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| Shirer H.N. — Nonlinear Hydrodynamic Modeling: A Mathematical Introduction |
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| Предметный указатель |
Parameters, physically interpreting unfolding 182 196
Parameters, response 131 140 156 292 293
Parameters, unfolding 174 175 182 196
Parseval property 496
Peixoto Theorem 397 398
Period of oscillation 51
Period, limiting 373 374
Period-doubling bifurcation 375 376
Period-doubling bifurcation, infinite sequence of 390 392 395 402 403 426 429 432
Period-doubling transitions 388 390 392 395 402 403
Periods, attractor 390 392 400 403
Periods, incommensurate 390
Perturbation of functional forms 164 165
Perturbation trajectory, linear evolution of 474 475
Perturbation, amplifying and unstable solution 34 70 71 362 364 365
Perturbation, behavior of and stability 70 79 359 367 465 468
Perturbation, damping of and stable solution 33 70 71 365
Perturbation, infinitesimal and stability 71 72
Perturbation, twist angle of and bifurcation 365 374 382
Pfaffian system 489
Phase angle of waveform 49
Phase space 11
Phase space volumes, evolution of 416 419 476
Phase space, appearance of waveforms in 49 54
Phase space, as closed and bounded 418
Phase space, deformation of 461
Phase space, dimension of 413
Phase space, in 48 54 460 461
Phase space, manifold in 84 486
Phase space, shifting origin in 167 271
Phase space, stable and unstable manifolds in 79 85 469 496
Phase-locked stationary solution 50
Phases of scientific study 2 3
Physical effects, representing by parameters 121 130 183 193
Physical space 48 49
Physical space, space 49 54
Physical systems, equivalent 182 191 193
Physical systems, reduced 446 447
Physical systems, unfolded 195
Picard Existence and Uniqueness, Theorem 492
Pitchfork branching form 44
Poincare disk or section 374 381 396 399 431
Poincare map 372 383 396 399 413 431
Poincare map for a strange attractor 413 414
Poincare map, definition of 373 396
Points at infinity 228 232
Poisson inequalities 451
Polynomial division 510
Polynomial equations and roots at infinity 228 236
Polynomial equations for branching periodic solutions 271 277 290
Polynomial equations for stationary solutions 112 118 167 172 177 178 225
Polynomial equations, finding common roots to 510 516
Polynomial map, definition of 226
Polynomial map, homogeneous 230
Polynomial map, homogenization of 228 236
Positive definite matrix 503 504
Power series method for branching for ordinary differential equations 326 341
Power series method for branching for partial differential equations 326 331 332 338
Power spectra and turbulence 386 389
Prandtl number, definition of 7 28 132 299 448
Prandtl number, solutions 119 121
Predictability 11 404 407 412 431 434 442 465
Predictability and Lyapunov exponents 417 431 434
Predictability problems and decision points 82 84 382 407 433
Predictability problems and initial errors 406 407
Predictability problems and unstable solutions 82 84
Predictability, gain or loss of 432
Predictability, limitations on 453 454
Predictability, solutions 404 407 431 432
Prediction limits 7 9 11
Preferred values of parameters, in convection 111 131 138
Preferred values of parameters, in convection with a background wind 292 295 296 304 308 314 324
Preferred values of parameters, in rotating convection 143 163
Pressure gradient force 25
Primary bifurcation point, definition of 87
Primary bifurcation point, double and nearby bifurcations 199 206 211 214 215 401
Primary branch, branch 197 198 204 206 210 218
Primary branch, definition of 87
Primary branch, expected stabilities for 92 100
Primary branch, solution 94 99
Primary branch, subcritical (definition) 87
Primary branch, supercritical (definition) 87
Principal frequencies in flow and transition to turbulence 384 390
Probabilistic expectation over an, ensemble 497
Projective space 234 236
Proper Orthogonal Decomposition Theorem and basis functions 496 504
Proper Orthogonal Decomposition Theorem, sketch of proof of 498 500
Pull-back of an unfolding 175 177
Pull-back operation and loss of clarity 176 177
Pull-back operation and loss of information 176 177
Quadratic polynomial equation for stationary solutions 115 116
Quartic polynomial equation for stationary solutions 116 118
Quasi-geostrophic flow in a channel 114 115 194 195 257
Quasi-geostrophic flow in atmosphere 496
Quasi-periodic flow 386 392
Quasi-periodic function 389 391
Quasi-periodic solution 377 380 396 399 403 424 427
Quasi-periodic two-torus 424 427
Quintic polynomial equation for amplitude of periodic solution 275 277
Quintic polynomial equation for stationary solutions 117 118 154 155 194 196
Random noise in system 423
Rank of a matrix 168
Rayleigh number, critical (see Critical Rayleigh number)
Rayleigh number, definition of 7 28 107 132 299
Rayleigh number, normalized 29
Rayleigh number, physical interpretation of 109
Rayleigh-Benard convection in presence of a height-dependent background wind (see Boussinesq system with background wind
