The thermocline as an internal boundary layer

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

Generalized two-layer models of ocean circulation

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

A two-layer Gulf Stream over a continental slope

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

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 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

Numerical solution of the two-layer shallow water equations with bottom topography

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

The shape of the main thermocline, revisited

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\u27s 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

Linear ocean circulation theory with realistic bathymetry

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 , in which the flow advecting includes a delta-function contribution at the ocean bottom in the sense of southward advection of along western continental slopes. This interpretation of the -equation helps to explain numerical solutions obtained with a finite-element model incorporating realistic North Atlantic bathymetry

A shallow water model conserving energy and potential enstrophy in the presence of boundaries

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

A simplified linear ocean circulation theory

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

Lattice Boltzmann solutions of the three-dimensional planetary geostrophic equations

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