMC4 is a third–generation wave model, which solves the wave energy transport equation explicitly without any a priori assumptions on the shape of the wave energy spectrum. The equation solved in the code reads in Cartesian coordinates as: ∂F ∂ ∂ ∂ ¡ F¢ ∂ + (cx F ) + (cy F ) + σ cσ + (cθ F ) = Stot (3.1) ∂t ∂x ∂y ∂σ σ ∂θ where F (t, x, y, σ, θ) is the wave energy spectrum, t is the time, σ is the intrinsic angular frequency, θ is the wave direction measured clockwise from the true north, c x and

34


cy are the propagation velocities in geographical space, and cσ and cθ are the propagation velocities in spectral space (frequency and directional space). The left–hand side of the above equation represents the local rate of change of wave energy density, propagation in geographical space, and shifting of frequency and refraction due to the spatial variation of the depth and current. The right–hand side represents all effects of generation and dissipation of the waves including wind input Sin , white capping dissipation Sds , non–linear quadruplet wave–wave interactions Snl and bottom friction dissipation Sbf . A detailed description of the WAMC4 model can be found in G¨ unther et al. 1992 and Komen et al. 1994. One can show that Equation (3.1) is equivalent to the action density conservation equation. This is important because in the presence of ambient current wave action is conserved and wave energy is not (for a more detailed explanation, see Chapter 4). WAMC4 is also a state–of–the–art third–generation spectral wave model specifically designed for global and shelf sea applications. It can run for deep or intermediate water and includes depth and current refraction (steady depth and current field only). It can be set up for any local or global grid with a prescribed data set, and grids may be nested for fine scale applications. Computationally, Equation (3.1) is solved in two parts. The propagation of the energy density is solved by discretisation of the left–hand side, setting the right–hand side equal to zero, into a first–order explicit upwind scheme. The time step for this part is limited by the Courant–Friedrichs–Lewy (CFL) stability condition, and the wave propagation may be calculated on either a Spherical or Cartesian grid. The source term contribution is then added using a semi–implicit forward time scheme.

3.2.2

Difficulties in using WAMC4 in high–resolution applications

Some enhancements were required before the standard WAMC4 model could be applied to coastal areas. The obstacles encountered may be placed in the following categories: propagation scheme, source terms, CPU demand, (I/O) and other. One of the main restrictions in the standard WAMC4 code was the fact that the source integration time step had to be shorter than or equal to the propagation time step. Keeping the number of bins in the frequency domain constant, the computational cost of the model grows as m x n3 when the wave directional resolution increases m times and the space resolution increases n times, because the propagation time step is limited by the CFL criterion. Consequently, operational use of WAMC4 in coastal regions is computationally very expensive, with 1 km resolution applications running at the order of real time on current workstations. A second propagation problem was encountered for waves moving nearly parallel and close to the coast. There is an unrealistic loss of energy at the boundary points caused by the large second–order diffusion error of the first–order upwind scheme. The “numerical” losses tend to spread rapidly through the geographical and spectral domain. WAMC4 only takes into account stationary current fields and water depths. The shallow coastal zone is a dynamic region

3.3. MODEL IMPLEMENTATIONS

35

where changing currents and depths due to tide and surge play an important role in modifying the wave field. Additional source terms were considered necessary to deal with the shallow water physics. The bottom friction dissipation term in WAMC4 is modelled by the empirical JONSWAP expression. This formulation does not contain information about the processes in the wave boundary layer and it is therefore difficult to extract bottom stresses. This is especially true in a combined wave–current field. Moreover, in extreme shallow water surf zone, the process of depth–induced wave breaking becomes dominant over all other processes. Also, triad wave–wave interactions might become important. These last two processes are not included as source terms in WAMC4. A coastal application will typically use a fine grid nested into a coarser grid. WAMC4 writes the boundary conditions for the finer grid to disk every propagation time step. Before the fine grid run, these boundary conditions need to be interpolated in space and time to the propar scales of the fine grid. In addition to being time– consuming in terms of (I/O), this procedure uses tremendous amount of disk space. Some other items, mostly programming details, needed to be solved as well. The WAMC4 model limits all time steps to be an integer multiple of 1 min. This strongly constrains applications of the model at high resolutions. If space resolution is better than 1.13 km and the minimum frequency is set at 0.04177 Hz (standard value), the propagation time step must be smaller than 1 min. In shallow water applications, the grid resolution can become a fraction of a degree. Problems arise when resolutions, given in degrees in the original WAMC4 code, are fractional numbers. In such cases, the accuracy for the locations of the grid points might be insufficient and the number of computational points computed by the program might not correspond with the number of points given by the user. Also, the accuracy in the locations of the I/O points can be insufficient so that the difference in location for an output point computed by WAMC4 and then introduced by the user in the post–processing modules of WAMC4 can become so large (relatively) that no model information is found. In local applications where it can be justified to keep the boundary conditions constant, steady state conditions can be reached. Continuing the run can lead to unnecessary use of the computational resources.

3.3

Model implementations

The Holderness area The WAM–PRO code was implemented on a 2.4–km resolution grid of the Holderness area. Holderness is situated on the east coast of England. The bathymetry and the output locations are given in Figure 3.1. For more details about this region, the reader is referred to Prandle et al. (2000). The wave spectra were modelled with 25 frequency and 12 directional bands. The directional resolution was therefore equal to 30o . The frequency bands were set to a logarithmic scale, with ∆f /f = 0.1. The lowest and highest frequencies were 0.04177 Hz and 0.4114 Hz, respectively.

36


Figure 3.1: The Holderness area. L1 to L17 are locations for model output

The North Sea area For the North Sea area test, two different implementations were used. The first is a three–level nested grid implementation using spherical coordinates. To account for swell generated in the Norwegian Sea, the coarse grid covers the area 48 o N– 70o N, 7o W–12o E and this has a resolution of 1/3o latitude and 2/3o longitude. The intermediate grid covers 50o N–52o N, 0o W–4o E with a resolution of 1/24o latitude and 1/12o longitude. The fine grid covers the area 51o N–51.50o N, 2.5o E–3.6o E with a resolution of 1/96o in both latitude and longitude, corresponding to a grid size of about 1.2 km. The second is a two–level nested grid implementation (resolution of 50 x 50 km 2 for the coarse and 10 x 10 km2 for the nested grid), which covers basically the same area as the coarse grid and intermediate spherical grid. A stereographic projection was applied to enable the use of Cartesian coordinates. The directional and frequency grids were identical to the Holderness implementation.

3.4 3.4.1

WAM–PRO for fine–scale coastal applications Introduction

The obstacles encountered while running WAMC4 in high–resolution applications were identified in Section 3.2.2. Modifications made to the code are described below, and the effects are illustrated with a number of examples compiled from various tests carried out in the North Sea or coastal scales. The areas of the model implementations

3.4. WAM–PRO FOR FINE–SCALE COASTAL APPLICATIONS

37

are described in Section 3.3. The illustrative examples and the discussion and interpretation of the results follow immediately after the theoretical background. Special attention is paid to the aspects of speed–up and improved I/O without compromising on output quality. The changes in the code required to obtain WAM–PRO from the standard WAMC4 code have been documented in a technical report (Monbaliu et al., 1998).

3.4.2

Code modification

PROPAGATION Improved stability for large depth gradients The first–order advection algorithm becomes unstable when large depth gradients are present. Advection is multi–dimensional in the same time step. In particular, for high–resolution applications of the model in shallow water, the advection time step frequently needs to be reduced for a few grid points where high gradients in the bathymetry are found. For a specific energy component of the spectrum Fj at time step n + 1, the first–order explicit advection equation, after being rearranged, takes the form: 0

0

0

Fjn+1 = (1 − α1 − α2 − . . . − αn )Fjn + α1 F 1nj + α2 F 2nj + . . . + αn F Njn

(3.2)

0

where αi and αi refer to the appropriate coefficients in the upwind numerical scheme and F 1nj , . . . F 2nj . . . F Njn refer to the upwind components of the energy advection (from different spectra but from the same frequency direction bin for latitude and longitude advection, and from the same spectrum but from a different direction– frequency bin for propagation in direction and frequency). The number of terms, N , depends on the specifications supplied by the user (deep or shallow water run, depth refraction and/or current refraction included). This can be represented by a straight line of slope (1 – S) and intercept B: Fjn+1 = (1 − S)Fjn + B

(3.3)

Making α1 , α2 , . . . αn < 1 does not assure stability. The value of S(= α1 + α2 + . . . αn ) also needs to be checked: if (0 < S ≤ 1) then stable, if (1 < S ≤ 2) then probably unstable. Figure 3.2 shows how the slope changes with the value of S and how negative and non–convergent values can be obtained, making the advection algorithm unstable. A new subroutine called in the initialisation phase of the wave model checks the stability for all grid points and directions at the lowest frequency. The run is stopped if α > 1 or S > 2. If S is between 1 and 2, the required time step for the grid point is computed and given in the output as user information.

Altering the source integration and propagation time steps The contributions from the propagation terms and from the source terms are calculated separately.

38


Figure 3.2: First–order upwind advection algorithm. For S>1, negative energy can be produced and the system can become unstable.

For a typical WAMC4 application, 75% of the CPU time is needed for the calculation of the source terms, 20% for the I/O operations and only about 5% for the propagation calculation. The CFL condition limits the maximum time step to be used in the propagation scheme. Clearly, the efficiency of the source term calculation is critical for decreasing model run time so that fine resolution implementations become feasible. The source terms are calculated as point processes, meaning that they only use spectral information from one location in space. The time scale of the physical processes associated with the source terms does not necessarily scale with grid resolution in the same way as the CFL condition imposes time step restrictions on the propagation. There is, therefore, no reason why, in fine resolution applications, the source term time step should not be greater than the propagation time step. The code has been modified to allow an integer number of propagation time steps to be performed for each source term time step. The discretised version of Equation 3.1 expresses the spectrum F n+1 at the level t (= tn + ∆t; ∆t is the source term time step) in terms of F n . First Equation 3.1 is solved without source terms (i.e. the right–hand side of Equation 3.1 is zero). Assume that one source term time step is split into m (an integer ≥ 1) propagation time steps. The increment in wave energy densiy due to propagation at time level n+1


39

tn+r/m (0 ≤ r ≤ m − 1) is expressed as: ¡ r+1 r ¢ r (3.4) ∆prop F n+ m = F n+ m − F n+ m prop · ¸ ∂ ∂ ∂ ∂ ¡ F¢ = − cσ + (cx F ) + (cy F ) + σ (cθ F ) ∆tmax p ∂x ∂y ∂σ σ ∂θ

The equation is evaluated using a split first–order upwind scheme. Here, m equals at the ratio of the source term time step ∆tint to the propagation time step ∆tmax p max the highest frequency. Every ∆tp can be split into several sub–propagation time steps, depending on frequency. After m propagation time steps (i.e. one source term time step), time level tn+1 is reached and the wave energy spectrum increment due to propagation thus becomes: (F n+1 − F n )prop =

m−1 X

r

∆prop F n+ m

(3.5)

i=0

Second, the source terms are integrated. Finally, the full spectrum at time level t is obtained by adding the propagation and source term contributions. Care should be taken in the use of this feature. In high growth situations, the time scale of variation of the non–linear interactions is small. Also in shallow water regions, the source term time step should not be so large that there is significant variation in the water depth between successive source term calculations. With a source term time step that can become larger than the propagation time step, the proportion of the total CPU time used in calculating the source terms, of course, reduced. A further increase in the computational speed can be obtained by altering the propagation time step to be a function of frequency. Higher frequencies have a smaller group velocity. A longer time step can be used for slower propagating wave components, while retaining numerical stability. Therefore, a split–frequency time step numerical scheme was developed for wave propagation, which allows the propagation time step to be frequency–dependent. Users only need to specify one propagation time step corresponding to any frequency other than the first one. The model will determine how many different propagation time steps are to be used according to the CFL conditions corresponding to different frequencies. In any case, the propagation time steps at different frequencies must be an integer multiple of the user specified propagation time step for a particular frequency. The ratio of source term time step to propagation time steps must be an integer number (only possible in the present scheme) or the reciprocal of an integer number (as in the original scheme). The modifications described in this subsection are illustrated with a simple example using the Holderness area set–up (see Section 3.3 on page 35). Four tests, numbered H1–H4, were carried out with different propagation and source term time step combinations (see Table 3.1). A uniform southerly wind of 18.45 ms−1 was used. Note that the time step used in H1 is limited by the original scheme of the WAMC4 model. The limiter used was that of Equation 13 from Luo and Sclavo (1997). n+1

40


Table 3.1: Four tests with different time steps for the case of a uniform wind blowing offshore of the Holderness region.