Rayleigh-Benard convection, classical (see Classical Rayleigh-Benard convection)
Rayleigh-Benard convection, rotating (see Rotating convection)
Rayleigh-Benard convection, spherical 408 409
Recurrence properties of trajectories and attractor dimensions 420 423
Recurrent point 464
Recurrent trajectory 405
Reduced partial differential equations for the atmosphere, condition for stability in 456 457
Reduced partial differential equations for the atmosphere, equations for 450
Reduced partial differential equations for the atmosphere, Jeffreys' Theorem for 452
Reduced partial differential equations for the atmosphere, Trapping Theorem for 452 (see also Canonical partial differential equations for the atmosphere)
Reduced spectral model for the atmosphere, basis functions for 459
Reduced spectral model for the atmosphere, equations for 465
Reduced spectral model for the atmosphere, instability process in 465 468
Reduced spectral model for the atmosphere, Jeffreys' Theorem for 464
Reduced spectral model for the atmosphere, optimum truncation for 501 504
Reduced spectral model for the atmosphere, Reduction of complexity of models 457 460 464 484 504
Reduced spectral model for the atmosphere, stability of nontrivial solution in 466 468
Reduced spectral model for the atmosphere, stability of trivial solution in 465 466
Reduced spectral model for the atmosphere, stationary solution in 465
Reduced spectral model for the atmosphere, Trapping Theorem for 465 503 504
Regime selection 7 9 453
Regime transition as a bifurcation 469
Regime transition as given by loss of stability 464 469
Regimes of flow in rotating annulus 8 9
Regimes of flow, convective 7 8 384 390
Regimes of flow, distinguishing in data 386 390
Regimes of flow, frequency-locking in 388 390
Regimes of flow, generic 8 384
Regimes of flow, modeling of 446 447
Regimes of flow, optimum spectral models for 496 500 501
Regular turning point in branching stationary solutions 98 100 104 117 118 122—125 129 130 132 155 163
Regular turning point in branching temporally periodic solutions 276 329 331 351
Regular turning point, physical relevance of minimizing values of Ra at 155 157
Repelling direction 79 85 416 420 468 469
Resonating terms 336
Response parameters 131 140 156 292 293
Restricted attractor 438 439
Return map (see Poincare map defined)
Reverse Jacobian matrix and tangent vectors of unstable manifold 476 477
Reverse Jacobian matrix, definition of 475 476
Reverse Jacobian matrix, relation with Jacobian matrix 476 477
Reynolds number for global atmospheric flows 456
Reynolds number, critical (see Critical Reynolds number)
| Reynolds number, definition of 7 280 299
Reynolds stress in partial differential system 281 282 300 453
Reynolds stress in spectral model 285 303 304 310
Richardson number 323 324
Roll axis 279
Roll circulation, expected sense of 122
Roll coordinate system 279 280 293 294
Roll energetics in partial differential equations 281 282 299 300
Roll energetics in spectral model 284 285 290 291 302 304 310 311
Roll energetics, effects of Coriolis terms on 296 300 303
Roll kinetic energy in partial differential system 299
Roll kinetic energy in spectral model 302 310
Rolls in atmosphere 292 294 Inflection Parallel and
Rolls in atmosphere in laboratory vessel 292
Root line 233
Roots at infinity 228 232 236 256 257 262 263
Roots at infinity, finite representation of 234
Rotating convection 140 163
Rotating convection, bifurcation diagram for 117 118 156 159
Rotating convection, bifurcation points in 142 146
Rotating convection, boundary conditions for 142
Rotating convection, expected branching direction for 153 163
Rotating convection, expected type of branching solution in 146 153
Rotating convection, finding appropriate spectral model for 326
Rotating convection, hypothetical larger model of 155 157
Rotating convection, partial differential equations for 140 141
Rotating convection, preferred wavelengths in 144 154
Rotating convection, spectral model for (see Veronis model)
Rotating convection, unfoldings of 194 195
Rotation number 396 398
Routes to chaos 389 403
Routes to chaos and attractor dimensions 424 426
Routes to chaos and Lyapunov exponents 424 428
Routes to chaos, classification of 400 404
Routes to chaos, Feigenbaum's conjecture for 390 395 403 426
Routes to chaos, Landau's hypothesis for 389 392 400 424 425
Routes to chaos, Ostlund et al.'s conjecture for 390 395 399 402 403 426—427
Routes to chaos, Ruelle and Takens conjecture for 391 392 395 401 403 425—426
Routes to chaos, Shirer and Wells conjecture for 392 400 404 427
Routes to chaos, subcritical bifurcation 390 427 428
Routes to chaos, summary of hypothesized 390
Ruelle matrix and Lyapunov exponents 478 479
Ruelle matrix, forward 473 477 478 488
Ruelle matrix, reverse 476 477 479 488
Ruelle — Takens conjecture 391 392 395 401 403 425 426
Ruelle — Takens turbulence 385
Ruelle's Theorem and finding optimum spectral models 487 496
