Elements of Physical Oceanography is a derivative of the Encyclopedia of Ocean Sciences, 2nd Edition and serves as an important reference on current physical oceanography knowledge and expertise in one convenient and accessible source. Its selection of articles-all written by experts in their field-focuses on ocean physics, air-sea transfers, waves, mixing, ice, and the processes of transfer of properties such as heat, salinity, momentum and dissolved gases, within and into the ocean. Elements of Physical Oceanography serves as an ideal reference for topical research.
- References related articles in physical oceanography to facilitate further research
- Richly illustrated with figures and tables that aid in understanding key concepts
- Includes an introductory overview and then explores each topic in detail, making it useful to experts and graduate-level researchers
- Topical arrangement makes it the perfect desk reference
Elements of Physical Oceanography is a derivative of the Encyclopedia of Ocean Sciences, Second Edition and serves as an important reference on current physical oceanography knowledge and expertise in one convenient and accessible source. Its selection of articles-all written by experts in their field-focuses on ocean physics, air-sea transfers, waves, mixing, ice, and the processes of transfer of properties such as heat, salinity, momentum and dissolved gases, within and into the ocean. Elements of Physical Oceanography serves as an ideal reference for topical research. References related articles in physical oceanography to facilitate further research Richly illustrated with figures and tables that aid in understanding key concepts Includes an introductory overview and then explores each topic in detail, making it useful to experts and graduate-level researchers Topical arrangement makes it the perfect desk reference
Front cover
1
Encylopedia of Ocean Sciences: Elements of Physical Oceanography
4
Copyright page
5
Contents
6
Elements of Physical Oceanography: Introduction 10
Editorial Advisory Board Members who helped in the production of this volume 11
References
11
Surface Waves, Tides, and Sea level
12
Surface Gravity and Capillary Waves 14
Introduction 14
Basic Formulations 14
Linear Waves 15
The Group Velocity 17
Second Order Quantities 17
Waves on Currents: Action Conservation 18
Nonlinear Effects 19
Resonant Interactions 19
Parasitic Capillary Waves 20
Wave Breaking 21
Further Reading 21
Wave Generation by Wind 23
Introduction 23
Theories of Wave Growth 23
Experiments and Observations 25
Numerical Modeling of the Wind Input 27
Conclusions 27
Further Reading 27
Relevant Website 28
Rogue Waves 29
Introduction 29
Surface Gravity Waves 30
Physical Mechanisms 31
Statistics of Large Waves 32
Experiments and Observations 35
Numerical Simulations 37
Conclusions 37
Further Reading 38
Waves on Beaches 40
Introduction 40
The Dynamics of Incident Waves 41
Radiation Stress: the Forcing of Mean Flows and Set-up 42
Nonlinear Incident Waves 43
Vertically Dependent Processes 43
2HD Flows - Circulation 43
Infragravity Waves and Edge Waves 44
Shear Waves 45
Conclusions 46
Wave Energy 48
Introduction 48
Wave Power: Resource and Exploitation 49
Economics of Wave Power Conversion 50
Concluding remarks 51
Further Reading 51
Whitecaps and Foam
52
Introduction 52
Spilling Wave Crests: Stage A Whitecaps 52
Decaying Foam Patches: Stage B Whitecaps 53
Wind-Dependence of Oceanic Whitecap Coverage 55
Stabilized Sea Foam 55
Global Implications 57
Further Reading 57
Breaking Waves and Near-Surface Turbulence 58
Introduction 58
Breaking Waves 58
Turbulence beneath Breaking Waves 60
Conclusion 63
Further Reading 64
Seiches 66
Introduction 66
History 66
Dynamics 67
Generating Mechanisms and Observations 69
Further Reading 72
Tsunami 73
Introduction 73
Historical and Recent Tsunamis 74
Tsunami Generation Mechanisms 75
Modeling of Tsunami Generation, Propagation, and Coastal Inundation 79
Tsunami Hazard Mitigation 84
Acknowledgment 85
Further Reading 85
Storm Surges 87
Introduction and Definitions 87
Storm Surge Equations 87
Generation and Dynamics of Storm Surges 88
Areas Affected by Storm Surges 89
