I Turbulence and cascade models
1
1 Introduction
to turbulence
3
1 The Navier-Stokes equation
5
1.2 Kolmogorov's 1941 Theory
(K41)
10
1.3 The spectral Navier-Stokes
equation
13
1.4 The spectral energy density
14
1.5 The spectral energy flux
16
1.6 The closure problem
18
1.7 The four-fifth law
20
1.8 Self-similarity of the energy
spectrum
24
k-e
models
27
1.9 The dissipative anomaly
29
1.10 Intermittency in turbulence 31
1.11 Finite time singularities
33
1.12 2D turbulence
34
2 Atmospheric turbulence
39
2.1 The governing equations
41
Potential temperature 43
Hydrostatic balance
43
2.2 Geostrophic wind and the
Rossby number
44
2.3 Stratification and rotational
Froude number
45
2.4 Quasi-geostrophy
46
2.5 Observations of the atmosphere
49
3 Shell
models
53
3.1
The
Obukhov shell model
55
3.2 Liouville's theorem
56
3.3 The Gledzer shell model
57
3.4 Scale invariance of the shell
model
58
3.5 The shell spacing and energy
conservation
59
3.6 Parameter space for the GOY
model
61
3.7 2D and 3D shell models
65
3.8 Other quadratic invariants
66
3.9 Triad interactions and
nonlinear fluxes
67
3.10 The special case epsilon =1
68
3.11 The Sabra shell model
71
4 Scaling and symmetries
75
4.1
The
nonlinear fluxes
75
4.2 3D GOY and Sabra models
77
4.3 Phase symmetries
83
4.4 Equivalent of the four-fifth
law
85
5 Chaotic dynamics
87
5.1
The
Lyapunov exponent
88
5.2 The attractor dimension}
93
5.3 The attractor for the shell
model
99
5.4 Predictability
100
5.5 Large scale predictability 102
5.6 The finite size Lyapunov
exponent
105
5.7 Limited predictability of the
small scales 107
6 Helicity
113
6.1 The helicity spectrum
114
6.2 Comparison with 2D turbulence
115
6.3 The helicity dissipation scale
116
6.4 Helicity in shell
models
120
6.5 A generalized helical GOY
model
125
7 Intermittency
131
7.1
Kolmogorov's
lognormal correction
132
7.2 The beta-model
135
7.3 The multi-fractal model
136
7.4 Intermittency in shell models
138
7.5 Probability densities and
intermittency
138
8 Equilibrium statistical mechanics
145
8.1
The
statistical ensemble
146
8.2 The partition function
148
8.3 Phase space geometry
152
8.4 Statistical equilibrium and
turbulence
154
8.5 Cascade or equilibrium
156
II Stochastic models and climate
data analysis
161
9 Climate theory
163
9.1 The Budyko-Sellers energy
balance model
165
9.2 One-component dynamical
systems
171
9.3 Fixed points and linear
stability analysis
172
9.4 The thermohaline circulation
173
10 Climate data analysis
181
10.1 Correlation times and power spectra
182
10.2 A few types of signals
185
Random signals modeled as white noise
187
Scaling noise
188
11 Stochastic dynamics
189
11.1
Separating
timescales
190
11.2 The Fokker-Planck equation
195
Brownian motion and coin tossing 195
The conditional probability 196
11.3 The fluctuation-dissipation
theorem
198
11.4 The stationary distribution 199
11.5 Stochastic resonance 202
12 Anomalous jumping 205
12.1
alpha-stable distributions
206
Addition of alpha-stable random variables
207
12.2 The spectral Fokker-Planck
equation
210
12.3 The potential 216
12.4 Waiting time 216
Gaussian noise and the Arrhenius formula 216
alpha-stable noise 218
12.5 The stationary distribution 220
12.6 Barrier penetration 222
13 Ice core records
225
13.1
Separating
timescales in the data record 228
13.2 The climatic noise and
intermittency 232
13.3 Exotic statistics 235
14 Climate shifts
243
14.1
Waiting
times for the climate shifts
245
14.2 The quasi-stationary climatic
states
253
14.3 The linear climate dynamics
254
14.4 Analysis of the data for the
two states
257
14.5 Comparison with present day's
climate
258
14.6 The saw-tooth shape of the
interstadials
260
15.1 Chemical tracers in the
climate
264
15.2 Factor analysis models
265
15.3 EOF analysis of chemical
tracers
266
15.4 A new receptor point model
267
15.5 Testing the model
271
15.6 The Akaike information
criterion (AIC)
274
15.7 Analysis of the ice core data
280
15.8 A posteriori validation
285
15.9 The correlation between
sources
288
15.10 A proposed scenario
291
15.11 A common precipitation
record
294
A Appendix
303
A.1 Velocity Fourier
transforms in 3D
303
A.2 Rotation of the velocity field
306
A.3 Product rules for Levi-Civita
symbols
307
A.5 The vorticity equation
309
A.6 Helicity
310
A.7 Energy balance between triads
311
A.8 The 2D case
312
A.9 Scaling consequence of
lognormal assumption
314
A.10 The Brownian motion increment
316
A.11 Solution to the Langevin
equation
318
References
319
List of Papers
339