SWAN SCIENTIFIC AND TECHNICAL DOCUMENTATION SWAN Cycle III version 41.31AB SWAN SCIENTIFIC AND TECHNICAL DOCUMENTATION by : TheSWANteam mail address : Delft University of Technology Faculty of Civil Engineering and Geosciences Environmental Fluid Mechanics Section P.O. Box 5048 2600 GA Delft The Netherlands e-mail : [email protected] homepage : http://www.swan.tudelft.nl Copyright (c) 1993-2021 Delft University of Technology. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back- Cover Texts. A copy of the license is available at http://www.gnu.org/licenses/fdl.html#TOC1. iv Contents 1 Introduction 1 1.1 Historicalbackground. 1 1.2 Purposeandmotivation . 2 1.3 Readership.................................... 2 1.4 Scopeofthisdocument............................. 2 1.5 Overview..................................... 4 1.6 Acknowledgements ............................... 4 2 Governing equations 7 2.1 Spectraldescriptionofwindwaves . .... 7 2.2 Propagationofwaveenergy . 10 2.2.1 Wavekinematics ............................ 10 2.2.2 Spectralactionbalanceequation. ... 10 2.3 Sourcesandsinks................................ 13 2.3.1 Generalconcepts ............................ 13 2.3.2 Input by wind (Sin)........................... 18 2.3.3 Dissipation of wave energy (Sds).................... 20 2.3.4 Nonlinear wave-wave interactions (Snl) ................ 27 2.3.5 First- and second-generation model formulations in SWAN . 33 2.4 Wavedampingduetovegetation . 35 2.5 Wavedampingduetoseaice. 38 2.6 Theinfluenceofambientcurrentonwaves . .... 39 2.7 Modellingofobstacles . .. .. .. .. ... .. .. .. .. .. ... .. 40 2.7.1 Transmission .............................. 40 2.7.2 Reflection ................................ 42 2.7.3 Freeboard dependent reflection and transmission . ....... 42 2.7.4 Diffraction................................ 43 2.8 Wave-inducedset-up .............................. 45 3 Numerical approaches 47 3.1 Introduction................................... 47 3.2 Discretization .................................. 48 3.2.1 Discretization in geographical space . ..... 49 v vi 3.2.2 Note on the choice of geographic propagation schemes . ...... 53 3.2.3 Discretizationinspectralspace . ... 54 3.2.4 Conservative elimination of negative energy densities ........ 55 3.3 Solutionalgorithm ............................... 57 3.4 Iterationprocessandstoppingcriteria . ....... 60 3.5 An illustrative explanation of the sweeping approach . ......... 63 3.6 Implementation of DIA within the four-sweep technique . .......... 66 3.7 Action density limiter and under-relaxation. ........ 67 3.7.1 Introduction............................... 67 3.7.2 Convergence-enhancingmeasures . .. 68 3.8 On the approximation of refraction in large-scale SWAN applications . 70 3.8.1 Introduction............................... 70 3.8.2 Energytransportalongwaverays . 71 3.8.3 The problem with refraction in non-stationary applications . 73 3.8.4 A historical overview of limitation on cθ ............... 78 3.8.5 The problem with refraction on coarse grids . .... 79 3.9 Governing equations in curvilinear co-ordinates . .......... 80 3.10 Computation of force in curvilinear co-ordinates . ........... 82 3.11 Numericaltreatmentofobstacles . .... 83 3.12 Crossingofobstacleandgridline . ..... 85 3.13 Integration over σ ................................ 86 3.14 Transformation from relative to absolute frequency . ........... 87 3.15 Interpolationofspectra. .... 88 3.16 Computationofbreakingsourceterm . ..... 89 4 Wave boundary and initial conditions 91 5 Implementation of 2D wave setup 93 5.1 Introduction................................... 93 5.2 Numericalapproach............................... 94 5.2.1 Discretizationofthe2Dsetupequation . .... 94 5.2.2 The iterative solver for the linear system . ..... 99 6 Iterative solvers 101 6.1 StronglyImplicitProcedure(SIP) . ..... 101 6.2 Successive Over Relaxation (SOR) technique . ....... 102 7 Parallel implementation aspects 103 7.1 Loadbalancing ................................. 103 7.2 Parallelization of implicit propagation schemes . ........... 104 vii 8 Unstructured mesh implementation 109 8.1 Descriptionofanunstructuredgrid . ..... 109 8.1.1 Definitions................................ 109 8.1.2 Relations between number of cells, vertices and faces ........ 110 8.1.3 Conditionsimposedtothegrid . 110 8.2 Somenotesongridgeneration . 111 8.3 Numericalmethod ............................... 111 8.3.1 Discretizationprocedure . 111 8.3.2 Thesweepingalgorithm . 116 8.4 Interpolation at user-defined locations. ........ 117 8.5 Computationofwave-inducedforce . .... 119 8.6 Calculationofdiffusion-liketerms . ...... 