NUMERICAL SIMULATION OF LARGE-SCALE WAVES AND CURRENTS ZHANG DAN NATIONAL UNIVERSITY OF SINGAPORE 2004 NUMERICAL SIMULATION OF LARGE-SCALE WAVES AND CURRENTS ZHANG DAN (B Eng., Ocean University, PRC) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF CIVIL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2004 Acknowledgement I wish to take this opportunity to acknowledge my supervisor Dr Lin Pengzhi for his keen guidance, support and invaluable advice during the course of this work Uncounted numbers of discussion lead me to learn more and quickly from him His personal assistance and understanding have made my study in National University of Singapore a memorable experience I am also indebted to my co-supervisor, Professor N Jothi Shankar, for sharing his ideas throughout my study period I would like to give my sincere appreciation to my friends for their insightful suggestion and fruitful discussion when I encountered problems In addition, I am thankful to the staff of the Hydraulics Laboratory for their friendly assistance and pleasant jokes Heartfelt thanks to my mother and brother Their selfless support and love have pushed and are still pushing me to go forward bravely Last but not least, I give my special gratitude to my wife, Ren Chunxia, for her steadfast encouragement and love i ABSTRACT For the past decades, wave models and ocean circulation models have been developed separately Wave models don’t recognize the vertical structure of ocean currents and ocean models neglect the effect of waves However, the waves and ocean currents can interact in many ways, one of which is through radiation stresses The conventional radiation stresses (2D) are defined in the vertically integrated form The coupling of wave model and ocean circulation model is usually accomplished by including the depthaveraged radiation stresses as the forcing term in the momentum equation Unfortunately, 2D radiation stresses can not properly represent the effect of waves on currents In this thesis, expressions for depth-dependent radiation stresses (3D) are derived in the Cartesian coordinates on the basis of linear wave theory After vertical integration, these expressions revert to the conventional 2D radiation stresses In viewpoint of physics, the effect of waves varies along the water depth, especially in deep water However, the conventional radiation stresses fail to reflect this phenomenon In contrast, 3D radiation stresses are able to explain the wave-current interaction The newly derived 3D radiation stresses are suitable for simulating the effect of waves on currents, such as wind-induced circulation in a water basin, wave induced cross-shore currents and long-shore currents The performance of the numerical model is demonstrated by comparison with theoretical results, experimental data and conceptual analysis They display a favorable match It is shown that the turbulence needs to be considered when we study wave-induced currents in the nearshore zone Compared with ii 2D radiation stresses, 3D radiation stresses have larger effect on ocean currents More work should be carried out about the wave-current interaction through depth-dependent radiation stresses Keyword: Wind waves, Ocean current, 3D radiation stresses, Wave-current interaction, Princeton Ocean Model, SWAN iii TABLE OF CONTENTS Acknowledgements i Abstract ii Table of Contents iv List of Figures vii CHAPTER ONE INTRODUCTION .1 1.1 HISTORICAL REVIEW OF WAVE MODEL 1.2 REVIEW OF OCEAN CIRCULATION MODEL 1.3 COUPLING OF WAVE MODEL AND OCEAN CIRCULATION MODEL CHAPTER TWO WAVE MODEL DESCRIPTION AND VERIFICATION 12 2.1 SWAN WAVE MODEL 12 2.1.1 Model Description 12 2.1.2 Governing Equation 13 2.1.3 SWAN Numerical Implementation 14 2.2 TESTING OF SWAN WAVE MODEL 16 2.2.1 Wave Refraction in Coastal Area 17 2.2.2 Wind Generated Waves 19 2.2.3 Effect of Currents and MWL Fluctuation on Wave Propagation 22 2.2.4 Selection of The Wave Breaking Coefficient 25 iv 2.3 SIMULATION OF WIND WAVES IN SOUTH CHINA SEA 27 2.3.1 The South China Sea 27 2.3.2 Wind Field 29 2.3.3 Numerical Simulation of Wind Induced Waves 33 2.4 SIMULATION OF WIND WAVES IN SINGAPORE WATERS 37 CHAPTER THREE DEPTH-DEPENDENT RADIATION STRESSES 42 3.1 DERIVATION OF DEPTH-DEPENDENT RADIATION STRESSES 42 3.1.1 Evolution of Radiation Stresses 42 3.1.2 Governing Equations for