Wolfgang Krauss:

Methoden und Ergebnisse der Theoretischen Ozeanographie
Methods and Results of Theoretical Oceanography

Volume 1: Dynamics of the homogeneous and quasihomogeneous ocean

1973. IX, 302 pages, 89 figures, 17x25cm, 900 g
Language: English

ISBN 978-3-443-01001-0, bound, price: 56.00 €

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Synopsis top ↑

The present volume deals with the derivation of the basic system of equations and boundary conditions, surface waves, inertial and planetary waves, and with ocean currents. Two-layered ocean models are included. We refer to them as quasi homogeneous models; the solutions obtained are basically the same as for a homogeneous ocean. The knowledge of mathematics assumed is that of introductory courses in advanced calculus and differential equations. Some basic relations from vector analysis are reviewed in the first three sections of chapter 1.

Chapter 1 of this volume is devoted to basic hydrodynamics. The boundary conditions are of major importance for oceanography, because so many of the forces to which the ocean responds, act at the sea surface. To give a unified theory of the field equations and the boundary conditions, which replace the field equations at an interface, they are all derived from the same general conservation law. Chapter 2 deals with periodic processes, perturbation theory and linear systems, which are needed for the subsequent chapters. The theory of gravity and capillary waves is treated in chapter 3. The theory of the development of wind-generated ocean waves has been excluded, because this seems to be one of the major stochastic processes in the ocean. It is, therefore, beyond the scope of this volume. Chapter 4 is devoted to inertial and planetary waves, chapter 5 deals with the theory of ocean currents.

