Fundamentals of Geophysical Hydrodynamics (Encyclopaedia of Mathematical Sciences Book 103)

This Newly-translated Book Takes The Reader From The Basic Principles And Conservation Laws Of Hydrodynamics To The Description Of General Atmospheric Circulation. Among The Topics Covered Are The Kelvin, Ertel And Rossby-obukhov Invariants, Quasi-geostrophic Equation, Thermal Wind, Singular Helmholtz Vortices, Derivation Of The Navier-stokes Equation, Kolmogorov's Flow, Hydrodynamic Stability, And Geophysical Boundary Layers. Generalizing V. Arnold's Approach To Hydrodynamics, The Author Ingeniously Brings In An Analogy Of Coriolis Forces Acting On Fluid With Motion Of The Euler Heavy Top And Shows How This Is Used In The Analysis Of General Atmospheric Circulation. This Book Is Based On Popular Graduate And Undergraduate Courses Given By F.v.dolzhansky At The Moscow Institute Of Physics And Technology, And Is The Result Of The Author's Highly Acclaimed Work In Moscow's Laboratory Of Geophysical Hydrodynamics. Each Chapter Is Full Of Examples And Figures, Exercises And Hints, Motivating And Illustrating Both Theoretical And Experimental Results. The Exposition Is Comprehensive Yet User-friendly In Engaging And Exploring The Broad Range Of Topics For Students And Researchers In Mathematics, Physics, Meteorology And Engineering. Main Principles And Laws Of Motion Of An Ideal Fluid. Equations Of Motion Of An Ideal Incompressible Fluid; Kelvin's Circulation Theorem -- Potential Vorticity And The Conservation Laws Of Energy And Momentum For A Stratified Incompressible Fluid -- Helicity; Equations Of Gas Dynamics; The Ertel Invariant -- The Rossby-obukhov Potential Vortex; Energy Momentum Of A Compressible Fluid; Hydrodynamic Approximation Of Equations Of Gas Dynamics -- Quasi-geostrophic Approximations Of The Equations Of Motion Of Rotating Barotropic And Baroclinic Fluids. Equations Of Motion Of A Rotating Fluid; The Notion Of A Geophysical Flow -- What Is Geophysical Hydrodynamics? -- The Obukhov-charney Equation; Rossby Waves -- Resonant Interaction Of Rossby Waves; Helmholtz And Obukhov Singular Vortices; The Kirchhoff Equations -- Equations Of Quasi-geostrophic Baroclinic Motion -- The Energy Balance, Available Potential Energy, And Rossby Waves In A Baroclinic Atmosphere --^ Important Remarks On The Description Of Baroclinic Geophysical Flows -- Hydrodynamic Stability And Atmospheric Dynamics. The Notion Of Dynamical Stability Via The Example Of Motion Of A Rigid Body With A Fixed Point -- Stating The Linear Stability Problem For Plane-parallel Flows Of Ideal Homogeneous And Nonhomogeneous Fluids -- The Method Of Normal Modes And Its Simplest Applications In The Theory Of Linear Stability Of Plane-parallel Flows -- The Taylor Problem Of Stability Of Motion Of A Stratified Fluid With A Linear Velocity Profile -- Applications Of Integral Relations And Conservation Laws In The Theory Of Hydrodynamic Stability -- Stability Of Zonal Flows Of A Barotropic Atmosphere; Th Enotion Of Barotropic Instability -- The Concept Of Baroclinic Instability; The Eady Model -- Friction In Geophysical Boundary Layers And Their Models. Equations Of A Motion Of A Viscous Fluid; The Boundary Conditions --^ Friction Mechanisms In Global Geophysical Flows; Quasi-geostrophic Equation For Transformation Of Potential Virticity -- Kolmogorov Flow And The Role Of Surface Friction -- Stability Of Quasi-two-dimensional Shear Flows With Arbitrary Velocity Profiles -- Friction In A Turbulent Boundary Layer -- Mechanical Prototypes Of Equations Of Motion Of A Rotating Stratified Fluid And A Toy Model Of Atmospheric Circulation. Hydrodynamic Interpretation Of The Euler Equations Of Motion Of A Classical Gyroscope And Their Invariants -- Mechanical Interpretation Of The Oberbeck-boussinesq Equations Of Motion Of An Incompressible Stratified Fluid In A Gravitational Field -- Motion Of Barotropic And Baroclinic Tops As Mechanical Prototypes For The General Circulation Of Barotropic And Baroclinic Inviscid Atmospheres -- Toy Models For General Circulation Of A Viscous Atmosphere -- Appendix A. On A Certain Boundary Condition -- Appendix B. Stability Of The Kolmogorov Flow With An External Friction. Felix V. Dolzhansky ; Translated By Boris Khesin. Includes Bibliographical References And Index. Front Matter....Pages I-XIV Front Matter....Pages 1-1 Equations of Motion of an Ideal Incompressible Fluid; Kelvin’s Circulation Theorem....Pages 3-11 Potential Vorticity and the Conservation Laws of Energy and Momentum for a Stratified Incompressible Fluid....Pages 13-21 Helicity; Equations of Gas Dynamics; The Ertel Invariant....Pages 23-30 The Rossby–Obukhov Potential Vortex; Energy and Momentum of a Compressible Fluid; Hydrodynamic Approximation of Equations of Gas Dynamics....Pages 31-39 Front Matter....Pages 41-41 Equations of Motion of a Rotating Fluid; The Notion of a Geophysical Flow....Pages 43-50 What is Geophysical Hydrodynamics?....Pages 51-60 The Obukhov–Charney Equation; Rossby Waves....Pages 61-70 Resonant Interaction of Rossby Waves; Helmholtz and Obukhov Singular Vortices; The Kirchhoff Equations....Pages 71-82 Equations of Quasi-geostrophic