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| Shirer H.N. — Nonlinear Hydrodynamic Modeling: A Mathematical Introduction |
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| Предметный указатель |
Continuation, Alexander-Yorke (see Alexander-Yorke Continuation)
Continuation, equation 239
Continuity equation 25
Control parameter and bifurcation 72 131 140 144
Control parameter and routes to chaos 401 403
Control parameter and unfoldings 165 166 177 178 187 194 196
Control parameter, cubic equations for critical value of 322 324
Control parameter, minimizing when two 322 324
Control parameter, minimum values of (convection with wind) 292 295 296 304 308 314 324
Control parameter, minimum values of (convection) 111 131 138
Control parameter, minimum values of (rotating convection) 143 163
Control parameter, number of crucial 166 174 182
Convection with background wind 278 282 298 300 308 309
Convection, boundary conditions for 30 31 281
Convection, cell broadening in 197 198 211 223
Convection, circulation patterns for 24 43 45 46 201 293
Convection, deduced properties of 31 37
Convection, depth of circulation in 23 281 293
Convection, essential features of 23 37
Convection, forced by horizontal and vertical heating 122 124 190 193 195
Convection, necessary conditions for 109 111 133
Convection, observed characteristics of 23 197 292 293 386 390
Convection, onset of 111
Convection, possible bifurcation diagrams for 34 156 159 199 206 211 223 410
Convection, preferred wavelengths of 131 140 144 154 222 223 292—293
Convection, rotating (see Rotating convection)
Convective roll 278 290 292 308 314 324
Convective solution in Chang and Shirer model 208 224
Convective solution in Lorenz model 42 45 77 79 92 342
Convective solution in Veronis model 153 160
Convective solution, existence of 109 111
Convective solution, expected 122 (see also Control parameter minimum
Convective turbulence as a three-dimensional problem 411
Coordinate map 473
Coordinates based on stable and unstable manifolds, simple example of 479 484
Coordinates based on unstable manifolds, constructing 494 495
Coordinates based on unstable manifolds, expressing spectral model in 485 496
Corank of matrix 170
Coriolis parameter (see Dimensionless Coriolis parameter)
Coriolis terms and routes to chaos 424 426
Coriolis terms, calculating from observations 422
Coriolis terms, Correlation dimension 420 422 443
Coriolis terms, definition of 423
Coriolis terms, effects on roll energetics 296 300 303
Correlation integral 422 423
Correlation tensor 498 499
Couette flow, transition to turbulence in 386 387
Critical parameter values and bifurcation 70
Critical parameter values and unfoldings 166 178 182
Critical Rayleigh number and Coriolis parameter 143 154 292
Critical Rayleigh number in rotating fluid 143
Critical Rayleigh number in shearing environment and Coriolis parameter 308
Critical Rayleigh number in shearing environment, definition of 287 294 305 308
Critical Rayleigh number in shearing environment, minimum value of 292 295 305 306 322 324
Critical Rayleigh number, definition of 36 112 133 199
Critical Rayleigh number, dependence on domain aspect ratio of 36 119 120 131 138 143 199—206
Critical Rayleigh number, effects of cell apsect ratio on 37 133 138 143 157 292 295 305 308 314 324
Critical Rayleigh number, minimum value of 119 133 140 143 144 150 157
Critical Rayleigh number, normalized 36
Critical Rayleigh number, physical interpretation of 110 111
Critical Reynolds number, definition of 306 307 312
Critical Reynolds number, effects of cell aspect ratio on 312 324
Critical Reynolds number, effects of Coriolis parameter on 306 307
Crucial parameter 129 130 164 196 411
Crucial parameter, identifying 174 182
Cubic characteristic equations, some properties of 78 161 163
Cubic polynomial equation for amplitude of periodic solutions 271 275 290
Cubic polynomial equation for stationary solutions 112 116 119 130 183 194 195
Curry model 408
Cusp 125 130
Cusp point 125 130 178 195 196
Cusp surface 125 130 183 188 190 258
Decision points and predictability 82 84 382 407 433
Degenerate system 234
Degree of a, monomial 233
Degree of a, scalar polynomial function 233
Degree of a, stationary solution 100
Denjoy Theorem 396 397
Determinant and bifurcation 89 92 210
Determinant and characteristic equation 41 74 92 143 285 286
Determinant and characteristic multipliers 363
Determinant and existence of nontrivial solutions 36
Diabatic heating rate 185 188 189 195 449
Differential matrix 72 89 168 178 269 466 470 473
Diffusive time scale 28
Dilatation factor 365
Dimension of attractor 412 426 457
Dimension of chaotic attractor 421 424
Dimension of phase space 413
Dimension of restricted attractor 438 439
Dimension of simple attractor 413 421 423 428