Fluid Influenced by Waves 43 3.1.3 Derivation of Depth-Dependent Radiation Stresses 45 3.2 PROFILE OF 3D RADIATION STRESSES WITH WATER DEPTH 50 CHAPTER FOUR 3D OCEAN CIRCULATION MODELLING 53 4.1 DESCRIPTION OF PRINCETON OCEAN MODEL 54 4.1.1 Sigma Coordinates 54 4.1.2 Governing Equations 54 4.1.3 Sigma Coordinate Representation 57 4.1.4 Momentum Equations Including Radiation Stresses 59 4.2 VERIFICATION OF 3D RADIATION STRESSES BY IPOM 60 4.2.1 The Steady-Wind-Driven Flow in a Closed Basin 60 4.2.2 Wave Set-up and Set-down 64 4.2.3 Numerical Simulation of Undertow 66 v 4.2.4 Wave-Induced Longshore Currents 71 4.3 WIND DRIVEN OCEAN CIRCULATION IN SOUTH CHINA SEA 74 4.4 SIMULATION OF OCEAN CIRCULATION IN SINGAPORE WATERS 93 CHAPTER FIVE CONCLUSIONS 94 BIBLIOGRAPHY …………………………………………………… 96 vi LIST OF FIGURES Figure 2.1: The propagation of obliquely incident wave from deep to shallow water 18 Figure 2.2: The propagation of obliquely incident wave from deep to shallow water 19 Figure 2.3: The comparison of significant wave height 21 Figure 2.4: The comparison of significant wave period 21 Figure 2.5: The comparison of significant wave height-1 24 Figure 2.6: The comparison of significant wave height-2 24 Figure 2.7: Selection of wave breaking coefficient 27 Figure 2.8: Topography of South China Sea 28 Figure 2.9: Map of South China Sea 29 Figure 2.10: Wind field in the SCS at 00 hrs, March 14, 2003 30 Figure 2.11: Wind field in the SCS at 00 hrs, March 15, 2003 30 Figure 2.12: Wind field in the SCS at 00 hrs, March 16, 2003 31 Figure 2.13: Wind field in the SCS at 00 hrs, March 17, 2003 31 Figure 2.14: Wind field in the SCS at 00 hrs, March 18, 2003 32 Figure 2.15: Wind field in the SCS at 00 hrs, March 19, 2003 32 Figure 2.16: Wind field in the SCS at 18 hrs, March 19, 2003 33 Figure 2.17: Wave field in the SCS at 12th hrs, March 14, 2003 34 Figure 2.18: Wave field in the SCS at 12th hrs, March 15, 2003 34 Figure 2.19: Wave field in the SCS at 12th hrs, March 16, 2003 35 Figure 2.20: Wave field in the SCS at 12th hrs, March 17, 2003 35 Figure 2.21: Wave field in the SCS at 12th hrs, March 18, 2003 36 Figure 2.22: Wave field in the SCS at 18th hrs, March 19, 2003 36 Figure 2.23: Map of Singapore surroundings 38 Figure 2.24: 3D geographical formations of Singapore Straits 38 vii Figure 2.25: Wind conditions at the Singapore Straits 40 Figure 2.26: Wave conditions at the Singapore Straits 41 Figure 3.1: Profile of the cumulative 3D radiation stress with water depth 51 Figure 4.1: The sigma coordinate system 57 Figure 4.2: Velocity profile when water depth is 5m 62 Figure 4.3: Velocity profile when water depth is 10m 63 Figure 4.4: Velocity profile when water depth is 40m 63 Figure 4.5: Velocity profile when water depth is 80m 64 Figure 4.6: Wave set-up and set-down 66 Figure 4.7: Experimental arrangement 67 Figure 4.8 (a): Variation of horizontal current velocity with depth 69 Figure 4.8 (b): Variation of horizontal current velocity with depth 70 Figure 4.9: The wave induced longshore currents 73 Figure 4.10: The profile of longshore currents 73 Figure 4.11: The profile of cross-shore currents 74 Figure 4.12: The average wind field in SCS 76 Figure 4.13: The wave conditions in SCS 77 Figure 4.14: The variation of the surface water level 77 Figure 4.15: The current pattern at the water surface when wind is considered only 79 Figure 4.16: The current pattern at the sigma layer 10 when wind is considered only 80 Figure 4.17: The current pattern at the sigma layer 18 when wind is considered only 80 Figure 4.18: The current pattern at the water surface as 2D stresses considered 81 Figure 4.19: The current pattern at the sigma layer 10 as 2D stresses considered 81 Figure 4.20: The current pattern at the sigma layer 18 as 2D stresses considered 82 Figure 4.21: The current pattern at the water surface as 3D stresses considered 82 viii 0.15 wind only 2D stress 3D stress current velocity (y-component, m/s) 0.1 0.05 -0.05 -0.1 -0.15 -0.2 100 105 110 115 120 125 (Deg) Longitude (Latitude=8 degree, N) Figure 4.27: The variation of current velocity (y-component) in the sigma layer σ=10 along longitude when latitude=8 degree, N 0.4 wind only 2D stress 3D stress current velocity (x-component, m/s) 0.3 0.2 0.1 -0.1 -0.2 -0.3 -0.4 100 105 110 115 120 125 (Deg) Longitude (Latitude=17 