Fundamentals of Geophysical Hydrodynamics

1. Front Matter

   Pages I-XIV

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2. Main Principles and Laws of Motion of an Ideal Fluid

   1. Front Matter

      Pages 1-1

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   2. Equations of Motion of an Ideal Incompressible Fluid; Kelvin’s Circulation Theorem

      • Felix V. Dolzhansky

      Pages 3-11

   3. Potential Vorticity and the Conservation Laws of Energy and Momentum for a Stratified Incompressible Fluid

      • Felix V. Dolzhansky

      Pages 13-21

   4. Helicity; Equations of Gas Dynamics; The Ertel Invariant

      • Felix V. Dolzhansky

      Pages 23-30

   5. The Rossby–Obukhov Potential Vortex; Energy and Momentum of a Compressible Fluid; Hydrodynamic Approximation of Equations of Gas Dynamics

      • Felix V. Dolzhansky

      Pages 31-39

3. Quasi-geostrophic Approximations of the Equations of Motion of Rotating Barotropic and Baroclinic Fluids

   1. Front Matter

      Pages 41-41

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   2. Equations of Motion of a Rotating Fluid; The Notion of a Geophysical Flow

      • Felix V. Dolzhansky

      Pages 43-50

   3. What is Geophysical Hydrodynamics?

      • Felix V. Dolzhansky

      Pages 51-60

   4. The Obukhov–Charney Equation; Rossby Waves

      • Felix V. Dolzhansky

      Pages 61-70

   5. Resonant Interaction of Rossby Waves; Helmholtz and Obukhov Singular Vortices; The Kirchhoff Equations

      • Felix V. Dolzhansky

      Pages 71-82

   6. Equations of Quasi-geostrophic Baroclinic Motion

      • Felix V. Dolzhansky

      Pages 83-92

   7. The Energy Balance, Available Potential Energy, and Rossby Waves in a Baroclinic Atmosphere

      • Felix V. Dolzhansky

      Pages 93-99

   8. Important Remarks on the Description of Baroclinic Geophysical Flows

      • Felix V. Dolzhansky

      Pages 101-106

4. Hydrodynamic Stability and Atmospheric Dynamics

   1. Front Matter

      Pages 107-107

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   2. The Notion of Dynamical Stability via the Example of Motion of a Rigid Body with a Fixed Point

      • Felix V. Dolzhansky

      Pages 109-115

   3. Stating the Linear Stability Problem for Plane-Parallel Flows of Ideal Homogeneous and Nonhomogeneous Fluids

      • Felix V. Dolzhansky

      Pages 117-123

   4. The Method of Normal Modes and Its Simplest Applications in the Theory of Linear Stability of Plane-Parallel Flows

      • Felix V. Dolzhansky

      Pages 125-131

   5. The Taylor Problem of Stability of Motion of a Stratified Fluid with a Linear Velocity Profile

      • Felix V. Dolzhansky

      Pages 133-139

   6. Applications of Integral Relations and Conservation Laws in the Theory of Hydrodynamic Stability

      • Felix V. Dolzhansky

      Pages 141-150

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. 

• geophysics
• hydrodynamics
• hydrogeology
• fluid- and aerodynamics

From the reviews:

“This is a clearly written and pedagogically well balanced book on numerous aspects of geophysical hydrodynamic processes that can be useful not only for teachers and graduate students but also for researchers as a good general reference literature.” (Vladimir Čadež, zbMATH, Vol. 1286, 2014)

“One may find a complete undergraduate and graduate course of geophysical hydrodynamics in this book. Moreover, the reading of this textbook is quite pleasant even from the literary viewpoint. … Every chapter ends with some very concrete exercises.” (Catherine Choquet, Mathematical Reviews, November, 2013)

• Inst. Atmospheric Physics, Russian Academy of Sciences, Moscow, Russia

  Felix V. Dolzhansky

Preface.- Part I Main Principles and Laws of Motion of an Ideal Fluid: 1 Equations of motion of an ideal incompressible fluid; Kelvin's circulation theorem.- 2 Potential vorticity and the conservation laws of energy and momentum for a stratified incompressible fluid.- 3 Helicity, equations of gas dynamics, and the Ertel invariant.- 4 The Rossby-Obukhov potential vortex; energy and momentum of a compressible fluid; hydrodynamic approximation of equations of gas dynamics.- Part II Quasi-geostrophic Approximations of the Equations of Motion of Rotating Barotropic and Baroclinic Fluids: 5 Equations of motion of a rotating fluid and the notion of a geophysical flow.- 6 What is geophysical hydrodynamics?.- 7 The Obukhov-Charney equation; Rossby waves.- 8 Resonant interaction of Rossby waves, Helmholtz and Obukhov singular vortices, and the Kirchhoff equations.- 9 Equations of quasi-geostrophic baroclinic motion.- 10 The energy balance, available potential energy, and Rossby waves in a baroclinic atmosphere.- 11 Important remarks on the description of baroclinic geophysical flows.- Part III Hydrodynamic Stability and Atmospheric Dynamics: 12 The notion of dynamical stability via the example of motion of a rigid body with a fixed point.- 13 Stating the linear stability problem for plane-parallel flows of ideal homogeneous and nonhomogeneous fluids.- 14 The method of normal modes and its simplest applications in the theory of linear stability of plane-parallel flows.- 15 The Taylor problem of stability of motion of a stratified fluid with a linear velocity profile.- 16 Applications of integral relations and conservation laws in the theory of hydrodynamic stability.- 17 Stability of zonal flows of a barotropic atmosphere; the notion of barotropic instability.- 18 The concept of baroclinic instability; the Eady model.- Part IV Friction in Geophysical Boundary Layers and Their Models: 19 Equations of motion of a viscous fluid; the boundary conditions.- 20 Friction mechanismsin global geophysical flows; quasi-geostrophic equation for transformation of potential vorticity.- 21 Kolmogorov flow and the role of surface friction.- 22 Stability of quasi-two-dimensional shear flows with arbitrary velocity profiles.- 23 Friction in a turbulent boundary layer.- Part V Mechanical Prototypes of Equations of Motion of a Rotating Stratified Fluid and a Toy Model of Atmospheric Circulation: 24 Hydrodynamic interpretation of the Euler equations of motion of a classical gyroscope and their invariants.- 25 Mechanical interpretation of the Oberbeck-Boussinesq equations of motion of an incompressible stratifed fluid in a gravitational field.- 26 Motion of barotropic and baroclinic tops as mechanical prototypes for the general circulation of barotropic and baroclinic inviscid atmospheres.- 27 Toy model for general circulation of a viscous atmosphere.- Part VI Appendices: A1 On a certain boundary condition.- A2 Stability of the Kolmogorov flow with an external friction.- Index

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