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  • 1
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    Large-scale semigeostrophic equations for use in ocean circulation models (1996)
    Cambridge University Press
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    Publication Date: 1996-07-01
    Description: Hamiltonian approximation methods yields approximate dynamical equations that apply to nearly geostrophic flow at scales larger than the internal Rossby deformation radius. These equations incorporate fluid inertia with the same order of accuracy as the semi-geostrophic equations, but are nearly as simple (in appropriate coordinates) as the equations obtained by completely omitting the inertia.
    Print ISSN: 0022-1120
    Electronic ISSN: 1469-7645
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
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  • 2
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    Practical use of Hamilton's principle (1983)
    Cambridge University Press
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    Publication Date: 1983-07-01
    Description: Hamilton's principle of mechanics has special advantages as the beginning point for approximations. First, it is extremely succinct. Secondly, it easily accommodates moving disconnecting fluid boundaries. Thirdly, approximations-however strong-that maintain the symmetries of the Hamiltonian will automatically preserve the corresponding conservation laws. For example, Hamilton’s principle allows useful analytical and numerical approximations to the equations governing the motion of a homogeneous rotating fluid with free boundaries. © 1983, Cambridge University Press. All rights reserved.
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  • 3
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    Analogous formulation of electrodynamics and two-dimensional fluid dynamics (2014)
    Cambridge University Press
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    Publication Date: 2014-11-18
    Description: A single, simply stated approximation transforms the equations for a two-dimensional perfect fluid into a form that is closely analogous to Maxwell's equations in classical electrodynamics. All the fluid conservation laws are retained in some form. Waves in the fluid interact only with vorticity and not with themselves. The vorticity is analogous to electric charge density, and point vortices are the analogues of point charges. The dynamics is equivalent to an action principle in which a set of fields and the locations of the point vortices are varied independently. We recover classical, incompressible, point vortex dynamics as a limiting case. Our full formulation represents the generalization of point vortex dynamics to the case of compressible flow. © 2014 Cambridge University Press.
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  • 4
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    An alternative view of generalized Lagrangian mean theory (2013)
    Cambridge University Press
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    Publication Date: 2013-02-19
    Description: If the variables describing wave-mean flow interactions are chosen to include a set of fluid-particle labels corresponding to the mean flow, then the generalized Lagrangian mean (GLM) theory takes the form of an ordinary classical field theory. Its only truly distinctive features then arise from the distinctive feature of fluid dynamics as a field theory, namely, the particle-relabelling symmetry property, which corresponds by Noether's theorem to the many vorticity conservation laws of fluid mechanics. The key feature of the formulation is that all the dependent variables depend on a common set of space-time coordinates. This feature permits an easy and transparent derivation of the GLM equations by use of the energy-momentum tensor formalism. The particle-relabelling symmetry property leads to the GLM potential vorticity law in which pseudo-momentum is the only wave activity term present. Thus the particle-relabelling symmetry explains the prominent importance of pseudo-momentum in GLM theory. © Cambridge University Press 2013.
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  • 5
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    Entropy budget and coherent structures associated with a spectral closure model of turbulence (2018)
    Cambridge University Press
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    Publication Date: 2018-10-29
    Description: We 'derive' the eddy-damped quasi-normal Markovian model (EDQNM) by a method that replaces the exact equation for the Fourier phases with a solvable stochastic model, and we analyse the entropy budget of the EDQNM. We show that a quantity that appears in the probability distribution of the phases may be interpreted as the rate at which entropy is transferred from the Fourier phases to the Fourier amplitudes. In this interpretation, the decrease in phase entropy is associated with the formation of structures in the flow, and the increase of amplitude entropy is associated with the spreading of the energy spectrum in wavenumber space. We use Monte Carlo methods to sample the probability distribution of the phases predicted by our theory. This distribution contains a single adjustable parameter that corresponds to the triad correlation time in the EDQNM. Flow structures form as the triad correlation time becomes very large, but the structures take the form of vorticity quadrupoles that do not resemble the monopoles and dipoles that are actually observed. © 2018 Cambridge University Press.