Storm Surge Prediction 92
Interactions with Wind Waves 94
Data Assimilation 95
Related Issues 96
Further Reading 96
Coastal Trapped Waves 98
Introduction 98
Formulation 98
Straight Unstratified Shelf 98
Other Geometry 99
Stratification 100
Friction 101
Mean Flows 102
Non-linear Effects 102
Alongshore Variations 103
Generation and Role of Coastal-trapped Waves 103
Summary 104
Further Reading 105
Tides 106
Introduction 106
Tidal Patterns 106
Gravitational Potential 106
The Equilibrium Tide 108
Tidal Analysis 108
Tidal Dynamics 109
Ocean Tides 111
Energy Fluxes and Budgets 112
Further Reading 113
Tidal Energy 114
Introduction 114
Energy of Tides 114
Extracting Tidal Energy: Traditional Approach 115
Extracting Tidal Energy: Non-traditional Approach 117
Utilizing Electric Energy from Tidal Power Plants 118
Conclusion 118
Further Reading 119
Sea Level Change
120
Introduction 120
Sea-Level Changes Since the Last Glacial Maximum 120
Observed Recent Sea-Level Change 120
Processes Determining Present Rates of Sea-Level Change 122
Projected Sea-Level Changes for the Twenty-first Century 124
Regional Sea-Level Change 124
Longer-term Changes 124
Summary 125
Further Reading 125
Sea Level Variations Over Geological Time 126
Introduction 126
Sea Level Change due to Volume of Water in the Ocean Basin 126
Sea Level Change due to Changing Volume of the Ocean Basin 128
Sea Level Change Estimated from Observations on the Continents 130
Summary 134
Further Reading 134
The Air-Sea Interface
136
Heat and Momentum Fluxes at the Sea Surface 138
Introduction 138
Measuring the Fluxes 138
Sources of Flux Data 140
Regional and Seasonal Variation of the Momentum Flux 141
Regional and Seasonal Variation of the Heat Fluxes 143
Accuracy of Flux Estimates 143
Further Reading 145
Sea Surface Exchanges of Momentum, Heat, and Fresh Water Determined by Satellite 146
Introduction 146
Flux Estimation Using Satellite Observations 146
Summary and Applications 152
Further Reading 154
Relevant Websites 155
Evaporation and Humidity 156
Introduction 156
History/Definitions and Nomenclature 156
Clausius-Clapeyron Equation 159
Tropical Conditions of Humidity 159
Latitudinal and Regional Variations 160
Vertical Structure of Humidity 160
Sublimation-Deposition 160
Sources of Data 160
Estimation of Evaporation by Satellite Data 161
Future Directions and Conclusions 161
Further Reading 162
Freshwater Transport And Climate 163
Introduction 163
Methods of Fresh Water Flux and Transport Estimation 165
Basin Balances 166
Interbasin Exchange 167
Global Budgets 169
Future Directions 169
Further Reading 170
Air-Sea Gas Exchange 171
Introduction 171
Theory 171
Experimental Techniques and Results 177
Outlook 180
Further Reading 180
Relevant Websites 180
Air-Sea Transfer: Dimethyl Sulfide, COS, CS2, NH4, Non-methane Hydrocarbons, Organo-halogens 181
Dimethylsulfide 182
Carbonyl Sulfide 183
Carbon Disulfide 183
Nonmethane Hydrocarbons 184
Ammonia 184
Organohalogens 185
Conclusions 186
Further Reading 186
Air-Sea Transfer: N2O, NO, CH4, CO 187
Introduction 187
Nitrous Oxide (N2O) 187
Nitric Oxide (NO) 189
Methane (CH4) 190
Carbon Monoxide (CO) 191
Air-Sea Exchange of Trace Gases 193
Further Reading 194
Gas Exchange in Estuaries 195
Introduction 195
Gas Solubility 195
Gas Exchange (Flux) Across the Air/Water Interface 195
Models of Gas Exchange 195
Direct Gas Exchange Measurements 196
Individual Gases 197
Conclusions 200
Further Reading 201
Penetrating Shortwave Radiation 203
Introduction 203
Albedo 203
Spectrum of Downward Irradiance 204
Modeled Irradiance 205
Parameterized Irradiance versus Depth 206
Further Reading 207
Radiative Transfer in the Ocean 209
Introduction 209
Terminology 209
Radiometric Quantities 209
Inherent Optical Properties 211
The Radiative Transfer Equation 213
Apparent Optical Properties 213
Optical Constituents of Seawater 214
Examples of Underwater Light Fields 215
Further Reading 218
Atmospheric Transport and Deposition of Particulate Material to the Oceans 219
Introduction 219
Aerosol Sources, Composition, and Concentrations 219
Aerosol Removal Mechanisms 221
Deposition of Aerosols to the Oceans 223
Conclusions 228
Further Reading 228