Propagation time step (s) H1 H2 H3 H4

60 for all frequencies 60 for the lowest frequency 120 for the remaining frequencies Idem H2 Idem H2

Source term time step (s) 60 360 720 1080

Figure 3.3 shows the growth of the significant wave height as a function of fetch for the four test runs along the fetch line L1–L17 (see Figure 3.1) after 48 h. One can see that nearly identical growth curves are obtained for runs H1, H2 and H3. For the run H4 with a source term time step of 1080 s, the difference with the original scheme is about 5% for significant wave height and 1% for the mean frequency (plot not shown). Figure 3.4 shows the significant wave height as a function of time for the four test cases at locations L5 and L17. Similar growth curves (not shown) were found at other locations. The waves are fully developed in a few hours. The growth curves of the runs H1, H2 and H3 are very close. The significant wave height difference between the H1 and the H4 run is of the order of 5%. The mean frequency difference between the H1 and the H4 run is less than 2%. Some oscillations (numerical noise) are visible for the run H4, but these are not amplified with time. The corresponding one–dimensional frequency spectra are shown in Figure 3.5 for runs H1 and H3. An overshoot is visible in the results of both runs and the energy difference at the peak frequency is less than 5%. One of the crucial objectives to be achieved in the study was the improvement in the computational efficiency. Figure 3.6 shows the relative CPU time used for the original propagation and source term integration scheme and the present scheme with different time steps combinations. Run H1 corresponds to the original scheme and limits the source term time step to be less than or equal to the propagation time step, set here to 60 s to ensure numerical stability. It is clear that the longer the source term integration time step, the more efficient the computation. Runs H3 and H4 illustrate that relaxing the constraint on the source term integration time step makes the computations nearly one order of magnitude more efficient. However, for runs with a source term time step longer than that in run H4, the drastic reduction in CPU time will disappear. The relative contribution of the source term integration to the total computational time is no longer dominant when the source time step exceeds a certain number of times the propagation time step.


41

Figure 3.3: Significant wave height as a function of fetch for uniform offshore wind. Test cases are given in Table 3.1. The law of diminishing returns applies. Moreover, the results from run H4 with a source term time step of 1080 s already start to deviate from the ‘true’ H1 run results, which is not the case for the results of the H2 and H3 runs. This suggests that it remains wise to limit the source term time step to less than 10 min, in accordance to what is needed from a physical point of view and according to the suggestion in the WAMC4 manual of 10 m as a maximum time step for shallow water applications. Generally, the physical conditions (wind, currents, depth, etc.) do not change more rapidly and one can obtain a good compromise between accurate results and computational efficiency.

An octant vs. a quadrant propagation scheme In order to reduce the physically unrealistic energy loss at the boundary points in conditions such as when the waves are propagating parallel and close to the coast, an alternative propagation coordinate system was introduced into the WAM–PRO model. The characteristics and the mathematical details of the octant scheme can be found in Cavaleri and Sclavo (1998). Geometrical interpretations of the octant (new)

42


Figure 3.4: Time series of significant wave height at locations L5 and L17 for the four test cases described in Table 3.1.

Figure 3.5: Comparison of the one–dimensional frequency spectrum for different time steps. The test cases are given in Table 3.1 (run H1, ∆tprop = ∆tint = 60 s; run H3, ∆tprop = 120 s, ∆tint = 720 s).


43

Figure 3.6: The relative CPU time (%) for different time step combinations. The test cases are given in Table 3.1. and the quadrant coordinate system (standard in WAMC4) are given in Figure 3.7. In the case of the octant advection scheme, eight possible propagation directions are defined instead of four for the quadrant scheme, the four cardinal directions plus the four diagonals, splitting the advection space into eight parts. The energy at point A at time tn is advected to the point P after one propagation time step. The vector AP is split into its two components AP1 and AP2 along the two axes AD (point D is consider as a land point) and AC (see Figure 3.7a), and the energy is considered to be advected as a whole to both P1 and P2 . Then, the redistribution of energy is: n+1 n EC = EA

AP1 ; AC

n+1 n ED = EA

AP2 ; AD

n+1 n+1 n+1 n EA = EA − EC − ED

(3.6)

The distribution of energy in the quadrant propagation scheme is along the axes AB and AD (see Figure 3.7b) and is therefore: n+1 n EB = EA

AP1 ; AB

n+1 n ED = EA

AP2 ; AD

n+1 n+1 n+1 n EA = EA − EB − ED

(3.7)

One can see immediately that the energy lost (equal to that removed) at the land point D in the case of octant scheme is less than in the quadrant scheme (the vector AP2 in Figure 3.7a is smaller than the corresponding one in Figure 3.7b). As an illustrative example, hindcast runs for the period of February 1993 were carried out for the North Sea area using the coarse grid implementation with stereographical projection (see Section 3.3). Two runs labelled C3 and C4 were retained. Except for the propagation scheme, the runs are identical. It was found that in

44


Figure 3.7: Geometrical interpretation of the two schemes for first–order advection. The region AP1 PP2 is influenced by one advection step. (a) octant scheme, (b) quadrant scheme. general, the model produces higher significant wave height when using the octant coordinate than when using the quadrant system. Figure 3.8 illustrates the significant wave height differences between run C4 (octant propagation scheme) and run C3 (quadrant propagation scheme) on February 21, 1993 at 00:00 GMT. At that moment, waves in the central North Sea were quite high ('8 m significant wave height). In the central part, differences up to about 15 cm can be observed. However, more importantly close to some coasts, e.g., along the coast of eastern Scotland and along the coast to the southeast of Orkney and the Shetlands, differences can be higher than 30 cm. This will have consequences on boundary conditions generated for more detailed assessment of the wave conditions in these areas.

Currents. To take into account the non–stationary current field and water level changes, the WAM–PRO was prepared for coupling with a storm–surge model in a two–way system. The model can accept hydrodynamic fields (currents and water level) from and return wave–related information (radiation and surface stresses) to the hydrodynamic model. In Chapter 4 the interaction mechanisms and their importance are explored.


45

Figure 3.8: The significant wave height difference between run with the octant (C4) and quadrant (C3) propagation scheme at 00:00 GMT February 21, 1993 (C4 minus C3).

46


SOURCE TERMS Numerical limiters In the standard WAMC4, the source term equation (Equation 3.1 without the advection terms) is solved by the following finite difference approximation: n+1 n Fi,j,k,l − Fi,j,k,l n+1 n + αSi,j,k,l (3.8) = (1 − α)Si,j,k,l ∆t where i and j denote the position in geographical space, k and l represent the position in the wave direction and relative frequency space and α is in the range [0,1]. Since the source functions depend non–linearly on the spectrum F , Taylor expansions were introduced. By disregarding the negligible off–diagonal contribution of the function derivatives (Komen et al., 1994) in the Taylor expansions, the increments in spectral energy density due to the source terms for one time step may be expressed as: ¤−1 £ n n (3.9) 1 − ∆tαΛni,j,k,l = ∆tSi,j,k,l ∆int Fi,j,k,l

where Λni,j,k,l is the diagonal matrix of the partial derivatives of the source function. A forward time splitting technique is used (α = 1) except for positive Λ ni,j,k,l when because of the obvious numerical instability, an explicit technique is used (α = 0). However, the explicit implementation is not generally stable (Press et al., 1994), so a limiter on the increments in wave energy was imposed in WAMC4. It is expressed as: n = 6.2 × 10−5 f −5 ∆t/1200 ∆int Fmax

(3.10)

Hersbach and Janssen (1999) found that for applications with a very small spatial grid such as in coastal regions, the old limiter (Equation 3.10) was so severe that the growth curves did not scale properly with the air friction velocity. They revised another limiter (further referred to as the HJ limiter) to solve the above problem: n = 3 × 10−7 g u ˜∗a fc f −4 ∆t ∆int Fmax

(3.11)

where fc is the cut–off frequency (the highest frequency in the spectrum), ∆t is the source term integration time step and u ˜∗a is determined by: ∗ /f); u ˜∗a = max(u∗a , gfPM

∗ = 5.6 × 10−3 fPM

(3.12)

with u∗a the air friction velocity and fP∗ M the dimensionless Pierson–Moskowitz peak frequency. Hersbach and Janssen (1999) show that the limiter in Equation 3.11 is scale–invariant because it can be written just in terms of non–dimensional quantities. It was released as a patch to WAMC4 (Hasselmann, 1996). According to Luo and Sclavo (1997), the HJ limiter may give rise to problems for low frequencies in relatively calm conditions since the value for u ˜ ∗a will be much larger in low frequencies than in the high frequencies. They use the mean frequency fm instead of the cut–off frequency fc [note that Hersbach and Janssen, (1999) also mention this possibility] n ∆int Fmax = 3 × 10−7 g u ˜∗a fm f −4 ∆t

(3.13)


47

where ∗ /f) u ˜∗a = max(u∗a , gfPM

(3.14)

However, when a small enough time step is used, WAM is numerically stable for fetch–limited growth cases even when no limiter is used. This demonstrates that the problem arises from the numerical integration scheme rather than the underlying physical equations for wave growth. The imposition of a limiter prevents convergence to the continuum solution as the time increment is decreased and so Hargreaves and Annan (1999) argue that a better approach would be to dispense with the limiter and improve the integration scheme. By constraining Λni,j,k,l to be always negative, the forward time scheme (α = 1) can always be used and this gives greatly improved results over the standard method (this method is referred to, hereafter, as the HA method). However, stability cannot be absolutely guaranteed for any computationally reasonable time step, since the integration of source terms is a ‘stiff’ problem (Press et al., 1994). The method is found to be stable for fetch–limited growth test cases up to high wind speeds as illustrated below, but when the model is run with the HA limiter in real applications, it is found that instabilities do sometimes occur. The model has been found to be well behaved in real applications when the HJ limiter is imposed on the highest frequencies (>0.25 Hz) while employing the HA method. The HA method will be included as an option in the disseminated WAM–PRO code. To illustrate the effect of limiters, the Holderness set–up is again used with a uniform southerly wind of 30 ms−1 to drive the model. When the limiter is completely removed from the WAMC4 coding with the standard integration scheme, the model is stable only for very small time steps. Therefore, reference runs were made running with a 10 s source term time step. Figure 3.9 shows the fetch–limited growth curves for model runs using the HA method and the HJ limiter, and the reference run for comparison. For a time step of 300 s, the HJ limiter and the HA method produce similar results. However, as the time step is decreased, the HA results converges to the reference run whereas the HJ limiter run does not. Figure 3.10 shows a further comparison of the results. This figure shows the fetch–limited growth wave spectra obtained for fetches equivalent to the Holderness wave buoys (see Prandle et al., 2000). The effect of the HA method only seems critical in high growth situations, and it only has a significant effect in small fetch situations near coasts, consistent with the remarks in Section 3.4.2 on page 37, that correct representation of rapid variations demands small time steps. As can be seen from Figure 3.9, all three runs produce almost identical results for fetches of 100 km.

Bottom friction and depth–induced breaking Four bottom friction formulations for pure wave conditions have been implemented in WAM–PRO. They are the formulations of Hasselmann and Collins (1968), Collins (1972), Madsen et al. (1988) and Weber (1991). A detailed discussion on these

48


12

10

hs(m)

8

6

4

2

0 0

20

40

60

80 100 fetch (km)

120

140

160

180

Figure 3.9: Fetch–limited growth curves. The solid line shows the reference run, the dotted line the run with a 10 s time step and HJ limiter, the dot–dashed line the run with a 200-s time step and HJ limiter and the dashed line shows the 300-s run with the HA method. This last method is indistinguishable from the reference run for a 10-s time step. formulations as well as on the empirical JONSWAP formulation, which is implemented in the standard WAMC4, can be found in Chapter 2. An illustration of their use in a practical situation can be found in Luo et al. (1996). Bottom friction formulations accounting for a combined wave–current field have been introduced as well. Several theoretical models for the bottom friction in combined wave–current flows have been developed and have advanced our knowledge of wave–current interactions (e.g., Grant and Madsen, 1979; Christoffersen and Jonsson, 1985 and Madsen, 1994). The formulations of Christoffersen and Jonsson (1985) and Madsen (1994) have been implemented in WAM–PRO, but they have not yet been fully tested. An interesting discussion on wave–current interaction observations in the Holderness area is given by Wolf and Prandle (1999) and Wolf (1999). To simulate depth–induced wave breaking, a source term based on the theory of Battjes and Janssen (1978) has been added in WAM–PRO. They proposed a bore– based model to calculate the total rate of energy dissipation of random breaking waves. Analyzing laboratory observations Beji and Battjes (1993) has shown that the shape of initially uni–modal spectra propagating across simple beach profile, is fairly insensitive to the presence of depth–induced breaking. For the mathematical expression of the source term and for a practical illustration, the reader is referred to van Vledder et al. (1995) and Luo (1995). Since depth–induced wave breaking is a dominant source term in shallow water applications, a more detailed analysis of breaking formulations should be carried out. An interesting study on the introduction of other model formulations for depth–induced wave breaking has been executed by Becq and Benoit (1996). They have compared the behaviour of four different formulations.


fetch=2.4km

fetch=12km

9

40

8

35

7

30 25 energy

energy

6 5 4

20 15

3

10

2

5

1 0 0

49

0.2 0.4 frequency (Hz)

0 0

0.6

fetch=31.2km


0.6

fetch=98.4km

100

200

80

60

energy

energy

150

40

100

50 20

0 0


0.6

0 0


0.6

Figure 3.10: Wave spectra at different fetches. The solid lines are the reference run with 10 s time step, the dot–dashed and dashed lines are for the Hersbach limiter and Hargreaves–Annan approach, respectively, both with a 300-s time step.

