# Hydrodynamics of Oceans and Atmospheres

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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

Rotation

Forces

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

Interpretations

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

Sea-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

Vorticity

Helmholtz and Bjerknes Vorticity Theorems

31. First-Order Cyclogenesis

No Rotation

Rotation

Rotation with Spherical Level Surfaces

Summary

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

Conclusions

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

Index

## 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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