Surface Films 229
Introduction 229
Orgin of Surface Films 229
Modifications of Air-Sea Interaction by Surface Films 230
Further Reading 231
Bubbles 232
Introduction 232
Sources of Bubbles 232
Dispersion and Development 233
Surfacing and Bursting 235
Acoustical and Optical Properties 236
Summary of Bubble Distribution 236
Further Reading 237
Boundary Layers: The Upper Ocean Boundary Layer
238
Upper Ocean Vertical Structure 240
Introduction 240
Major Features of the Upper Ocean Vertical Structure 240
Definitions 242
Variability in Upper Ocean Vertical Structure 244
Other Properties That Define the Upper Ocean Vertical Structure 247
Conclusions 247
Further Reading 247
Wind- and Buoyancy-Forced Upper Ocean 248
Introduction 248
Air-Sea Interaction 248
The Seasonal Cycle 253
Conclusion 255
Further Reading 256
Upper Ocean Space And Time Variability 257
Introduction 257
Turbulence and Mixing 257
Langmuir Circulation and Convection 257
Internal Waves 258
Fronts and Eddies 259
Wind-Forced Currents 260
Seasonal Cycles 261
Climatic Signals 261
Conclusion 262
Further Reading 262
Upper Ocean Mean Horizontal Structure 263
Introduction 263
Horizontal Property Fields 263
The Mixed Layer and Seasonal Thermocline 266
The Barrier Layer 268
The Subtropical Gyres and the Permanent Thermocline 269
The Equatorial Region 270
The Polar Regions 271
Further Reading 272
Upper Ocean Structure: Responses to Strong Atmospheric Forcing Events
273
Introduction 273
Atmospheric Forcing 274
Air-Sea Parameters 275
Gulf of Mexico Basin 278
Oceanic Response 280
Summary 287
Acknowledgments 290
Further Reading 290
Relevant Website 291
Upper Ocean Mixing Processes 292
Introduction 292
Convection 294
Wind Forcing 295
Effects of Precipitation 297
Ice on the Upper Ocean 297
Parameterizations of Upper Ocean Mixing 298
Further Reading 298
Langmuir Circulation and Instability 299
Introduction 299
Description of Langmuir Circulation 299
Theory 301
Field Observations 305
Further Reading 307
Upper Ocean Heat and Freshwater Budgets 308
Introduction 308
Governing Processes 308
Measurements 310
Distributions 311
Severe Storms 316
Reactions to Climate Change 317
Future Developments 318
Further Reading 318
Relevant Websites 318
Boundary Layers: The Benthic Boundary Layer
320
Turbulence In The Benthic Boundary Layer 322
Introduction 322
The Ekman Layer 322
Viscous Sublayer 324
The Wall Layer 324
Observations 325
Discussion 327
Further Reading 327
Benthic Boundary Layer Effects 328
Introduction 328
Organisms of the Benthic Boundary Layer 328
BBL Flow Adaptations 330
Life History Adaptations 330
Suspension-feeding Adaptations 331
Adaptations to Resist Shear Stress 332
Aggregation as an Adaptation 333
Conclusions 334
Further Reading 335
Boundary Laers: Under-Ice Boundary Layer
336
Under-ice Boundary Layer 338
Introduction 338
History and Basic Concepts 338
Turbulence in the Under-ice Boundary Layer 340
Outstanding Problems 344
Further Reading 345
Ice-ocean Interaction 346
Introduction 346
Drag and Characteristic Regions of the Under-ice Boundary Layer 346
Heat and Mass Balance at the Ice-Ocean Interface: Wintertime Convection 349
Effects of Horizontal Inhomogeneity: Wintertime Buoyancy Flux 351
Effects of Horizontal Inhomogeneity: Summertime Buoyancy Flux 353
Internal Waves and Their Interaction with the Ice Cover 355
Outstanding Issues 356
Further Reading 356
Internal Waves
358
Internal Waves 360
Introduction 360
Interfacial Waves 360
Internal Waves 362
Other Aspects 366
Conclusions 366
Further Reading 367
Internal Tides 368
Introduction 368
Modes and Beams 368
Observations 371
Implications for Energetics and Mixing 374
Further Reading 375
Processes of Diapycnal Mixing
376
Three-dimensional (3d) Turbulence 378
Introduction 378
The Mechanics of Turbulence 378
Stationary, Homogeneous, Isotropic Turbulence 379
Turbulence in Geophysical Flows 382
Length Scales of Ocean Turbulence 383
Further Reading 385
Laboratory Studies of Turbulent Mixing 386
Introduction 386
Experiments 386
Continuous Stratification 389
Summary 391
Further Reading 391
Internal Tidal Mixing 392
Introduction 392
Stirring and Mixing 392
The Battle for Spatial Resolution 392
