Hydrodynamics of Oceans and Atmospheres

Hydrodynamics of Oceans and Atmospheres

1st Edition - January 1, 1960

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  • Author: Carl Eckart
  • eBook ISBN: 9781483149561

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Hydrodynamics of Oceans and Atmospheres is a systematic account of the hydrodynamics of oceans and atmospheres. Topics covered range from the thermodynamic functions of an ideal gas and the thermodynamic coefficients for water to steady motions, the isothermal atmosphere, the thermocline, and the thermosphere. Perturbation equations, field equations, residual equations, and a general theory of rays are also presented. This book is comprised of 17 chapters and begins with an introduction to the basic equations and their solutions, with the aim of illustrating the laws of dynamics. The nonlinear equations of thermodynamics and hydrodynamics are analyzed using the methods of perturbation theory, with emphasis on the zero-order solution; zero-order states of an ideal gas; the first-order equations; the additive barotropic terms; and boundary conditions. The following chapters focus on the steady component of atmospheric pressure; free steady motion with or without rotation; field equations and general theorems relating to such equations; and the stratification of the Earth's atmosphere, oceans, and lakes. The next two chapters present calculations concerning the isothermal atmosphere, with particular reference to plane level surfaces with or without rotation. The final chapter looks at spherical level surfaces with rotation. This monograph will be of interest to physicists, oceanographers, atmospheric scientists, and meteorologists.

Table of Contents

  • Preface

    Chapter I. The Basic Equations

    1. Introduction

    2. Thermodynamics

    3. The Thermodynamic Functions of an Ideal Gas

    4. The Thermodynamic Coefficients for Water

    5. Hydrodynamics

    Gravitational Potential



    Accession of Heat, Advection

    Elimination of Entropy

    Conservation of Energy

    Chapter II. The Perturbation Equations

    6. Introduction

    7. The Zero-Order Solution

    Barotropic States

    The Gradients

    8. Zero-Order States of an Ideal Gas

    The Isothermal Atmosphere

    The Isentropic Atmosphere

    Atmosphere with Constant Temperature Gradient

    General Case

    9. The First-Order Equations


    Vertical Displacement

    10. The Additive Barotropic Terms

    11. Boundary Conditions

    Chapter III. Steady Motions

    12. Introduction

    Mean Pressure

    The Zonal Component

    The Tesseral Component

    The Oceans

    The Atmosphere and Oceans as a Heat Engine

    The Equations

    13. Free Steady Motion, No Rotation

    Plane Level Surfaces

    Spherical Level Surfaces

    14. Second-Order Instability; The Secular Equation

    Secular Equations

    Relation to the Vorticity Theorem

    A First Integral of the Secular Equation

    Relation to the Bernoulli Theorem

    15. Free Steady Motion with Rotation

    The Geostrophic Equation

    Relation between Density and Pressure

    Planetary Vorticity

    Plane Level Surfaces

    Spherical Level Surfaces

    16. Pure Convection, No Rotation

    The Vertical Velocity

    Boundary Conditions

    Consequences of the Conservation of Matter

    Explicit Solution of the Equations

    Plane Level Surfaces

    Spherical Level Surfaces

    17. Pure Convection, with Rotation

    Thermobaric Motion

    Density and Pressure

    Plane Level Surfaces

    Spherical Level Surfaces

    18. Hadley's Hypothesis of Zonal Heating

    No Rotation

    Instability of the Hadley Vortices

    19. Analysis of the Earth's Permanent Pressure Field

    The Zonal Component of Pressure

    The Tesseral Component of pressure

    The Effects of Terrain

    Chapter IV. The Field Equations

    20. Introduction

    21. The External and Thermobaric Energies

    22. The Field Variables

    Oceanic Case

    Isothermal Atmosphere

    Atmosphere with Constant Temperature Gradient

    23. The Field Equations

    24. Significance of the Coefficients N and T

    25. Special Formula for the Coefficients

    Perfect Gas

    Fresh Water


    Chapter V. The Earth's Atmosphere, Oceans and Lakes

    26. Introduction

    27. The Stratification of the Oceans

    Large-scale Averages

    Small-scale Averages

    28. The Stratification of Freshwater Lakes

    29. The Stratification of the Earth's Atmosphere

    30. Planetary Rotation and Cyclogenesis

    Early Ideas

    Observations of Redfield and Reid: Cyclones

    Tracy's Theory

    Hann's Theory

    Cyclones and Anticyclones


    Helmholtz and Bjerknes Vorticity Theorems

    31. First-Order Cyclogenesis

    No Rotation


    Rotation with Spherical Level Surfaces


    Chapter VI. General Theorems concerning the Field Equations

    32. Introduction

    33. The Eigensolutions

    34. The Expansion Theorem

    Expansion Theorem (for a Finite Volume)

    Expansion Theorem (for an Infinite Volume)