degree, N) Figure 4.28: The variation of current velocity (x-component) at the surface along longitude when latitude=17 degree, N 86 0.2 current velocity (y-component,m/s) 0.15 wind only 2D stress 3D stress 0.1 0.05 -0.05 -0.1 -0.15 -0.2 100 105 110 115 120 125 (Deg) Longitude (Latitude=17 degree, N) Figure 4.29: The variation of current velocity (y-component) at the surface along longitude when latitude=17 degree, N 0.1 current velocity (x-component, m/s) 0.08 wind only 2D stress 3D stress 0.06 0.04 0.02 -0.02 -0.04 -0.06 -0.08 -0.1 100 105 110 115 120 125 (Deg) Longitude(Latitude=17 degree, N) Figure 4.30: The variation of current velocity (x-component) in the sigma layer σ=10 along longitude when latitude=17 degree, N 87 0.15 current velocity (y-component,m/s) 0.1 wind only 2D stress 3D stress 0.05 -0.05 -0.1 -0.15 -0.2 100 105 110 115 120 125 (Deg) Longitude (Latitude=17 degree, N) Figure 4.31: The variation of current velocity (y-component) in the sigma layer σ=10 along longitude when latitude=17 degree, N Figures 4.33-4.38 display the profile of current velocity along the water depth These three points are shown on Figure 4.32 u and v represent the x-component and ycomponent of the current velocity, respectively Again, 2D and 3D mean the radiation stresses are two-dimensional or three-dimensional As mentioned above, the current pattern is fundamentally determined by the wind since the wind force is much stronger than the wave’s influence It is shown that 2D radiation stresses have barely transformed the currents, but 3D radiation stresses have somehow altered the current profile Since the wave fields around these points are changing much more rapidly in the latitudinal direction than in the longitudinal direction, the current profile is transformed more in the y-component than in the x-component In principal, when it is in shallow water or in the intermediate depth water, the waves will more or less influence the currents in the whole domain But if it is in deep water, the effect of 88 waves on currents is almost restricted to the water surface In this sense, 3D radiation stresses are correct to represent the physical process More work should be carried out in the future for the wave-current interaction through depth-dependent radiation stresses 25 TAIWAN 15 ON TI N EN T LU ZO N AS IAN C D1 10 D2 A AL M Latitude (degree) 20 D3 IA YS -5 100 RN BO SU MA TR A 105 110 EO 115 120 125 Longitude (degree) Figure 4.32: The location of three points in South China Sea 89 u-wind only u-3D stress u-2D stress water depth (m) -400 -800 -1200 -1600 -2000 -2179 -0.1 -0.05 0.05 0.1 current speed (m/s) Figure 4.33: The profile of x-component of current velocity at point D1 v-wind only Pom 3D stress v-3D stress 2D stress v-2D stress water depth (m) -400 -800 -1200 -1600 -2000 -2179 -0.2 -0.15 -0.1 -0.05 current speed (m/s) Figure 4.34: The profile of y-component of current velocity at point D1 90 u-wind only Pom 3D stress u-3D stress 2D stress u-2D stress -20 water depth (m) -40 -60 -80 -100 -120 -140 -0.2 -0.15 -0.1 -0.05 0.05 0.1 current speed (m/s) Figure 4.35: The profile of x-component of current velocity at point D2 -20 v-wind only Pom 3D stress v-3D stress 2D stress v-2D stress water depth (m) -40 -60 -80 -100 -120 -140 -0.2 -0.15 -0.1 -0.05 current speed (m/s) Figure 4.36: The profile of y-component of current velocity at point D2 91 u-wind only u-3D stress u-2D stress water depth (m) -10 -20 -30 -40 -50 -60 -0.1 -0.05 0.05 0.1 current speed (m/s) Figure 4.37: The profile of x-component of current velocity at point D3 water depth (m) -10 v-wind only Pom 3D stress v-3D stress 2D stress v-2D stress -20 -30 -40 -50 -60 -0.1 -0.05 current speed (m/s) Figure 4.38: The profile of y-component of current velocity at point D3 92 4.4 SIMULATION OF OCEAN CIRCULATION IN SINGAOPRE WATERS Because of geographic and economic importance, Singapore Straits has attracted much attention for its hydrodynamic characteristics In 1979, a joint team was established to investigate the tides and tidal currents in the Straits of Malacca and Singapore The harmonic parameters were given out for tidal currents and tides From then on, the geography of Singapore waters has experienced a huge change because of land reclamation and coastal construction It is necessary to evaluate the hydrodynamic and mass transport