Maintaining the Stratification 393
Tidal Dissipation: The Astronomic Evidence 393
Boundary Layer Dissipation Versus Scatter 393
Satellite Altimetry to The Rescue 394
Discussion 395
Further Reading 395
Estimates of Mixing 396
Introduction 396
Approaches to Quantifying Mixing 397
Large-Scale Estimates 398
Fine- and Microscale Estimates 399
Summary 405
Further Reading 406
Energetics of Ocean Mixing 407
Introduction 407
The Global Ocean’s Energy Budget 408
The Traditional Paradigm of Ocean Mixing: The Abyssal Ocean 409
An Alternative Paradigm of Ocean Mixing: The Permanent Pycnocline 413
Conclusion 415
Further Reading 416
Fossil Turbulence 417
Introduction 417
History of Fossil Turbulence 419
Intermittency of Oceanic Turbulence and Mixing 421
Turbulence and Fossil Turbulence Definitions 422
Formation and Detection of Stratified Fossil Turbulence 423
Quantitative Methods 423
Further Reading 424
Open Ocean Convection 425
Introduction 425
Phenomenology 425
Penetrative Convection 427
Relative Contributions of Convection and Shear Stress to Turbulence 427
Convection and Molecular Sublayers 429
Diurnal and Seasonal Cycles of Convection 430
Conclusions 431
Further Reading 432
Deep Convection 433
Introduction 433
Plumes - the Mixing Agent 435
Temperature and Salinity Variability 436
Restratification 437
Discussion 439
Further Reading 441
Double-Diffusive Convection 442
Introduction 442
Salt Fingers 442
Diffusive Convection 446
Intrusions 448
Global Importance 448
Further Reading 450
Differential Diffusion 451
Introduction 451
What Is Differential Diffusion? 451
Laboratory Evidence for Differential Diffusion 452
Numerical Simulation of Differential Diffusion 452
Oceanic Values of Diffusivity Ratio 454
Other Observational Evidence for Differential Diffusion? 455
Does Differential Diffusion Matter? 456
Further Reading 458
Dispersion and Diffusion in the Deep Ocean 459
Introduction 459
The Thermohaline Circulation 459
Deep-sea Observations of Mixing 459
Summary 465
Further Reading 466
Horizontal Dispersion, transport, and Ocean Properties
468
Vortical Modes 470
Introduction 470
Potential Vorticity 470
Basin Scales 471
Mesoscale 472
Fine-scale 472
Generation Mechanisms 472
Observational Challenge 473
Conclusions 474
Further Reading 474
Intrusions 475
Introduction 475
Observational Studies 477
Theoretical Studies 478
Summary 479
Further Reading 479
Dispersion in Shallow Seas 480
Introduction 480
Fundamentals - The Fluid Mechanics of Dispersion 480
Dispersion Phenomena 483
Further Reading 485
Dispersion From Hydrothermal Vents 486
Introduction 486
The Rising Plume 486
Mesoscale Flow and Vortices 488
Large-scale flow 490
Discussion 492
Further Reading 494
Nepheloid Layers 495
Introduction 495
Optics of Nephelometers: What They ’See’ 495
Nepheloid Layer Features 498
Separated Mixed-Layer Model 499
Decay of Concentration: Aging of Particulate Populations 499
Chemical Scavenging by Particles in Nepheloid Layers 500
The Turbidity Minimum 502
Concentration and Spreading in the Atlantic and Indian Oceans 503
Boundary Mixing, INLs, and Inversions 503
Trenches and Channels 504
Further Reading 504
Heat Transport and Climate 506
Introduction: The Global Heat Budget 506
Air-Sea Heat Exchange 508
Distribution of Ocean Heat Transport 508
Eddy Heat Transport 510
Future Developments 511
Further Reading 512
El Nintildeo Southern Oscillation (enso) 513
Introduction 513
The Tropical Pacific Ocean-Atmosphere System 515
Mechanisms of ENSO 520
Interannual Variations in Climate 520
Impacts 523
ENSO and Seasonal Predictions 524
Further Reading 525
North Atlantic Oscillation (nao) 526
Introduction 526
What is the NAO? 526
Impacts of the NAO 528
What are the Mechanisms that Govern NAO Variability? 532
Further Reading 533
Water Types And Water Masses 534
Introduction 534
What is a Water Mass? 534
Descriptive Tools: The TS Curve 534
Global Water Mass Distribution 536
Summary TS Relationships 540
Discussion and Conclusion 541
Further Reading 542
Neutral Surfaces and the Equation of State 543
Introduction 543
Requirements for a Neutral Surface to Exist 544
The Helical Nature of Neutral Trajectories 544
Neutral Density Surfaces Compared with Potential Density Surfaces 545
