Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Equilibrium thermodynamics of sea water
- 3 Balance equations
- 4 Molecular flux laws
- 5 The gravitational potential
- 6 The basic equations
- 7 Dynamic impact of the equation of state
- 8 Free wave solutions on a sphere
- 9 Asymptotic expansions
- 10 Reynolds decomposition
- 11 Boussinesq approximation
- 12 Large-scale motions
- 13 Primitive equations
- 14 Representation of vertical structure
- 15 Ekman layers
- 16 Planetary geostrophic flows
- 17 Tidal equations
- 18 Medium-scale motions
- 19 Quasi-geostrophic flows
- 20 Motions on the f-plane
- 21 Small-scale motions
- 22 Sound waves
- Appendix A Equilibrium thermodynamics
- Appendix B Vector and tensor analysis
- Appendix C Orthogonal curvilinear coordinate systems
- Appendix D Kinematics of fluid motion
- Appendix E Kinematics of waves
- Appendix F Conventions and notation
- References
- Index
8 - Free wave solutions on a sphere
Published online by Cambridge University Press: 03 February 2010
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Equilibrium thermodynamics of sea water
- 3 Balance equations
- 4 Molecular flux laws
- 5 The gravitational potential
- 6 The basic equations
- 7 Dynamic impact of the equation of state
- 8 Free wave solutions on a sphere
- 9 Asymptotic expansions
- 10 Reynolds decomposition
- 11 Boussinesq approximation
- 12 Large-scale motions
- 13 Primitive equations
- 14 Representation of vertical structure
- 15 Ekman layers
- 16 Planetary geostrophic flows
- 17 Tidal equations
- 18 Medium-scale motions
- 19 Quasi-geostrophic flows
- 20 Motions on the f-plane
- 21 Small-scale motions
- 22 Sound waves
- Appendix A Equilibrium thermodynamics
- Appendix B Vector and tensor analysis
- Appendix C Orthogonal curvilinear coordinate systems
- Appendix D Kinematics of fluid motion
- Appendix E Kinematics of waves
- Appendix F Conventions and notation
- References
- Index
Summary
In this chapter we discuss free linear waves on a sphere, again for an ideal fluid. Waves are a basic mechanism by which a fluid adjusts to changes and propagates momentum and energy. Many of the approximations that are applied to the basic equations of oceanic motions and that form the remainder of this book can be characterized by the wave types that they eliminate. Free linear waves, as opposed to forced waves, are solutions of homogeneous linear equations. They can be superimposed. Mathematically, these homogeneous linear equations form an eigenvalue problem. The eigenfunctions determine the wave form, the eigenvalues the dispersion relation. Specifically, we consider linear waves on a stably stratified motionless background state. The linearized equations have separable eigensolutions. The vertical eigenvalue is characterized by the speed of sound, gravitation, and stratification. The horizontal eigenvalue problem is characterized by rotation and the spherical geometry of the earth. We then classify the various wave solutions into:
sound (or acoustic) waves;
surface and internal gravity waves; and
barotropic and baroclinic Rossby waves;
and consider the limit of short-wave solutions. This chapter owes much to Kamenkovich (1977) where more details and proofs can be found.
Basic kinematic concepts of waves, such as propagating and standing waves, and the geometric optics approximation are covered in Appendix E.
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- Chapter
- Information
- The Equations of Oceanic Motions , pp. 84 - 104Publisher: Cambridge University PressPrint publication year: 2006