Euler Boussinesq Equations Research Articles

On Nonlinear Baroclinic Waves and Adjustment of Pancake Dynamics

Three-dimensional nonhydrostatic Euler–Boussinesq equations are studied for Bu=O(1) flows as well as in the asymptotic regime of strong stratification and weak rotation. Reduced prognostic equations for ageostrophic components (divergent velocity potential and geostrophic departure/thermal wind imbalance) are analyzed. We describe classes of nonlinear anisotropic ageostrophic baroclinic waves which are generated by the strong nonlinear interactions between the quasi-geostrophic modes and inertio-gravity waves. In the asymptotic regime of strong stratification and weak rotation we show how switching on weak rotation triggers frontogenesis. The mechanism of the front formation is contraction in the horizontal dimension balanced by vertical shearing through coupling of large horizontal and small vertical scales by weak rotation. Vertical slanting of these fronts is proportional to μ−1/2 where μ is the ratio of the Coriolis and Brunt–Vaisala parameters. These fronts select slow baroclinic waves through nonlinear adjustment of the horizontal scale to the vertical scale by weak rotation, and are the envelope of inertio-gravity waves. Mathematically, this is generated by asymptotic hyperbolic systems describing the strong nonlinear interactions between waves and potential vorticity dynamics. This frontogenesis yields vertical “gluing” of pancake dynamics, in contrast to the independent dynamics of horizontal layers in strongly stratified turbulence without rotation.

Dispersive barotropic equations for stratified mesoscale ocean dynamics

Dispersive effects induced by weak hydrostatic imbalance in the presence of topography and stratification are incorporated into a new model of barotropic (vertically integrated) mesoscale ocean dynamics. This barotropic model is obtained by first expanding the solutions of three dimensional Euler-Boussinesq equations in a regular perturbation expansion in terms of the several small dimensionless parameters appropriate to mesoscale ocean dynamics. Vertically integrating from a fixed bottom topography to the free surface interface with the atmosphere and balancing orders in the expansion at fourth order in the small aspect ratio parameter then yields a system of reduced barotropic equations. These reduced barotropic equations are considerably more tractable than the starting equations and have appropriate limits to known dispersive wave equations such as the forced Kadomtsev-Petviashvili equation in one limit, and the rotating shallow water equations in another. This new model of barotropic ocean dynamics may be of use in developing numerical algorithms for global ocean circulation modeling.