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78
The fully compressible semigeostrophic system from meteorology.
 Arch. Rational Mech. Anal.
, 2003
"... Abstract The fully compressible semigeostrophic system is widely used in the modelling of largescale atmospheric flows. In this paper, we prove rigorously the existence of weak Lagrangian solutions of this system, formulated in the original physical coordinates. In addition, we provide an alterna ..."
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Abstract The fully compressible semigeostrophic system is widely used in the modelling of largescale atmospheric flows. In this paper, we prove rigorously the existence of weak Lagrangian solutions of this system, formulated in the original physical coordinates. In addition, we provide an alternative proof of the earlier result on the existence of weak solutions of this system expressed in the socalled geostrophic, or dual, coordinates. The proofs are based on the optimal transport formulation of the problem and on recent general results concerning transport problems posed in the Wasserstein space of probability measures.
Mesoscale subduction at the AlmeriaOran front: Part 1: Ageostrophic flow
 Journal of Marine Systems
, 2001
"... This paper presents a detailed diagnostic analysis of hydrographic and current meter data from three, rapidly repeated, finescale surveys of the AlmeriaOran front. Instability of the frontal boundary, between surface waters of Atlantic and Mediterranean origin, is shown to provide a mechanism for ..."
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This paper presents a detailed diagnostic analysis of hydrographic and current meter data from three, rapidly repeated, finescale surveys of the AlmeriaOran front. Instability of the frontal boundary, between surface waters of Atlantic and Mediterranean origin, is shown to provide a mechanism for significant heat transfer from the surface layers to the deep ocean in winter. The data were collected during the second observational phase of the EU funded OMEGA project on RRS Discovery cruise 224 during December 1996. High resolution hydrographic measurements using the towed undulating CTD vehicle, SeaSoar,. traced the subduction of Mediterranean Surface Water across the AlmeriaOran front. This subduction is shown to result from a significant baroclinic component to the instability of the frontal jet. The Qvector formulation of the omega equation is combined with a scale analysis to quantitatively diagnose vertical transport resulting from mesoscale ageostrophic circulation. The analyses are presented and discussed in the presence of satellite and airborne remotely sensed data; which provide the basis for a thorough and novel approach to the determination of observational error.
Rearrangements of vector valued functions, with application to atmospheric and oceanic ows
"... This paper establishes the equivalence of four de nitions of two vector valued functions being rearrangements, and gives a characterisation of the set of rearrangements of a prescribed function. The theory of monotone rearrangement ofavector valued function is used to show the existence and uniquene ..."
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This paper establishes the equivalence of four de nitions of two vector valued functions being rearrangements, and gives a characterisation of the set of rearrangements of a prescribed function. The theory of monotone rearrangement ofavector valued function is used to show the existence and uniqueness of the minimiser of an energy functional arising from a model for atmospheric and oceanic ow. At each xed time solutions are shown to be equal to the gradient of a convex function, verifying the conjecture of Cullen, Norbury and Purser. Key words Rearrangement of functions, semigeostrophic, variational problems, generalised solution.
Diracbracket approach to nearly geostrophic Hamiltonian balanced models
"... Dirac's theory of constrained Hamiltonian systems is applied to derive the Poisson structure of a class of balanced models describing the slow dynamics of geophysical flows. Working with the Poisson structure, instead of the canonical Hamiltonian structure previously considered in this conte ..."
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Dirac's theory of constrained Hamiltonian systems is applied to derive the Poisson structure of a class of balanced models describing the slow dynamics of geophysical flows. Working with the Poisson structure, instead of the canonical Hamiltonian structure previously considered in this context, allows the standard Eulerian description of fluids to be used, with no need for Lagrangian variables, and leads to completely explicit balanced equations of motion. The balanced models are derived for a class of multilayer, isentropic or isopycnal, hydrostatic models by constraining the velocity field to be an arbitrary pseudodi #erential function of the mass field. Particularization to the geostrophic constraint and a slight modification thereof provides the Poisson formulation of (a multilayer version of) Salmon's L1 model and of the semigeostrophic model, respectively. A higherorder balanced model is also derived using a constraint that is more accurate than geostrophy. Key words: Balance, geostrophy, atmospheric dynamics, Poisson structure PACS: 47.32.y, 45.20.Jj, 92.10.Ei, 92.60.Bh # J. Vanneste, Department of Mathematics and Statistics, University of Edinburgh, Mayfield Road, Edinburgh, EH9 3JZ, U.K. Email addresses: vanneste@maths.ed.ac.uk (J. Vanneste), o.bokhove@math.utwente.nl (O. Bokhove). Preprint submitted to Elsevier Science 31 August 2001 1
Potential vorticities in semigeostrophic theory
 Q. Jl R. Met. Soc
, 1996
"... We devise a family of kinematically possible motions in shallow water versions of semigeostrophic theory, with variable Coriolis parameter, for which two distinct measures of potential vorticity take the same value. We describe conditions for which this value is conserved following the actual motio ..."
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We devise a family of kinematically possible motions in shallow water versions of semigeostrophic theory, with variable Coriolis parameter, for which two distinct measures of potential vorticity take the same value. We describe conditions for which this value is conserved following the actual motion of a particle. We relate the results to the existing literature, in particular for a constant Coriolis parameter. 1
On the stability of boundary layers of incompressible Euler equations
, 1998
"... In this paper we investigate the stability and instability of boundary layers of incompressible Euler equations. ..."
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In this paper we investigate the stability and instability of boundary layers of incompressible Euler equations.