Baroclinic Motion....Pages 83-92 The Energy Balance, Available Potential Energy, and Rossby Waves in a Baroclinic Atmosphere....Pages 93-99 Important Remarks on the Description of Baroclinic Geophysical Flows....Pages 101-106 Front Matter....Pages 107-107 The Notion of Dynamical Stability via the Example of Motion of a Rigid Body with a Fixed Point....Pages 109-115 Stating the Linear Stability Problem for Plane-Parallel Flows of Ideal Homogeneous and Nonhomogeneous Fluids....Pages 117-123 The Method of Normal Modes and Its Simplest Applications in the Theory of Linear Stability of Plane-Parallel Flows....Pages 125-131 The Taylor Problem of Stability of Motion of a Stratified Fluid with a Linear Velocity Profile....Pages 133-139 Applications of Integral Relations and Conservation Laws in the Theory of Hydrodynamic Stability....Pages 141-150 Stability of Zonal Flows of a Barotropic Atmosphere; The Notion of Barotropic Instability....Pages 151-157 The Concept of Baroclinic Instability; The Eady Model....Pages 159-167 Front Matter....Pages 169-169 Equations of Motion of a Viscous Fluid; The Boundary Conditions....Pages 171-179 Friction Mechanisms in Global Geophysical Flows; Quasi-geostrophic Equation for Transformation of Potential Vorticity....Pages 181-188 Front Matter....Pages 169-169 Kolmogorov Flow and the Role of Surface Friction....Pages 189-194 Stability of Quasi-two-dimensional Shear Flows with Arbitrary Velocity Profiles....Pages 195-204 Friction in a Turbulent Boundary Layer....Pages 205-213 Front Matter....Pages 215-215 Hydrodynamic Interpretation of the Euler Equations of Motion of a Classical Gyroscope and Their Invariants....Pages 217-224 Mechanical Interpretation of the Oberbeck–Boussinesq Equations of Motion of an Incompressible Stratified Fluid in a Gravitational Field....Pages 225-233 Motion of Barotropic and Baroclinic Tops as Mechanical Prototypes for the General Circulation of Barotropic and Baroclinic Inviscid Atmospheres....Pages 235-244 Toy Model for General Circulation of a Viscous Atmosphere....Pages 245-257 Back Matter....Pages 259-272 This newly-translated book takes the reader from the basic principles and conservation laws of hydrodynamics to the description of general atmospheric circulation. Among the topics covered are the Kelvin, Ertel and Rossby-Obukhov invariants, quasi-geostrophic equation, thermal wind, singular Helmholtz vortices, derivation of the Navier-Stokes equation, Kolmogorov's flow, hydrodynamic stability, and geophysical boundary layers. Generalizing V. Arnold's approach to hydrodynamics, the author ingeniously brings in an analogy of Coriolis forces acting on fluid with motion of the Euler heavy top and shows how this is used in the analysis of general atmospheric circulation. This book is based on popular graduate and undergraduate courses given by F.V. Dolzhansky at the Moscow Institute of Physics and Technology, and is the result of the author's highly acclaimed work in Moscow's Laboratory of Geophysical Hydrodynamics. Each chapter is full of examples and figures, exercises and hints, motivating and illustrating both theoretical and experimental results. The exposition is comprehensive yet user-friendly in engaging and exploring the broad range of topics for students and researchers in mathematics, physics, meteorology and engineering. This book is based on popular graduate and undergraduate courses given by F.V. Dolzhansky at the Moscow Institute of Physics and Technology, and is the result of the author's highly acclaimed work in Moscow's Laboratory of Geophysical Hydrodynamics. Each chapter is full of examples and figures, exercises and hints, motivating and illustrating both theoretical and experimental results. The exposition is comprehensive yet user-friendly in engaging and exploring the broad range of topics for students and researchers in mathematics, physics, meteorology and engineering This newly-translated book takes the reader from the basic principles and conservation laws of hydrodynamics to the description of general atmospheric circulation. Among the topics covered are the Kelvin, Ertel and Rossby-Obukhov invariants, quasi-geostrophic equation, thermal wind, singular Helmholtz vortices, derivation of the Navier-Stokes equation, Kolmogorov's flow, hydrodynamic stability, and geophysical boundary layers. Generalizing V. Arnold's approach to hydrodynamics, the author ingeniously brings in an analogy of Coriolis forces acting on fluid with motion of the Euler heavy top and shows how this is used in the analysis of general atmospheric circulation. This book is based on popular graduate and undergraduate courses given by F.V. Dolzhansky at the Moscow Institute of Physics and Technology, and is the result of the author's highly acclaimed work in Moscow's Laboratory of Geophysical Hydrodynamics. Each chapter is full of examples and figures, exercises and hints, motivating and illustrating both theoretical and experimental results. The exposition is comprehensive yet user-friendly in engaging and exploring the broad range of topics for students and researchers in mathematics, physics, meteorology and engineering.-- Source other than Library of Congress