The mathematical structure of theories of semigeostrophic type

Roulstone, Ian; Sewell, M. J.

doi:10.1098/rsta.1997.0144

Cited by 11 publications

(12 citation statements)

References 27 publications

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“…The conditions under which these properties exist (a Monge{Amp ere equation with constant coe¯cients) may appear a little restrictive from the point of view of the hydrodynamics, but the identi cation of these geometries has, in turn, facilitated an approach to studying the duality structure of the models as described in x 4 c and x 4 d. A study with less restrictive integrability conditions is in progress. The duality properties of semi-geostrophic theory have been studied extensively elsewhere (see, for example, Roulstone & Sewell (1997) and references cited therein), and such features are of practical importance too, noting both the utility of balanced dynamics in understanding mid-latitude weather systems and the attempts to improve the representation of such weather systems in numerical models via, for example, the use of adaptive techniques in data-assimilation schemes (see, for example, Desroziers 1997).…”

Section: Discussionmentioning

confidence: 99%

“…Finally, we show how the contact structure of the semi-geostrophic equations, which is perhaps the most important mathematical property of these equations because it facilitates practical solution strategies (see Purser (1993) and references therein; see also Roulstone & Sewell (1997) and Sewell & Roulstone (1994)), generalizes to other near-local balanced models.…”

Section: (D ) a Contact Transformationmentioning

confidence: 99%

“…For a thorough discussion of this transformation, including its Legendre and contact properties, see Roulstone & Sewell (1997). The coordinates (X; Y ) are called geostrophic coordinates, because, when f is a constant, _ X = u g , _ Y = v g .…”

Section: Shallow-water Theory and Balanced Dynamicsmentioning

confidence: 99%

“…The reason is that semi-geostrophy has certain useful and elegant mathematical properties, in particular Hamiltonian structure, Legendre duality, and the convexity of certain potential functions (Chynoweth & Sewell 1989Roulstone & Norbury 1994;Roulstone & Sewell 1997). These permit both robust numerical integration| even in cases where frontal discontinuities form|and a rather complete knowledge of mathematical properties, such as existence and uniqueness of solutions (Cullen & Purser 1984;Purser & Cullen 1987;Cullen et al 1991).…”

Section: Introductionmentioning

confidence: 99%

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Holomorphic structures in hydrodynamical models of nearly geostrophic flow

Roubtsov

Roulstone

2001

Proc. R. Soc. Lond. A

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We study complex structures arising in Hamiltonian models of nearly geostrophic ®ows in hydrodynamics. In many of these models an elliptic Monge{Amp ere equation de nes the relationship between a`balanced' velocity eld, de ned by a constraint in the Hamiltonian formalism, and the materially conserved potential vorticity. Elliptic Monge{Amp ere operators de ne an almost-complex structure, and in this paper we show that a natural extension of the so-called geostrophic momentum transformation of semi-geostrophic theory, which has a special importance in theoretical meteorology, de nes K ahler and special K ahler structures on phase space. Furthermore, analogues of the`geostrophic momentum coordinates' are shown to be special Lagrangian coordinates under conditions which depend upon the physical approximations under consideration. Certain duality properties of the operators are studied within the framework of the K ahler geometry.

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Section: Discussionmentioning

confidence: 99%

Section: (D ) a Contact Transformationmentioning

confidence: 99%

Section: Shallow-water Theory and Balanced Dynamicsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Holomorphic structures in hydrodynamical models of nearly geostrophic flow

Roubtsov

Roulstone

2001

Proc. R. Soc. Lond. A

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“…In the semi-geostrophic approximation (cf Salmon (1988), Roulstone and Norbury (1994), Roulstone and Sewell (1997)), the shallow-water equations (2.5)-(2.8) are replaced by In contrast to corresponding result for the two-dimensional Euler equations (cf section 1), Ω is not the vorticity. It is however related to Belrami's material vorticity (cf Casey and Naghdi (1991), Viúdez (2001)).…”

Section: Semi-geostrophic Shallow-water Equationsmentioning

confidence: 99%

Vorticity and symplecticity in Lagrangian fluid dynamics

Bridges¹,

Hydon²,

Reich³

2005

J. Phys. A: Math. Gen.

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The relationship between potential vorticity (PV) and the symplectic form is explored, for the shallow-water equations governing Lagrangian particle paths. Starting with the symplectic form, the PV is found by the pullback operation to the reference space. At first sight, the encoding of PV in the symplectic form appears to be independent of the particle relabelling symmetry. The analysis is carried a step further in two ways. Using the 'conservation of symplecticity' as a starting point, the fluxes of symplecticity arise as differential forms, and a complete pull back of the flux forms leads to a geometric description of PV conservation. Secondly, symmetry methods are used to give a rigorous connection between particle relabelling, symplecticity and PV conservation. Generalizations of these issues to semi-geostrophic flow and three-dimensional Lagrangian fluid flows, and the connection with Ertel's theorem are also discussed.

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Generalized semi-geostrophic theory on a sphere

et al. 2005

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Douglas, Robert; Cullen, M.J.P.; Roulston, I.; Sewell, M.J., (2005) 'Generalized semi-geostrophic theory on a sphere', Journal of Fluid Mechanics 531 pp.123-157 RAE2008It is shown that the solution of the semi-geostrophic equations for shallow-water flow can be found and analysed in spherical geometry by methods similar to those used in the existing $f$-plane solutions. Stable states in geostrophic balance are identified as energy minimizers and a procedure for finding the minimizers is constructed, which is a form of potential vorticity inversion. This defines a generalization of the geostrophic coordinate transformation used in the $f$-plane theory. The procedure is demonstrated in computations. The evolution equations take a simple form in the transformed coordinates, though, as expected from previous work in the literature, they cannot be expressed exactly as geostrophic motion. The associated potential vorticity does not obey a Lagrangian conservation law, but it does obey a flux conservation law, with an associated circulation theorem. The divergence of the flow in the transformed coordinates is primarily that naturally associated with geostrophic motion, with additional terms coming from the curvature of the sphere and extra ?curvature? resulting from the variable Coriolis parameter in the generalized coordinate transformation. These terms are estimated, and are found to be very small for normal data. The estimate is verified in computations, confirming the accuracy of the local $f$-plane approximation usually made with semi-geostrophic theory.Peer reviewe

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The mathematical structure of theories of semigeostrophic type

Cited by 11 publications

References 27 publications

Holomorphic structures in hydrodynamical models of nearly geostrophic flow

Holomorphic structures in hydrodynamical models of nearly geostrophic flow

Vorticity and symplecticity in Lagrangian fluid dynamics

Generalized semi-geostrophic theory on a sphere

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