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Imperial2006年A Mathematical Theory of Large-scale Atmosphere ocean Flow
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Preface vii 2 . The governing equations and asymptotic approximations to them 11 2.1 The governing equations . . . . . . . . . . . . . . . . . . . . 11 2.2 Key asymptotic regimes . . . . . . . . . . . . . . . . . . . . 14 2.3 Derivation of the semi-geostrophic approximation . . . . . . 18 2.4 Various approximations to the shallow water equations . . . 21 2.4.1 The shallow water equations . . . . . . . . . . . . . . 21 2.4.2 Key parameters . . . . . . . . . . . . . . . . . . . . . 22 2.4.3 General equations for slow solutions . . . . . . . . . 26 2.4.4 Slow solutions on small scales . . . . . . . . . . . . . 30 2.4.5 Quasi-geostrophic solutions . . . . . . . . . . . . . . 33 2.4.6 Slow solutions on large scales . . . . . . . . . . . . . 36 2.5 Various approximations to the three-dimensional hydrostatic Boussinesq equations . . . . . . . . . . . . . . . . . . . . . . 41 2.5.1 The hydrostatic Boussinesq equations . . . . . . . . . 41 2.5.2 Key parameters . . . . . . . . . . . . . . . . . . . . . 42 2.5.3 General equations for slow solutions . . . . . . . . . 43 2.5.4 Slow solutions on with large aspect ratio . . . . . . . 46 2.5.5 Quasi-geostrophic solutions . . . . . . . . . . . . . . 48 2.5.6 Slow solutiolls with small aspect ratio . . . . . . . . 53 3 . Solution of the semi-geostrophic equations in plane geometry 57 xii Contents 3.1 The solution as a sequence of minimum energy states . . . . 57 3.1.1 The evolution equation for the geopotential . . . . . 57 3.1.2 Solutions as minimum energy energy states . . . . . 59 3.1.3 Physical meaning of the energy minimisation . . . . 61 3.2 Solution as a mass transportation problem . . . . . . . . . . 64 3.2.1 Solutioil by change of variables . . . . . . . . . . . . 64 3.2.2 The equations in dual variables . . . . . . . . . . . . 66 3.2.3 Consequences of the duality relation . . . . . . . . . 70 3.3 The shallow water semi-geostrophic equations . . . . . . . . 76 3.3.1 Solutions as minimum energy states . . . . . . . . . . 76 3.3.2 Solution by change of variables . . . . . . . . . . . . 79 3.3.3 The equations in dual variables . . . . . . . . . . . . 80 3.3.4 Consequences of the duality relation . . . . . . . . . 81 3.4 A discrete solution of the semi-geostrophic equations . . . . 84 3.4.1 The discrete problem . . . . . . . . . . . . . . . . . . 84 3.4.2 Example: frontogenesis . . . . . . . . . . . . . . . . . 90 3.4.3 Example: outcropping . . . . . . . . . . . . . . . . . 95 3.5 Rigorous results on existence of solutions . . . . . . . . . . 98 3.5.1 Solutions of the mass transport problem . . . . . . . 98 3.5.2 Existence of semi-geostrophic solutions in dual variables105 3.5.3 Solutions in physical variables . . . . . . . . . . . . . 111 4 . Solution of the semi-geostrophic equations in more general cases 117 4.1 Solution of the semi-geostrophic equations for compressible flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 The compressible equations in Cartesian geometry . 4.1.2 The solution as a sequence of minimum energy states 4.1.3 Solution by change of variables . . . . . . . . . . . . 4.1.4 The equations in dual variables . . . . . . . . . . . . 4.1.5 Rigorous weak existence results . . . . . . . . . . . . 4.2 Spherical semi-geostrophic theory . . . . . . . . . . . . . . . 4.3 The shallow water spherical semi-geostrophic equations . . 4.3.1 Solution procedure . . . . . . . . . . . . . . . . . . . 4.3.2 Demonstration of the solution procedure . . . . . . . 4.4 The theory of almost axisymmetric flows . . . . . . . . . . . 4.4.1 Minimum energy states for axisymmetric flows . . . 4.4.2 Theories for non-misymmetric flow . . . . . . . . . . ... Contents xlll 5 . Properties of semi-geostrophic solutions 163 5.1 The applicability of semi-geostrophic theory . . . . . . . . . 163 5.1.1 Error estimates . . . . . . . . . . . . . . . . . . . . . 163 5.1.2 Experimental verification of error estimates . . . . . 170 5.2 Stability theorems for semi-geostrophic flow . . . . . . . . . 175 5.2.1 Extremising the energy by rearrangement of the po- tentid density . . . . . . . . . . . . . . . . . . . . . . 175 5.2.2 Properties of rearrangements . . . . . . . . . . . . . 179 5.2.3 Analysis of semi-geostrophic shear flows . . . . . . . 184 5 -3 Numerical methods for solving the semi-geost rophic equations 190 5.3.1 Solutions using the geostrophic coordinate transfor- mation . . . . . . . . . . . . . . . . . . . . . . . . . . 190 5.3.2 The geometric method . . . . . . . . . . . . . . . . . 193 5.3.3 Finite difference methods . . . . . . . . . . . . . . . 194 6 . Application of semi-geostrophic theory to the predictability of atmospheric flows 20 1 6.1 Application to shallow water flow on various scales . . . . . 201 6.2 The Eady wave . . . . . . . . . . . . . . . . . . . . . . . . . 206 6.3 Simulations of baroclinic waves . . . . . . . . . . . . . . . . 213 6.4 Semi-geostrophic flows on the sphere . . . . . . . . . . . . . 219 6.5 Orographic flows . . . . . . . . . . . . . . . . . . . . . . . . 223 6.6 Inclusion of friction . . . . . . . . . . . . . . . . . . . . . . . 228 6.7 Inclusion of moisture . . . . . . . . . . . . . . . . . . . . . . 234 7 . Summary 243 Bibliography 245 Index 25 5 |
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