Table of Contents top ↑

Preface XI
1. Physical and mathematical foundations of oceanography 1
11.Field functions, equation of state and mass conservation law 2
111.Scalar fields 2
112.Vector fields 5
113.Tensor fields 8
114.Lagrangian and Eulerian description of a field 12
115.General form of the conservation law for field functions 13
116.The equation of state 15
117.The equation of continuity 15
118.The equation of diffusion 16
12.Hydrodynamic equations of motion in an absolute frame of reference 17
121.The general field of forces (stress tensor) 17
122.The general equation of motion (conservation law for momentum) 20
123.Equation of motion for a perfect fluid (Eulerian equation of motion) 22
124.The molecular viscosity coefficient 22
125.Navier-Stokes equation of motion 23
126.The gravitational potential 24
127.The tidal potential 25
13.Hydrodynamic equation of motion in a coordinate system fixed to the earth 32
131.The Coriolis force 32
132.The gravitational force and the natural coordinate system 34
133.The equation of motion in vector notation 36
134.The equation of motion in a rectangular coordinate system (f plane) 36
135.The equation of motion in spherical coordinates 37
136.The equation of motion in the ß plane 38
14.Energy conservation law and equation of heat flux 39
141.The law of energy conservation 39
142.The equation of heat flux 40
143.The entropy equation and the equation of heat conduction 41
144.Bernoulli's equation 42
15.Specifying conditions 43
151.The incompressibility condition 44
152.The incompressible homogeneous and the barotropic ocean 44
153.The incompressible inhomogeneous ocean without mixing 45
16.Boundary conditions 45
161.The general conservation law at a surface of discontinuity (Kotchine's theorem) 45
162.The kinematic boundary condition 47
163.The boundary condition for salinity (partial mass) 49
164.The dynamic boundary condition 49
165.The boundary condition for temperature 53
166.The equilibrium conditions for v, S and T 53
17.Irrotational and rotational motion 53
171.Potential flow 54
172.Two-dimensional flow and stream function 55
173.Rotational flow 57
174.Ertel's vorticity theorem 58
175.Ertel's conservation theorem 60
176.Rossby's potential vorticity theorem 61
177.Helmholtz' vorticity theorem 62
178.Bjerknes' and Kelvin's circulation theorems 64
179.Diffusion of vorticity 66
2. Linearization of the system of hydrodynamic equations and representation of
hydrodynamic processes by means of waves 68
21.Periodic processes 70
211.Harmonic oscillations in real and complex notation 70
212.Superposition of harmonic oscillations 71
213.Phase and group velocity of progressive waves 73
214.Standing waves 75
215.Waves propagating in arbitrary directions 75
216.Damped waves and unstable waves 77
22.Perturbation theory 77
221.The hydrodynamic basic state 79
222.Hydrodynamic perturbation equations 81
223.Perturbation equations for the basic state v(0) = 0 83
224.Perturbation equations for the basic state) v(0) != 0 84
225.Boundary and initial conditions for perturbation equations when
v(0) = 0 85
226.Boundary and initial conditions for perturbation equations when v(0) != 0 87
23.Linear systems 88
231.The FOURIER transformation 88
232.The LAPLACE transformation 91
233. Transfer function, response function and Green's function for linear
systems with constant coefficients 93
3. Gravity waves and capillary waves 95
31.Gravity waves in an ocean of constant depth (f = 0) 97
311.The system of equations 97
312.Free plane sine waves 98
313.Long plane waves 102
314.Short plane waves 104
315.Group velocity, energy and energy transmission for plane progressive waves 105
316.Three-dimensional waves 109
317.Plane waves in a rectangular canal forced by the tidal potential or by atmospheric pressure fluctuations 117
318.Co-oscillating waves in an open canal 121
32.Long waves in an ocean of variable depth (f = 0) 122
321.Linear equations for long waves in an ocean of variable depth 123
322.Edge waves 125
323.Seiches in an arbitrarily formed narrow canal and co-oscillating waves 129
324.Frictional influences upon seiches and co-oscillating tides 134
33.The influence of the earth's rotation on long gravity waves 136
331.The linear system of equations 137
332.Kelvin waves, double Kelvin waves and boundary waves 138
333.Poincaré waves 143
334.Standing edge waves in a canal closed at one end 144
335.Amphidromic systems 145
336.Waves in a circular basin 147
337.Kelvin waves in a rectangular basin 149
338.Waves on a rotating globe 156
34.Forced tidal waves 161
341.The equilibrium tide of a water-covered earth 161
342.LAPLACE's tidal equation 162
343.Some solutions of the tidal equation for a water-covered earth 164
344.Tides on a hemisphere bounded by meridians and canal theories 166
345.Numerical integration of the tidal equations 166
35.Gravity waves of finite amplitude and non-linear effects 169
351.Lagrangian, Eulerian and Stokes velocity 169
352.STOKES waves of permanent form in deep water 170
353.Standing waves of finite amplitude in deep water 175
354.Non-linear interaction among gravity waves 176
355.Cnoidal waves and solitary waves 182
356.Relations between phase velocity, energy and momentum for permanent waves of finite amplitude 189
36.Capillary waves 191
361.Linear theory of capillary waves 191
362.The influence of viscosity upon capillary and gravity waves 196
363.Capillary waves of finite amplitude 201
37.Radiation stress of surface waves and non-linear effects 201
371.Radiation stress due to progressive gravity waves 201
372.Radiation stress due to standing gravity waves 203
373.Momentum balance 205
374.The influence of a wave group on the mean sea level 206
38.Gravity waves at an interface separating water of different densities and velocities 207
381.Gravity waves at the free surface of inhomogeneous, horizontally moving water 208
382.Waves at an interface between flowing water of different densities 210
4.Geostrophic currents, inertial waves and planetary waves 217
41.Equilibrium conditions 217
411.The ocean at rest 217
412.Geostrophic currents 217
413.Magnitude of forces in geostrophic currents 218
42.Inertial waves 220
421.Inertial oscillations of infinite wave length and the circle of inertia 220
422.Inertial waves as an exact solution of the hydrodynamic equation of motion 220
43.Planetary waves 223
431.Approximate equations for waves of the second class 223
432.Non-divergent planetary waves 225
433.Reflection of planetary waves 227
434.Planetary waves in an enclosed rectangular basin 228
5.Theory of ocean currents 231
51.Systems of averaged equations 233
511.Relations and postulates for averaged fields 233
512.Averaged equation of motion and continuity 235
513.Averaged equation ofdiffusion and heat conduction, equation of state .. 238
514.Averaged boundary conditions 239
52.Plane ocean currents in a homogeneous ocean due to wind and air pressure fluctuations 243
521.First-order perturbation equations 243
522.The Ekman spiral 243
523.The bottom layer of frictional influence in geostrophic currents produced
by air pressure fluctuations 246
524.The elementary current system resulting from a geostrophic wind and pressure field 248
525.FREDHOLM's solution of the unsteady EKMAN problem and inertial waves 249
526.Response functions and spectra of wind-driven ocean currents 253
53.Equations for horizontal mass transport due to wind and air pressure fluctuations 256
531.Relations for vertically averaged fields 256
532.Vertically integrated equations of motion and continuity 257
533.Vertically integrated equations of diffusion and heat conduction; the
equation of state 259
534.Zero- and first-order perturbation equations 260
54.The large-scale wind-driven horizontal mass transport 262
541.Differential equation for the stationary transport function on the beta-plane 262
542.The SVERDRUP Regime 263
243.The western boundary currents, linear theories 264
544.Non-linear effects and westward intensification 268
545.Time-dependent solutions on an infinite beta-plane 271
546.Time-dependent solutions and westward intensification 274
547.The influence of bottom topography on ocean currents 275
548.Numerical investigations of the oceanic general circulation 279
549.Numerical investigations of storm surges 280
Some physical properties of the ocean and the air-sea interface 282
Glossary of principal symbols 287
Subject Index 295
Author Index 301