Equation of State 547
Summary 548
Further Reading 549
Relevant Website 549
Ice
550
Sea Ice: Overview 552
Introduction 552
Extent 552
Geophysical Importance 552
Properties 554
Drift and Deformation 555
Trends 557
Further Reading 559
Sea Ice Dynamics 561
Introduction 561
Drift Ice Medium 561
Equation of Motion 566
Numerical Modeling 569
Concluding Words 569
Further Reading 570
Relevant Websites 571
Sea Ice 572
Introduction 572
Sea Ice Extent 572
Sea Ice Thickness 582
Further Reading 588
Polynyas 590
Introduction 590
Physical Processes within the Two Polynya Types 591
Remote Sensing Observations 593
Physical Importance 593
Biological Importance 594
Conclusions 595
Acknowledgments 595
Further Reading 595
Processes in Coastal and Shelf Seas
596
Beaches, Physical Processes Affecting 598
Introduction 598
Beaches 598
Wave-dominated Beaches 598
Tide-modified Beaches 604
Tide-dominated Beach 606
Beach Modification 606
Further Reading 608
Shelf Sea and Slope Sea Fronts
609
Introduction
609
Freshwater Fronts in Shelf Seas
609
Tidal Mixing Fronts in Shelf Seas
610
Shelf Slope Fronts
615
Summary
617
Further Reading
617
Relevant Websites
618
Appendices
620
Appendix 1. SI Units and Some Equivalences 622
Appendix 6. The Beaufort Wind Scale anD Seastate
625
Index
628
Surface Gravity and Capillary Waves
W.K. Melville Scripps Institution of Oceanography, University of California, San Diego, La Jolla, USA
Introduction
Ocean surface waves are the most common oceanographic phenomena that are known to the casual observer. They can at once be the source of inspiration and primal fear. It is remarkable that the complex, random wave field of a storm-lashed sea can be studied and modeled using well-developed theoretical concepts. Many of these concepts are based on linear or weakly nonlinear approximations to the full nonlinear dynamics of ocean waves. Early contributors to these theories included such luminaries as Cauchy, Poisson, Stokes, Lagrange, Airy, Kelvin and Rayleigh. Many of the current challenges in the study of ocean surface waves are related to nonlinear processes which are not yet well understood. These include dynamical coupling between the atmosphere and the ocean, wave–wave interactions, and wave breaking.
For the purposes of this article, surface waves are considered to extend from low frequency swell from distant storms at periods of 10 s or more and wavelengths of hundreds of meters, to capillary waves with wavelengths of millimeters and frequencies of O(10) Hz. In between are wind waves with lengths of O(1–100) m and periods of O(1–10) s. Figure 1 shows a spectrum of surface waves measured from the Research Platform FLIP off the coast of Oregon. The spectrum, Φ, shows the distribution of energy in the wave field as a function of frequency. The wind wave peak at approximately 0.13 Hz is well separated from the swell peak at approximately 0.06 Hz.
Ocean surface waves play an important role in air–sea interaction. Momentum from the wind goes into both surface waves and currents. Ultimately the waves are dissipated either by viscosity or breaking, giving up their momentum to currents. Surface waves affect upper-ocean mixing through both wave breaking and their role in the generation of Langmuir circulations. This breaking and mixing influences the temperature of the ocean surface and thus the thermodynamics of air–sea interaction. Surface waves impose significant structural loads on ships and other structures. Remote sensing of the ocean surface, from local to global scales, depends on the surface wave field.
Basic Formulations
The dynamics and kinematics of surface waves are described by solutions of the Navier-Stokes equations for an incompressible viscous fluid, with appropriate boundary and initial conditions. Surface waves of the scale described here are usually generated by the wind, so the complete problem would include the dynamics of both the water and the air above. However, the density of the air is approximately 800 times smaller than that of the water, so many aspects of surface wave kinematics and dynamics may be considered without invoking dynamical coupling with the air above.