Dimension of strange attractor 413 414 418 425 428
Dimension, correlation (see Correlation dimension)
Dimension, fractal 413 414 420 422 436
Dimension, geometric 412 414 421 438
Dimension, information 414 421 422
Dimension, Lyapunov (see Lyapunov dimension)
Dimensional forms of equations for convection 25 26
Dimensional forms of equations for convection with wind 280
Dimensional forms of equations for rotating convection with wind 298
Dimensionless Coriolis parameter and critical Rayleigh number 143 154 292 308
Dimensionless Coriolis parameter and critical Reynolds number 306 307
Dimensionless Coriolis parameter, definition of 141 299
Dimensionless forms of equations for convection 27 31 61 107 185 190 192
Dimensionless forms of equations for convection with wind 280
Dimensionless forms of equations for difference fields 453
Dimensionless forms of equations for global atmosphere 448
Dimensionless forms of equations for Navier — Stokes equation 450
Dimensionless forms of equations for rotating convection 141
Dimensionless forms of equations for rotating convection with wind 298
Dimensions and system dynamics 414
Dimensions and transition to turbulence 424 426
Dimensions as measures of chaos 442 443
Dimensions, box-counting 414
Dimensions, data coverage for 422 423
Dimensions, properties of 424 434
Dimensions, relations among 414 420 422 436 442
Directional derivative 489 492
Discriminant 36
Dissipation terms in partial differential system 32 33 106 111 281 300 449—450
Dissipation terms in spectral model 39 41 66 285 303 310 311
Dissipative system 70 101 106 111 267 415 417
Divergence of system 415
Divergence of trajectories (see Sensitive dependence on initial conditions)
Divergence theorem 57 449 460 502
Domain aspect ratio for double-valued bifurcation point 202 206
Domain aspect ratio, definition of 29 132 299
Domain aspect ratio, effects on critical Rayleigh number 36 119 120 131 138 143 199—206
Domain, dimensional 27 132 136 138 280 281
Domain, dimensionless 27 38 57 61 64 142 191 281 448 449
Double root 125
Dual vectors 488 490
Dynamical system as spectral model 415 416 421 487 488
Dynamical system, decoupling 484 487 488 495 496
Dynamical system, hierarchical 495 496
Dynamical system, routes to chaos in 424 428
Dynamical system, stable and unstable manifolds in 473 479 487 489
Earth System model 3 6
Earth System Science 4 5
Earth System, generic model of 5
Earth System, spatial scales for 10
Earth System, temporal scales for 4
Eddy thermometric conductivity 26 107 278
Eddy viscosity 26 107 278 449
Eigenfunctions in Fourier series 47
Eigenfunctions, spatial 55 60
Eigenvalue problems and basis functions 55 60 457 459
Eigenvalue problems and optimum basis functions 498 501
Eigenvalues of Ruelle matrices 478 479
Eigenvalues, spatial 56 60
| Eigenvalues, temporal 74 466 468
Eigenvectors of forward Ruelle matrix 473 477 478 488
Eigenvectors of linear stability problem 468
Eigenvectors of pure modes 74 468
Eigenvectors of reverse Ruelle matrix 476 477 479 488
Ekman depth, dependence of preferred cell geometry on 319 321
Ekman wind profile 319 324
Eleven-coefficient model 429 439 443
Eliminants, definition of 514
Eliminants, method of 322 514 516
Embedding inequality 455
Empirical orthogonal functions and explanation of variance 496 501
Empirical orthogonal functions, calculated from observations 496 497
Empirical orthogonal functions, produced by numerical simulation 500 501
Energetically steady state 109 111 290
Energy conversion term 108
Energy method and origin of instability 453 454 465
Energy method and stability of temporally periodic solution 355 357
Energy method and uniqueness of solutions 107 111 454 455
Energy method, solutions 455 457
Energy of difference fields 453 457
Energy sources for cloud street modes 300
Energy sources for convection 31 33
Energy transfer by nonlinear terms 198 206
Energy, available potential (see Available potential energy)
Energy, dissipation terms (see Dissipation terms)
Energy, generation terms (see Generation terms)
Energy, kinetic (see Kinetic energy)
Energy, longitudinal kinetic (see Longitudinal kinetic energy)
Energy, roll kinetic (see Roll kinetic energy)
Energy, total (see Total energy)
Ensemble of flow realizations as data for empirical orthogonal functions 497
Ensemble of flow realizations as data for optimum basis functions 500 502
Ensemble of flow realizations, and number of empirical orthogonal functions 499
Equation of state 25
Equations of motion for atmosphere 445 (see also Canonical partial differential equations for the atmosphere and Reduced partial differential equations for the atmosphere)
Error growth as phase error 406 407
Euclidean algorithm 125 322 510 513
Euclidean algorithm in singular cases 513
Euclidean algorithm, summarized 511
Euler method for integration 240
