We {\textquoteright}derive{\textquoteright} 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.

}, keywords = {2-dimensional turbulence, approximation, dynamics, homogeneous turbulence, mechanics, physics, turbulence theory, turbulent flows}, isbn = {0022-1120}, doi = {10.1017/jfm.2018.778}, url = {A single, simply stated approximation transforms the equations for a two-dimensional perfect fluid into a form that is closely analogous to Maxwell{\textquoteright}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.

}, keywords = {mathematical foundations, model, sound, vortex, vortex dynamics, wave-turbulence interactions, waves}, isbn = {0022-1120}, doi = {10.1017/jfm.2014.642}, url = {Ordinary two-dimensional turbulence corresponds to a Hamiltonian dynamics that conserves energy and the vorticity on fluid particles. This paper considers coupled systems of two-dimensional turbulence with three distinct governing dynamics. One is a Hamiltonian dynamics that conserves the vorticity on fluid particles and a quantity analogous to the energy that causes the system members to develop a strong correlation in velocity. The other two dynamics considered are non-Hamiltonian. One conserves the vorticity on particles but has no conservation law analogous to energy conservation; the other conserves energy and enstrophy but it does not conserve the vorticity on fluid particles. The coupled Hamiltonian system behaves like two-dimensional turbulence, even to the extent of forming isolated coherent vortices. The other two dynamics behave very differently, but the behaviours of all four dynamics are accurately predicted by the methods of equilibrium statistical mechanics.

}, keywords = {dynamics, homogeneous turbulence, turbulence theory}, isbn = {0022-1120}, doi = {10.1017/jfm.2013.422}, url = {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{\textquoteright}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.

}, keywords = {flow interaction, fluid, mathematical foundations, variational methods, wave-action, waves in rotating fluids}, isbn = {0022-1120}, doi = {10.1017/jfm.2012.638}, url = {We apply equilibrium statistical mechanics based upon the conservation of energy and potential enstrophy to the mass-density distribution within the ocean, using a Monte Carlo method that conserves the buoyancy of each fluid particle. The equilibrium state resembles the buoyancy structure actually observed. (C) 2011 Elsevier B.V. All rights reserved.

}, keywords = {flow, models, Monte Carlo method, ocean circulation, Statistical mechanics, turbulence}, isbn = {1007-5704}, doi = {10.1016/j.cnsns.2011.05.044}, url = {Using the Monte Carlo method of statistical physics, we compute the equilibrium statistical mechanics of the shallow water equations, considered as a reduced-gravity model of the ocean{\textquoteright}s upper layer in a square ocean that spans the equator. The ensemble-averaged flow comprises a westward drift at low latitudes, associated with the poleward deepening of the main thermocline, and a more intense compensating eastward flow near the latitudes at which the layer depth vanishes. Inviscid numerical simulations with a model that exactly conserves mass, energy, and potential enstrophy support the theoretical prediction.

}, keywords = {circulation, equations, equilibrium statistical-mechanics, flow, ocean, shallow-water model, topography}, isbn = {0022-2402}, doi = {10.1357/002224010794657182}, url = {We extend a previously developed method for constructing shallow water models that conserve energy and potential enstrophy to the case of flow bounded by rigid walls. This allows the method to be applied to ocean models. Our procedure splits the dynamics into a set of prognostic equations for variables (vorticity, divergence, and depth) chosen for their relation to the Casimir invariants of mass, circulation and potential enstrophy, and a set of diagnostic equations for variables that are the functional derivatives of the Hamiltonian with respect to the chosen prognostic variables. The form of the energy determines the form of the diagnostic equations. Our emphasis on conservation laws produces a novel form of the boundary conditions, but numerical test cases demonstrate the accuracy of our model and its extreme robustness, even in the case of vanishing viscosity.

