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A variational derivation of the geostrophic momentum approximation

Published online by Cambridge University Press:  16 June 2014

Marcel Oliver
Marcel Oliver*
Affiliation:
School of Engineering and Science, Jacobs University, 28759 Bremen, Germany
*
†Email address for correspondence: oliver@member.ams.org
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Abstract

This paper demonstrates that the shallow water semigeostrophic equations arise from a degenerate second-order Hamilton principle of very special structure. The associated Euler–Lagrange operator factors into a fast and a slow first-order operator; restricting to the slow part yields the geostrophic momentum approximation as balanced dynamics. While semigeostrophic theory has been considered variationally before, this structure appears to be new. It leads to a straightforward derivation of the geostrophic momentum approximation and its associated potential vorticity law. Our observations further affirm, from a different point of view, the known difficulty in generalizing the semigeostrophic equations to the case of a spatially varying Coriolis parameter.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Rotating flows Geophysical and Geological Flows: Shallow water flows
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 751 , 25 July 2014 , R2
Copyright
© 2014 Cambridge University Press 

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References

Ambrosio, L., Colombo, M., De Philippis, G. & Figalli, A. 2012 Existence of Eulerian solutions to the semigeostrophic equations in physical space: the 2-dimensional periodic case. Commun. Part. Diff. Equ. 37, 2209–2227.Google Scholar
Andrews, D. G. & McIntyre, M. E. 1978 An exact theory of waves on a Lagrangian-mean flow. J. Fluid Mech. 89, 609–646.CrossRefGoogle Scholar
Arnold, V. I. & Khesin, B. A. 1998 Topological Methods in Hydrodynamics. Springer.Google Scholar
Benamou, J.-D. & Brenier, Y. 1998 Weak existence for the semigeostrophic equations formulated as a coupled Monge–Ampère/transport problem. SIAM J. Appl. Maths 58, 1450–1461.Google Scholar
Blumen, W. 1981 The geostrophic coordinate transformation. J. Atmos. Sci. 38, 1100–1105.Google Scholar
Bretherton, F. P. 1970 A note on Hamilton’s principle for perfect fluids. J. Fluid Mech. 44, 19–31.Google Scholar
Bridges, T., Hydon, P. & Reich, S.2001 Vorticity and symplecticity in Lagrangian fluid dynamics. Preprint.Google Scholar
Cullen, M. & Gangbo, W. 2001 A variational approach for the 2-dimensional semi-geostrophic shallow water equations. Arch. Rat. Mech. Anal. 156, 241–273.Google Scholar
Cullen, M. J. P. 2006 A Mathematical Theory of Large-Scale Atmosphere/Ocean Flow. Imperial College Press.Google Scholar
Cullen, M. J. P. 2008 Analysis of the semi-geostrophic shallow water equations. Physica D 237, 1461–1465.Google Scholar
Cullen, M. J. P., Douglas, R. J., Roulstone, I. & Sewell, M. J. 2005 Generalized semi-geostrophic theory on a sphere. J. Fluid Mech. 531, 123–157.CrossRefGoogle Scholar
Cullen, M. J. P., Norbury, J., Purser, R. J. & Shutts, G. J. 1987 Modelling the quasi-equilibrium dynamics of the atmosphere. Q. J. R. Meteorol. Soc. 113, 735–757.Google Scholar
Cullen, M. J. P. & Purser, R. J. 1984 An extended Lagrangian theory of semigeostrophic frontogenesis. J. Atmos. Sci. 41, 1477–1497.2.0.CO;2>CrossRefGoogle Scholar
Cullen, M. J. P. & Purser, R. J. 1989 Properties of the Lagrangian semigeostrophic equations. J. Atmos. Sci. 46, 2684–2697.Google Scholar
Eliassen, A. 1948 The quasi-static equations of motion with pressure as an independent variable. Geophys. Publ. 17, 1–44.Google Scholar
Eliassen, A. 1962 On the vertical circulation in frontal zones. Geophys. Publ. 24, 147–160.Google Scholar
Figalli, A. 2013 Sobolev regularity for the Monge–Ampère equation, with application to the semigeostrophic equations. Zap. Nauchn. Sem. POMI 411, 103–118; Representation Theory, Dynamical Systems, Combinatorial Methods. Part XXII.Google Scholar
Holm, D. D. 2002 Averaged Lagrangians and the mean effects of fluctuations in ideal fluid dynamics. Physica D 170, 253–286.CrossRefGoogle Scholar
Hoskins, B. J. 1975 The geostrophic momentum approximation and the semi-geostrophic equations. J. Atmos. Sci. 32, 233–242.2.0.CO;2>CrossRefGoogle Scholar
Jian, H.-Y. & Wang, X.-J. 2007 Continuity estimates for the Monge–Ampère equation. SIAM J. Math. Anal. 39, 608–626.CrossRefGoogle Scholar
Lychagin, V. V., Rubtsov, V. N. & Chekalov, I. V. 1993 A classification of Monge–Ampère equations. Ann. Sci. Ècole Norm. Sup. (4) 26, 281–308.Google Scholar
Marsden, J. E. & Ratiu, T. S. 1994 Introduction to Mechanics and Symmetry. Springer-Verlag.CrossRefGoogle Scholar
Marsden, J. E. & Shkoller, S. 2001 Global well-posedness for the Lagrangian averaged Navier–Stokes (LANS- $\alpha $ ) equations on bounded domains. Phil. Trans. R. Soc. Lond. A 359, 1449–1468.Google Scholar
McIntyre, M. E. & Roulstone, I. 2002 Are there higher-accuracy analogues of semigeostrophic theory? In Large-scale Atmosphere–Ocean Dynamics: II: Geometric Methods and Models (ed. Roulstone, I. & Norbury, J.), Cambridge University Press.Google Scholar
Oliver, M. 2006 Variational asymptotics for rotating shallow water near geostrophy: a transformational approach. J. Fluid Mech. 551, 197–234.CrossRefGoogle Scholar
Oliver, M. & Vasylkevych, S. 2013 Generalized LSG models with varying Coriolis parameter. Geophys. Astrophys. Fluid Dyn. 107, 259–276.Google Scholar
Phillips, N. A. 1963 Geostrophic motion. Rev. Geophys. 1, 123–176.CrossRefGoogle Scholar
Reznik, G. M., Zeitlin, V. & Ben Jelloul, M. 2001 Nonlinear theory of geostrophic adjustment. Part 1. Rotating shallow-water model. J. Fluid Mech. 445, 93–120.Google Scholar
Salmon, R. 1983 Practical use of Hamilton’s principle. J. Fluid Mech. 132, 431–444.Google Scholar
Salmon, R. 1985 New equations for nearly geostrophic flow. J. Fluid Mech. 153, 461–477.Google Scholar
Salmon, R. 1988 Semi-geostrophic theory as a Dirac-bracket projection. J. Fluid Mech. 196, 345–358.Google Scholar
Salmon, R. 1998 Lectures on Geophysical Fluid Dynamics. Oxford University Press.Google Scholar
Salmon, R. 2013 An alternative view of generalized Lagrangian mean theory. J. Fluid Mech. 719, 165–182.Google Scholar
Vanneste, J. & Bokhove, O. 2002 Dirac-bracket approach to nearly geostrophic Hamiltonian balanced models. Physica D 164, 152–167.CrossRefGoogle Scholar
Yudin, M. I. 1955 Invariant quantities in large-scale atmospheric processes. Tr. Glav. Geofiz. Obser. 55, 3–12; (in Russian).Google Scholar