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  • 6
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    Variational treatment of inertia–gravity waves interacting with a quasi-geostrophic mean flow (2016)
    Cambridge University Press
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    Publication Date: 2016-11-14
    Description: The equations for three-dimensional hydrostatic Boussinesq dynamics are equivalent to a variational principle that is closely analogous to the variational principle for classical electrodynamics. Inertia-gravity waves are analogous to electromagnetic waves, and available potential vorticity (i.e. the amount by which the potential vorticity exceeds the potential vorticity of the rest state) is analogous to electric charge. The Lagrangian can be expressed as the sum of three parts. The first part corresponds to quasi-geostrophic dynamics in the absence of inertia-gravity waves. The second part corresponds to inertia-gravity waves in the absence of quasi-geostrophic flow. The third part represents a coupling between the inertia-gravity waves and quasi-geostrophic motion. This formulation provides the basis for a general theory of inertia-gravity waves interacting with a quasi-geostrophic mean flow. © 2016 Cambridge University.
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  • 7
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    Weakly dispersive nonlinear gravity waves (1985)
    Cambridge University Press
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    Publication Date: 1985-08-01
    Description: The equations for gravity waves on the free surface of a laterally unbounded inviscid fluid of uniform density and variable depth under the action of an external pressure are derived through Hamilton's principle on the assumption that the fluid moves in vertical columns. The resulting equations are equivalent to those of Green & Naghdi (1976). The conservation laws for energy, momentum and potential vorticity are inferred directly from symmetries of the Lagrangian. The potential vorticity vanishes in any flow that originates from rest; this leads to a canonical formulation in which the evolution equations are equivalent, for uniform depth, to Whitham's (1967) generalization of the Boussinesq equations, in which dispersion, but not nonlinearity, is assumed to be weak. The further approximation that nonlinearity and dispersion are comparably weak leads to a canonical form of Boussinesq's equations that conserves consistent approximations to energy, momentum (for a level bottom) and potential vorticity. © 1985, Cambridge University Press. All rights reserved.
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  • 8
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    New equations for nearly geostrophic flow (1985)
    Cambridge University Press
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    Publication Date: 1985-04-01
    Description: 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 order-one 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. © 1985, Cambridge University Press. All rights reserved.
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  • 9
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    The equilibrium statistical mechanics of simple quasi-geostrophic models (1976)
    Cambridge University Press
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    Publication Date: 1976-06-25
    Description: We have applied the methods of classical statistical mechanics to derive the inviscid equilibrium states for one- and two-layer nonlinear quasi-geostrophic flows, with and without bottom topography and variable rotation rate. In the one-layer case without topography we recover the equilibrium energy spectrum given by Kraichnan (1967). In the two-layer case, we find that the internal radius of deformation constitutes an important dividing scale: at scales of motion larger than the radius of deformation the equilibrium flow is nearly barotropic, while at smaller scales the stream functions in the two layers are statistically uncorrelated. The equilibrium lower-layer flow is positively correlated with bottom topography (anticyclonic flow over seamounts) and the correlation extends to the upper layer at scales larger than the radius of deformation. We suggest that some of the statistical trends observed in non-equilibrium flows may be looked on as manifestations of the tendency for turbulent interactions to maximize the entropy of the system. © 1976, Cambridge University Press. All rights reserved.
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  • 10
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    Semigeostrophic theory as a Dirac-bracket projection (1988)
    Cambridge University Press
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    Publication Date: 1988-11-01
    Description: This paper presents a general method for deriving approximate dynamical equations that satisfy a prescribed constraint typically chosen to filter out unwanted high-frequency motions. The approximate equations take a simple general form in arbitrary phase-space coordinates. The family of semigeostrophic equations for rapidly rotating flow derived by Salmon (1983, 1985) fits this general form when the chosen constraint is geostrophic balance. More precisely, the semigeostrophic equations are equivalent to a Dirac-bracket projection of the exact Hamiltonian fluid dynamics onto the phase-space manifold corresponding to geostrophically balanced states. The more widely used quasi-geostrophic equations do not fit the general form, and are instead equivalent to a metric projection of the exact dynamics on to the same geostrophic manifold. The metric, which corresponds to the Hamiltonian of the linearized dynamics, is an artificial component of the theory, and its presence explains why the quasi-geostrophic equations are valid only near a state with fiat isopycnals. © 1988, Cambridge University Press. All rights reserved.
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