    Chapter VII. Formulation of the Major Mathematical Problems

    35. Introduction

    36. The Case of no Rotation

    Plane Level Surfaces

    Residual Equations

    Separation of Variables: The Two-dimensional Wave Equation

    Spherical Level Surfaces

    Residual Equations

    Separation of Variables

    Wave Equation for a Spherical Surface

    37. Rotation with Plane Level Surfaces

    The Traditional Approximations

    Separation of Variables

    Two-Dimensional Wave Equation

    Calculation without the Approximation

    Separation of Variables

    38. Rotation with Spherical Level Surfaces

    The Traditional Approximation

    Separation of Variables: Laplace's Tidal Equation

    Omission of the Traditional Approximations

    39. Complex Vectors and the Hodograph

    No Rotation

    Rotation: Traditional Approximation

    Rotation: Without Approximation

    Chapter VIII. The Isothermal Atmosphere: Plane Level Surfaces without Rotation

    40. Introduction

    41. Lamb's Waves

    42. Other Eigensolutions; Simple Waves

    43. The Propagation Surface; Phase Velocity

    44. Rays and Group Velocity

    45. The Pressure-Entropy Impedance

    46. The Flow and Partition of Energy in Simple Waves

    47. The Eigensolutions

    The Phase Diagram

    The Hodograph

    48. The Gravity Waves and the Fluctuating Wind

    Chapter IX. The Isothermal Atmosphere: Plane Level Surfaces with Rotation

    49. Vertical Axis of Rotation

    50. Lamb's Waves

    51. Simple Waves and Eigensolutions

    52. Sub-Critical Stability

    53. Inclined Axis of Rotation

    Lamb's Waves

    The Laws of Reflection


    Chapter X. Oceans with Constant Coefficients

    54. Introduction

    55. Theory of an Homogeneous Compressible Ocean

    56. Theory of a Stratified but Incompressible Ocean

    57. The General Case

    58. A Simple Approximation for the Internal Gravity Modes

    59. The Modes of a Rectangular Tank

    60. Other Lateral Boundaries

    Chapter XI. General Theory of Rays

    61. Introduction

    62. The Hamilton-Jacobi Equation

    63. Plane Level Surfaces: Vertical Axis

    General and Complete Solutions

    The Hamilton-Jacobi Function

    64. The Rays and Group Velocity

    Derivation of the Rays

    The Group Velocity

    Interpretation of the Ray-Point

    Explicit Equations for the Rays

    General Properties of the Rays

    Limiting Forms of the Rays

    65. Spherical Level Surfaces: no Rotation

    The Complete Solution

    The Rays; The Rays are Plane Curves

    Altitude of the Rays

    66. Spherical Level Surfaces with Rotation: Traditional Approximation

    The Tracks of the Rays

    Concerning the Approximation

    The Period of the Ray Tracks

    Altitude of the Rays

    Chapter XII. The Thermocline

    67. Formulation of the Problem

    68. Preliminary Discussion of the Rays

    69. The Sound Waves of Area II

    70. The Gravity Waves of Area III

    71. The Waves of Areas IV and V

    72. The Residual Equations

    Modification When the Thermocline Channel Extends to Surface or Bottom

    73. Analytic Solution of the Residual Equations

    74. Further Application of the W-K-B- Approximation

    The Acoustic Modes of Area II

    The Internal Gravity Modes of Area III

    75. The Two-Layer Model

    Chapter XIII. The Thermosphere

    76. Introduction

    77. The Case of no Rotation

    Formal Calculation of the Rays

    Interpretation of the Results

    78. Vertical Axis of Rotation

    79. Solution of the Residual Equations

    80. The W-K-B Approximation

    Chapter XIV. General Theory of the Residual Equations

    81. Introduction

    82. Canonic Form of the Residual Equations

    The Canonic Variables

    The Canonic Equations

    Constant Coefficients

    The Phase Diagram

    83. General Theorems concerning the Phase Paths

    84. Sturm's Comparison Theorem

    Sturm's Formula

    The Oscillation Theorems

    Dependence on the Parameters

    85. The W-K-B Approximation

    Chapter XV. Applications of the General Theory

    86. The Thermosphere

    87. The Modal Curves and the Comparison Theorem

    88. An Atmosphere with a Single Temperature Minimum

    89. The Modal Equation for an Ocean of Constant Depth

    Chapter XVI. The Wave Equation for a Spherical Surface

    90. Introduction

    91. The Legendre Functions

    Separation of Variables

    The Legendre Equation

    The Legendre functions

    Expressions for the Velocities

    Natural Boundary Conditions

    The Phase Diagrams

    92. Segmental Ocean

    Chapter XVII. Spherical Level Surfaces with Rotation

    93. Introduction

    94. The First-Order Tidal Equations

    95. The Zonal Oscillations

    Connection with Spheroidal Wave Functions

    The Functions Sml(h,τ)

    The Phase Paths

    96. The Solutions near the Poles

    97. The Tidal Equations in Canonic Form

    The Signatures

    The Phase Paths

    The Modal Curves

    Transformation to the (χ, ω) Plane

    98. The High-Frequency Limit

    99. The Semi-Diurnal Oscillations

    100. Oscillations of the Second Kind, and the "Long Waves"

    The Eigensolutions

    The Phase Paths

    Positive Speed

    The Rossby Waves

    Appendix: Mercator Co-ordinates


Product details

  • No. of pages: 302
  • Language: English
  • Copyright: © Pergamon 1960
  • Published: January 1, 1960
  • Imprint: Pergamon
  • eBook ISBN: 9781483149561

About the Author

Carl Eckart

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