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78
Surface quasigeostrophic dynamics
, 1995
"... The dynamics of quasigeostrophic flow with uniform potential vorticity reduces to the evolution of buoyancy, or potential temperature, on horizontal boundaries. There is a formal resemblance to twodimensional flow, with surface temperature playing the role of vorticity, but a different relationshi ..."
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Cited by 117 (3 self)
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The dynamics of quasigeostrophic flow with uniform potential vorticity reduces to the evolution of buoyancy, or potential temperature, on horizontal boundaries. There is a formal resemblance to twodimensional flow, with surface temperature playing the role of vorticity, but a different relationship between the flow and the advected scalar creates several distinctive features. A series of examples are described which highlight some of these features: the evolution of an elliptical vortex; the startup vortex shed by flow over a mountain; the instability of temperature filaments; the ‘edge wave’ critical layer; and mixing in an overturning edge wave. Characteristics of the direct cascade of the tracer variance to small scales in homogeneous turbulence, as well as the inverse energy cascade, are also described. In addition to its geophysical relevance, the ubiquitous generation of secondary instabilities and the possibility of finitetime collapse make this system a potentially important, numerically tractable, testbed for turbulence theories.
Upper Ocean Turbulence from HighResolution 3D Simulations
 JOURNAL OF PHYSICAL OCEANOGRAPHY
, 2008
"... The authors examine the turbulent properties of a baroclinically unstable oceanic flow using primitive equation (PE) simulations with high resolution (in both horizontal and vertical directions). Resulting dynamics in the surface layers involve large Rossby numbers and significant vortical asymmetri ..."
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Cited by 33 (5 self)
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The authors examine the turbulent properties of a baroclinically unstable oceanic flow using primitive equation (PE) simulations with high resolution (in both horizontal and vertical directions). Resulting dynamics in the surface layers involve large Rossby numbers and significant vortical asymmetries. Furthermore, the ageostrophic divergent motions associated with smallscale surface frontogenesis are shown to significantly alter the nonlinear transfers of kinetic energy and consequently the time evolution of the surface dynamics. Such impact of the ageostrophic motions explains the emergence of the significant cyclone–anticyclone asymmetry and of a strong restratification in the upper layers, which are not allowed by the quasigeostrophic (QG) or surface quasigeostrophic (SQG) theory. However, despite this strong ageostrophic character, some of the main surface properties are surprisingly still close to the surface quasigeostrophic equilibrium. They include a noticeable shallow (�k �2) velocity spectrum as well as a conspicuous local spectral relationship between surface kinetic energy, sea surface height, and density variance over a large range of scales (from 400 to 4 km). Furthermore, surface velocities can be remarkably diagnosed from only the surface density using SQG relations. This suggests that the validity of some specific SQG relations extends to dynamical regimes with large Rossby numbers. The interior dynamics, on the
Hamiltonian ODE’s in the Wasserstein space of probability measures
, 2000
"... In this paper we consider a Hamiltonian H on P2(R 2d), the set of probability measures with finite quadratic moments on the phase space R 2d = R d × R d, which is a metric space when endowed with the Wasserstein distance W2. We study the initial value problem dμt/dt +∇·(Jdvt μt) = 0, where Jd is th ..."
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Cited by 29 (7 self)
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In this paper we consider a Hamiltonian H on P2(R 2d), the set of probability measures with finite quadratic moments on the phase space R 2d = R d × R d, which is a metric space when endowed with the Wasserstein distance W2. We study the initial value problem dμt/dt +∇·(Jdvt μt) = 0, where Jd is the canonical symplectic matrix, μ0 is prescribed, vt is a tangent vector to P2(R 2d) at μt, and belongs to ∂H(μt), the subdifferential of H at μt. Two methods for constructing solutions of the evolutive system are provided. The first one concerns only the case where μ0 is absolutely continuous. It ensures that μt remains absolutely continuous and vt = ∇H(μt) is the element of minimal norm in ∂H(μt). The second method handles any initial measure μ0. If we furthermore assume that H is λ–convex, proper and lower semicontinuous on P2(R 2d), we prove that the Hamiltonian is preserved along any solution of our evolutive system: H(μt) = H(μ0).
Geometric Integration and Its Applications
 in Handbook of numerical analysis
, 2000
"... This paper aims to give an introduction to the relatively new eld of geometric integration. ..."
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Cited by 21 (2 self)
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This paper aims to give an introduction to the relatively new eld of geometric integration.
New equations for nearly geostrophic flow
 J. Fluid Mech
, 1985
"... I have used a novel approach based upon Hamiltonian mechanics to derive new equations for nearly geostrophic motion in a shallow homogeneous fluid. The equations have the same order accuracy as (say) the quasigeostrophic equations, but they allow orderone variations in the depth and Coriolis parame ..."
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Cited by 18 (1 self)
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I have used a novel approach based upon Hamiltonian mechanics to derive new equations for nearly geostrophic motion in a shallow homogeneous fluid. The equations have the same order accuracy as (say) the quasigeostrophic equations, but they allow orderone variations in the depth and Coriolis parameter. My equations exactly conserve proper analogues of the energy and potential vorticity, and they take a simple form in transformed coordinates. 1.
The EulerPoincaré equations in geophysical fluid dynamics
 IN PROCEEDINGS OF THE ISAAC NEWTON INSTITUTE PROGRAMME ON THE MATHEMATICS OF ATMOSPHERIC AND OCEAN DYNAMICS
, 2002
"... Recent theoretical work has developed the Hamilton’sprinciple analog of LiePoisson Hamiltonian systems defined on semidirect products. The main theoretical results are twofold: 1. Euler–Poincaré equations (the Lagrangian analog of LiePoisson Hamiltonian equations) are derived for a parameter depe ..."
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Cited by 17 (13 self)
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Recent theoretical work has developed the Hamilton’sprinciple analog of LiePoisson Hamiltonian systems defined on semidirect products. The main theoretical results are twofold: 1. Euler–Poincaré equations (the Lagrangian analog of LiePoisson Hamiltonian equations) are derived for a parameter dependent Lagrangian from a general variational principle of Lagrange d’Alembert type in which variations are constrained; 2. an abstract Kelvin–Noether theorem is derived for such systems. By imposing suitable constraints on the variations and by using invariance properties of the Lagrangian, as one does for the Euler equations for the rigid body and ideal fluids, we cast several standard Eulerian models of geophysical fluid dynamics (GFD) at various levels of approximation into EulerPoincaré form and discuss their corresponding Kelvin–Noether theorems and potential vorticity conservation laws. The various levels of GFD approximation are related
Feldman: Lagrangian solutions of semigeostrophic equations in physical space
 J. Math. Anal
"... The semigeostrophic equations are a simple model of largescale atmosphere/ocean flows, where ’largescale ’ is defined to mean that the flow is rotationdominated, [4]. They are also accurate in the case where one horizontal scale becomes small, allowing them to describe weather fronts and jet stre ..."
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Cited by 17 (3 self)
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The semigeostrophic equations are a simple model of largescale atmosphere/ocean flows, where ’largescale ’ is defined to mean that the flow is rotationdominated, [4]. They are also accurate in the case where one horizontal scale becomes small, allowing them to describe weather fronts and jet streams. Previous work by J.D. Benamou and