Dynamics of Surface Waves in Coastal Waters (Wave-Current-Bottom Interaction) 7040250616, 9787040250619

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Hu Huang

Dynamics of Surface Waves in Coastal Waters Wave-Current-Bottom Interactions

HuHuang

Dynamics of Surface Waves in Coastal Waters Wave-Current-Bottom Interactions

With10 figures

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HIGHER EDUCATION PRESS

fl Springer

Author. Prof. Hu Huang IShanghm Inslitute of Applied MathematIcsand Mechamcsl IShanghm Umverslly Shanghai 200072, P.R.China ~-mail: [email protected]

IISBN 978-7-04-025061-9 IHigher Education Press, Beijing IISI:lN lJ7~-3-54U-~~~3U-7 e ISBN 978-3-540-88831-41 ISpnnger Dordrecht HeIdelbergLondon New York (LIbrary 01 CongressControlNumber: 2008937491

@Higher Education Press, Beijing and Springer- Verlag Berlin Heidelberg IIhiS work

IS

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To my youthfulfather and mother '(levoting all their lives to The Xinjiang Production andl ronstruction Corps in China since September25th 1949

Preface

Water waves hold up a great mirror to one man and let him look at himself inl his billowing blue skyl

IWave motion surrounds us-from the most secret, profound waves of quantuml mechanics to the grand waves of the ocean surfaceJ Ocean waves, or water waves, may be divided into deep- and shallow- water (coastal) waves. From an advance pomt of VIew, coastal waves are not studIed a~ thoroughly as deep-water waves due to a complIcated seabed topography on the former but not on the latter. Therefore, in conjunction with the effects of ubiquij tous ambient currents, wave-current-bottom interactions make up the most funj damental, widespread dynamical mechanism in coastal waters manifesting itselfj as refraction, diffraction, scattering, and resonant wave interactions involved in energyexchanges·1 Apparently, It IS essentIal to obtam a fuII, clear explanation and descnptIon of coastal waves for the development of broad offshore, coastal and harbor enj gineering, and also for having a better understanding of the evolutionary mechj anism of deep-water waves. In fact, a commanding view on long-term investIgatmg water waves IS to whoIIy and umformly treat and descnbe deep- and shaIIow-water waves, thus promoung the present rapid exploratIon and devel1 opment of global oil and gas fields in deep waters of the oceans. The aforementioned views, ideas, judgments, all that I have thought and done over the last ten years, were compiled by me in this book. The book consists ofj nine chapters and appendices from A to H, depicting the fundamental paradigms of weakly nonlInear water waves.

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Preface

Chapter 1 makes a concise, historical review of water wave theories with the emphasis on current mild-slope equations and Boussinesq-type equations. An outline of three kinds of fundamental formulations of the surface water wave Iproblems then follows. Chapter 2, on the Liu-Dingemans evolution equations, gives a modification to that, and extensions of that, from third- to fourth-order with stability analysis, and from pure waves to wave-current interactions. Chapter 3 deals with resonant wave-current-bottom interactions, concentrating on subharmonic resonance concerning the modulated wave groups and second-order long waves I Chapter 4 rs dedIcated to SIX kmds of the mIld-slope equations dependmg on the variation of ambient currents and bottom topography. In particular, a new type of conservative quantity, the product of phase velocity and group velocityJ is put forward, and an operator indicating wave-current interaction is introduced to develop a hierarchy of the mild-slope equationsJ Chapter 5 addresses linear gravity waves over rigid, porous bottoms by for1 mulatmg two dIfferent models mvolvmg the mIld-slope equations and the gen1 eral variation of the wave continuity. Chapter 6 leads to a type of nonlinear unified equations suitable for an unevenl Ibottom, containing four kinds of major equations and theories as special cases. Chapter 7, taking into account the Coriolis force and moving bottoms, bears on a generalIzed mean-flow theory for waves on currents, thus creatmg three new kinds of wave actions Chapter 8, on Hamiltonian stratified wave-current interactions, covers three Ikinds of stratified fluid systems with a number of the canonically conjugate