}, keywords = {equations, general-method, scheme, Vorticity}, isbn = {0022-2402}, url = {This paper offers a simple, entirely prognostic, ocean circulation model based on the separation of the complete dynamics, including sound waves, into elementary Poisson brackets. For example, one bracket corresponds to the propagation of sound waves in a single direction. Other brackets correspond to the rotation of the velocity vector by individual components of the vorticity and to the action of buoyancy force. The dynamics is solved by Strang splitting of the brackets. Key features of the method are the assumption that the sound waves propagate exactly one grid distance in a time step and the use of Riemann invariants to solve the sound-wave dynamics exactly. In these features the method resembles the lattice Boltzmann method, but the flexibility of more conventional methods is retained. As in the lattice Boltzmann method, very short time steps are required to prevent unrealistically strong coupling between the sound waves and the slow hydrodynamic motions of primary interest. However, the disadvantage of small time steps is more than compensated by the model{\textquoteright}s extreme simplicity, even in the presence of very complicated boundaries, and by its massively parallel form. Numerical tests and examples illustrate the practicality of the method.

}, keywords = {energy, equations, flows, general-method, potential-enstrophy, system}, isbn = {0022-3670}, doi = {10.1175/2009jpo4134.1}, url = {The shallow-water equations may be posed in the form dF/dt = {F, H, Z}, where H is the energy, Z is the potential enstrophy, and the Nambu bracket {F, H, Z} is completely antisymmetric in its three arguments. This makes it very easy to construct numerical models that conserve analogs of the energy and potential enstrophy; one need only discretize the Nambu bracket in such a way that the antisymmetry property is maintained. Using this strategy, this paper derives explicit finite-difference approximations to the shallow-water equations that conserve mass, circulation, energy, and potential enstrophy on a regular square grid and on an unstructured triangular mesh. The latter includes the regular hexagonal grid as a special case.

}, keywords = {dynamics, equations, scheme}, isbn = {0022-4928}, doi = {10.1175/jas3837.1}, url = {Nambu proposed a generalization of Hamiltonian dynamics in the form dF/dt = {F, H, Z}, which conserves H and Z because the Nambu bracket {F, H, Z} is completely antisymmetric. The equations of fluid dynamics fit Nambu{\textquoteright}s form with H the energy and Z a quantity related to potential vorticity. This formulation makes it easy, in principle, to construct numerical fluid-models that conserve analogues of H and Z; one need only discretize the Nambu bracket in such a way that the antisymmetry property is preserved. In practice, the bracket may contain apparent singularities that are cancelled by the functional derivatives of Z. Then the discretization must be carried out in such a way that the cancellation is maintained. Following this strategy, we derive numerical models of the shallow-water equations and the equations for incompressible flow in two and three dimensions. The models conserve the energy and an arbitrary moment of the potential vorticity. The conservation of potential enstrophy - the second moment of potential vorticity - is thought to be especially important because it prevents the spurious cascade of energy into high wavenumbers.

}, keywords = {dynamics, enstrophy, mechanics, shallow-water equations}, isbn = {0951-7715}, doi = {10.1088/0951-7715/18/5/r01}, url = {This paper derives a set of two-dimensional equations describing a thin inviscid fluid layer flowing over topography in a frame rotating about an arbitrary axis. These equations retain various terms involving the locally horizontal components of the angular velocity vector that are discarded in the usual shallow water equations. The obliquely rotating shallow water equations are derived both by averaging the three-dimensional equations and from an averaged Lagrangian describing columnar motion using Hamilton{\textquoteright}s principle. They share the same conservation properties as the usual shallow water equations, for the same energy and modified forms of the momentum and potential vorticity. They may also be expressed in noncanonical Hamiltonian form using the usual shallow water Hamiltonian and Poisson bracket. The conserved potential vorticity takes the standard shallow water form, but with the vertical component of the rotation vector replaced by the component locally normal to the surface midway between the upper and lower boundaries. (c) 2005 American Institute of Physics.