Ruelle's Theorem and refined unstable manifolds 495 496
Ruelle's Theorem, finding stable manifolds with 473 477 478 487 488 493—494
Ruelle's Theorem, finding unstable manifolds with 474 477 479 487 488
Ruelle's Theorem, simple example of applying 479 484
Ruelle's Theorem, statement of 473
Ruelle's Theorem, time reversal in 474 477 479 488
Saltzman model 37 39 198 206 408
Sard's theorem 237 238
Scale selection 446
Schwarz inequality 452 456 462
Scientific questions, significant 7
Secondary bifurcation point and double primary bifurcation point 204 206 211 214 215 401
Secondary bifurcation point, definition of 87
Secondary branch, branch 197 198 204 206 210 218 401 408 411
Secondary branch, definition of 87
Secondary branching 197 224
Secondary branching and catalytic terms 207 208 211
Secondary branching, asymmetric and cell broadening 218 223
Secondary branching, chaos 402 403
Secondary branching, diagnosing possible 198 206
Secondary branching, originating from Jacobian term 210 211
Selection rules 64 198
Self-adjoint system 339
Semi-periodicity 393
Sensitive dependence on initial and Lorenz attractor 404 407
Sensitive dependence on initial, as property of turbulence 384
Sensitive dependence on initial, quantifying 417 420 423 425
Sensitive dependence on initial, represent turbulence 390 398
Separable solutions, to eigenvalue problems 58 59
Shallow Boussinesq equations 23 31
Shape of attractor 84
Shear production term 282
Shifting the trident branching form for stationary solutions 119 122
Shifting the trident branching form for temporally periodic solutions 274 275
Shirer 198
Shirer and Wells conjecture 392 400 404 427
Shirer and Wells model 191 193
Shirer, bifurcation points in 305 308 322 324
Shirer, characteristic equation for 286
Shirer, energetics of 302 304
Shirer, equations for 301 302
Shirer, expansions for 301
Shirer, instability case in 305 306 314 321
Shirer, mixed parallel and thermal mode in 307 308 322 324
Shirer, parallel instability case in 306 307 314 321
Shirer, periodic solution in 287
Shirer, temporally periodic solutions in 288 291
Singular points and exchange of stability 128 130
Singular points, as bifurcation points 128 172 194
Singular points, definition of 125
Singular points, types of 128 178
Singularity, codimension of 178 196
Singularity, unfoldings of 174 182
Snap-through bifurcation 96 97 100 117 330 331
Solution surface 119 121 125 130
Solutions and unstable manifolds 469 470 483 488
Solutions, in grid-point and spectral models 219
Solutions, N-dimensional approximations of 446 447 460 461
Solutions, space 267
Spectral bands 5 10
Spectral expansions, finding 55 60
Spectral expansions, general form of for convection 61 62
Spectral models as dynamical systems 415 416 421 487 488
Spectral models, advantages of using 11 48
Spectral models, appropriate truncation level for 326
Spectral models, as projections on N-space 447
Spectral models, comparisons of solutions with those of a grid-point model 219
Spectral models, existence of solutions in 462 463
Spectral models, forming 60 69 457 461
Spectral models, forming optimum 485 504
Spectral models, general forms of 72 87 166 359 460 465 473 487 488
Spectral models, hierarchical 495 496
Spectral models, optimum (see Optimum spectral models)
Spectral models, reduction of (see Optimum spectral models)
Splitting the trident branching form 122 130 164 165 183
Spurious roots 510 511
Stability analysis and secondary bifurcation 198 206 209 218
Stability analysis for branching, energy method for 355 357
Stability analysis for branching, integration method for 357 372
Stability analysis for branching, power series method for 326 341
Stability analysis for stationary solutions 70 79 92 100 465 469
Stability analysis, as first step in nonlinear analysis 34 35 70
Stability change, conditions for 75 92
Stability exchange and expected branching solution 138 139
Stability exchange, periodic solutions 372 383
Stability exchange, solutions 92 105 269 466 469
Stability portrait (or profile) 95 100
Stability properties of the conductive and bifurcation 70 109 111
Stability properties of the conductive and secondary branching 199 206
Stability properties of the conductive, in Boussinesq system 35 37 109 119 199 206
Stability properties of the conductive, in Chang and Shirer model 208 212 222
Stability properties of the conductive, in Lorenz model 41 42 77 92 112
Stability properties of the conductive, in Stensrud and Shirer model 312 321
Stability properties of the conductive, in Veronis model 142 163
Stability properties of the conductive, model 306 308 314 324
Stability properties, general comments on 100 105
Stability transfer, branches 204 206 211 218
Stability, asymptotic orbital (see Asymptotic orbital stability)
Stability, concept of for stationary solutions 70 71
Stability, exchange of at bifurcation point 34 37 77 92 100 128 130 269 325 373 383
Stability, global 72 108 111
Stability, infinitesimal 72
Stability, neutral 71
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