李群分析在地球物理流体动力学中的应用 英文版2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

李群分析在地球物理流体动力学中的应用 英文版PDF格式电子书版下载

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PartⅠ Internal Waves in Stratified Fluid1

1 Introduction3

2 Governing Equations5

2.1 Stratification6

2.2 Linear model for small disturbances7

2.2.1 Linearization of the boundary conditions10

2.2.2 Linear boundary value problem11

2.3 The Boussinesq approximation for nonlinear internal waves in continuously stratified ocean13

2.3.1 Two-dimensional nonlinear Boussinesq equations15

2.3.2 Dispersion relation and anisotropic property of internal waves17

3 Two Model Examples25

3.1 Generation of internal waves25

3.1.1 Harmonic tidal flow over a corrugated slope26

3.1.2 Discussion about the radiation condition29

3.2 Reflection of internal waves from sloping topography33

3.2.1 The problem of internal waves impinging on a sloping bottom34

3.2.2 Direct answer to the question36

3.2.3 Latitude anomaly as an alternative answer39

PartⅡ Introduction to Lie Group Analysis43

4 Calculus of Differential Algebra47

4.1 Definitions47

4.1.1 Main variables47

4.1.2 Total differentiations48

4.1.3 Differential functions48

4.1.4 Euler-Lagrange operator49

4.2 Properties49

4.2.1 Divergence test49

4.2.2 One-dimensional case51

4.3 Exact equations53

4.3.1 Definition53

4.3.2 First-order equations53

4.3.3 Second-order equations54

4.3.4 Linear second-order equations56

4.4 Change of variables in the space？57

4.4.1 One independent variable57

4.4.2 Several independent variables59

5 Transformation Groups61

5.1 Preliminaries61

5.1.1 Examples from elementary mathematics61

5.1.2 Examples from physics64

5.1.3 Examples from fluid mechanics66

5.2 One-parameter groups69

5.2.1 Introduction of transformation groups69

5.2.2 Local one-parameter groups71

5.2.3 Local groups in canonical parameter74

5.3 Infinitesimal description of one-parameter groups75

5.3.1 Infinitesimal transformation75

5.3.2 Lie equations76

5.3.3 Exponential map79

5.4 Invariants and invariant equations82

5.4.1 Invariants82

5.4.2 Invariant equations83

5.4.3 Canonical variables85

5.4.4 Construction of groups using canonical variables89

5.4.5 Frequently used groups in the plane90

6 Symmetry of Differential Equations91

6.1 Notation91

6.1.1 Differential equations91

6.1.2 Transformation groups92

6.2 Prolongation of group generators92

6.2.1 Prolongation with one independent variable92

6.2.2 Several independent variables94

6.3 Definition of symmetry groups95

6.3.1 Definition and determining equations95

6.3.2 Construction of equations with given symmetry96

6.3.3 Calculation of infinitesimal symmetry98

6.4 Lie algebra99

6.4.1 Definition of Lie algebra99

6.4.2 Examples of Lie algebra100

6.4.3 Invariants of multi-parameter groups103

6.4.4 Lie algebra L2 in the plane:Canonical variables106

6.4.5 Calculation of invariants in canonical variables107

7 Applications of Symmetry111

7.1 Ordinary differential equations111

7.1.1 Integration of first-order equations111

7.1.2 Integration of second-order equations114

7.2 Partial differential equations116

7.2.1 Symmetry of the Burgers equation116

7.2.2 Invariant solutions117

7.2.3 Group transformations of solutions120

7.3 From symmetry to conservation laws121

7.3.1 Introduction121

7.3.2 Noether's theorem123

7.3.3 Theorem of nonlocal conservation laws125

PartⅢ Group Analysis of Internal Waves131

8 Generalities135

8.1 Introduction135

8.1.1 Basic equations135

8.1.2 Adioint system136

8.1.3 Formal Lagrangian136

8.2 Self-adjointness of basic equations137

8.2.1 Adjoint system to basic equations138

8.2.2 Self-adjointness139

8.3 Symmetry139

8.3.1 Obvious symmetry139

8.3.2 General admitted Lie algebra141

8.3.3 Admitted Lie algebra in the case f=0142

9 Conservation Laws143

9.1 Introduction143

9.1.1 General discussion of conservation equations143

9.1.2 Variational derivatives of expressions with Jacobians145

9.1.3 Nonlocal conserved vectors146

9.1.4 Computation of nonlocal conserved vectors147

9.1.5 Local conserved vectors149

9.2 Utilization of obvious symmetry150

9.2.1 Translation ofν150

9.2.2 Translation ofρ151

9.2.3 Translation ofψ151

9.2.4 Derivation of the flux of conserved vectors with known densities152

9.2.5 Translation of x153

9.2.6 Time translation153

9.2.7 Conservation of energy154

9.3 Use of semi-dilation156

9.3.1 Computation of the conserved density157

9.3.2 Conserved vector158

9.4 Conservation law due to rotation159

9.5 Summary of conservation laws159

9.5.1 Conservation laws in integral form160

9.5.2 Conservation laws in differential form160

10 Group Invariant Solutions163

10.1 Use of translations and dilation163

10.1.1 Construction of the invariant solution163

10.1.2 Generalized invariant solution and wave beams166

10.1.3 Energy of the generalized invariant solution167

10.1.4 Conserved density P of the generalized invariant solution168

10.2 Use of rotation and dilation172

10.2.1 The invariants172

10.2.2 Candidates for the invariant solution173

10.2.3 Construction of the invariant solution174

10.2.4 Qualitative analysis of the invariant solution175

10.2.5 Energy of the rotationally symmetric solution176

10.2.6 Comparison with linear theory177

10.3 Concluding remarks180

A Resonant Triad Model183

A.1 Weakly nonlinear model184

A.2 Two questions188

A.3 Solutions to the resonance conditions189

A.4 Resonant triad model192

A.4.1 Utilization of the GM spectrum196

A.4.2 Model example:Energy conservation for two resonant triads198

A.4.3 Model example:Resonant interactions between 20000 internal waves202

A.5 Stability of the GM spectrum and open question on dissipation modelling205

References209

Index213