}, keywords = {2-fluid system, conservation-laws, deep atmospheres, earths angular velocity, hamiltonian-structure, horizontal component, layer model, mesoscale ocean, model, quasi-hydrostatic equations, solar tachocline}, isbn = {1070-6631}, doi = {10.1063/1.21156747}, url = {Arakawa and Lamb discovered a finite-difference approximation to the shallow-water equations that exactly conserves finite-difference approximations to the energy and potential enstrophy of the fluid. The Arakawa Lamb (AL) algorithm is a stunning and important achievement-stunning, because in the shallow-water case, neither energy nor potential enstrophy is a simple quadratic, and important because the simultaneous conservation of energy and potential enstrophy is known to prevent the spurious cascade of energy to high wavenumbers. However, the method followed by AL is somewhat ad hoc, and it is difficult to see how it might be generalized to other systems. In this paper, the AL algorithm is rederived and greatly generalized in a way that should permit still further generalizations. Beginning with the Hamiltonian formulation of shallow-water dynamics, its two essential ingredients-the Hamiltonian functional and the Poisson-bracket operator-are replaced by finite-difference approximations that maintain the desired conservation laws. Energy conservation is maintained if the discrete Poisson bracket retains the antisymmetry property of the exact bracket, a trivial constraint. Potential enstrophy is conserved if a set of otherwise arbitrary coefficients is chosen in such a way that a very large quadratic form contains only diagonal terms. Using a symbolic manipulation program to satisfy the potential-enstrophy constraint, it is found that the energy- and potential-enstrophy-conserving schemes corresponding to a stencil of 25 grid points contain 22 free parameters. The AL scheme corresponds to the vanishing of all free parameters. No parameter setting can increase the overall accuracy of the schemes beyond second order, but 19 of the free parameters may be independently adjusted to yield a scheme with fourth-order accuracy in the vorticity equation.

}, keywords = {scheme}, isbn = {0022-4928}, doi = {10.1175/1520-0469(2004)061<2016:pattco>2.0.co;2}, url = {We present a simple, robust numerical method for solving the two-layer shallow water equations with arbitrary bottom topography. Using the technique of operator splitting, we write the equations as a pair of hyperbolic systems with readily computed characteristics, and apply third-order-upwind differences to the resulting wave equations. To prevent the thickness of either layer from vanishing, we modify the dynamics, inserting an artificial form of potential energy that becomes very large as the layer becomes very thin. Compared to high-order Riemann schemes with flux or slope limiters, our method is formally more accurate, probably less dissipative, and certainly more efficient. However, because we do not exactly conserve momentum and mass, bores move at the wrong speed unless we add explicit, momentum-conserving viscosity. Numerical solutions demonstrate the accuracy and stability of the method. Solutions corresponding to two-layer, wind-driven ocean flow require no explicit viscosity or hyperviscosity of any kind; the implicit hyperdiffusion associated with third-order-upwind differencing effectively absorbs the enstrophy cascade to small scales.

}, keywords = {2-layer, advection, boundary, circulation, continental-slope, driven, flows, Ocean models, schemes}, isbn = {0022-2402}, doi = {10.1357/002224002762324194}, url = {We use the lattice Boltzmann method as the basis for a three-dimensional, numerical ocean circulation model in a rectangular basin. The fundamental dynamical variables are the populations of mass- and buoyancy-particles with prescribed discrete velocities. The particles obey collision rules that correspond, on the macroscopic scale, to planetary geostrophic dynamics. The advantages of the model are simplicity, stability, and massively parallel construction. By the special nature of its construction, the lattice Boltzmann model resolves upwelling boundary layers and unsteady convection. Solutions of the model show many of the features predicted by ocean circulation theories.

}, keywords = {convective-adjustment}, isbn = {0022-2402}, doi = {10.1357/002224099321514079}, url = {We construct a lattice Boltzmann model of a single-layer, "reduced gravity" ocean in a square basin, with shallow water or planetary geostrophic dynamics, and boundary conditions of no slip or no stress. When the volume of the moving upper layer is sufficiently small, the motionless lower layer outcrops over a broad area of the northern wind gyre, and the pattern of separated and isolated western boundary currents agrees with the theory of Veronis (1973). Because planetary geostrophic dynamics omit inertia, lattice Boltzmann solutions of the planetary geostrophic equations do not require a lattice with the high degree of symmetry needed to correctly represent the Reynolds stress. This property gives planetary geostrophic dynamics a significant computational advantage over the primitive equations, especially in three dimensions.

