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Semigeostrophic theory as a Dirac-bracket projection

  • Rick Salmon (a1)


This paper presents a general method for deriving approximate dynamical equations that satisfy a prescribed constraint typically chosen to filter out unwanted high frequency motions. The approximate equations take a simple general form in arbitrary phase-space coordinates. The family of semigeostrophic equations for rotating flow derived by Salmon (1983, 1985) fits this general form when the chosen constraint is geostrophic balance. More precisely, the semigeostrophic equations are equivalent to a Dirac-bracket projection of the exact Hamiltonian ønto the phase-space manifold corresponding to geostrophically balanced states. The more widely used quasi-geostrophic equations do not fit the general form and are instead equivalent to a metric projection of the exact dynamics on to the same geostrophic manifold. The metric, which corresponds to the Hamiltonian of the linearized dynamics, is an artificial component of the theory, and its presence explains why the quasi-geostrophic equations are valid only near a state isopycnals.



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Dirac, P. A. M. 1950 Generalized Hamiltonian dynamics. Can. J. Math. 2, 129148.
Dirac, P. A. M. 1958 Generalized Hamiltonian dynamics. Proc. R. Soc. Lond. A 246, 326332.
Dirac, P. A. M. 1964 Lectures on Quantum Physics. Belfer.
Hanson, A., Regge, T. & Teitelboim, C. 1976 Constrained Hamiltonian systems. Accademia Nazionale dei Lincei, contribution 22.
Hoskins, B. J. 1975 The geostrophic momentum approximation and the semi-geostrophic equations. J. Atmos. Sci. 32, 233242.
Hoskins, B. J., McIntyre, M. E. & Robertson, A. W. 1985 On the use and significance of isentropic potential vorticity maps. Q. J. R. Met. Soc. 111, 877946.
Leith, C. E. 1980 Nonlinear normal mode initialization and quasi-geostrophic theory. J. Atmos. Sci. 37, 958968.
Salmon, R. 1983 Practical use of Hamilton's principle. J. Fluid Mech. 132, 431444.
Salmon, R. 1985 New equations for nearly geostrophic flow. J. Fluid Mech. 153, 461477.
Salmon, R. 1988 Hamiltonian fluid mechanics. Ann. Rev. Fluid Mech. 20, 225256.
Schutz, B. F. 1980 Geometrical Methods of Mathematical Physics. Cambridge University Press.
Sudarshan, E. C. G. & Mukunda, N. 1983 Classical Dynamics: a Modern Perspective. Krieger.
Sundermeyer, K. 1982 Constrained Dynamics. Springer.
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Semigeostrophic theory as a Dirac-bracket projection

  • Rick Salmon (a1)


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