Reconstructing the ocean’s interior from surface data
 J. Phys. Oceanogr
, 2013
"... A newmethod is proposed for extrapolating subsurface velocity and density fields from sea surface density and sea surface height (SSH). In this, the surface density is linked to the subsurface fields via the surface quasigeostrophic (SQG) formalism, as proposed in several recent papers. The subsurfa ..."
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A newmethod is proposed for extrapolating subsurface velocity and density fields from sea surface density and sea surface height (SSH). In this, the surface density is linked to the subsurface fields via the surface quasigeostrophic (SQG) formalism, as proposed in several recent papers. The subsurface field is augmented by the addition of the barotropic and first baroclinic modes, whose amplitudes are determined by matching to the sea surface height (pressure), after subtracting the SQG contribution. An additional constraint is that the bottom pressure anomaly vanishes. The method is tested for three regions in the North Atlantic using data from a highresolution numerical simulation. The decomposition yields strikingly realistic subsurface fields. It is particularly successful in energetic regions like the Gulf Stream extension and at high latitudes where the mixed layer is deep, but it also works in less energetic eastern subtropics. The demonstration highlights the possibility of reconstructing threedimensional oceanic flows using a combination of satellite fields, for example, sea surface temperature (SST) and SSH, and sparse (or climatological) estimates of the regional depthresolved density. Themethod could be further elaborated to integrate additional subsurface information, such as mooring measurements. 1.
ConstrainedHamiltonian shallowwater dynamics on the sphere, in
, 2003
"... Abstract. Salmon’s nearly geostrophic model for rotating shallowwater flow is derived in full spherical geometry. The model, which results upon constraining the velocity field to the height field in Hamilton’s principle for rotating shallowwater dynamics, constitutes an important prototype of Hami ..."
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Abstract. Salmon’s nearly geostrophic model for rotating shallowwater flow is derived in full spherical geometry. The model, which results upon constraining the velocity field to the height field in Hamilton’s principle for rotating shallowwater dynamics, constitutes an important prototype of Hamiltonian balanced models. Instead of Salmon’s original approach, which consists in taking variations of particle paths at fixed Lagrangian labels and time, Holm’s approach is considered here, namely variations are taken on Lagrangian particle labels at fixed Eulerian positions and time. Unlike the classical quasigeostrophic model, Salmon’s is found to be sensitive to the differences between geographic and geodesic coordinates. One consequence of this result is that the β plane approximation, which is included in Salmon’s original derivation, is not consistent for this class of model. Key words: Hamilton’s principle, shallow water, balance, sphere 1.
WEAK STABILITY OF LAGRANGIAN SOLUTIONS TO THE SEMIGEOSTROPHIC EQUATIONS
, 901
"... Abstract. In [8], Cullen and Feldman proved existence of Lagrangian solutions for the semigeostrophic system in physical variables with initial potential vorticity in L p, p> 1. Here, we show that a subsequence of the Lagrangian solutions corresponding to a strongly convergent sequence of initial ..."
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Abstract. In [8], Cullen and Feldman proved existence of Lagrangian solutions for the semigeostrophic system in physical variables with initial potential vorticity in L p, p> 1. Here, we show that a subsequence of the Lagrangian solutions corresponding to a strongly convergent sequence of initial potential vorticities in L 1 converges strongly in L q, q < ∞, to a Lagrangian solution, in particular extending the existence result of Cullen and Feldman to the case p = 1. We also present a counterexample for Lagrangian solutions corresponding to a sequence of initial potential vorticities converging in BM. The analytical tools used include techniques from optimal transportation, Ambrosio’s results on transport by BV vector fields, and Orlicz spaces. 1. Semigeostrophic equations in physical and in dual variables Semigeostrophic equations are simplified models for largescale geophysical flows. These systems were introduced by Hoskins in [12], as part of a family of models for geophysical flows under approximate geostrophic balance, i.e. where Coriolis forces and horizontal gradients of pressure nearly balance. We refer the reader to [7] for
Spontaneous imbalance and hybrid vortex–gravity structures
"... After reviewing the background, this note discusses the recently discovered examples of hybrid propagating structures consisting of vortex dipoles and comoving gravity waves undergoing wave capture. It is shown how these examples fall outside the scope of the Lighthill theory of spontaneous imbalan ..."
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After reviewing the background, this note discusses the recently discovered examples of hybrid propagating structures consisting of vortex dipoles and comoving gravity waves undergoing wave capture. It is shown how these examples fall outside the scope of the Lighthill theory of spontaneous imbalance and, concomitantly, outside the scope of shallowwater dynamics. Besides the fact that going from shallowwater to continuous stratification allows disparate vertical scales – small for inertia–gravity waves and large for vortical motion – the key points are (1) that by contrast with cases covered by the Lighthill theory the wave source feels a substantial radiation reaction, when Rossby numbers R � 1, so that the source cannot be prescribed in advance, (2) that examples of this sort may supply exceptions to the general rule that spontaneous imbalance is exponentially small in R; and (3) that unsteady vortical motion in continuous stratification can stay close to balance thanks to three quite separate mechanisms. These are first the nearsuppression, by the Lighthill mechanism, of largescale imbalance (inertia–gravity waves of large horizontal scale) where “large” means large relative to a Rossby deformation length LD characterizing the vortical motion, second the flaccidity hence nearsteadiness of LDwide jets that meander and form loops, GulfStreamlike, on streamwise scales ≫ LD, and third the dissipation of smallscale imbalance by wave capture leading to wave breaking, generically probable in an environment of random shear and straining. Shallowwater models include the first two mechanisms but exclude the third.