}, keywords = {equation}, isbn = {0022-2402}, doi = {10.1357/002224099764805174}, url = {The linear equations governing stratified, wind-driven flow in an ocean of arbitrary shape may be combined into a single "advection-diffusion" equation for the pressure phi, in which the "flow" advecting phi includes a delta-function contribution at the ocean bottom in the sense of southward "advection" of phi along western continental slopes. This interpretation of the phi-equation helps to explain numerical solutions obtained with a finite-element model incorporating realistic North Atlantic bathymetry.

}, keywords = {models}, isbn = {0022-2402}, doi = {10.1357/002224098321667396}, url = {Yoon and Liu{\textquoteright}s analysis of long gravity waves in water of slowly varying depth is modified to allow for conservation of potential vorticity in place of their (incorrect) conservation at conventional vorticity.

}, isbn = {0165-2125}, doi = {10.1016/s0165-2125(96)00039-x}, url = {We consider a uniform-density ocean in which the depth-independent horizontal velocity is driven by a two-gyre wind. Numerical solutions of the governing vorticity equation reveal that the solutions with a flat ocean bottom differ greatly from those in which a continental shelf and slope are present along the western boundary. In the ocean basin with a western continental shelf, steady inertial circulations readily lose stability to unsteady inertial circulations that spawn eddies. However, eddies do not form in the corresponding solutions for the flat-bottom ocean. Reducing the friction leads to hat-bottom solutions that shed eddies far offshore, but still differ significantly from the corresponding continental-slope solutions, where eddies pinch off near the western boundary.

}, keywords = {flows, wind-driven circulation}, isbn = {0022-2402}, doi = {10.1357/0022240973224418}, url = {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.

}, keywords = {flow, intermediate models, principle, semi-geostrophic equations}, isbn = {0022-1120}, doi = {10.1017/s0022112096007045}, url = {We offer a simple model for studying the joint effect of baroclinicity and relief (jebar) on large-scale ocean circulation, based upon the planetary geostrophic equations. Applying a Galerkin approximation to the buoyancy equation, and asssuming that the temperature diffusion and vertical stratification are weak, we obtain a simple relation between the ocean temperature and the streamfunction psi for the vertically-averaged horizontal transport. Substituting this relation back into the vertically-averaged vorticity equation yields a single, generally nonlinear equation for psi, in which jebar corresponds to a clockwise {\textquoteright}advection{\textquoteright} of Jr along the continental slope (for the realistic case of temperature increasing with psi). Numerical solutions resemble those obtained by Salmon (1994) using a more accurate model, and provide a physically transparent explanation for the northward excursion of the Gulf Stream along the western continental slope observed in the previous study.

}, isbn = {0022-2402}, doi = {10.1357/0022240953213250}, url = {The assumption that surfaces of constant temperature and potential vorticity coincide leads to an exact, time-dependent reduction of the ideal thermocline equations in an ocean basin of arbitrary shape. After modifications to include forcing, dissipation, and the presence of the equator, these reduced equations form the basis for numerical models that are both more realistic and easier to solve than the conventional two-layer model.

}, keywords = {2-layer, boundary-layer, gyres, thermocline equations, wind-driven, world ocean}, isbn = {0022-2402}, doi = {10.1357/0022240943076939}, url = {We estimate the time-averaged velocity field in the North Atlantic from observations of density, wind stress and bottom topography. The flow is assumed geostrophic, with prescribed Ekman pumping at the surface, and no normal component at the bottom. These data and dynamics determine velocity to within an arbitrary function of (Coriolis parameter)/(ocean depth), which we call the {\textquoteright}{\textquoteright}dynamical free mode.{\textquoteright}{\textquoteright} The free mode is selected to minimize mixing of potential density at mid-depth. This tracer-conservation criterion serves as a relatively weak constraint on the calculation. Estimates of vertical velocity are particularly sensitive to variations in the free mode and to errors in density. In contrast, horizontal velocities are relatively robust. Below the thermocline, we predict a strong O(1 cm/sec) westward flow across the entire North Atlantic, in a narrow range of latitude between 25N and 32N. This feature supports the qualitative (and controversial) conjecture by Wust (1935) of flow along the {\textquoteright}{\textquoteright}Mediterranean Salt Tongue.{\textquoteright}{\textquoteright} Along continental margins and at the Mid-Atlantic Ridge, predicted bottom velocity points along isobaths, with shallow water to the right. These flows agree with many long-term current measurements and with notions of the circulation based on tracer distributions. The results conflict with previous oceanographic-inverse models, which predict mid-depth flows an order of magnitude smaller and often in opposite directions. These discrepancies may he attributable to our relatively strong enforcement of the bottom boundary condition. This involves the plausible, although tenuous, assertion that the flow {\textquoteright}{\textquoteright}feels{\textquoteright}{\textquoteright} only the large-scale features of the bottom topography. Our objective is to investigate the consequences of using this hypothesis to estimate the North Atlantic circulation.

