Hamilton’s principle for quasigeostrophic motion

Holm, Darryl D.; Zeitlin, Vladimir

doi:10.1063/1.869623

Cited by 32 publications

(31 citation statements)

References 26 publications

(54 reference statements)

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“…The changes to be made to obtain the variational principle in a rotating system are also known: a vector-potential for the angular velocity Ω should be introduced (see Landau & Lifshitz 1976 and, for example, Holm & Zeitlin 1998 in the hydrodynamical context), by 'augmenting' the Lagrangian L X,Ẋ :…”

Section: Variational Principle For the Boussinesq Equationsmentioning

confidence: 99%

Consistent shallow-water equations on the rotating sphere with complete Coriolis force and topography

et al. 2014

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International audienceConsistent shallow-water equations are derived on the rotating sphere with topography retaining the Coriolis force due to the horizontal component of the planetary angular velocity. Unlike the traditional approximation, this 'non-traditional' approximation captures the increase with height of the solid-body velocity due to planetary rotation. The conservation of energy, angular momentum and potential vorticity are ensured in the system. The caveats in extending the standard shallow-water wisdom to the case of the rotating sphere are exposed. Different derivations of the model are possible, being based, respectively, on (i) Hamilton's principle for primitive equations with a complete Coriolis force, under the hypothesis of columnar motion, (ii) straightforward vertical averaging of the 'non-traditional' primitive equations, and (iii) a time-dependent change of independent variables in the primitive equations written in the curl ('vector-invariant') form, with subsequent application of the columnar motion hypothesis. An intrinsic, coordinate-independent form of the non-traditional equations on the sphere is then given, and used to derive hyperbolicity criteria and Rankine-Hugoniot conditions for weak solutions. The relevance of the model for the Earth's atmosphere and oceans, as well as other planets, is discussed

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Section: Variational Principle For the Boussinesq Equationsmentioning

confidence: 99%

Consistent shallow-water equations on the rotating sphere with complete Coriolis force and topography

et al. 2014

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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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“…Stating variational principles for QG in both Lagrangian and Eulerian representations, Holm and Zeitlin (1998) arrive at a momentum equation (see their equation (3.6)) that can be written in dimensional form for the f plane as…”

Section: The Variational Momentum Equation For Qgmentioning

confidence: 99%

“…As a sequel to Mohebalhojeh (2002), this note is intended to compare the variational and standard momentum representations for the quasi-geostrophic (QG) model. Holm and Zeitlin (1998) present variational principles and a momentum equation for QG on the β plane, which will be referred to as the variational momentum equation for QG. There is also a standard, conventional momentum representation for QG based on small Rossby-number expansions.…”

Section: Introductionmentioning

confidence: 99%

“…The indeterminacy referred to above gives us the freedom to close the QG equations and thus use them in infinitely many ways for applications like the initialization of a primitive-equation model through QG, initialization of QG as a balanced model, and what is known as wave-vortex decomposition. In this note, the indeterminacy is exploited to show that the variational momentum equation derived by Holm and Zeitlin (1998) for QG corresponds to a particular way of closing the standard, conventional momentum equations for QG. This note is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the momentum equation for the quasi‐geostrophic model

Mohebalhojeh

2009

Quart J Royal Meteoro Soc

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ABSTRACT:The momentum equation for the quasi-geostrophic (QG) model derived based on the conventional Rossbynumber expansions does not uniquely determine the QG motion up to first order in the Rossby number. There are infinitely many ways of closing the equations. The momentum equation for QG derived by Holm and Zeitlin in 1998 based on a variational formulation for QG is compared with that for the conventional Rossby-number expansions. The underlying assumption in the construction of the variational formulation is geostrophic velocity for the particles. It is shown that the variational momentum equation corresponds to a particular way of closing the conventional momentum equation for QG. The numerical results for potential vorticity (PV) inversion on a circular vortex indicate a smaller range of applicability and loss of accuracy for the variational momentum equation for QG when compared with the QG one that sets the first-order linearized potential vorticity to zero.

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Hamilton’s principle for quasigeostrophic motion

Cited by 32 publications

References 26 publications

Consistent shallow-water equations on the rotating sphere with complete Coriolis force and topography

Consistent shallow-water equations on the rotating sphere with complete Coriolis force and topography

Riemannian Geometry of the Contactomorphism Group

On the momentum equation for the quasi‐geostrophic model

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