Asymptotic Modeling of Atmospheric Flows

1. Introduction.- 2. The Equations.- 2.1 The Euler Equations.- 2.2 The Tangent Plane Approximation.- 2.3 The So-called ss-Plane Approximation.- 2.4 Different Forms of the Euler Equations.- 2.5 The Non-dimensional Non-adiabatic Equations.- 3. Internal Waves and Filtering.- 3.1 The Case of $$d{bar T_infty }/d{bar z_infty } equiv 0$$. The Wave Equation.- 3.2 The Vertical Structure of the Internal Waves.- 3.3 Filtering.- 3.4 Conclusions and Bibliographical References.- 4. Rossby Waves.- 4.1 An Evolution Equation for Rossby Waves.- 4.2 Rossby Waves in Linear Theory.- 4.3 Rossby Waves in a So-called Barotropic Atmosphere.- 4.4 On the Problem of Hydrodynamic Instability.- 4.5 Conclusions and Bibliographical References.- 5. A Presentation of Asymptotic Methods.- 5.1 The Matched Asymptotic Expansions Method.- 5.2 The Multiple-Scale Method.- 6. Some Applications of the MMAE and MSM.- 6.1 Application of the MMAE to Adiabatic Flows with Small Kibel Numbers.- 6.2 Double-Scale Structure of the Boussinesq Waves: Linear Theory.- 6.3 Various Hydrostatic Limiting Processes.- 6.4 A Triple-Deck Structure Related Local Model.- 7. The Quasi-static Approximation.- 7.1 The Exact Quasi-static Equations.- 7.2 Asymptotic Analysis of the Primitive Equations.- 7.3 The Boundary Layer Phenomenon and the Primitive Equations.- 7.4 Simplified Primitive Equations.- 7.5 The Hydrostatic Balance Adjustment Problem (in an Adiabatic Atmosphere).- 7.6 Complementary Remarks 1.- 7.7 Complementary Remarks 2.- 8. The Boussinesq Approximation.- 8.1 The Boussinesq Equations.- 8.2 Some Considerations concerning the Singular Nature of the Boussinesq Approximation.- 8.3 Three New Forms of the Boussinesq Equations.- 8.4 Concerning a Linear Theory of the Boussinesq Waves $$left( {{rm{Ro}}not equiv infty }right)$$.- 8.5 The Problem of Adjustment to the Boussinesq State.- 8.6 Complementary Remarks.- 9. The Isochoric Approximation.- 9.1 The Isochoric Equations.- 9.2 Some Considerations concerning the Singular Nature of