}, keywords = {atlas, boundary, current-meter, deep, general-circulation, labrador sea-water, mean circulation, model, variability, western, world ocean}, isbn = {0022-2402}, doi = {10.1357/0022240933223855}, url = {We consider the two-layer form of the planetary geostrophic equations, in which a simple Rayleigh friction replaces the inertia, on a western continental slope. In the frictionless limit, these equations can be written as characteristic equations in which the potential vorticities of the top and bottom layers play the role of Riemann invariants. The general solution is of two types. In the first type, the characteristics can cross, and friction is required to resolve the resulting shocks. In the second type, one of the two Riemann invariants is uniform, the remaining characteristic is a line of constant f/H, and the solutions take a simple explicit form. A solution resembling the Gulf Stream can be formed by combining three solutions of the second type. Compared to the corresponding solution for homogeneous fluid, the Gulf Stream and its seaward countercurrent are stronger, and the latter is concentrated in a thin frictional layer on the eastern edge of the Stream.

}, keywords = {abaco, bahamas, currents, East, thermocline}, isbn = {0022-2402}, doi = {10.1357/002224092784797610}, url = {We apply symmetry group methods to find the group of transformations of the dependent and independent variables that leave the thermocline equations unchanged. These transformations lead to an optimal subset of sixteen forms of similarity solution. Each form obeys an equation with one fewer dependent variable than the original thermocline equations. Previously obtained similarity solutions, which are based solely upon scaling symmetries, are special cases of just three of these forms. Two of the sixteen forms lead to linear, two-dimensional, advection-diffusion equations for the temperature, Bernoulli functional or potential vorticity. Simple exact solutions contain "internal boundary layers" that resemble the thermocline in subtropical gyres.

}, isbn = {0022-2402}, doi = {10.1357/002224091784995909}, url = {In this paper, we analyze one-, two- and three-dimensional numerical solutions of a simple, inertia-less ocean circulation model. The solutions, which all approach a steady state, demonstrate that, in the limit of vanishing thermal diffusivity κ, a front of thickness κ1/2, identifiable with the thermocline, spontaneously appears at a location anticipated by simple arguments that treat the front as an "internal boundary layer." The temperature and velocity are generally discontinuous across the front, but the velocity component normal to the front is zero. In the asymptotic limit of vanishing diffusivity, the temperature has no vertical variation within the layer above the front, and the potential vorticity is correspondingly zero. The appearance of a front seems to require that the horizontal advection terms cancel in the temperature equation, i.e., that the horizontal velocity be directed along the isotherms on level surfaces. When the surface boundary conditions are specially chosen to prevent this cancellation, the front does not appear. However, in the more realistic cases in which the flow determines its own surface temperature, the cancellation occurs spontaneously and appears to be generically associated with the front.

}, isbn = {0022-2402}, url = {A simple method yields discrete Jacobians that obey analogues of the differential properties needed to conserve energy and enstrophy in 2-dimensional flow. The method is actually independent of the type of discretization and thus applies to an arbitrary representation in gridpoints, finite elements, or spectral modes, or to any mixture of the three. We illustrate the method by deriving simple energy- and enstrophy-conserving Jacobians for an irregular triangular mesh in a closed domain, and for a mixed gridpoint-and-mode representation in a semi-infinite channel.

