On the differential geometry approach to geophysical flows

Zeitlin, Vladimir; Pasmanter, R.A.

doi:10.1016/0375-9601(94)90818-4

Cited by 13 publications

(21 citation statements)

References 8 publications

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“…We begin with the case of f -plane dynamics, for which equations (1.2) and (1.6) for potential vorticity convection may be interpreted as geodesic equations on the group of symplectic diffeomorphisms in the same way as for the classical 2d Euler equations discussed in Arnold [17]. The difference is in the choice of metric which is determined by the relation between the stream-function and the (potential) vorticity [5], [18]. Without giving rigorous differential-geometric and functional-analytic meanings to this interpretation we shall simply discuss it heuristically, by using the guideline that in formulating a geodesic variational principle on a manifold it is convenient to deal with two sets of variables: those belonging to the linear (co-)tangent space (the dual of the Lie algebra of planar divergenceless vector fields in our case); and those belonging to the configuration manifold itself (the Lie group of area-preserving mappings in our case).…”

Section: Preliminaries: Variational Principles For Euler Dynamicsmentioning

confidence: 99%

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Hamilton’s principle for quasigeostrophic motion

Holm

Zeitlin

1998

Physics of Fluids

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We show that the equation of quasigeostrophic (QG) potential vorticity conservation in geophysical fluid dynamics follows from Hamilton's principle for stationary variations of an action for geodesic motion in the f -plane case or its prolongation in the β -plane case. This implies a new momentum equation and an associated Kelvin circulation theorem for QG motion. We treat the barotropic and two-layer baroclinic cases, as well as the continuously stratified case.

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Section: Preliminaries: Variational Principles For Euler Dynamicsmentioning

confidence: 99%

“…In particular, they are Hamiltonian with a Lie-Poisson bracket given in Weinstein [4] {F, G} = − dx 1 dx 2 q δF δµ , δG δµ , (1.3) where µ ≡ q − f . In terms of the variable µ, the Hamiltonian for QG is expressed as [4], [5].…”

Section: Introductionmentioning

confidence: 99%

Hamilton’s principle for quasigeostrophic motion

Holm

Zeitlin

1998

Physics of Fluids

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“…As mentioned earlier, in the absence of the tensor fields a and when l is the kinetic energy metric, the basic Euler-Poincaré equations are the geodesic spray equations for geodesic motion on the diffeomorphism group with respect to that metric. See, e.g., Arnold [1966a], Ovsienko and Khesin [1987], Zeitlin and Kambe [1993], Zeitlin and Pasmanter [1994], Ono [1995aOno [ , 1995b and Kouranbaeva [1997] for details in particular applications of ideal continuum mechanics.…”

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confidence: 99%

The Euler–Poincaré Equations and Semidirect Products with Applications to Continuum Theories

Holm

Marsden

Raţiu

1998

Advances in Mathematics

869

1,343

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We study Euler Poincare systems (i.e., the Lagrangian analogue of Lie Poisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler Poincare equations for a parameter dependent Lagrangian by using a variational principle of Lagrange d'Alembert type. Then we derive an abstract Kelvin Noether theorem for these equations. We also explore their relation with the theory of Lie Poisson Hamiltonian systems defined on the dual of a semidirect product Lie algebra. The Legendre transformation in such cases is often not invertible; thus, it does not produce a corresponding Euler Poincare system on that Lie algebra. We avoid this potential difficulty by developing the theory of Euler Poincare systems entirely within the Lagrangian framework. We apply the general theory to a number of known examples, including the heavy top, ideal compressible fluids and MHD. We also use this framework to derive higher dimensional Camassa Holm equations, which have many potentially interesting analytical properties. These equations are Euler Poincare equations for geodesics on diffeomorphism groups (in the sense of the Arnold program) but where the metric is H 1 rather than L 2 .

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“…We may rescale the metric on M so that the Reeb field has a different constant length α, and in this case the momentum takes the form m = α 2 f − f . Thus the Euler-Arnold equation on the quantomorphism group of M is the quasigeostrophic equation in f -plane approximation on N , as in Holm and Zeitlin (1998) and Zeitlin and Pasmanter (1994); here α 2 is the Froude number. An alternative approach to the quantomorphism group is to view it as a central extension of the group D Ham (N ) of Hamiltonian diffeomorphisms of the symplectic manifold N ; this approach is used in Ratiu and Schmid (1981) and is also taken in the references Tronci (2009), Gay-Balmaz andVizman (2012) and GayBalmaz and Tronci (2012).…”

Section: Corollary 43mentioning

confidence: 99%

Riemannian Geometry of the Contactomorphism Group

Ebin

Preston

2014

Arnold Math J.

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Given an odd-dimensional compact manifold and a contact form, we consider the group of contact transformations of the manifold (contactomorphisms) and the subgroup of those transformations that precisely preserve the contact form (quantomorphisms). If the manifold also has a Riemannian metric, we can consider the L 2 inner product of vector fields on it, which by restriction gives an inner product on the tangent space at the identity of each of the groups that we consider. We then obtain right-invariant metrics on both the contactomorphism and quantomorphism groups. We show that the contactomorphism group has geodesics at least for short time and that the quantomorphism group is a totally geodesic subgroup of it. Furthermore we show that the geodesics in this smaller group exist globally. Our methodology is to use the right invariance to derive an "Euler-Arnold" equation from the geodesic equation and to show using ODE methods that it has solutions which depend smoothly on the initial conditions. For global existence we then derive a "quasi-Lipschitz" estimate on the stream function, which leads to a Beale-Kato-Majda criterion which is automatically satisfied for quantomorphisms. Special cases of these Euler-Arnold equations are the Camassa-Holm equation (when the manifold is one-dimensional) and the quasi-geostrophic equation in geophysics.

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On the differential geometry approach to geophysical flows

Cited by 13 publications

References 8 publications

Hamilton’s principle for quasigeostrophic motion

Hamilton’s principle for quasigeostrophic motion

The Euler–Poincaré Equations and Semidirect Products with Applications to Continuum Theories

Riemannian Geometry of the Contactomorphism Group

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