50

3.4.3


I/O and some other peculiarities

Boundary conditions Boundary conditions are now interpolated in time within the WAM–PRO model. When the standard WAMC4 is run in a nested mode, the boundary conditions for a finer grid are written to the computer disk at every coarse grid propagation time step. Those boundary conditions needed to be interpolated in space and time for the finer grid (the time–space resolution depends on the characteristics of the finer grid), and stored for the whole period on the computer disk. For high–resolution applications, this meant that nested grid cases demanded very high disk storage capacity. In order to avoid these disk storage problems, two changes were made to the code. Firstly, the interpolation of the boundary conditions in time and space is now executed in the nested grid, drastically reducing the disc space needed and the computing time. This was done by incorporating the stand–alone boundary interpolation routine into the new WAM–PRO code. Secondly, the user can now define the time step to write boundary conditions for a finer grid application. The statistical parameters of the spectrum do not change much in the time span of a few minutes. In order to compare the differences in disk usage and results (wave heights and periods) between the original WAMC4 and the two disk usage optimisations implemented, the three–level nested grid system in spherical coordinates described in Section 3.3 was used. For this test, three cases were carried out. The first case used the standard WAMC4 procedure. This means that the boundary conditions were saved every propagation time step for the coarse and intermediate grids. These boundary conditions then need to be interpolated in time (every propagation time step in the nested grid) and space, and all the interpolated information need to be stored on disk. In the second case, the boundary conditions were interpolated in the nested runs themselves (intermediate and fine grids). In the third case, not only was the interpolation of boundary conditions in time and space done inside the nested runs (as in the second case), but the output time step for the boundary conditions from the coarse and intermediate grid was also user–defined. The wave model was run in the third case with an output time step for the boundary conditions of 40 min for the coarse and 20 min for the intermediate grid. Note that the propagation time step was 20 min for the coarse and 4 min for the intermediate grid. The savings in disk storage space are enormous (see Figure 3.11). This is of considerable practical importance for operational modelling. The outputs of the model for the different cases are almost identical (comparing second against third case) as can be seen in Figure 3.12 for significant wave height and Figure 3.13 for the zero up crossing wave period. The maximum difference for H s is 0.02 m for the maximum Hs of 3.7 m. For Tm02 the maximum difference is 0.06 s for a maximum period of 7.4 s.

The minimum time step The constraint in the original scheme of WAMC4, which limits all time steps to be


51

100

90

80

Relative disk usage [%]

70

60

50

40

30 20% 20

10

0

5%

1

2 case

3

Figure 3.11: Relative disk usage for different optimizations in output boundary conditions. a multiple of 1 min, has been removed. All the dates are now represented by a 12– digit character number (with the format yymmddhhmmss, where yy indicates the year, mm is for month, dd for day, hh for hour, mm for minute and ss for second). Therefore, all model time steps (wind I/O time steps, source term integration time step, propagation time step and all output time steps) can be specified as integer in seconds or hours (if greater than 1 h). They do not have to be in multiples of 1 min as limited by the original scheme of WAMC4.

High spatial resolution To avoid problems with fractional numbers, the dimension and location of the computational grid must now be given in arc seconds. To solve rounding problems when comparing the location of grid points, these are now defined with accuracy better than 1 s.

Steady state In a small area covered by a high–resolution grid nested into a coarse grid, a steady state situation can be reached before boundary conditions are changed. In this case, the model will continue integrating in time unnecessarily. An option has been installed in the code so that consecutive wave fields are compared and a warning message is printed if a steady state is found.

52


0

s

∆ H [m] × 10

−3

−5

−10

−15

(BC 20 : 4 min) − (BC 40 : 20min) −20 16

18

20

22 Feb 93 (days)

24

26

28

Figure 3.12: Absolute differences of significant wave height with two different input boundary condition time steps.

The second moment period Tm02 The second moment period Tm02 is quite often recorded by buoys. In order to compare the model data to these measurement data, a subroutine to compute that parameter was added in the WAM–PRO model code. Its expression is

Tm02

¸− 1 ¸− 1 ·R R 2 ·R R 2 ω F (σ, θ)dσdθ 2 ω F (ω, θ)dωdθ 2 R R R R = 2π = 2π F (ω, θ)dωdθ F (σ, θ)dσdθ

(3.15)

where ω is the absolute radian frequency, determined by the Doppler shifted dispersion relation.

The radiation stress Also the calculation of the radiation stresses is now included. This is of interest when the wave model is coupled to a hydrodynamic model. Its expression can be given in term of the wave spectrum F as calculated by the wave model: Rij = ρg

Z

2π 0

Z

0

∞·

c g ki kj + c k2

µ

¶ ¸ cg 1 − δij × F (σ, θ)dσdθ c 2

(3.16)

where ki and kj are the components of the wavenumber vector in x–, y–direction, δij is the delta function.

53

3.5. FURTHER WORK

0.06

0.04

∆T

m02

[s]

0.02

0

−0.02

−0.04 (BC 20 : 4 min) − (BC 40 : 20min) 16

18

20

22 Feb 93 (days)

24

26

28

Figure 3.13: Absolute differences of zero up crossing wave period (Tm02 with two different input boundary condition time steps.

3.5

Further work

Nonlinear interactions in shallow water, the so–called triads, are not yet accounted for in the model. Young and Eldeberky (1998) have seen some evidence of triad interactions at fairly high values of the relative depth (kp h, where kp is the wave number at the spectral peak and h is the water depth) when analyzing data from experiments in a shallow lake. They suggest that the effects of such interactions might not be significant only in the shoaling region, but also may need to be introduced in transitional water depths found on many continental shelves. Although several formulations for bottom friction energy dissipation have been coded, which one to use in operational applications is far from clear3 . Little or no effort has been made up to now to assimilate data (wave, wind, bathymetry, etc.) in high–resolution wave model applications. For coastal applications, the data needs are very demanding, since the details of the directional energy distribution are very important. Moreover, memory in the system is short–lived. Waves entering the system at the seaward boundary arrive at the coast in a period of a few hours and dissipate most, if not all, of their energy.

3.6

Summary and conclusions

New features have been added, and several changes to the standard WAMC4 model code were needed in order to run it efficiently in shallow water applications. The restriction of having a source term integration time step smaller than or equal to the 3

Analyzing the different expressions for the bottom friction dissipation, in Chapter 5, an attempt is made to select a formulation to be used in depth–limited situations.

54


propagation time step was relaxed. This change enabled the source term integration time step to be much longer than the propagation time step and, therefore, the CPU time for high–resolution applications could be reduced considerably. Additional reduction of the CPU time was obtained by the introduction of a split–frequency time step. Besides the changes to speed up computation, the propagation coordinates system can be optionally chosen to be octant or quadrant. The role of the limiter in the source term integration was discussed. Different bottom friction formulations and the Battjes–Janssen expression for the energy dissipation due to depth–induced wave breaking were added to the WAM code in the hope of obtaining a more reliable evaluation of the energy decay in shallow water areas. The procedure of making nested runs was changed in order to save disk space. As a side–benefit, time spent on I/O was also decreased. Other minor changes were made to enhance the model performance in high–resolution applications, and an option to output additional wave parameters was included. As a result of all these changes, applications of the WAM–PRO model code in shallow water areas, where high–spatial resolution is needed, has become feasible and economical compare to the original WAMC4. The WAM–PRO code provides a powerful tool to further explore the modelling of wave spectra in coastal regions. As such, it forms an important part of a pre–operational framework of mathematical modelling tools necessary to quantify sediment transport in the coastal and nearshore zone. It is hoped that the improved description of the wave field and of the tides and surge field will lead to more sophisticated parameterization of the hydrodynamic forces in sediment transport modelling.

List of symbols cg cx , c y cσ , c θ d E f F fc fm fP∗ M n Fmax g h Hs

Propagation velocity of the wave energy [ms−1 ] Wave propagation velocities in geographical x–, y–space [ms−1 ] Wave propagation velocities in spectral σ, θ–space [s−2 , rad s−1 ] Water depth [m] Total energy of the wave spectrum [m2 ] Frequency [Hz] Energy density spectrum [m2 s rad−1 ] Cut-off frequency [Hz] Mean frequency of the wave spectrum [Hz] Dimensionless Pierson–Moskowitz peak frequency [–] Maximum wave energy for a given spectral component at at time (n) [m−2 s rad−1 ] Acceleration due to gravity [ms−2 ] Total water depth [m] Significant wave height [m]

3.7. REFERENCES

k k m N s Stot Sbf Sbk Sds Snl t Tm02 Tp U U u ˜∗a x, y Delta θ Λni,j,k,l σ ω ∇

3.7

55

Wavenumber vector [m−1 ] Wavenumber [m−1 ] Space coordinate perpendicular to the wave propagation direction Wave action density [m2 s2 rad−1 ] Space coordinate in the wave propagation direction Represents all effects of generation and dissipation of wave energy [m−2 s rad−1 ] Dissipation of wave energy by bottom friction [m2 s rad−1 ] Dissipation due to depth–induced wave breaking [m2 s rad−1 ] Dissipation by wave friction [m2 s rad−1 ] Non-linear wave–wave interactions [m2 s rad−1 ] Time [s] Second moment period [s] Wave peak period [s] Mean current velocity [ms−1 ] Mean current speed Air friction velocity [ms−1 ] Horizontal coordinates [m] Integration time step [s] Wave direction [rad] Diagonal matrix of the (time) partial derivatives of the source function [m−2 rad−1 ] Intrinsic frequency [Hz] Radian frequency [rad s−1 ] Gradient operator in the geographical space [-]

References

Battjes J.A. and Janssen J.P.F.M., 1978. Energy loss and set-up due to breaking of random waves. Proc. 16th Int. Conf. Coastal Eng., 569–587. Battjes J.A., 1994. Shallow water wave modelling. Proc. Int. Symp. Waves – Phys. Numer. Modell., University of British Columbia, Vancouver I, 1–23. Becq F. and Benoit M., 1996. Implementation et comparaison de différents modèles de houle dans la zone de déferlement. EDF, Report HE–42/96/037/A, in French. Beji S. and Battjes J.A., 1993. Experimental investigation of wave propagation over a bar. Coastal Eng., 19, 151–162. Booij N., Ris R.C. and Holthuijsen L.H., 1999. A third-generation wave model for coastal regions: 1. Model description and validation. J. Geophys. Res., 104 (C4), 7649–7666.

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Cavaleri L., Sclavo M., 1998. Characteristics of the quadrant and octant advection schemes in wave models. Coastal Eng., 34, 221–242. Collins J.I., 1972. Prediction of shallow water spectra. J. Geophys. Res., 93 (C1), 491–508. Christoffersen J.B. and Jonsson I.G., 1985. Bed friction and dissipation in a combined current and wave motion. Ocean Eng. 12, 387–423. Grant W.D. and Madsen O.S., 1979. Combined wave and current interaction with a rough bottom. J. Geophys. Res., 84 (C4), 1797–1808. G¨ unther H., Hasselmann S. and Janssen P.A.E.M, 1992. The WAM model Cycle 4. Report No. 4, Hamburg. G¨ unther H. and Rosenthal W., 1997. Shallow water wave modelling with non-linear dissipation. Dtsch. Hydrogr. Z. 49, 431–444. Hargreaves J.C. and Annan J.D., 1999. Integration of source terms in a thirdgeneration wave model. J. Phys. Oceanogr., Submitted. Hersbach H. and Janssen P., 1999. Improvements of the short fetch behaviour in the WAM model. J. Atmos. Oceanic Technol. 16, 884–892. Hasselmann K. and Collins J.I., 1968. Spectral dissipation of finite–depth gravity waves due to turbulent bottom friction. J. Mar. Res., 26, 112. Hasselmann S., 1996. Personal communication. Holthuijsen L.H., Booij N., Ris R.C., Haagsma IJ.G., Kieftenburg A.T.M.M., Kriezi E. E., 2000. SWAN Cycle III version 40.11, User Manual. Delft University of Technology, the Netherlands. Komen G.J., Cavaleri L., Donelan M., Hasselmann K., Hasselmann S., Janssen P.A.E.M., 1994. Dynamics and Modelling of Ocean Waves. Cambridge Univ. Press, Cambridge. Luo W., 1995. Wave modelling in shallow water. PhD Thesis, Civil Engineering Department, Katholieke Universiteit Leuven, Belgium. Luo W., Monbaliu J., Berlamont J., 1996. Bottom friction dissipation in the Belgian coastal waters. Proc. 25th Int. Conf. Coastal Eng., 836–849. Luo W., Sclavo M., 1997. Improvement of the third-generation WAM Model (Cycle 4) for applications in the nearshore regions. Proudman Oceanographic Laboratory, Internal Document No. 116. Madsen O.S., Poon Y.K., Graber H.C., 1988. Spectral wave attenuation by bottom friction: theory. Proc. 21st Conf. Coastal Eng., 492–504. Madsen O.S., 1994. Spectral wave–current bottom boundary layer flows. Proc. 24th Conf. Coastal Eng., 384–398.

