Institute for Mathematics and Its Applications
Darryl Holm, Los Alamos National Lab
One might suppose the ocean's mean circulation, or `climate,' should be described by some sort of mean of the primitive equations (PE). (The PE describe an incompressible stratified fluid in hydrostatic balance in a rotating frame.) We will discuss various candidates for these mean PE.
The nonlinear, nondissipative part of each model is obtained by applying asymptotic expansions, two-timing, Taylor expansions and (sometimes) Lagrangian mean averaging in Hamilton's principle for the primitive equations. Before adding viscosity, we shall discuss some of the nondissipative properties of these mean PE, such as their Kelvin circulation theorems and their related potential vorticity conservation laws. We also compare the results of numerical simulations for some of these models with the PE solution for an unstable baroclinic jet.
Their linear, viscous part is obtained by identifying the appropriate momentum that should be diffused by viscosity. For one of these mean equations, called the ``alpha model,'' we test the natural candidate for the viscous term by using it to derive a new closure model for the mean velocity and Reynolds stress in channel turbulence. Agreeably, the analytical solution of this model compares well with experimental data.