characteristics of the new environmental system Various numerical ocean models have been developed for this purpose Chan (1991) developed a 2D hydraulic model to simulate tidal elevation and associated currents in Singapore waters The numerical results from curvilinear and Cartesian system are compared with field measurements Chao (2000) developed a 3D multi-level turbulence model to simulate tidal motions in Singapore waters Another 3D model is carried out by Zhang and Gin (2000) to simulate the circulation driven by local tide forcing and seasonal wind forcing Recently, the hydrodynamic characteristics of Singapore Straits are further tested by researchers in National University of Singapore (Ma, 2002; Didier, 2003) in terms of POM and COHERENS Since the leading driving force is the variation of water level for Singapore waters, these ocean models are driven by the time-dependent tidal elevation on the boundary By contrast with field measurements of surface elevation and surface current speed (Ma, 2002), the numerical results are quite accurate The current pattern alternates between the high tide and the low tide 93 CHAPTER FIVE CONCLUSIONS For the last decades, wave models and ocean circulation models have been developed separately Actually, waves and ocean currents can interact in many ways, one of which is through radiation stresses The conventional radiation stresses (2D) are developed in the vertically integrated form They are suitable for two-dimensional problems, such as wave set-up, longshore currents, and so on However, for the purpose of recognizing the vertical distribution of the wave-induced nearshore process, such as undertow, 2D radiation stresses are inapplicable It is necessary to develop a new kind of radiation stresses that can represent the effect of waves in the threedimensional space In this thesis, the depth-dependent radiation stresses (3D) are developed on the premise of the linear wave theory Since the wave’s effect attenuates as water goes deeper, it is more appropriate to use the 3D radiation stresses as the driving force for the waveinduced nearshore process Moreover, 3D radiation stresses can accurately represent the effect of waves on ocean currents The depth-dependent radiation stresses are useful not only for two-dimensional questions but also for three-dimensional questions It is shown in chapter that both 3D and 2D radiation stresses can be used to simulate the change of mean water level for a normally incident wave However, only 3D 94 radiation stresses are available to simulate the undertow because 2D radiation stresses can’t properly represent the change of radiation stresses along water depth For waveinduced longshore currents, 3D radiation stresses can provide the vertical structure of current velocity, which is important for sediment transport According to the numerical results, the current field changes somehow when 3D radiation stresses are taken as the driving force The reason may be simple Compared with the 2D radiation stresses, more wave energy is concentrated on the top of the water body in terms of the distribution of 3D radiation stresses, so stronger currents could be generated at the surface Through the vertical mixing process and advection, the momentum is conveyed to the whole water body Field measurements are needed to verify the explanation The above radiation stresses are derived on the basis of Airy wave theory However, when waves advance on a sloping sea bottom, it is pointed out that Biesel theory (1952) is better to calculate wave motion In the future work, wave models and ocean circulation models could be coupled in terms of more precise three-dimensional radiation stresses Bottom friction can also be incorporated to achieve better results As for wave induced nearshore currents, turbulence is necessary to be included Further study should be carried out about the wave-current 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Singapore coastal waters, Coastal Engineering, 39, 71-95 2000 100 [...]