Asymptotic Modeling of Atmospheric Flows, Softcover reprint of the original 1st ed. 1990

The present work is not exactly a "course", but rather is presented as a monograph in which the author has set forth what are, for the most part, his own results; this is particularly true of Chaps. 7-13. Many of the problems dealt with herein have, since the school year 1975-76, been the subject of a series of graduate lectures at the "Universire des Sciences et Techniques de Lille I" for students preparing for the "Diplome d'Etudes Ap­ profondies de Mecanique (option fluides)". The writing of this book was thus strongly influenced by the author's own conception of meteorology as a fluid mechanics discipline which is in a privi­ leged area for the application of singular perturbation techniques. It goes without saying that the modeling of atmospheric flows is a vast and complex problem which is presently the focal point of many research projects. The enonnity of the topic explains why many important questions have not been taken up in this work, even among those which are closely related to the subject treated herein. Nonetheless, the author thought it worthwhile for the development of future research on the modeling of atmospheric flows (from the viewpoint of theoretical fluid mechanics) to bring forth a book specifying the problems which have already been resolved in this field and those which are, as yet, unsolved.

1. Introduction.- 2. The Equations.- 2.1 The Euler Equations.- 2.1.1 Steady Flows.- 2.2 The Tangent Plane Approximation.- 2.3 The So-called ß-Plane Approximation.- 2.4 Different Forms of the Euler Equations.- 2.4.1 The Euler Equations for $$\bar u,\,\bar v,\,\bar w,\,\pi ,\,\omega ,\,{\rm{and}}\,\vartheta$$.- 2.4.2 The Euler Equations for $$\bar u,\,\bar v,\,\bar w,\,\bar \Pi ,\,{\rm{and}}\,\bar \theta$$.- 2.4.3 The So-called Primitive 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.- 3.3.1 Quasi-static Filtering.- 3.3.2 Filtering of Waves with Frequency ?g q?s.- 3.3.3 “Boussinesq” Filtering.- 3.3.4 Isochoric Filtering (Quasi-Incompressible).- 3.3.5 Deep Convection Filtering (“Anelastic”).- 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.- 8.3.1 Taking into Account the Shearing of a Basic Wind; the So-called Long Equation.- 8.3.2 Generalized Boussinesq-Type Equations.- 8.3.3 Quasi-static Boussinesq Equations; the Problem of Meso-scale Circulations.- 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 the Isochoric Approximation.- 9.3 The Relation Between the Isochoric and Boussinesq Approximations.- 9.4 Wave Phenomena in the Isochoric Flows.- 9.4.1 The Long Wave Theory.- 9.4.2 The Short Wave Theory.- 9.4.3 Solitary Internal Waves.- 9.5 Complementary Remarks.- 10. The Deep Convection Approximation.- 10.1 The “Anelastic” Equations of Ogura and Phillips.- 10.2 The Deep Convection Equations According to Zeytounian.- 10.2.1 The Quasi-static Deep Convection Equations.- 10.2.2 A New Approach for the Derivation of the Deep Convection Equations (Case of the Adiabatic Atmosphere).- 10.3 The Relation Between the Boussinesq and the Deep Convection Approximations.- 10.4 Complementary Remarks.- 11. The Quasi-geostrophic and Ageostrophic Models.- 11.1 The Classical Quasi-geostrophic Model.- 11.2 The Adjustment to Geostrophy.- 11.3 The Ekman Steady Boundary Layer and the Ackerblom Problem.- 11.4 The So-called “Ageostrophic” Model.- 11.4.1 The Equation for the Ageostrophic Model.- 11.4.2 The Problem of the Unsteady Ekman Boundary Layer. Adjustment to the Ackerblom Model.- 11.4.3 The Problem of Adjustment to Ageostrophy.- 11.4.4 The Second Approximation Steady Ekman Problem.- 11.5 Complementary Remarks.- 12. Models Derived from the Theory of Low Mach Number Flows.- 12.1 The So-called Classical “Quasi-nondivergent” Model and Its Limitations.- 12.1.1 Analysis of Singularities Related to the Monin-Charney Limiting Process.- 12.2 The Generalized Quasi-nondivergent Model and Its Limitations.- 12.3 Analysis of Guiraud and Zeytounian’s Recent Results.- 12.4 The Problem of Adjustment to the Quasi-nondivergent Flow.- 12.5 Complementary Remarks.- 13. The Models for the Local and Regional Scale Atmospheric Flows.- 13.1 The Free Circulation Models.- 13.1.1 Inner Degeneracies.- 13.1.2 Outer Degeneracies.- 13.1.3 Matching. Formulation of the Free Circulation Problem.- 13.2 The Models for the Asymptotic Analysis of Lee Waves.- 13.2.1 Emergence of the Vertical Structure. Condition for z ? +?.- 13.2.2 The General Requirement for Trapped Lee Waves.- 13.2.3 Non-linear Models for Two-Dimensional Steady Lee Waves.- 13.2.4 Asymptotic Interpretation of the Long Model in the Troposphere.- 13.2.5 Asymptotic Representation of Three-Dimensional Linearized Lee Waves in the Lower Atmosphere.- 13.3 Modeling of the Interaction Phenomenon Between Free and Forced Circulations.- 13.3.1 Formulation of the Regional Boundary Layer Problem.- 13.3.2 The Interaction Model.- 13.4 Complementary Remarks.- 13.4.1 A Model for the Local Winds of Slopes and Valleys.- 13.4.2 Double Layer Periodic Slope (or Valley) Winds.- 13.4.3 Low Mach Number Flow over a Relief.- 13.4.4 Asymptotic Formulation of the Rayleigh-Bénard Problem via the Boussinesq Approximation for Expansible Liquids.- Appendix. The Hydrostatic Forecasting Equations for Large-Synoptic-Scale Atmospheric Processes.- A.1 The Governing Equations.- A.2 The Hydrostatic Model Equations.- A.3 The Large-Scale, Synoptic, Boundary Layer Equations.- References.