3.7. REFERENCES

57

Monbaliu J., Zhang M.Y., de Backer K., Hargreaves J., Luo W., Flather R., Carretero J.C., Gomez Lahoz M., Lozano I., Stawartz M., Gunther H., Rosenthal W. and Ozer J., 1997. WAM model intercomparisons – North Sea. Proudman Oceanographic Laboratory, Report No. 47. Monbaliu J., Hargreaves J.C., Carretero J.C., Gerritsen H., Flather R., 1999. Wave modelling in the PROMISE project. Coastal Eng., 37, 379–407. Monbaliu J., Padilla R., Osuna P., Hargreaves J., Flather R. and Carretero J.C., 1998. Shallow water version WAM–C4–S.01 documentation. Proudman Oceanographic Laboratory, Report No. 52. Prandle D., 2000. Operational oceanography in coastal waters. Coastal Eng., 41, 3–12. Prandle D., Hargreaves J.C., McManus J.P., Campbell A.R., Duwe K., Lane A., Mahnke P., Shimwell S. and Wolf, J., 2000. Tide, wave and suspended sediment modelling on an open coast Holderness. Coastal Eng., 41, 237–269.. Press W.H, Teukolsky S.A., Vetterling W.T. and Flannery B.P., 1994. Numerical Recipes in FORTRAN: The Art of Scientific Computing. Cambridge Univ. Press. Ris R.C., 1997. Spectral modelling of wind waves in coastal areas. PhD Thesis, Delft University of Technology, the Netherlands, 160 pp. Ris R.C., Holthuijsen L.H. and Booij, N., 1999. A third-generation wave model for coastal regions: 2. Verification. J. Geophys. Res., 104 (C4), 7667–7681. Schneggenburger C., 1998. Spectral wave modelling with non–linear dissipation. PhD Thesis, University of Hamburg, GKSS External Report 98/E/42. Schneggenburger C., G¨ unther H. and Rosenthal W., 2000. Spectral wave modelling with non–linear dissipation: validation and applications in a coastal tidal environment. Coastal Eng., 41, 201–235. van Vledder G.Ph., de Ronde J.G. and Stive M.J.F., 1995. Performance of a spectral wind-wave model in shallow water. Proc. 24th Int. Conf. Coastal Eng., 761–774. Weber S.L., 1991. Eddy viscosity and drag law models for random ocean wave dissipation. J. Fluid Mech., 232, 73–98. Wolf J. and Prandle D., 1999. Coastal Eng., 37, 471–485.

Some observations of wave–current interaction.

Wolf J., 1999. The estimation of shear stresses from near-bed turbulent velocities for combined wave–current flows. Coastal Eng., 37, 529–543. Young I.R., Eldeberky, Y., 1998. Observations of triad coupling of finite depth wind waves. Coastal Eng., 33, 137–154.

58


Chapter 4

Tide, surge and waves interactions 1

Abstract

Two existing pre–operational numerical models, a wave model and a hydrodynamic model, were incorporated into a coupling framework that allows an efficient exchange of information between them. Minimal adaptation of the models was needed. The module has then been implemented and applied to the North Sea, then a series of experiments were performed to investigate the sensitivity of computed waves and surges to coupling. These experiments show that the sensitivity of the modelled waves to coupling increases from deep to shallow water. The sensitivity of surges is more uniformly distributed over the area. The tidal currents produce a modulation, at the tidal period, of the peak period and of the significant wave height. The former is largely derived from the way the Doppler shift is computed in the wave model. Young wind sea gives rise to drag coefficients larger than the values of the empirical coefficients for the same wind speed.

4.1

Introduction

From the review of existing operational oceanography services (Flather, 2000), it is clear that for the time being, wave and storm surge predictions in the North Sea are still made separately in most operational centers. There are, however, several known mechanisms through of which each component of the total motion affects the others. Heaps (1983) had already identified the need for a wave model to improve the 1

This chapter was adapted from the article published as “A coupling module for tides, surges and waves”, by Ozer, J., Padilla–Hern´ andez, R., Monbaliu, J., Alvarez Fanjul, E., Carretero Albiach, J.C., Osuna, P., Yu, J.C.S., Wolf, J., Coastal Eng., 41, 95–124, 2000.

59

60

CHAPTER 4. TIDE, SURGE AND WAVES INTERACTIONS

specification of wind–stress in surge models. Various interaction mechanisms (e.g. the surface drag) were identified as potentially important (Wolf et al., 1988). Some results from early attempts at coupling are given in Wu and Flather (1992). Tolman (1990) concluded from his investigation into the effects of tides and storm surges on wind waves that “both the instationarity and the inhomogeneity of depth and current play a significant role in wave–tide interaction” and recommended further investigations into the effects of wave–tide interactions on wave heights. Waves can influence the sea surface elevation in three different ways: by changing the wind surface stress, through the spatial gradients of the radiation stress and by enhancing the bottom friction. Mastenbroek et al. (1993) clearly show the influence of a wave–dependent surface drag coefficient on surge elevations. Even if these surge elevations can be reproduced with an appropriate “tuning” of the parameter in more conventional wind stress formulations (the dimensionless constant α in the Charnock relation (Charnock, 1955) in this case), they argue that “a wave-dependent drag is to be preferred for storm surge modelling”. Davies and Lawrence (1994) notice a significant change of the tidal amplitude and phase in shallow near–coastal regions due to enhanced of frictional effects associated with wind–driven flow and wind wave turbulence. A summary of contributions to coupling up to the end of the WAM project are given in Burgers et al. (1994) and Cavaleri et al. (1994). The main purpose of this chapter is to describe how the module has been developed and to report on a series of experiments dealing with the sensitivities of both models to coupling. In Section 4.2, the basic tools are briefly described, the modifications imposed by the coupling are summarized and the way each component may influence the other is discussed. In Section 4.3, the implementation of the models on the North Sea is presented followed by a short discussion on the atmospheric forcing during the test period (February 1993) and on the model results (when run separately). Section 4.4 deals with the presentation of the experiments performed to investigate the sensitivity of the models to coupling and with a detailed investigation of the results of these experiments. A summary is given and conclusions are drawn in the last section. A list of symbols is given at the end.

4.2. THE MODULE FOR COMBINED TIDE, SURGE AND WAVES

4.2

61

The Module for Combined Tide, Surge and Waves

The following steps have been performed, with the intention of preparing for dissemination, a tool which enables the combined modelling of tides, surges and waves at the North Sea scale and in shallow water. Firstly, two models were chosen. Secondly, model equations were adapted, where necessary, to account for interactions between processes. Finally, model codes were modified for an efficient and correct exchange of information. An overview of the two models with a discussion of the modifications imposed by coupling at the level of model equations is given below. Coding aspects are subsequently addressed.

4.2.1

Overview of the models

The wave model The rate of change of action density for irregular waves is described by the spectral action conservation equation. In its general form in Cartesian coordinates reads: ∂ ∂ ∂ ∂ Stot ∂N + (cx N ) + (cy N ) + (cω N ) + (cθ N ) = ∂t ∂x ∂y ∂ω ∂θ σ

(4.1)

Where N (t, x, y, ω, θ) is the action density spectrum, t is the time, ω is the absolute frequency, θ is the wave direction measured clockwise from the true north, c x , cy are the propagation velocities in the geographical space, cω , and cθ are the propagation velocities in the spectral space (frequency and directional space). The first term in the left–hand–side of this equation represents the rate of change of action density in time. The second and third term represent rectilinear propagation of action in geographical space (with propagation velocities cx , cy ). The fourth term represents the effects of shifting of relative frequency due to variation in depths and currents (with propagation velocity cω ). The fifth term represents propagation in directional space due to current and/or depth-induced refraction (with propagation velocity cθ ). In the right hand side of the Equation 4.1, Stot represents all effects of generation and dissipation of the waves, generally written as the sum of a number of separate source terms, each representing a different type of process: S = Sin + Sds + Snl

(4.2)

Sin represents the generation of wave energy by wind, Sds the dissipation of wave energy due to whitecapping Swc , wave–bottom interaction Sbf and in very shallow water depth-induced wave breaking Sbk , and Snl is the wave energy transfer due to conservative nonlinear wave–wave interactions (both quadruplet Snl4 , and triad Snl3 , interactions). The expected relative importance of each of the typical process mentioned above for different regions is given in Battjes (1994) who argues that depth–induced refraction/shoaling, depth–induced wave breaking and triad–wave interactions are dominant in shallow waters. Using linear theory and conservation of

62


wave crests, the propagation velocities, in geographical and spectral spaces can be obtained from the kinematics of a wave train (e.g., LeBlond and Mysac, 1978; Mei, 1983; Tolman, 1990): · ¸ 1 dx 2kd σk +U (4.3) = cg + U = 1+ dt 2 sin 2kd k 2 dθ = cθ dt dσ = cσ dt

· ¸ 1 ∂σ ∂d ∂U +k· k ∂d ∂m ∂m ¸ · ∂σ ∂d ∂U = + U · ∇d − cg k · ∂d ∂t ∂s

=

−

(4.4) (4.5)

Where the term (cg + U ) is the propagation velocity of the wave energy in geographical x-, y–space, cσ and cθ are the propagation velocities in spectral σ-, θ– space respectively, d is the total depth (H + η: mean depth + surface elevation), s is the space coordinate in the wave propagation direction θ, and m is a spatial coordinate perpendicular to s, the operator d/dt denotes the total derivative along a spatial path of energy propagation. However, in the original formulation of the model, current and water depth were assumed to be time–independent, the term ∂d/∂t in Equation 4.5 is not included. In shallow water areas, with tidal amplitude in the order of O(2 m) and tidal currents running mainly perpendicularly to the depth gradients, (∂d/∂t) (of the order of 10−4 ms−1 ) should be significantly greater than u · ∇d. The absolute or angular frequency ω in a fixed referenced frame is given by the dispersion relation which includes the water depth d and the current vector U. p ω = gk tanh kd + k · U (4.6)

k is the modulus of the wave number vector k. Equation 4.1 has the advantage that the absolute frequency is conserved for stationary currents. Therefore d (ω) = 0 dt

(4.7)

and consequently the term

∂ (cω N ) = 0 (4.8) ∂ω drops out in Equation 4.1. In the other hand one of the problems solving the Equation 4.1 is the computation of the wave number modulus k from the dispersion relation 4.6. In case of strong opposing current for high frequencies two solutions exist and the group velocity for certain frequencies become zero (wave blocking). If an alternative coordinate is taken for the wave spectrum, like the intrinsic frequency, σ which is observed in a reference frame moving with the current U, determined by p σ = gk tanh kd (4.9) the advantage of this expression is that the relation between the intrinsic frequency and the wave number modulus is unequivocal, and we can express the action density


63

Equation 4.1 in terms of energy density because both frequencies (Equations 4.6 and 4.9) are connected by the Doppler term k·U, and the action density N and the energy density F are connected by the intrinsic frequency N=

F σ

(4.10)

To overcome the problem of strong opposing currents the WAM model uses internally the intrinsic frequency σ as coordinate. Equation 4.1 can be multiplied by σ and the partial derivatives with respect to t, x, y, and θ can be exchanged with the multiplication. Using Equation 4.10) the action density Equation 4.1 transforms to the following balance equation for the energy density spectrum. ∂F ∂ ∂ ∂ ¡ F¢ ∂ + (cx F ) + (cy F ) + σ (cθ F ) = Stot cσ + ∂t ∂x ∂y ∂σ σ ∂θ

(4.11)

where F (t, x, y, σ, θ) is the wave energy spectrum. Because the relation (Equation 4.9) between intrinsic frequency and wave number modulus k is unequivocally, this transformation is valid for all currents. In contrast to the ω term in Equation 4.1 which is zero already for stationary currents the σ term in Equation 4.11 only drops out if the current speed is zero. In this case both frequencies are the same and therefore the resulting energy density spectrum is given in absolute frequencies. If a current is present the spectrum has to be transformed from σ to ω frequencies using the Doppler term k · u. This transformation is carried out in the WAM model for all outputs. Equation 4.11 takes into account implicitly the interaction of waves with the mean flow through the term ∂ ¡ F¢ cσ (4.12) σ ∂σ σ Handling this equation carrying out the derivatives one can obtain that µ ¶ F 1 ∂ ∂ F cσ = σ (cσ F ) − cσ (4.13) ∂σ σ σ ∂σ σ where

¸ · ∂σ ∂d dσ = + U · ∇d − k · (cg · ∇)U cσ = dt ∂t ∂t

(4.14)

The first term in the right side of Equation 4.13 is the flux of energy in σ space, and the second one, according to Phillips (1977), describes the exchange of the wave energy with the mean flow. The frequency spectrum is transformed from relative to absolute frequency at all outputs to account for the Doppler effect (ω = σ+k·u). The WAM equations have not been modified for model runs in coupled mode, which means that the time derivative of the depth is still not included (Equation 4.5). For these runs, new current and water depth fields are introduced into the model at regular time intervals. Using these new fields, some arrays (e.g, wave number and group velocity) that were only computed

64


once in the pre–processing of WAM are updated. Note that for those depth–dependent parameters and/or coefficients, their evolution in time due to tide and surge is not as smooth as it should be. Indeed, their values are only known for a discrete set of depth values. As time passes, it is the pre–computed value corresponding to the depth closest to the actual depth that is used. Almost all the terms in the governing equation are affected in one way or another by time and space varying currents and water depths. These modify the propagation velocity in the geographical and spectral spaces. Interactions between the wave field and the mean flow can, locally, be a source or a sink of wave energy. Current and depth gradients redistribute energy density in the spectral space. Through the dispersion relationship, the wave numbers become time dependent. This makes the analysis of the influence of coupling on the wave model results particularly difficult.