YaIi.ahJ.es. Chapter 9 ISthe longest and devoted to short -crested waves (SCW s), perhaps the SImplest genumely three-dImensIOnal water waves. Ihe mcompatlbIlIty m1 volving standing waves and SCWs is found and effectively solved. Then, linear, first-, second- and third-order theories for pure gravity and capillary-gravity SCWs with or without ambient currents are respectively established in greatl detail, involving wave forces essential to engineering applications I I'hls book IS mmed at researchers and graduate students speclalIzmg m coastal and ocean engmeenng, phYSIcal oceanography, flUId mechamcs and ap1 Iplied mathematics, and at all those who wish to obtain a proper understanding of coastal dynamic processesJ Naturally, this book wiII not always stay "up-to-date" in terms of this rej search, but It can make a baSIC contnbutlOn to coastal wave mvestIgatlOns. Prac1 lIcally, what the monograph, actmg as a couner, wants to become ISnothmg but

!preface

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my first battle call for the two great things not sought or conjectured by Imj manuel Kant in 1788: the starry heavens above me and the moral law within mEJ

'{lu Huang

Shanghai Apri1200~

Acknowledgements

I am deeply indebted to my late father, Zhizhong Huang, and my 76-year-old mother, Guifang Che, for all their eternal love and expectation, to my three memorable teachers in primary and junior middle schools, Guohua Zhang, XH aoshl Yuan, and Lanfang Sheng, for all their smcenty and youth, and to my wife, Xiuhong Lv, and daughter, Huan Huang, for all their understanding and support. I would like to thank my wife for our 21 years of marriage, I am happy to express my gratitude to my PhD and postdoctoral advisers, Pro1 fessors Xlreng Zhou and DeJun Li at lIanJin Omverslty and Professor Pmgxmg Ding at the East China Normal University, for their invaluable support, to Professor Hong Zhang at Shanghai Library for keeping me abreast with the latest foreign books, to JIanbo LIUat the Higher Education Press for her constructive suggestions and cheerfully puttmg up with my long delayed wnung, to Jle Jm at the Shanghai University Library for her warm-heartedly delivering valuable literature to me at my requests, to my student Zhengzhi Deng for his editing and typmg the bulk of the manuscnpt, and to all my students for their contmued assistance. My sincere appreciation is also extended to librarians, classmates and staffj for their kmd heIpl Ihe monograph has been financed by the NatIOnal Natural SCience Special Foundation of China for the Publishing of Scientific Achievement Monograph (Grant No. 50424913), the FoundatIOn for the Author of National Excellentl Doctoral Dissertation of Chma (Grant No. 200428), the SCientific Research Inj novation Project of the Shanghai Education Committee (Grant No. 08YZ05)j the Open Foundation of the State Key Laboratory of Ocean Engineering at the

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Acknowledgements'

Shanghai Jiao Tong University (Grant No. 0501), and the Shanghai Leading AcademIc DIsclplme Project (Grant No. YOI03).

About the Author

Hu Huang was born on March 2, 1964 in Shihezi, the Xinjiang Uygur AU1 tonomous Region of China, belonging to the second generation of the Xinjiang Production and Construction Corps. Huang

IS

a professor of flUid mechamcs and coastal surface wave dynamiCs

at Shanghai University. He was an undergraduate student in irrigation and water conservancy engineering at the Shihezi Institute of Agriculture (1982-1986), and received his Master of engineering in experimental mechanics in 1993 and hiS PhD m coastal engmeenng m 1998, both at hanJm Omverslty. He was a National Science Foundation Postdoctoral Fellow at the State Key Laboratory of Estuarine and Coastal Research, the East China Normal University (1998-1 2000). He had prevIOusly taught at the XmJlang Shlhezl Secondary SpeCIalized School of Land ReclamatIOn and CultivatIOn(1986-1990), and hanJm Omver1 sity (1993-1998). Huang's research interests focus on mechanics and symmetry, Hamiltonian mechamcs, the dynamiCs of coastal and ocean surface waves, and applied mathematics. Among Huang's honors and recognitions are: the first recipient of the Second pnze m Natural SCience of the Mmlstry of Education of Chma m 2002 (No.1 2002-063), a recipient of the National Excellent Doctoral Dissertation of Chma in 2004 (No. 2004028), and the Special Bonus of the State Council of China in 2004 (No. (02) 9310004).