120 8.7 Conservationofaction . 121 9 The overall solution algorithm 123 Bibliography 125 Index 143 viii Chapter 1 Introduction The main goal of the SWAN model is to solve the spectral action balance equation without any a priori restrictions on the spectrum for the evolution of wave growth. This equation represents the effects of spatial propagation, refraction, shoaling, generation, dissipation and nonlinear wave-wave interactions. The basic scientific philosophy of SWAN is identical to that of WAM cycle 3. SWAN is a third-generation wave model and it uses the same formulations for the source terms. Whereas the WAM model considers problems on oceanic scales, with SWAN wave propaga- tion is calculated from deep water to the surf zone. Since, WAM makes use of explicit propagation schemes in geographical and spectral spaces, it requires very small grid sizes in shallow water and is thus unsuitable for applications to coastal regions. For that reason, SWAN employs implicit schemes, which are more robust and economic in shallow water than the explicit ones. Note that SWAN may be less efficient on oceanic scales than WAM. 1.1 Historical background Over the past two decades, a number of advanced spectral wind-wave models, known as third-generation models, has been developed such as WAM (WAMDI Group, 1988), WAVE- WATCH III (Tolman, 1991), TOMAWAC (Benoit et al., 1996) and SWAN (Booij et al., 1999). These models solve the spectral action balance equation without any a priori re- strictions on the spectrum for the evolution of wave growth. Based on the wave action balance equation with sources and sinks, the shallow water wave model SWAN (acronym for Simulating WAves Nearshore) is an extension of the deep water third-generation wave models. It incorporates the state-of-the-art formulations for the deep water processes of wave generation, dissipation and the quadruplet wave-wave interactions from the WAM model (Komen et al., 1994). In shallow water, these processes have been supplemented with the state-of-the-art formulations for dissipation due to bottom friction, triad wave-wave interactions and depth-induced breaking. SWAN is fully spectral (in all directions and frequencies) and computes the evolution of wind waves in coastal regions 1 2 Chapter 1 with shallow water and ambient current. SWAN is developed at Delft University of Technology and is freely available at http://www.swan.tudelft.nl. It is used by many goverment authorities, research institutes and consultants worldwide. The feedback has widely indicated the reliability of SWAN in different experiment and field cases. Initially, the SWAN cycle 1 was formulated to be able to handle only stationary condi- tions on a rectangular grid. Later on, SWAN cycle 2 model has been developed. This is considered as the second step in the development of SWAN models. Cycle 2 of SWAN is stationary and optionally nonstationary. It can compute the wave propagation not only on a regular rectangular grid, but also on a curvilinear grid. Previous official versions 30.62, 30.75, 40.01 and 32.10 belong to the cycle 2 of SWAN. This section is under preparation. 1.2 Purpose and motivation The purpose of this document is to provide relevant information on the mathematical models and numerical techniques for the simulation of spectra of random short-crested, wind-generated waves in coastal regions. Furthermore, this document explains the essential steps involved in the implementation of various numerical methods, and thus provides an adequate reference with respect to the structure of the SWAN program. 1.3 Readership This document is, in the first place, addressed to those, who wish to modify and to ex- tend mathematical and numerical models for shallow wind-wave problems. However, this material is also useful for those who are interested in the application of the techniques discussed here. The text assumes the reader has basic knowledge of analysis, partial dif- ferential equations and numerical mathematics and provides what is needed both in the main text and in the appendices. 1.4 Scope of this document SWAN is a third-generation wave model for obtaining realistic estimates of wave parameters in coastal areas, lakes and estuaries from given wind, bottom and current conditions. However, SWAN can be used on any scale relevant for wind-generated surface gravity waves. The model is based on the wave action balance equation (or energy balance in the absence of currents) with sources and sinks. Good introductory texts on the background of SWAN are Young (1999) and Booij et al. (1999). Introduction 3 The following wave propagation processes are represented in SWAN: • propagation through geographic space, •