The tide–surge model The tide–surge model is a revised version of an existing operational model used to forecast storm surges in the North Sea (Van den Eynde et al., 1995). It is a conventional, vertically integrated, two–dimensional “shallow water wave equations” model. The model state variables are the depth–averaged current velocity and the elevation of the free surface with respect to mean sea level. The model is forced by the tide (four semi–diurnal and four diurnal tidal constituents) and the inverse barometric effect along the open boundaries, the atmospheric pressure gradients and the wind stress in the area. A zero normal flux is imposed along the solid boundaries. Conventional quadratic laws are used to compute surface and bottom stresses. The equation for the surface stress is: τs = ρa Cs kWkW

(4.15)

where ρa is the air density (1.23 kgm−3 ), W is the wind velocity at 10 m above the sea surface and Cs is the surface drag coefficient. Various surface drag coefficients are proposed in the literature. We generally used the one proposed by Heaps (1965) Cs

=

Cs

=

Cs

=

0.565 × 10−3

(−0.12 + 0.137W ) × 10−3 2.513 × 10

−3

for for for

W ≤ 5 ms−1

5 < W < 19.22 ms−1 W ≥ 19.22 ms

(4.16)

−1

where W is the wind speed. Bottom friction is computed according to: τ b = ρw Cb kuku − mτ s

(4.17)

where, ρw is the water density (1023 kg m−3 ), u is depth mean current velocity and Cb is the bottom drag coefficient with a constat value of 0.00243 ms−1 .


65

It is not unusual in 2–D models of storm surges (e.g., Groen and Groves, 1962; Heaps, 1967; Ronday, 1976) to modify the bottom stress so that there is a component directly related to the wind stress (a crude way to account for the vertical structure of the wind–driven current). The coefficient m is generally set equal to 0.1. Note that this term is introduced into the computation of surface stress instead of the bottom friction in the computer code (see Section 4.4). Waves can influence the mean flow in three different ways: through the spatial gradients of the radiation stress, by changing the wind stress, and by affecting the bottom friction.

Radiation stress The radiation stress represents the contribution of the wave motions to the mean horizontal flux of horizontal momentum. It is expressed in terms of the wave spectrum. The spatial derivatives of the radiation stress were introduced in the momentum equation. The computation of the radiation stress and its implementation in the model equations follows that of Mastenbroek et al. (1993) (see Equation 3.16).

Surface stress The variation of the surface drag with wind speed as shown in Equation (4.16) is an empirical concept that reflects the increase of the sea surface roughness with increasing wind speed. The wave field largely determines the change of sea surface roughness with wind speed. There is experimental evidence of a certain dependency between wind stress and wave age (e.g., Maat et al., 1991; Monbaliu, 1994). In recent years, different parameterizations for computing the surface stress as function of wind and waves have been proposed (Makin and Chalikov, 1986; Janssen, 1991). Janssen’s theory is implemented in WAM Cycle–4 (see Section 4.4.4). In the tide–surge model the surface stress is no longer computed according to Equation (4.15) but directly transferred from WAM to the model at regular time intervals.

Bottom friction In shallow waters, the waves interact with the bottom and the orbital motion of the low frequency gravity waves cause an alternating current in its vicinity. This current originates from a thin boundary layer (typically a few centimetres) in which the level of turbulence is increased causing an enhancement of the bottom stress felt by the current (Chirstoffersen and Jonsson, 1985; Gross et al. 1992). In WAM Cycle–4, bottom dissipation due to the combined effect of current and waves can be computed either following the approach proposed by Madsen (1994) or following the approach proposed by Christoffersen and Jonsson (1985). While it is certainly an interesting subject for further research, the influence of combined waves and currents on bottom dissipation (for waves and/or for currents) has not been investigated in the experiments reported here. The routines that allow

66


the modelling of bottom dissipation due to waves and currents still need to be tested in a realistic configuration.

4.2.2

Coupling procedure

A general purpose framework has been specifically developed during the course of PROMISE with the intention of preparing for dissemination, tools that allow the combined modelling of tides, surges and waves at the North Sea scale and in shallow water. This framework is presented on Figure 4.1.

The Structure of the CHIEF module

INITIAL STAGE

CHIEF

Surge Model

WAM

CHECK

CHIEF

Time = 0.0

W2S

WAM

Surge Model

W2S

CHIEF

Coupled time-step

COUPLED STAGE

S2W

S2W

END of the Coupling Figure 4.1: The PROMISE coupling framework.

In the PROMISE coupling framework, the two models (i.e., the tide–surge model and the wave model) are taken as subroutines of a main program (CHIEF) that controls the execution. It takes care of the initialization of both models and verifies their status at the start time of coupling. During the coupled mode, it calls the subroutines needed to transfer the information between the two model grids. All North Sea applications have been made with this coupling framework.

67

4.3. NORTH SEA APPLICATIONS

4.3 4.3.1

North Sea Applications Implementation

For the North Sea applications, the two models have been implemented on a relatively coarse grid covering approximately the region between 48o –71o N and between 12o W–12o E. The bottom topography and coastlines in this area are presented on Figure 4.2. The horizontal resolution is equal to 1/2o longitude and to 1/3o latitude. The bottom topography is taken from the Northeast Atlantic model developed by Flather (1981). Such an implementation has to be seen as a first step in the development of an operational model for the forecast of waves, tides and surges in coastal areas. Indeed, it is not unusual to start with such an implementation in a wave forecasting system. The desired horizontal resolution in the coastal area of interest is then obtained through successive nesting. The same nesting procedure for the combined model had been developed recently by Osuna (2002). Depth [m]

PROMISE AREA

1000

70

00 500

0 10 50

10

0

50

20

00

500

10

2500

0

10

50

00

1000

500

10 60

20 100

1500

20

10 0 20

50

2500

1000

20 WEH

100

1000

500

50 GER 20 K13 MPN

50

10

500

50 20

00

−10

2000

AUK

50

55

3000

500

0 100

50

Latitude [deg]

50

50

3500 20

65

0

50

10

10

00

4000

−5

0 5 Longitude [deg]

10

0

Figure 4.2: Model area for the North Sea sensitivity study. Depths are given in meters. The position of the five stations used for the analysis of time series is also given. In Figure 4.2, the positions of the five stations at which model results are investigated in more detail are also shown. The exact geographical locations of these

68


stations as well as information on the mean depth and on the characteristics of the tidal range are given in Table 4.1.

Table 4.1: Location, mean depth and tidal range at the five stations considered in this study. Note that water depths are taken from bathymetric data used by the two models at the nearest grid point.

Station

Latitude

Longitude

Auk Ger K13 Mpn Weh

56o 23’59” 54o 30’00” 53o 13’01” 52o 16’26” 51o 22’56”

2o 03’56” 7o 45’00” 3o 13’12” 4o 17’46” 2o 26’20”

Mean depth [m] 80 21 31 18 31

Tidal range [m] 0.8 1.6 1.2 1.2 2.8

Tidal currents at Auk are rather weak. At K13, the tidal ellipse is nearly circular and the currents are of the order of 0.5 ms−1 . At the station Ger, the tidal ellipse is rather flat with nearly a west–east orientation. Tidal currents are also of the order of 0.5 ms−1 . A southwest–northeast orientation of tidal currents is observed at both stations Weh and Mpn. Tidal currents at station Weh (0.70 ms−1 ) are nearly twice as strong as those at Mpn. In the tide–surge model, the amplitude and phase of the eight constituents used to define the tidal forcing are also taken from Flather’s Northeast Atlantic model. The model equations are integrated with a time step equal to 75 s. The frequency grid is also logarithmic with f increasing successively by a factor of 1.1. It starts at 0.04 Hz and has 25 values. A resolution of 30o is used in the directional space. The source term integration time step and the propagation time step are both equal to 600 s. Propagation is computed in a quadrant coordinate system (see also Chapter 3).

4.3.2

Results from an uncoupled mode run

Reference run In a first experiment, the models were run without exchanging any information. This run will be referred to as the reference run. Both models were called by the main program which controls the timing for the coupling system. The results from both models were compared with the results obtained by the same models but out of the coupling system (offline). This has to be done because every time step for coupling, the modules (wave and tide–surge models) have to re-start. The period for the simulation is the month of February 1993. The same period has been used for the PROMISE North Sea WAM model intercomparison (Monbaliu et al., 1997). In this reference run

69


the water level used for the wave model is the mean sea level and not currents are introduced. An overview of the atmospheric forcing during that month is given first. Model results are discussed afterwards.

Atmospheric conditions The atmospheric forcing is taken from the UK Met. Office forecast routinely received for storm surge predictions (Van den Eynde et al., 1995). For the model runs in hindcast mode, atmospheric pressure and wind speed are available at 6–h intervals, on 1.25o latitude/longitude grid. Each day, at 00:00 GMT and 12:00 GMT, the information corresponds to a “nowcast” (i.e., a previous model forecast corrected by assimilation of in situ observations). At 06:00 GMT and 18:00 GMT, a model forecast is used. A spatial interpolation is performed to obtain the atmospheric pressure and the wind speed at the grid–nodes. A linear interpolation is made at each time step in the tide–surge model while the wind speed is kept constant during 6 h in the wave model (see Section 4.4). Time series of wind speed at the five stations are presented in Figure (4.3). Winds are relatively weak, O(5 ms−1 ), between the 5th and 15th of February. Three relatively important wind events, one with wind speed up to 25 ms−1 , occur between the 15th and the 25th February. Each event last about 2 days. During these events, the winds are stronger in the central North Sea (station Auk) than in the southern Bight (stations K13 and Weh) and close to the coasts (stations Ger and Mpn). Wind speed at the five locations 25 AUK GER K13 MPN WEH 20

W10 [m/s]

15

10

5

0

5

10

15 Days of February 1993

20

25

Figure 4.3: Time series of the wind speed at five stations from 5th –25th February 1993. Comparisons between these winds and in-situ measurements (Ovidio et al., 1995)

70


Atmos. pressure and Wind velocity

21/02/1993 at 00 GMT W 24.1 [m/s]

70

max

100

101

20

0

60

0

101

AUK

0 100

1020

Latitude [deg]

0

10 65

GER

30

10

55

K13 MPN WEH

50

1000

10

20

−10

−5

0 Longitude [deg]

101

0

5

10

Figure 4.4: Atmospheric situation at 00:00 GMT 21st February 1993. Atmospheric pressure in hPa and wind speed. indicate that the quality of these predicted winds is quite good during that time (a bias less than 0.5 ms−1 and a scatter index around 0.2). The descriptions of the three wind events are nearly the same. A relatively high and stable pressure system exists in the southwestern part of the area. A low pressure enters the north of the region through the western boundary. It traverses the North Sea eastward and then turns to southeast. It leaves the region over Germany. The atmospheric situation at 00:00 GMT 21st February is depicted in Figure 4.4. These three “northwest” storms generate northwesterly winds over the North Sea.

Model results Time series of significant wave height (Hs ), peak period (Tp ) and surge elevation (ηs ) at the five stations are presented in Figure 4.5.

71


The results of the WAM reference run are discussed at length in Monbaliu et al. (1997, 1999). Waves are relatively small during the first part of the month. Higher waves are observed during the stormy period (16th – 23rd February). At that time, waves are significantly higher in the central North Sea than in the Southern Bight and coastal areas. Comparison with in situ measurements indicates that the model tends to underpredict wave height (at station Auk, the bias, for the whole month, is of the order of 0.4 m; at station K13, it is of the order of 0.1m).

