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
19 - Quasi-geostrophic flows
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
Geostrophic flows require that the Rossby and Ekman numbers are small, Ro, Eh, Ev ≪ 1, so that the momentum balance reduces to the geostrophic balance. Quasi-geostrophic motions (or small-scale geostrophic motions) are geostrophic motions for which additionally:
the horizontal length scale is much smaller than the radius of the Earth, γ := L/r0 ≪ 1; and
the vertical displacement Z of isopycnals is much smaller than the vertical scale D, σ := Z/D ≪ 1.
The second condition is equivalent to requiring the vertical strain to be smaller than one. The quasi-geostrophic theory also disregards temperature–salinity effects and assumes a one-component fluid. The smallness of γ is exploited by applying the midlatitude beta-plane approximation. We will carry out the perturbation expansion with respect to all of the small parameters explicitly since the zeroth order is degenerate and one has to go to the first order. The evolution of quasi-geostrophic flows is again governed by the potential vorticity equation. The quasi-geostrophic vorticity consists of the relative vorticity, planetary vorticity, and a vertical strain contribution. The boundary conditions also include zeroth and first order contributions. Because of the inherent approximations, the quasi-geostrophic equations cannot include all dissipation and forcing processes. Linearization of the quasigeostrophic potential vorticity equation about a motionless stratified background state yields the Rossby wave solutions discussed in Chapter 8.
- Type
- Chapter
- Information
- The Equations of Oceanic Motions , pp. 201 - 212Publisher: Cambridge University PressPrint publication year: 2006