Fluid Dynamics in Astrophysics and Geophysics

Title......Page 1
Lectures in Applied Mathematics - Serie Contents......Page 2
Contents ......Page 6
Preface......Page 7
Historical notes ......Page 8
The nature of the fluid ......Page 9
2. Derivation of equations ......Page 11
Geophysical scaling ......Page 13
3. Fundamental solutions ......Page 15
Steady flow......Page 16
Buoyancy driven flow ......Page 21
Dissipation ......Page 23
Western boundary currents ......Page 26
Slow baroclinic waves ......Page 27
Nonlinearity ......Page 28
Barotropic Rossby waves ......Page 29
Steady planetary flow over a ridge ......Page 32
Steady shear ......Page 35
Shear-oscillatory in time ......Page 37
Shear oscillatory in time and space ......Page 38
Steady strain ......Page 40
Stochastic case ......Page 41
Waves on an adverse current ......Page 44
Expulsion of tracer gradients from a gyre ......Page 45
5. Geostrophic turbulence ......Page 47
Two-dimensional turbulence ......Page 48
Evolution of unforced turbulence ......Page 53
Inertial ranges ......Page 54
References ......Page 61
2. Preliminary considerations ......Page 64
3. Theoretical development ......Page 67
4. The persistence of longwave packets in deep fluids ......Page 73
5. The effect of shear ......Page 75
6. The effect of rotation ......Page 77
7. Two-space-dimensional waves ......Page 78
8. Effect of variable depth ......Page 81
References ......Page 82
2, General formulation and basic assumptions ......Page 84
3. The nonlinear and viscous critical layer regime: R_{cl} = O(1) ......Page 87
3.1. Outer expansion ......Page 88
3.2. Inner expansion ......Page 90
3.3. The amplitude equation ......Page 91
4. The viscous critical layer regime: R_{cl} << 1 ......Page 94
5. The nonlinear critical layer regime: R_{cl} = /infty ......Page 96
6. The nonlinear And "slightly viscous" critical layer regime: R_{cl} = O(/epsilon^{-1}) ......Page 97
7. Long Rossby waves ......Page 98
References ......Page 100
II. Astrophysical fluid dynamics ......Page 102
1. Introduction ......Page 104
2.1. The perturbation equations ......Page 105
2.2. The normal mode problem ......Page 106
2.4. A variational principle for the eigenvalues ......Page 107
2.5, How dynamical instability sets in ......Page 108
2.6. Conserved quantities ......Page 109
2.7. Secular instability: Lagrangian and Eulerian dissipation ......Page 111
2.8 The Mactaurin spheroids ......Page 113
2.10. Exciting the radiation-driven secular instability in magnetic stars ......Page 115
2.11. Exciting the radiation-driven secular instability in a collapsed stellar core by stirring the envelope ......Page 116
2.13. The continuous spectrum ......Page 117
2.14. Numerical studies ......Page 118
2.15. Nonlinear development ......Page 120
3.1. Are static stars spherical? ......Page 124
3.3. Nonaxisymmetric steady states and their bifurcation from axisymmetric models ......Page 125
3.4. Numerical calculations of the structure of axbynmtetric stars ......Page 126
4. A little differential geometry ......Page 127
4.1. Vectors and one-forms (covectors, covariant vectors, dual vectors) ......Page 128
4.2. Lie derivatives ......Page 131
4.3. Differential forms and exterior derivatives ......Page 135
4.4. Conservation of vorticity ......Page 137
5.2. Lagrangian changes in Eulerian quantities ......Page 139
5.4. Second-order perturbations ......Page 140
5.5. The energy to second order ......Page 141
5.6. Trivial perturbations: the relation of the first-order Eulerian and Lagrangian perturbations ......Page 142
5.8. Novel approaches to hydrodynamics ......Page 143
1, Introduction ......Page 146
2. Main events during protostellar collapse ......Page 151
3. Numerical problems in collapse calculations ......Page 155
4. Solutions in one space dimension ......Page 157
5. Solutions in two space dimensions ......Page 165
6. Solutions in 3 space dimensions ......Page 171
7, Summary ......Page 178
References ......Page 180
1. Introduction ......Page 184
2. Fundamental equations ......Page 185
3. Basic states and perturbations ......Page 187
4. Local dispersion relations ......Page 189
5. Sheared waves ......Page 194
6. Eigenmodes of WKBJ type ......Page 198
Appendix. Potential-density relation for short wavelength perturbations ......Page 204
References ......Page 207
III. Mathematical technique ......Page 210
1. Introduction ......Page 212
2. Bifurcation in the presence of symmetry ......Page 213
3. Equivariant bifurcation problems for O(3) ......Page 214
Busse's solutions ......Page 216
The case l = 2: method of Golubitsky and Schaeffer ......Page 217
4. Bifurcation of time periodic solutions ......Page 220
Stability of the bifurcating solutions ......Page 226
References ......Page 228
An introduction to chaotic motion and strange attractors by John Guckenheimer ......Page 230
1. The standard theory Axiom A systems ......Page 231
2. One dimensional mappings ......Page 233
3. Experimental data ......Page 235
References ......Page 237
1. Introduction ......Page 240
2. The model equations ......Page 242
3. Notation and preliminary results ......Page 245
4. Analytic comparison of the two model equations ......Page 250
Scaling and longer term comparisons ......Page 253
5. Further comparisons of the model equations ......Page 260
References ......Page 271