... D1 90 Figure 4.34: The profile of y-component of current velocity at point D1 90 Figure 4.35: The profile of x-component of current velocity at point D2 91 Figure 4.36: The profile of y-component of current velocity at point D2 91 Figure 4.37: The profile of x-component of current velocity at point D3 92 Figure 4.38: The profile of y-component of current velocity at point D3 92 ix CHAPTER... is the application of modified version of the NavierStokes equations, which have been developed for the study of fluid dynamics over centuries Because the ocean circulation is often assessed on the large scale, elements which are negligible in a small domain can not be excluded from the large- scale simulation, such as Coriolis force The inclusion of the effects of rotating earth and some appropriate... interaction through surface and bottom stresses Waveinduced wind stress increases currents both at the sea surface and near the seabed But wave-induced bottom stress reversely weakens currents both at the sea surface and near the seabed The net effect of wind waves depends on the relative importance of wave-induced wind stress and bottom stress Since waves can directly influence the currents by radiation... equation, σ and θ denote wave relative frequency and wave direction, respectively The first term in the left-hand side of the above equation represents the local rate of change of wave action density spectrum in 13 time The second and third terms represent propagation of wave action in geographical space with velocities Cx and Cy in x and y- directions, respectively The fourth term represents shifting of the... COUPLING OF WAVE MODEL AND OCEAN CIRCULATION MODEL The mutual influence of waves and currents has been recognized for a long time and research has been conducted on this subject (Longuet-Higgins et al., 1961; Kantardgi et al., 1993; Madden et al., 1998) Currents can deflect wave direction, stretch wave length or change wave celerity Surface waves can influence the current by the gradient of radiation... changing the wind stress and by affecting the bottom friction (Ozer et al., 2000) In the past decades, although great progress has been made in the understanding and numerical modeling of ocean surface waves and ocean circulation, the two streams have never been syncretized Usually, wave models can not directly calculate the ocean currents and ocean circulation models assume waves having no influence... differences between the ocean circulation and ordinary fluid dynamics Since the chief objective of ocean circulation is to explain and predict the flow in light of fundamentals of fluid dynamics, associated elements like density, pressure, salinity and temperature are also necessary for the motion of oceans The basic equations of ocean circulation are composed of mass conservation, momentum equations... the coastal areas of Singapore are low lands, the danger from storm tide can not be ignored 1 Based on these conditions, the purpose of this dissertation is to present a wave model and improve an ocean circulation model to simulate wind-induced waves and the storm surge, which could be valuable in forecasting the occurrence of flooding 1.1 HISTORICAL REVIEW OF WAVE MODEL The principles of wave prediction... function, consisting of a superposition of the energy input by wind, Sin, normally represented as the sum of a Phillips’ (1957) and Miles’ (1957) term, the nonlinear transfer Snl due to resonant wave-wave interactions, and the dissipation Sds by means of whitecapping, bottom friction, depth-induced wave breaking, and so on The wind input term is commonly represented as the summation of a linear and exponential... circulation is the persistent pattern of flow on the scale of basins It is heated by the sun and driven by the wind and circulates endlessly on the earth Over the past decades, the study of ocean circulation has attracted more attention from governments and industries because utilization of ocean resource is becoming increasingly important for human beings A knowledge of ocean circulation is useful for .. .NUMERICAL SIMULATION OF LARGE-SCALE WAVES AND CURRENTS ZHANG DAN (B Eng., Ocean University, PRC) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF CIVIL ENGINEERING... 4.33: The profile of x-component of current velocity at point D1 90 Figure 4.34: The profile of y-component of current velocity at point D1 90 Figure 4.35: The profile of x-component of current... 4.36: The profile of y-component of current velocity at point D2 91 Figure 4.37: The profile of x-component of current velocity at point D3 92 Figure 4.38: The profile of y-component of current