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Turbulence and Shell Models
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Contents Preface ix 1 Introduction to turbulence 1 1.1 The Navier–Stokes equation 3 1.2 Kolmogorov’s 1941 theory (K41) 7 1.3 The spectral Navier–Stokes equation 9 1.4 The spectral energy density 10 1.5 The spectral energy flux 12 1.6 The closure problem 13 1.7 The four-fifth law 14 1.8 Self-similarity of the energy spectrum 18 1.9 The dissipative anomaly 23 1.10 The vorticity equation 23 1.11 Intermittency in turbulence 24 1.12 Finite time singularities 25 1.13 Problems 26 2 2D turbulence and the atmosphere 30 2.1 2D turbulence 30 2.2 Atmospheric turbulence 34 2.3 The governing equations 35 2.4 Geostrophic wind and the Rossby number 38 2.5 Stratification and rotational Froude number 38 2.6 Quasi-geostrophy 39 2.7 Observations of the atmosphere 41 2.8 Problems 44 3 Shell models 46 3.1 The Obukhov shell model 47 3.2 Liouville’s theorem 48 3.3 The Gledzer shell model 50 v vi Contents 3.4 Scale invariance of the shell model 51 3.5 The shell spacing and energy conservation 52 3.6 Parameter space for the GOY model 53 3.7 2D and 3D shell models 56 3.8 Other quadratic invariants 57 3.9 Triad interactions and nonlinear fluxes 58 3.10 The special case ? = 1 60 3.11 The Sabra shell model 61 3.12 Problems 63 4 Scaling and symmetries 65 4.1 The nonlinear fluxes 65 4.2 3D GOY and Sabra models 67 4.3 Phase symmetries 71 4.4 Equivalent of the four-fifth law 72 4.5 Problems 73 5 Chaotic dynamics 74 5.1 The Lyapunov exponent 75 5.2 The attractor dimension 79 5.3 The attractor for the shell model 83 5.4 Predictability 85 5.5 Large scale predictability 86 5.6 The finite size Lyapunov exponent 87 5.7 Limited predictability of the small scales 89 5.8 Problems 93 6 Helicity 95 6.1 The helicity spectrum 95 6.2 Comparison with 2D turbulence 97 6.3 The helicity dissipation scale 97 6.4 Helicity in shell models 100 6.5 Ageneralized helical GOY model 105 6.6 Problems 108 7 Intermittency 110 7.1 Kolmogorov’s lognormal correction 110 7.2 The β-model 113 7.3 The multi-fractal model 114 7.4 Intermittency in shell models 115 7.5 Probability densities and intermittency 116 7.6 Problems 120 8 Equilibrium statistical mechanics 121 8.1 The statistical ensemble 121 8.2 The partition function 123 Contents vii 8.3 Phase space geometry 127 8.4 Statistical equilibrium and turbulence 128 8.5 Cascade or equilibrium 130 8.6 Problems 133 Appendix 135 A.1 Velocity Fourier transforms in 3D 135 A.2 Rotation of the velocity field 137 A.3 Product rules for Levi–Civita symbols 138 A.4 Divergence theorem (integration by parts) 139 A.5 The vorticity equation 140 A.6 Helicity 141 A.7 Energy balance between triads 142 A.8 The 2D case 142 A.9 Scaling consequence of lognormal assumption 144 References 147 Index 151 |
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