Hs [m]

10

AUK GER K13

MPN WEH

a)

5

0

5

10

15

20

25

12 b)

Tp [s]

10 8 6 4 2

5

10

15

20

25

c)

ηs [m]

2 1 0 5

10


20

25

Figure 4.5: Time series of (a) Hs , (b) Tp and (c) ηs at the five stations for the reference run 5th –25th February 1993. The results of the tide–surge model (Figure 4.5c) are explained below. The negative surge observed (Figure 4.5c) at all stations during the first part of the month (5th –15th February) is due to the atmospheric pressure (inverse barometric effect along the open boundary, horizontal pressure gradients inside the area). The wind has a very small influence during that time. During the storm events, the wind blowing towards south–southeast generates positive surge elevations that exceed 2 m in the southern North Sea (stations Mpn and Weh). From a comparison with observations at one station along the Belgian coast, one can say that the model also seems to underpredict the surge elevations (at the peak of the third surge, the surge elevation is underestimated by about 0.15 m). While any model intended for operational use needs an in–depth comparison with in situ measurements and a proper calibration, this is considered outside the scope of this study. The focus is primarily on the study of the sensitivity of the models to coupling. Interesting discussions on wave–current interaction observations in the

72


Holderness area are given by Wolf (1999) and by Wolf and Prandle (1999). An illustration of observed tidal modulation at the station Weh is given in Monbaliu et al. (1998).

4.4 4.4.1

North Sea Sensitivity Analysis Description of the experiments

To assess the sensitivity of the model results to various degrees of coupling, a series of model runs have been completed. Only some of them are discussed in detail in this chapter. Others will be briefly mentioned when appropriate. All these experiments follow the same scheme. The starting time of coupling is at 06:00 GMT 1st February 1993. The tide–surge model, starting with the sea at rest, is run only with tide for a few days before the coupling starts. WAM starts with an initial wave field at 00:00 GMT 1st of February and the integration runs over 6 h before coupling. The exchange of information between the models is done every 20 min. Going from an exchange every 20 min to one every 60 min just slightly reduces the details of the tide–induced modulation of wave parameters (Monbaliu et al., 1998). The highest level of coupling is when each model has an influence on the other (two–way coupling). The tide–surge model is then driven by the surface stress computed by WAM (replacing the surface stress as computed using Equation 4.15), and by the atmospheric pressure and the tide. Tide– and wind–induced current and water levels from the tide–surge model are, in turn, transferred to the wave model. This experiment will be referred to as WH. Apart from this experiment in fully coupled mode, others have been made in which information is just passed from one model to the other (one–way coupling). Those dealing with the sensitivity of waves to tide and surge are described in Table 4.2. The experiment dealing with the sensitivity of surge to waves are summarized in Table 4.3. The analysis will mainly be based on time series of differences between model results in one experiment and those obtained in the reference run Section 4.3.2, for three parameters (significant wave high, peak period and surge elevation) at the five stations listed in Table 4.1. Computing the value of a global estimator assesses the order of magnitude of the differences between two experiments. The global estimator we have used is defined by s n P 1 (yi (tk ) − yr (tk ))2 n Ei (y) = 100

k=1

s

1 n

n P

k=1

(4.18)

yr2 (tk )

4.4. NORTH SEA SENSITIVITY ANALYSIS

73

Table 4.2: Description of the experiments in which the information is just passed from the tide–surge model to the wave model. Experiment Description D2W The tide, atmospheric pressure and wind stress computed according to Equation 4.15 drive the tide–surge model. The tide– and wind–induced water levels are transferred to WAM. Time varying currents are not transferred and therefore have no influence on the wave model. TC2W The tide–surge model is driven by tide only. The tide–induced currents are transferred to WAM. Time varying water levels are not transferred. H2W The tide–surge model is driven by the tide, atmospheric pressure and wind stress computed according to Equation 4.15. Tide– and wind–induced current and water levels are transferred to WAM.

where y is one model parameter, n is the number of values in the time series, y i (tk ) and yr (tk ) are the values of y at time tk in the experiment i and in the reference run, respectively. Hourly sampled values of model results will be used for the computation of E.

Table 4.3: Description of the experiment in which the information is just passed from the wave model to the tide–surge model. Experiment Description W2H The tide–surge model is driven by the tide, atmospheric pressure and wind stress computed by the wave model

4.4.2

Sensitivity of waves, tides and surges to coupling

While Tolman (1990) mainly investigated the influence of tides and surges on waves and Mastenbroek et al. (1993) looked at the influence of a wave–dependent surface stress on surges, both effects are combined in our experiment in a fully coupled mode (WH). Therefore, the results from this experiment are analyzed first. Differences in the model variables between WH and the reference run are presented in Figure 4.6. The values of E for significant wave height, peak period and surge elevation at the five stations computed from 00:00 GMT 5th February–00:00 GMT 25th February are listed in Table 4.4. From Table 4.4, one can say that the coupling introduces a change of less than 5% in significant wave height, and a change of less than 10% in peak period and surge

74


0.4 [m]

a)

0

∆ Hs

WH−REF

0.2

−0.2 5

10

15

20

25 b)

0 −1

p

∆ TWH−REF [s]

1

AUK GER K13

−2 −3 5

10

15

20

[m]

25 c)

0.2 0.1 0

∆ ηs

WH−REF

MPN WEH

−0.1 −0.2 5

10


20

25

Figure 4.6: Time series of differences in Hs (top), Tp (middle) and ηs (bottom) between the fully coupled run (WH) and the reference run (REF) at the five stations, 5–25th February 1993. elevation. As expected, and to some extent hoped, the coupling does not have a strong influence on the model results. After all, both types of model are used independently by several operational centers and the reliability of the information they deliver does not need to be demonstrated again. However, as discussed below, the differences are interesting in terms of model behavior and understanding of physics. Sensitivity of wave parameters to coupling increases from deep to shallow water. This fact indicates the need to explore the effects of coupling in shallow coastal waters where modelling with high spatial resolution is needed. Peak period is more sensitive than significant wave height. Tidal modulation of both wave parameters at all stations is clearly visible in the time series of differences almost all the time (Figure 4.6). During the stormy period, wave heights are clearly influenced by coupling. At station Mpn, significant wave height is increased by 38 cm during the third storm. Tide and surge effects on waves are further discussed in Section 4.4.3. Between the 10th – 15th February, differences between the wave parameters over relatively short time intervals resulting from both experiments are as large as those observed during the stormy period. These are noticeable in the time series of differences in Hs and Tp at stations Ger, K13 and Mpn (see Figure 4.6). During these “events”, if significant wave height increases, peak period diminishes (see station Ger)

75

4.4. NORTH SEA SENSITIVITY ANALYSIS

Table 4.4: Mean value of H [m] ,Tp [s] and ηs [m] for the reference run and values of the global estimator, EW H , for the differences between the experiment in fully coupled mode (WH) and the reference run at five stations computed over the period 5th –25th February.

Station Auk Ger K13 Mpn Weh

hHs i 2.02 1.30 1.36 1.01 0.86

EWH (Hs ) 0.6 2.8 2.0 4.7 3.8

hTp i 6.79 5.90 5.72 6.01 5.56

EWH (Tp ) 1.3 5.2 3.9 6.8 7.7

hηs i 1, where δw is the WBBL thickness. Another reason (see Section 7.4.2) to involve ripples (through kr ) instead of kg in the total roughness height in the

156

CHAPTER 7. THE WAVE BOTTOM BOUNDARY LAYER

12 u *2.5 u*r u *fm

10

u

=5.8 [cm/s]

*crsf

6

*

u [cm/s]

8

4

u*crsl=3.49 [cm/s]

2

0

u*crin=1.46 [cm/s]

0

5

10

15 Time [days]

20

25

30

Figure 7.3: Comparison between friction velocities: (i) grain–related u∗2.5 , (ii) corrected u∗r and (iii) total roughness height–related u∗f m .

7.4. VERIFICATION OF CJ85 MODEL

157

WBBL model is the fact that using only the grain–roughness, one faces the problem that the ripple height that have been computed by the ripple geometry predictor, using the output from the WBBL model, is much larger than the WBBL thickness. Equation (7.10) includes two dimensionless parameters (r and β) to compute the WBBL thickness (δw ). CJ85 analytically determined the r–value equal to 0.45. However according to CJ85, the method to determine r is not the only one. Using slightly different requirements to solve their equations, they also find an r–value equal to 0.925. CJ85 suggested a value for r equal to 0.45 without strong analytical, numerical or experimental arguments. Figure (7.4) shows the relative WBBL thickness (δw /kr ) for the entire period of LA98 experiment. Using the value r = 0.45 the relative WBBL thickness is sometimes smaller than unity at the time when should be larger. The thick line at the bottom of Figure (7.4) indicates times where the Shields parameter is larger than its critical value for sediment movement (θcrin = 0.04; Amos et al., 1988). At least at those times the fluid is interacting with the sediment and the wave bottom boundary layer thickness should be larger than ripple roughness height. That is achieved in case when an r–value of 0.925 unlike when the value of 0.45 is used. For instance, between the days 9–10 the friction velocity is above the threshold of motion (1.46 cm/s, see Figure 7.2) and there is a ripple field in evolution, thus the ratio δw /kr should be larger than unity. Using a value of 0.925 for r seems more appropriate for the flow regime and the roughness height present in the field. The interruption of both curves around the first and eleventh day is due to the sheet flow regime, where the ripples are wiped out (kr = 0). For all of the results shown in this chapter (as in Section 7.2) the value r = 0.925 was used. The difference in ripple geometry using r = 0.45 or r = 0.925 is about 13.5%, 6.6% and 10.4% for ripple height, length and total roughness height respectively, being larger for r = 0.925. Change in the r–value from 0.45 to 0.925 represents a change of 106% for the WBBL thickness. From here it is suggested to use the value for r = 0.925 in the CJ85 model.

The apparent bed roughness height Figure (7.5) shows the modelled relative apparent roughness height (kA /kr ) against the relative orbital velocity (uwbm /u∗f m ) at the bottom, where kA is the apparent roughness height computed (using Equation 7.15) for the LA98 entire period of measurements in comparison with existing measurements in laboratory (see Table 7.1 for more details). Because in the laboratory experiments no sediments were present (in this case k g and kbl are zero), the system WBBL–ripple predictor was run using kr as an input to the CJ85 model. The results from the present study lie in the same range of values as the laboratory measurements. Although there is a large scatter in the data some authors have proposed relations to express the relative apparent roughness (k A /kr ) as

158


r = 0.925 r = 0.45 1

δw/kr

10

0

10

−1

10

0

5

10

15 Time [days]

20

25

30

Figure 7.4: Time evolution of the ratio of wave bottom boundary layer thickness δw and ripple roughness height kr for two r–values, which is involved in the computation of wave bottom boundary layer thickness δw . The discontinuous straight line indicates where, at least, the ratio δw /kr should be larger than the unity.

159


1

kA/kr

10

Kemp and Simons, 1982 Kemp and Simons, 1983 Asano and Iwaki 1984 Asano et al., 1986 Mathisen and Madsen, 1996 Mathisen and Madsen, 1996 Fredsoe et al., 1999 Present study

0

10 0 10

1

10 uwbm/u*fm

Figure 7.5: The relative apparent roughness kA /kr against the relative wave orbital velocity uuwb /u∗f m . The symbols for present study represents all the modelled data at the time of LA98 data set.

160


1

k /k

A r

10

0

10 −4 10

−3

Γ

10

Figure 7.6: The relative apparent roughness (kA /kr ) against the parameter Γ [= (u2∗f m λ)/(U uwbm A)]. Using the modelled data in which the friction velocity is in the range u∗crin < u∗f m < u∗crsf . The black line is the function that fits best the data set in a least square sense. This function is kA /kr = 26.1707 − 0.7923Γ−.3916 . a function of the relative free stream velocity (uwbm /u∗f m ) (Nielsen, 1992; Fredsøe, 1999). Figure (7.6) shows relative apparent roughness as function of several wave, current and ripple parameters. Under this space–parameter the data set concentrates on a line with little scatter. The thick line in the figure represents the function that fits the data set, in a least square sense. The function is given by kA = 26.1707 − 0.7923Γ−.3916 kr where Γ is given by Γ=

u2∗f m λ U uwbm A

(7.25)

(7.26)

The cloud of points in Figure (7.6), which were taken to compute the Γ–function, corresponds to data in which the friction velocity (u∗f m ) is larger than its critical value for initiation of motion and smaller than its critical value for sheet flow. For the study site of LA98, u∗crin =1.46 cm/s and u∗crsf = 5.8 cm/s, where the median grain size d50 is equal to 0.34 mm. The minimum current speed (U ) measured was 1.28 cm/s. When U tends to zero, the ratio kA /kr tends to 20.