The influence of viscosity is represented by the Reynolds number of the flow, Re = UL/μ, where U is a characteristic velocity, L a characteristic length scale, and v = μ/P is the kinematic viscosity, where μ is the viscosity and ρ the density of the fluid. The Reynolds number is the ratio of inertial forces to viscous forces in the fluid and if Re>>1, the effects of viscosity are often confined to thin boundary layers, with the interior of the fluid remaining essentially inviscid (v = 0). (This assumes a homogeneous fluid. In contrast, internal waves in a continuously stratified fluid are rotational since they introduce baroclinic generation of vorticity in the interior of the fluid). Denoting the fluid velocity by u = (u, v, w), the vorticity of the flow is given by =∇×u If =0, the flow is said to be irrotational. From Kelvin’s circulation theorem, the irrotational flow of an incompressible (∇.u = 0) inviscid fluid will remain irrotational as the flow evolves. The essential features of surface waves may be considered in the context of incompressible irrotational flows.
For an irrotational flow, u = ∇ϕ where the scalar ϕ is a velocity potential. Then, by virtue of incompressibility, ϕ satisfies Laplace’s equation
2ϕ=0
[1]
We denote the surface by z =(x, y,t), where (x, y) are the horizontal coordinates and t is time. The kinematic condition at the impermeable bottom at z = −h, is one of no flow through the boundary:
ϕ∂z=0atz=−h
[2]
There are two boundary conditions at z = η:
η∂t+u∂η∂x+υ∂η∂y=w
[3]
ϕ∂t+12u2+gη=(pa−p)/ρ
[4]
The first is a kinematic condition which is equivalent to imposing the condition that elements of fluid at the surface remain at the surface. The second is a dynamical condition, a Bernoulli equation, Which is equivalent to stating that the pressure p_ at z = η_, an infinitesimal distance beneath the surface, is just a constant atmospheric pressure, pa, plus a contribution from surface tension. The effect of gravity is to impose a restoring force tending to bring the surface back to z = 0. The effect of surface tension is to reduce the curvature of the surface.
Although this formulation of surface waves is considerably simplified already, there are profound difficulties in predicting the evolution of surface waves based on these equations. Although Laplace’s equation is linear, the surface boundary conditions are nonlinear and apply on a surface whose specification is a part of the solution. Our ability to accurately predict the evolution of nonlinear waves is limited and largely dependent on numerical techniques. The usual approach is to linearize the boundary conditions about z= 0.
Linear Waves
Simple harmonic surface waves are characterized by an amplitude a, half the distance between the crests and the troughs, and a wavenumber vector k with |k|= k=2π/λ, where λ is the wavelength. The surface displacement, (unless otherwise stated, the real part of complex expressions is taken)
=aei(k.x−σt)
[5]
where σ = 2π/T is the radian frequency and T is the wave period. Then ak is a measure of the slope of the waves, and if ak <<1, the surface boundary conditions can be linearized about z = 0.
Following linearization, the boundary conditions become
η∂t=w
[6]
ϕ∂t+gη=Γρ(∂2η∂x2+∂2η∂y2)atz=0
[7]
where the linearized Laplace pressure is
a−p_=Γ(∂2η∂x2+∂2η∂y2)
[8]
where Γ is the surface tension coefficient.
Substituting for η and satisfying Laplace’s equation and the boundary conditions at z= 0 and -h gives
=ig′acoshk(z+h)σcoshkh
[9]
Where
2=g′ktanhkh
[10]
and
′=g(1+Γk2/ρ)
[11]
Equations relating the frequency and wavenumber, =σ(k),are known as dispersion relations, and for linear waves provide a fundamental description of the wave kinematics. The phase speed,
=σ/k=(g′ktanhkh)1/2
[12]
is the speed at which lines of constant phase (e.g., wave crests) move.
For waves propagating in the x-direction, the velocity field...
| Erscheint lt. Verlag | 26.8.2009 |
|---|---|
| Mitarbeit |
Chef-Herausgeber: John H. Steele, Steve A. Thorpe, Karl K. Turekian |
| Sprache | englisch |
| Themenwelt | Sachbuch/Ratgeber |
| Naturwissenschaften ► Geowissenschaften ► Hydrologie / Ozeanografie | |
| Technik | |
| ISBN-10 | 0-12-375721-5 / 0123757215 |
| ISBN-13 | 978-0-12-375721-0 / 9780123757210 |
| Haben Sie eine Frage zum Produkt? |
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