Rick Salmon

The equilibrium statistical mechanics of simple quasi-geostrophic models

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.

Baroclinic instability and geostrophic turbulence

Abstract 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.

Two-layer quasi-geostrophic turbulence in a simple special case

Abstract 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 contributi...

New equations for nearly geostrophic flow

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.

Weakly dispersive nonlinear gravity waves

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 Hamiltons 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 Whithams (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 Boussinesqs equations that conserves consistent approximations to energy, momentum (for a level bottom) and potential vorticity.

The lattice Boltzmann method as a basis for ocean circulation modeling

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 upperlayer 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 signie cant computational advantage over the primitive equations, especially in three dimensions.

The North Atlantic circulation: combining simplified dynamics with hydrographic data

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 “dynamical free mode.” 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 0(1 cm/set) 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 Whist (1935) of flow along the “Mediterranean Salt Tongue.” 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 be attributable to our relatively strong enforcement of the bottom boundary condition. This involves the plausible, although tenuous, assertion that the flow “feels” 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.

H theorems in statistical fluid dynamics

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.

Shallow water equations with a complete Coriolis force and topography

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’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...

Semigeostrophic theory as a Dirac-bracket projection

This paper presents a general method for deriving approximate dynamical equations that satisfy a prescribed constraint typically chosen to filter out unwanted highfrequency 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 flat isopycnals.