161

Table 7.1: Legend for Figure 7.5. Test Conditions for the existing data on apparent roughness. (From Fredsøe et al., 1999). α: angle between waves and currents; FC: following current; OC: opposing current. Authors Sym. α A/kr λ/A η/λ Shape of the roughness Kemp and ◦ FC 0.5–0.9 1–2 0.28 Triangular section, Simons 0.5 cm high, 1.8 cm (1982) length Kemp and + OC 0.5–0.9 1–2 0.28 Triangular section, Simons 0.5 cm high, 1.8 cm (1983) length Asano and ¦ OC 2.5 0.3 0.13 2x2 mm square secIwaki (1984) tion stripe elements Case III with length of 15 mm Asano and ∇ OC 1.8 0.3 0.13 2x2 mm square secIwaki (1984) tion stripe elements Case IV with length of 15 mm Mathisen . FC 0.2–0.4 1.2–2.4 0.15 Triangular bars, 1.5 and Madsen cm high, 10 cm (1996a,b) length Mathisen / FC 0.3–0.6 1.3 0.075 Triangular bars, 1.5 and Madsen cm high, 20 cm (1996a,b) length Fredsøe et al. ∗ FC 1.2 2.5 0.16 Ripples with 3.5 cm (1999) high, 0.88 cm length Present · – 4.0–40 0.65 0.07–0.22 Modelled ripples study Present ? FC 7.0–16 0.65 0.11–0.17 Modelled ripples study

162

7.5


Summary and Conclusions

A modified version of the ripple predictor model of Nielsen (MN81) was coupled with the model of Christoffersen and Jonsson (1985), which supplies the friction velocity to the MN81 to compute the new ripple field. In turn MN81 supplies the new ripple field to the CJ85 model to compute the parameters inside the WBBL. Observations in the field and in the laboratory by several investigators were used to verify the WBBL model of Christoffersen and Jonsson (1985) for combined wave–current flows. In general the WBBL model, using the ripple geometry from the MN81 model, reproduces the different flow regimes observed by LA98 in the field, viz. (i) the non–transport, (ii) the saltation/suspension and (iii) the sheet flow. According to the results from several investigators and from the results obtained in the present study, modelling the skin–friction using the grain roughness height (k g ) is not appropriate to reproduce the different flow regimes that were observed in the field. The friction velocity reaches the same value as those obtained and shown by LA98 using the model of Grant and Madsen (1986) if only kg is used. Computing the WBBL parameters using kg and then correct them by an enhancement factor to take into account the presence of ripples does not give quantitatively nor qualitatively good results. The corrected friction velocity is only able to predict when the bedload flow regime starts according to the criterion for the critical friction velocity for initiation of the bed load regime of Bagnold (1957). Through an iteration process between the MN81 and CJ85 models it is possible to obtain good results even if the measured flow parameters are sparse in time. That can be done using a seed roughness height and letting the numerical system (WBBL– ripple predictor) iterate until an equilibrium is reached. Results of the system WBBL– ripple predictor suggest that the value for r, which is involved in the computation of the WBBL thickness, should be larger than the value proposed by CJ85. From results using two different values for r (both proposed by CJ85), preference should be given to an r–value of 0.925 to compute the wave boundary layer thickness δ w . In the ripple regime the apparent roughnesses computed according to the CJ85 model, for the field conditions of LA98, agree with laboratory measurements. The relative apparent roughness can be expressed as function of flow (wave–current) and bed (ripples) parameters, with small scatter.

Appendix Here the calculation procedure for Model I of CJ85 is given. This procedure is based on the one given by CJ85. From measurements or simulations of the wave–current situation the following quantities are known, U, δ − α, h, A and kb . For the first iteration the bottom roughness height has the value of the grain roughness. From those quantities, A, uwbm , ωr , and ωa can be determined. This gives four dimensionless

7.5. SUMMARY AND CONCLUSIONS

quantities:

163

Aωr kb U uwbm = , , , and (δ − α) kb ωa kb ωa h uwbm

which determine the friction factors fc and fw , and everything else connected with the CBBL and WBBL. Based on those parameters the MN81 is called to compute the ripple geometry and the corresponding total roughness height which in turn will be used in the subsequent call to the CJ85 model. This procedure is follow up to the end of the simulations. The calculation scheme given below is only for Model I, but for Model II it is basically the same. 1. Compute fc from (7.14) with kA = kb (pure current). 2. Compute J and fw from (7.11) and (7.13) with m = 1 (pure waves). 3. Keeping fc fixed, iterate through (7.9), (7.7), (7.11) and (7.13) until sufficient accuracy is obtained for σ, m, J and fw . 4. Compute δw and kA from (7.10) and (7.15) (with r = 0.925 and β = 0.0747). (Note that CJ85 suggest an r–value of 0.450). 5. Compute a new fc value from (7.14). 6. Repeat steps 2–6 until sufficient accuracy has been obtained for fc . After this iteration, it is easy to calculate the shear stresses and other parameters by using the appropriate equations.

Notation A CBBL CJ85 d50 e fc fw g h Hs k k kA kb

Wave particle amplitude just outside the WBBL, Equation (7.22) [m] Current bottom boundary layer Reference and model of Christofferssen and Jonsson (1985) Median grain size [mm] = exp(1) = 2.718 . . . Current friction factor, Equation (7.3) [–] Wave friction factor, Equation (7.4) [–] Gravity acceleration [ms-2 ] Mean water depth [m] Significant wave height [m] Wave number vector [m-1 ] Magnitude of the wave number vector [m-1 ] Apparent bed roughness [m] Bottom roughness height [m]

164

kbl kg kr kbt LA98 m MN81 p pc pw r s t Tp u u∗crbl u∗crin u∗crsf u∗crsl uwbm u∗f m u∗r u∗2.5 u100 U U w WBBL x z α β δ δw η θ θbm θcrin κ λ π


Bedload roughness height [m] Grain roughness height [m] Ripple roughness height [m] Total bottom roughness height, Equation (7.21) [m] Li and Amos, 1998 Factor that indicates the strength of the wave–current interactions, Equation (7.9) [–] Modified model of Nielsen, 1981 Total pressure [kgm-2 ] Steady part (current–related) of the pressure [kgm-2 ] Oscillating part (wave–related) of the pressure [kgm-2 ] Numerical constant (= 0.925) [–] Relative density of sediment [–] Time coordinate [s] Peak wave period [s] Total horizontal particle velocity [ms-1 ] Critical shear velocity for bed load [ms-1 ] Critical shear velocity for initiation of motion [ms-1 ] Critical shear velocity for sheet flow [ms-1 ] Critical shear velocity for saltation/suspension of sediment [ms-1 ] Amplitude of the wave particle velocity at the bed [ms-1 ] Maximum friction velocity, Equation (7.12) [ms-1 ] Friction velocity corrected due the presence of ripples, Equation (7.24) [ms -1 ] Grain-related friction velocity [ms-1 ] Current measured at 1 m above the sea bed (Taken as the mean current velocity). [ms-1 ] Mean current velocity [ms-1 ] Magnitude of the mean current velocity [ms-1 ] Vertical particle velocity [ms-1 ] Wave bottom boundary layer (= x1 , x2 ) Horizontal Cartesian coordinate vector [m] Vertical Cartesian coordinate [m] Angle from x1 –axis to the wave orthogonal [rad] Turbulence constant(=0.0747) [–] Current direction respect to the x1 –axis [rad] Wave bottom boundary layer thickness [m] Ripple height [m] Shields parameter for wave shear stress [–] Shields parameter for maximum wave stress, Equation (7.19) [m] Critical Shields parameter for initiation of motion [–] von Kármán constant (=0.40) [–] Ripple length [m] = 3.14159 . . . [–]

7.6. REFERENCES

ρs ρw σ τ τb τ bm τcb τ wb τwbm φb ϕ ψ ω ωa ωr ∇

7.6

165

Sediment density [kgm-3 ] Water density [kgm-3 ] Ratio between current and wave shear stress, Equation (7.7) [–] Total shear stress [Nm-2 ] Total shear stress at the bed [Nm-2 ] The maximum bed shear stress, Equation (7.8) [Nm-2 ] Steady part(current–related) of the total shear stress at the bed [Nm-2 ] Oscillatory part (wave–related) of the total shear stress at the bed [Nm -2 ] Amplitude of the τwb [Nm-2 ] Phase lead of τ wb relative to the wave particle velocity [rad] Phase function [rad] Wave mobility number, Equation (7.23) [–] Angular frequency [rad] Wave absolute angular frequency [rad] Wave relative angular frequency [rad] Horizontal gradient operator [rad]

References

Amos C.L., Bowen A.J., Huntley D.A. and Lewis C.F.M., 1988. Ripple generation under the combined influences of wave and currents on the Canadian continental shelf, Cont. Shelf Res., 8, No. 10, 1129–1153. Asano T. and Iwagaki Y., 1984. Bottom turbulent boundary layer in wave-current co-existing systems, Proc. 19th Int. Conf. on Coastal Eng., Chap.161. Houston, TX, USA, 2397–2413. Asano T. Nakagawa M. and Iwagaki Y., 1986. Changes in current properties due to wave superimposing, Proc. 20th Int. Conf. on Coastal Eng., Chap. 70. Taipie, Taiwan, 925–939. Bagnold R.A., 1956. The flow of cohesionless grains in fluids, Philosophical Transition of Royal Society London, No. 964, Vol 249, 235–297. Christoffersen J.B. and Jonsson I.G., 1985. Bed friction and dissipation in a combined current and wave motion, Oc. Eng., 12, No.5, 387–423. Du Toit C.G. and Sleath J.F.A., 1981. Velocity measurements close to ripple beds in oscillatory flow, J. of Fluid Mech., 112, 71–96. Fredsøe J., Andersen K.H. and Summer B.M., 1999. Wave plus current over ripple– covered bed, Coastal Eng., 38, 177–221. Grant W.D. and Madsen O.S., 1986. The continental shelf bottom boundary layer, Annual Rev. of Fluid Mech., 18, 265–305.

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Kemp P.H. and Simons R.R., 1982. The interactions between waves and a turbulent current: waves propagating with the current, J. of Fluid Mech., 116, 227–250. Kemp P.H. and Simons R.R., 1983. The interactions between waves and a turbulent current: waves propagating against the current, J. of Fluid Mech., 130, 73–89. Komar P.D. and Miller M.C., 1975. The initiation of oscillatory ripple marks and the development of plane–bed at high shear stresses under waves, J. of Sed. Petrol. 45, 697–703. Li M.Z. and Amos C.L., 1998. Predicting ripple geometry and bed roughness under combined waves and currents in a continental shelf environment, Cont. Shelf Res., 18, 941–970. Li M.Z. and Amos C.L., 1999. Field observations of bedforms and sediment transport thresholds of fine sand under combined waves and currents, Mar. Geology, 158, 147–160. Mathisen P.P. and Madsen O.S., 1996a. Waves and currents over a fixed ripple bed: 1. Bottom roughness experienced by waves in the presence and absence of currents, J. of Geophys. Res., 101, 16533–16542. Mathisen P.P. and Madsen, O.S., 1996b. Waves and currents over a fixed ripple bed: 1. Bottom roughness experienced by waves in the presence of currents, J. of Geophys. Res., 101, 16543–16550. Nielsen P., 1981. Dynamics and geometry of wave–generated ripples, J. of Geophys. Res., 86, 6467–6472. Nielsen P., 1986. Suspended sediment concentration under waves, Coastal Eng., 10, 23–31. Nielsen P., 1992. Coastal Bottom Boundary Layers and Sediment Transport, Advanced Series on Ocean Engineering. Vol 4, World Scientific. Singapore, 324 pp. Soulsby R.L., Hamm L., Klopman G., Myrhaug D., Simmons R.R. and Thomas G.P., 1993. Wave–current interaction within and outside the bottom boundary layer, Coastal Eng., 21, 41-69.

