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
18 - Medium-scale motions
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
Medium-scale motions are defined here as motions whose horizontal space scales L are much smaller than the Earth's radius r0 but still large enough that they are not directly affected by molecular diffusion. They are the middle component of our triple decomposition into large-, medium-, and small-scale motions that arises from two Reynolds decompositions. The first decomposition separates large- from medium- and small-scale motions. The second decomposition separates large- and medium-scale motions from small-scale motions. Large-scale motions must thus parametrize the eddy fluxes caused by medium- and small-scale motions. Mediumscale motions must prescribe the large-scale fields and parametrize the eddy fluxes caused by small-scale motions. Small-scale motions must prescribe the large- and medium-scale motions. Diffusion is molecular. Medium-scale motions thus face two closure problems, one with respect to larger scale motions and one with respect to smaller scale motions. The only simplification is one of geometry. The smallness of the parameter γ := L/r0 allows the spherical geometry to be approximated by a variety of “planar” geometries. These approximations are only valid locally and include:
midlatitude beta-plane approximation;
equatorial beta-plane approximation;
f-plane approximation; and
polar plane approximation.
These are geometric approximations that are similar to the spherical approximation that relied on the smallness of Earth's eccentricity d20 /r20 and of the parameter H/r0, and led to the introduction of pseudo-spherical coordinates.
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- Chapter
- Information
- The Equations of Oceanic Motions , pp. 193 - 200Publisher: Cambridge University PressPrint publication year: 2006