}, isbn = {0021-9991}, doi = {10.1016/0021-9991(89)90118-6}, url = {In this paper, we show that numerical solutions of the single-layer quasigeostrophic equation in a beta-plane basin approach the state predicted by equilibrium statistical mechanics when the forcing and dissipation are (unrealistically) zero. This equilibrium state, which we call Fofonoff flow, consists of a quasi-steady uniform westward interior flow closed by inertial boundary layers. When wind stress and bottom drag are switched on, we find that the nonlinear terms in the quasigeostrophic equation still try to drive the flow toward Fofonoff flow, but their success at this depends strongly on the geometry of the wind stress. If the prescribed wind stress exerts a torque with the right sign to balance the bottom-drag torque around every closed streamline of the Fofonoff flow, then solutions to the wind-driven quasigeostrophic equation are energetic, Fofonoff-like, and nearly steady. If, on the other hand, the wind opposes Fofonoff flow, the wind-driven solutions are turbulent, with small mean flows, and much less energy. Our results suggest that integral conservation laws (on which the equilibrium statistical mechanics is solely based) largely define the role of the nonlinearities in the quasigeostrophic equation. To support this viewpoint, we demonstrate a resemblance between the solutions of the quasigeostrophic equation and the solutions of a stochastic model equation. The stochastic model equation, in which the advected vorticity is replaced by a random variable, has only gross conservation laws in common with the quasigeostrophic equation.

}, isbn = {0022-2402}, doi = {10.1357/002224089785076235}, url = {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 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 {\o}nto 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 isopycnals.

}, isbn = {0022-1120}, doi = {10.1017/s0022112088002733}, url = {The equations for a relativistic perfect fluid result from the requirement that the total mass-energy be stationary with respect to variations δxα(a, b, c, s) in the space-time location of the fluid particle identified by Lagrangian labels (a, b, c) at the point s on its world-line. By considering variations of the Lagrangian labels that leave the specific volume and entropy unchanged, we obtain a general covariant statement of vorticity conservation. The conservation laws for circulation, potential vorticity, and helicity are simple corollaries. This Noether-theorem derivation shows that the vorticity laws have no analogues in particle mechanics, where the corresponding particle labels cannot be continuously vaned.

}, isbn = {0309-1929}, doi = {10.1080/03091928808213624}, url = {The linear theory of the wind- and thermally-driven ocean circulation simplifies considerably if the traditional Laplacian viscosity and thermal diffusivity are replaced by a linear-decay friction and heat diffusion. Solutions of the simplified equations display all the physically important features of the standard model.

}, isbn = {0022-2402}, doi = {10.1357/002224086788401602}, url = {We present a new method for estimating the three-dimensional field of geostrophic velocity from hydrographic station data. Very simply, we ask for the smoothest velocity field (in the sense of an arbitrarily defined norm) which is consistent with the data and with selected approximate dynamical constraints to within prescribed misfits, which, we will argue, should never be zero. The misfits represent errors in the data and in the approximate dynamical constraints. By varying the misfits relatively to one another, we explore the full envelope of physically plausible estimates of the average geostrophic flow. We illustrate the method by application to hydrographic measurements in the Labrador Sea.

}, isbn = {0022-2402}, doi = {10.1357/002224086788460175}, url = {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.

}, isbn = {0022-1120}, doi = {10.1017/s0022112085001343}, url = {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{\textquoteright}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{\textquoteright}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{\textquoteright}s equations that conserves consistent approximations to energy, momentum (for a level bottom) and potential vorticity.

}, isbn = {0022-1120}, doi = {10.1017/s0022112085002488}, url = {Hamilton{\textquoteright}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 {\textendash} however strong {\textendash} that maintain the symmetries of the Hamiltonian will automatically preserve the corresponding conservation laws. For example, Hamilton{\textquoteright}s principle allows useful analytical and numerical approximations to the equations governing the motion of a homogeneous rotating fluid with free boundaries.

}, isbn = {0022-1120}, doi = {10.1017/s0022112083001706}, url = {The mean ocean density field resembles the state of maximum entropy with given values for the upper water mass, total energy and potential enstrophy. Numerical experiments confirm that the thermocline of a freely evolving two-layer model spontaneously assumes the observed double-lobe shape.