Contents!

Preliminaries 11.1 WaterWave r heones III Histoncal Perspectlve 1.1.1 The Mild-Slope EquatIOns 1.1.2 I'he Boussillesq- rype Equatlons 11.2 The GovernmgEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 LagrangianFormulation II 4 Hamiltonian Formulation lReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3

Weakly Nonlinear Water Waves Propagating over Uneven IBottoms 12.1 Modified Third-OrderEvolution Equations of Liu andl Dingemans 12.2 Fourth-Order EvolutIOn Equations and Stability AnalYSIS 12.3 Third-Order Evolution Equations for Wave-Current Interactions lReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Resonant InteractIOns Between Weakly Nonlinear Stokes Waves land Ambient Currents and Uneven Bottoms 3 I Introduction 3.2 GovermngEquations and WKBJ PerturbatIOn Expansion . . . .. 3.3 Subharmonic Resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.4 DynamicalSystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. lReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

I 1 2 31 41

51 51

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9 ~

151 26 351

371 371 3~

40 46 51

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4

Contents

The Mild-Slope Equations IntroductIOn........................................... tl.2 Three-DimensIOnal Currents over Mildly Varymg Topography. tl.3 Two-DimensIOnalCurrents over Rapidly Varymg Topography. tl.4 Three-DimensIOnal Currents over Rapidly Varymg Topography f'I.5 Two-Dimensional Currents over Generally Varying Iopography f'I.6 A Hierarchy for Two-Dimensional Currents over Generally Varying Topography lReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ~.1

5

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5] 54 5~

651

70 7~

771

Linear Gravity Waves over Rigid, Porous Bottoms . . . . . . . . . . . ..

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5.1 Introduction........................................... 5.2 A Rapidly Varymg Bottom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5.3 Generally Varying Bottom

79 80 851

IReferences

9]

6

Nonlinear Umfied Equations over an Uneven Bottom. . . . . . . . . .. 951 6 I Introductjon 951 6.2 Nonlinear Unified Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 951 6.3 ExpliCit Special Cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 971 6.3.1 Generalized Nonlinear Shallow-Water Equations of Airy 9l 6.3.2 Generalized Mild-Slope Equation. . . . . . . . . . . . . . . . . .. 9§ 6.3.3 Stokes Wave I heory 9~ 6.3.4 Higher-Order Boussmesq-Type Equations . .. . . . . . . . .. 9~ IReferences 102

7

Generalized Mean-Flow Theory 17.1 IntroductIOn........................................... 17.2 Governing Equations and Boundary Conditions. . . . . . . . . . . . .. 17.3 Averaged Equations of Motion 17.4 Generahzed Wave Action ConservatIOn EquatIOnand Its Wave Aclions IReferences

8

Hamdtoman DeSCrIption of Stratified Wave-Current Interactions 8 I Introductjon 8.2 Two-Layer Wave-Current Interactions 8.3 n-Layer Pure Waves 8.4 n-Layer Wave-Current Interactions over Uneven Bottoms lReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

1O~

10]

104 1051 109 110 I I .~ 11 ]

114 11 ~

122 126

Contents

9

Surface Capillary-Gravity Short-Crested Waves with a Current 1m Water of i'mlte Depth 19 I Introduction 19.2 An Incomplete Match and Its SolutIOn 19.3 Linear Capillary-Gravity Short-Crested Waves 9.3.1 System Formulation 9.3.2 AnalytIcal SolutIOns and Klllematlc and Dynamlcall Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9.3.3 Special Cases 19.4 Second-Order CapIllary-GraVIty Short-Crested Waves 19.5 ThIrd-Order GravIty Short-Crested Waves 9.5.1 The System Equations and the Perturbation Method 9.5.2 Third-Order Solution 9.5.3 SpeCIal Cases 9.5.4 Short-Crested Wave Quantities 9.5.5 Short-Crested Wave Forces on Vertical Walls 19.6 Third-Order Pure Capillary-Gravity Short-Crested Waves 9.6.1 Formulation 9 6 2 SolutIOn 9.6.3 Kinematical and Dynamical Variables IReferences

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1271 128 12~

132 132 134 136 13~

146 146 14