Existence of solutions in partial differential equations 106 107 109 111 444 445 447 450
Existence of solutions in truncated spectral models 462 463
Expectation over an ensemble 497
Explanation of variance, concentrating into a few functions 496 501 504
Exponential decay of waveforms 49 50
Exponential of a matrix 467
Exponents, characteristic (see Characteristic exponents)
Exponents, Lyapunov (see Lyapunov exponents)
Family of low-order models 198 199 218 224
Fastest growing wave 131 138 140 212
Favored response of fluid 131 144
Feigenbaum conjecture 390 395 403 426
Finite models and infinite-dimensional systems 84 85 444 447 457 484
Finite models, reduction of (see Optimum spectral models)
Finite-amplitude instability 96 97 117 122 330 331
Finite-amplitude transition to chaos 390 427 428
First law of thermodynamics 6 447 450 457
Fixed point 49 50 79 85 413—421 425 428 464
Flow regimes (see Regimes of flow)
Flow, decoupling into stable and unstable parts 486 488
Fluid configuration, favored 131 150 152
Fluid flow characteristics 8 9
Fold point 125 130 178
Forced dissipative system 70 101 106 111 131 267 415 417
Forcing parameter 72 110 131 132 182 196
Forcing, external 106 184 185 189
Forcing, internal 106 184 185 189
Fourier expansion, dominant part of 497 500 501
Fourier series 47 457 458
Fractal dimension 413 414 420 422 436
Fredholm alternative, application of 339 340 348 351
Fredholm alternative, motivation for 336 338
Fredholm alternative, theorem for 338
Frequencies, incommensurate 387 392
Frequencies, irrationally related 387 392
Frequencies, principal (see Principal frequencies in flow)
Frequency of oscillation of waveform 51 54 264 266
Frequency, cell 146 290
Frequency, limiting (see Limiting frequency of branching periodic solution)
Frobenius Theorem and existence of manifolds 490
Frobenius Theorem for finding coordinates along unstable manifolds 490 495
Frobenius Theorem, sketched proof of 492 493
Frobenius Theorem, statement of 492
Functional forms, nearby 164 165
Functional forms, small perturbations of 164 165
Fundamental theorem of algebra 162
Fundamental Theorems of Atmospheric Science 450 452 461 464
Galerkin representation, numerically optimizing 496 504
Galerkin technique, four steps in 47
Generation terms in partial differential system 39 108 281 300
Generation terms in spectral model 40 41 66 285 303
Geometric dimension of an attractor 412 414 421 438
Geophysical fluid dynamics 6 12
Global attractor 464
Global stability 72 108 111
Goal form for bifurcation diagram 209 218
Grashof number 457
Gravest mode 137
Grid-point model as a projection on N-space 447
Grid-point model of a spectral model 219
Growth rate, wavelengths 138 140
Hadley number as unfolding parameter 191 193 195
Hadley number, definition of 190
Harmonic bifurcation 375 376 430
Heat flux, three-dimensional 453
Heat flux, vertical 109 111 144 150 318 450
Heating coefficients, Newtonian 194 195
Heating field, effects of horizontally varying 451
Heating rate, diabatic 185 188 189 195 449
Heating rate, horizontal 122 124 190 193 195
Heating rate, latent 122 185
Heating rate, vertical 23 122 124 184 185 195
Heaviside function 422
Helmholtz problem for finding basis functions 55 60
Homeomorphism 473
Homogeneous system 336 338
Hopf bifurcation 264 277 372 383
Hopf bifurcation and Hurwitz Theorem 78 79 145 146
Hopf bifurcation and transition to turbulence 390 392 399 403 408 411 424 428
Hopf bifurcation and trident branching form 272 275
Hopf bifurcation and truncation level 145 408 411
Hopf bifurcation point, false 148
Hopf bifurcation point, finding 78 79 269 285 288 333
Hopf bifurcation point, in Chang and Shirer model 435 436
Hopf bifurcation point, in Lorenz model 78 79 341
Hopf bifurcation point, in Stensrud and Shirer model 312 314
Hopf bifurcation point, in Veronis model 144 146 148 150 151
Hopf bifurcation point, model 305 308 322 324
Hopf bifurcation point, preferred value for 148 151 154 305 308 312 314 322 324
Hopf bifurcation point, triple 392 401 403 427
Hopf bifurcation point, well diagram for 150 153
Hopf bifurcation theorem 269
Hopf bifurcation, degenerate case of 288
Hopf bifurcation, infinite sequence of 389 391
Hopf bifurcation, motivating 264 268 372 375
Hopf bifurcation, signal for 269 270 287 375
Hopf bifurcation, solution 146 148 269 270 287 307 308 312 322 327 333 373 383
Hopf bifurcation, symmetry requirements for 270 277 330
Hopf bifurcation, to a torus 375 382
Horizontal heating rate 122 124 190 193 195
Hurwitz polynomial 508
Hurwitz theorem 508
Hurwitz Theorem and bifurcation points 78 145 508 509
Hydrodynamic problems, well-posed 106
Hyperbolic sink 428
Hyperbolic stationary solution 469 474
Hypercube 413 416
Hyperellipse 463 503 504
Hysteresis 96 97 100 122
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