}, isbn = {0022-3670}, doi = {10.1175/1520-0485(1982)012<1458:tsotmt>2.0.co;2}, url = {It is demonstrated that the second-order Markovian closures frequently used in turbulence theory imply an H theorem for inviscid flow with an ultraviolet spectral cut-off. That is, from the inviscid closure equations, it follows that a certain functional of the energy spectrum (namely entropy) increases monotonically in time to a maximum value at absolute equilibrium. This is shown explicitly for isotropic homogeneous flow in dimensions d\>or=2, and then a generalised theorem which covers a wide class of systems of current interest is presented. It is shown that the H theorem for closure can be derived from a Gibbs-type H theorem for the exact non-dissipative dynamics.

}, isbn = {0305-4470}, doi = {10.1088/0305-4470/14/7/026}, url = {I examine the geostrophic turbulence field in equilibrium with a horizontally uniform mean zonal flow driven by solar heating. The equilibrium mean vertical shear is highly supercritical, and the turbulence field has its maximum in kinetic energy at wavenumbers smaller than the wavenumbers of fastest growth predicted by linear stability theory. Wavenumber spectra obtained by averaging lengthy numerical integrations of the two-level quasi-geostrophic equations agree well with the predictions of a simple Markovian turbulence model. Analysis of the turbulence model suggests that the most energetic wavenumbers equilibrate from scattering of the temperature perturbations into higher wavenumbers by the barotropic adverting field. In the higher unstable wavenumbers, including the most supercritical, linear instability is offset chiefly by local rotations of the unstable structures by larger, more energetic eddies.

}, isbn = {0309-1929}, doi = {10.1080/03091928008241178}, url = {In the case of equal layer depths and uniform vertical energy density, the quadratic integral invariants of two-layer rotating flow are close analogs of the corresponding invariants of two-dimensional turbulence. A simple theory based on the invariants and on the selection rules governing triad interactions qualitatively explains the major features of forced equilibrium flow. The general physical picture is very similar to that of Rhines (1977). In the geophysically interesting case. net baroclinic energy is produced at low wavenumbers and moves toward hisher wavenumbers in relatively nonlocal triad interactions which are unhampered by the constraint to conserve enstrophy. The energy converts to barotropic mode and moves back toward low wavenumbers in more local interactions which are similar to those in two-dimensional turbulence. Equilibrium wavenumber spectra are obtainable from a simple Markovian turbulence closure model in which the estimate of turbulent scramhling rate includes a contribution from vortex stretching along the axis of rotation. Numerical experiments with the closure model confirm the qualitative predictions and demonstrate the sensitivity of the flow at low wavenumbers to changes in the forcing and dissipation.

}, doi = {10.1080/03091927808242628}, author = {Salmon, R.} } @article {36261, title = {The equilibrium statistical mechanics of simple quasi-geostrophic models}, journal = {Journal of Fluid Mechanics}, volume = {75}, number = {JUN25}, year = {1976}, note = {n/a}, pages = {691-703}, type = {Article}, abstract = {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.

}, isbn = {0022-1120}, doi = {10.1017/s0022112076000463}, url = {We have coupled Lorenz{\textquoteright}s (1960) two-layer atmospheric model to a {\textquotedblleft}copper plate{\textquotedblright} ocean to obtain a simple model that can be used to study the effects of large-scale sea surface temperature (SST) anomalies on the dynamics of the atmosphere. With 164 degrees of freedom, the atmospheric model mimics the observed northern hemisphere energy cycle in fair detail. In three experiments, each lasting one year, we consider an ocean (a) with fixed temperature that depends only on latitude; (b) with fixed large-amplitude SST anomalies; and (c) with temperature determined by heat exchange with the air. Only in (b) are significant dynamical effects observed. These consist of a weak monsoon response in the subtropics and a tendency for storms to intensify over warm water at higher latitude. Experiment (c) developed SST anomalies that resemble the observed anomalies; however, the atmosphere and ocean in (c) differ insignificantly from the atmosphere and re-computed {\textquotedblleft}slave ocean{\textquotedblright} in (a), even when the flows are averaged in reference frames moving with the anomalies. Our results suggest that the atmosphere may be too noisy to be much affected by SST anomalies on time scales over which the anomalies are themselves predictable.

}, isbn = {0040-2826}, doi = {10.1111/j.2153-3490.1976.tb00671.x}, url = {