Chapter 8

Summary and conclusions “Dios no juega a los dados, pero que bien juega a las escondidas”

8.1

Recapitulation

This dissertation addresses dynamic aspects of wind waves using numerical modelling as a main tool to simulate and understand the evolution of wind–waves in the coastal zone. One of the tools is the WAM model which is the prototype numerical model to simulate and forecast waves in deep and intermediate water. In this study the WAM wave model has been improved in order to make it efficient for high–resolution applications as is demanded in shallow water zones. Two aspects were considered for its improvement: the computational efficiency and the representation of physical processes involved in the evolution of waves in shallow water areas. As a result of all the changes and additions, the WAM–PRO model has become a feasible, an economical and a powerful tool to explore the wave spectra evolution in coastal areas compared with the original WAM model. Other wave model used in this study is the SWAN wave model. This model has become a very useful tool to simulate waves in shallow water areas. For this study the eddy–viscosity model of Weber for bottom friction dissipation was incorporated into the SWAN code. Results using this expression were compared and confronted with results from other formulations already coded in SWAN using measurements that represent a nearly idealized situation. Interaction between waves and ambient currents (driven by tides and surges), which is more pronounced in coastal areas than in open ocean, is a very interesting and an important process. In this study, with the participation of several institutions,

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a module that enables combined modelling of tides, surges and waves in shallow water at the North Sea scale has been developed. From numerical experiments using the coupling module it was concluded that the larger influence of coupling on wave parameters is partly due to the horizontal resolution that has been used. In the Southern Bight, tidal currents have a stronger influence than tidal elevations. These tidal currents produce a modulation, at tidal period, of the spectrum at all wave frequencies, largely derived from the Doppler shift, and modulation of the significant wave height due to the exchange of energy with the mean flow. During storms, the change in mean depth due to the surge elevation modifies the dissipation of energy due to bottom friction. Higher waves are observed when the increase of the total depth due to surge elevation is taken into account. The storm surge model is very sensitive to the wind stress parameterizations. Conventional quadratic laws can produce surge elevations similar to those obtained with a wave–dependent surface stress. However, it is necessary that the drag coefficient can accurately reproduce the increase of the surface roughness with increasing wind speed. Since this roughness is strongly correlated with the wave field, the wave–dependent surface stress should be preferred in storm surge modelling. While model results in the Southern North Sea, using a finer grid, were not very sensitive to coupled or uncoupled boundary information, the increasing importance of coupling when going towards shallower areas has been confirmed by the investigations made by others in the Holderness and the Sylt–Rømø Bight. In the shallow water region the wave–bottom interaction is another important process that is responsible for drastic changes in the wave field and, in a way, it determines the interactions between waves and other phenomena, including those between wave and current. Although many formulations for bottom friction wave energy dissipation exist (this reflects the complexity of the phenomenon), it is far from clear which one should be used in operational applications. In this study an attempt is made to search for evidence to determine which friction formulation performs best or is more consistent in shallow water regions. For this purpose four different expressions for wave energy dissipation by bottom friction are intercompared: the JONSWAP expression , the drag law friction model of Collins, the eddy–viscosity model of Madsen and the eddy– viscosity model of Weber. The first three are already in the SWAN wave model (Version 40.01). The eddy–viscosity model of Weber was incorporated into the SWAN code. The data obtained from Lake George (Australia) in water of limited depth provide a nearly ideal situation to test and analyze those bottom friction formulations. According to the results Weber’s model showed the best performance in the cases of depth– and fetch–limited wave growth. In the case of depth–limited wave growth the fit of the calculated curve for the non–dimensional peak frequency to the one obtained by Young and Verhagen (1996) from observations is as good as perfect. Running the SWAN model using Weber’s formula with different roughness height

8.1. RECAPITULATION

169

suggests that in the equations for depth– and fetch–limited wave growth the effect of bottom roughness should be included. Formulations for dissipation by bottom friction, such as the model by Madsen or Weber, which take explicitly physical parameters for bottom roughness into account, should be preferred in wave modelling in shallow water areas. They offer the possibility to adapt the dissipation rate according to the changing roughness under different wave or wave–current conditions. However, it is very common in wave modelling to consider the bottom roughness height as a constant once its value is chosen for a particular storm (tuning its value approaching a particular data set), but in order to make reasonable predictions under a wide variety of conditions the physical processes at the bottom, such as evolution of the ripples, should be represented. In order to remove the constrains of a constant roughness height to be constant in the bottom friction formulations, a new ripple geometry predictor (MN81) has been proposed. It is based on the model of Nielsen but includes findings from other published investigations. The proposed model was coupled with the model of Christofferson and Jonsson (1985) (CJ85) for wave–current bottom boundary layer (WCBBL). The wave bottom boundary layer model supplies the friction velocity to the ripple geometry model to compute the new ripple field. In turn MN81 supplies the new ripple field to run the CJ85 model to compute the parameters inside the WBBL. A set of measurements (Li and Amos, 1996) in the field were used to verify the ripple predictor MN81. Without making specific assumptions for the different stages, which is very common in ripple geometry predictor models, the proposed MN81 model was able to fit the measurements in all ripple regimes viz, (i) the formation of the ripple field from the plane bed, (ii) during the spin up of the storms, (iii) the break–off range, (iv) the sheet flow (v) the formation of the ripple field after the peak of the storms. In the entries (ii) and (v) large ripples were measured in the field and they were modelled without explicit assumptions. The numerical results show that large ripples can develop before the peak of the storm independently of how fast this peak is reached. The WBBL model (CJ85), using the ripple geometry from the MN81 model, reproduces the different flow regimes observed in the field, viz. (i) the non–transport, (ii) the saltation/suspension and (iii) the sheet flow. The apparent roughnesses computed according to the CJ85 model, for the field conditions of Li and Amos (1998) measurements, agree with laboratory measurements. In the ripple regime the relative apparent roughness can be expressed as function of flow (wave–current) and bed (ripples) parameters, with small scatter.

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8.2

Recommendations

Although the present version of the WAM–PRO model is adapted for applications with high spatial resolution, there are still some outstanding tasks. Those tasks can be divided in several categories such as: • Triad interactions. Because the importance of the triad interactions in shallow water areas, it is recommended to include their expression in the WAM– PRO model. • Interpolation of spectra. For the time being WAM–PRO model is able to give a required output spectra in the closest node of the computational grid to the desired output location. This has been problematic in cases where a comparison with measurements should be done and the output location and the desired location have not the same depth. • Grids rotation. In some cases where the coast line it is not oriented with the meridians or parallels a big portion of the computational grid lies on land. In the cases with nested grids, due to the restrictions of grid orientation, a larger than the required computational grid should be used and as a result more computational resources are used. It is recommended to introduce a method to rotate the computational grids in order to exclude most of the points that lie on land. • Bottom friction formulation for wave–current situations. To take advantage of a full coupling system between the wave model and the hydrodynamic model, expressions for the wave energy dissipation at the bottom in wave–current situation had been introduced in the WAM–PRO model. They should be tested. • Wave–current interaction at the bottom. In shallow water the waves interact with the bottom and the orbital motion of the low frequency gravity waves cause a turbulence in its vicinity. This turbulence inside the thin boundary layer causes an enhancement of the bottom stress felt by the tide–surge current. This is the so–called apparent bed roughness. It is recommended to include in the coupling system (comprised by the wave and the tide–surge models) the wave–current interactions at the bottom. • Mobility of the bed. Despite the fact that some models for wave energy dissipation by bottom friction are very elaborate and have some very fine qualities, they are a function of the roughness height, which most of the time, at least in wave forecasting, remains unknown for a given wave, current and sediment characteristics. The implementation of the couple system WBBL–Ripple geometry in a wave model is recommended.

8.2. RECOMMENDATIONS

171

• Wave–current–bedforms interaction. To progress further in modelling the bed shear stress in wave–current situations, the inclusion of the mobility of the bed is required. It is hoped that the improved description of the evolution of the roughness height as function of the flow field which in turn is modified by the ripples will help to understand the evolution of the wind–waves in the coastal zone and lead to more sophisticated parameterizations of the hydrodynamic forces in sediment transport modelling.

Acknowledgements I would like to thank the Mexican people who made my stay in Belgium possible by means of scholarship granted by Consejo Nacional de Ciencia y Tecnolog´ıa (CONACYT, México). I sincerely express my gratitude to my supervisor Prof. Dr. Ir. Jaak Monbaliu for all those stimulating and fruitful discussions. My gratitude to all those who participate in the PROMISE project. This work has benefited by this project. I am grateful to my second promotor Prof. No¨ el Vandenberghe. My appreciation is extended to the jury members Prof. Dr. Ir. Jean Berlamont, Dr. Francisco J. Ocampo–Torres and especial thanks to Prof. Dr. Ir. J. Battjes. The Hydraulics laboratory should by acknowledge for their support through national and international projects. I thank to the SWAN Team for the numerical code of SWAN. Thanks WL Delft Hydraulics for the SBMSWAN (Suite 40.01.a of the Benchmark Test for the Shallow Water Wave Model SWAN Cycle 2, version 40.01), thanks to Dr. Ian Young, University of New South Wales, Canberra, Australia for making the data of Lake George publicly available . Thanks to my colleagues in the Hydraulics Laboratory, especially Jaak and Pedro for their friendship and for all those talks behind a pintje. Quiero agradecer y a mis hijos, Adrián y Andrea ya que sin su ayuda hubiese sido posible terminar antes esta tesis (por supuesto, siempre hay que mantener algunas prioridades) y a Lilia por el balance. Al grupo de latinos, gracias por su amistad en la u ´ltima etapa de mi estad´ıa en Leuven. Gracias a mis amigos Luis Zavala y Ernesto Garc´ıa por su amistad e inspiración. A mis padres y hermanos por su amor y a mi otra familia, la Familia Crisóstomo Vázquez, por su apoyo.

Curriculum Vitae Roberto Padilla-Hernández was born in León, Guanajuato (México) on 23th November, 1962. He graduated in the Universidad Aut´ onoma de Baja California in the Faculty of Marine Science in 1987 with a thesis entitled “Power and direction of wind– waves and its exploitation”, under the supervision of MSc Oscar Delgado. In 1993, he completed the Master in Science program in Centro de Investigaci´ on Cient´ıfica y de Educacion Superior de Ensenada (CICESE). The subject of the thesis was “Wind– waves prediction with the Spectrum–Angular Density Model”, under the supervision of MSc. Cuauhtémoc Nava. From 1993 to 1995, he worked as a Research Assistant in CICESE in the project “Use of the oceanographic satellite data to improve the wave prediction in Mexican Seas”. In the period from 1995 to 1997, he worked as Adjoint Researcher, in the Faculty of Civil Engineering (Fluid Mechanics Section) of the Technical University of Delft (the Netherlands), developing the numerical wave model SWAN and implementing the second generation wave model, HISWA into SWAN. From 1997 to 1998 he worked at the Hydraulics Laboratory of the Katholieke Universiteit Leuven (KULeuven) on the project “PRe–Operational Modelling In the Seas of Europe (PROMISE)” as a Researcher. Since 1998 he begun the Ph.D. program in KULeuven, under the supervision of Prof. Dr. Ir. Jaak Monbaliu.

List of Selected Publications 1. Padilla–Hern´ andez R. and Monbaliu J., –, Coupled models for the wavecurrent flow and the bedform dynamics. Part 1. Ripple Geometry. Submitted to Journal of Geophysical Research. 2. Padilla–Hern´ andez R. and Monbaliu J., –, Coupled models for the wavecurrent flow and the bedform dynamics. Part 2. The wave bottom boundary layer. Submitted to Journal of Geophysical Research. 3. Padilla–Hern´ andez R. and Monbaliu J., 2001, Energy balance of wind waves as a function of the bottom friction formulation. Journal of Coastal Engineering, 43, 131-148. 4. Monbaliu, J., R. Padilla–Hern´ andez, J.C. Hargreaves, J.-C. Carretero, W. Luo, M. Sclavo and H. G¨ unther, 2000. The spectral wave model, WAM, adapted for applications with high spatial resolution. Journal of Coastal Engineering. 41, 41-62. 5. Ozer, J., Padilla–Hern´ andez R., Monbaliu J., Alvarez Fanjul E., Carretero Albiach J.C., Osuna P., Yu, C.S.J., Wolf J., 2000. A coupling module for tides, surges and waves. Journal of Coastal Engineering. 41, 95-124.

6. L.H. Holthuijsen, N. Booij, Ris R.C, Haagsma IJ. C., Kieftenburg A.T.M.M. and R. Padilla–Hern´ andez, 1999. SWAN Cycle 2 User Manual – Simulation of Waves in the Nearshore Zone, Delft University of Technology, the Netherlands. 7. Monbaliu, J., Padilla, R., Osuna, P., Hargreaves, J., Flather, R. and Carretero, J.–C., 1998. Shallow water version WAM–C4–S.01 - documentation, POL (Proudman Oceanographic Laboratory, U.K.) report 52. 8. Padilla R., J. Monbaliu, P. Osuna, L. Holthuijsen, 1998. Intercomparing Third-Generation Wave Model Nesting. Proceedings of the 5th International Workshop on Wave Hindcasting and Forecasting. 26-30 January, 1998. Melbourne, Fla. USA. 9. Booij N., Holthuijsen L., Padilla–Hern´ andez R., A Non-stationary Parametric Coastal Wave Model. Int. Conf. on Coastal Research through Large Scale Experiments. Coastal Dynamics ’97. 23–27 June, 1997. Plymouth, UK. 10. Holthuijsen, L.H., N. Booij and R. Padilla–Hern´ andez, A curvi–linear, thirdgeneration coastal wave model. Int. Conf. on Coastal Research through Large Scale Experiments. Coastal Dynamics ’97. 23–27 June, 1997. Plymouth, UK.