# An Explicit Potential-Vorticity Conserving Approach to Modelling Three-Dimensional Boussinesq Flows

Wednesday, February 13, 2002 - 9:30am - 10:30am

Keller 3-180

David Dritschel (University of St. Andrews)

The Boussinesq equations are used to describe the dynamical behaviour of a rotating, stratified fluid, a prime example being the oceans. These equations consist of momemtum and mass conservation, together with the condition of incompressibility. As normally written, they obscure the underlying material conservation of a quantity called potential vorticity, given by the product of the absolute vorticity (including the background rotation of the Earth) and the gradient of the density, itself materially conserved. For a stably-stratified fluid, density decreases monotonically with height everywhere, a situation typical of most of the oceans. Then, material conservation of potential vorticity amounts to the conservative advection of potential vorticity on surfaces of constant density (isopycnals).

This mathematical result is well known but, in practise, little exploited. Using potential vorticity explicitly poses two major problems: (1) it forces one to solve a nonlinear diagnostic equation for one of the primitive variables (velocity, density, pressure or a combination thereof); and (2) numerical methods are traditionally not suited for conservative advection. In this talk, a new approach is presented which overcomes these two problems. Theoretically, the equations are reformulated in a mathematically convenient way, revealing the existence of an underlying Monge-Ampere equation, a nonlinear diagnostic equation for one of the primitive variables. The reformulation uses the ageostophic horizontal vorticity, a first-order estimate for the imbalanced (wave-part) of the flow. Numerically, explicit potential vorticity conservation is handled by contour advection, which tracks potential vorticity contours in a grid-free way on density surfaces. These contours are converted to gridded values for the purpose of solving the Monge-Ampere equation, and the remaining part of the numerical algorithm uses conventional methods (e.g. pseudo-spectral).

An example of a strongly anticyclonic vortex is presented. We focus on the behaviour of a notoriously difficult field, the vertical velocity, which is typically 10,000 times weaker than the horizontal velocity. Our solutions are shown to be highly accurate, as judged indirectly by comparison with the vertical velocity diagnosed from an approximate balance relation (the quasi-geostrophic omega equation). The method also appears able to accurately quantify the radiation of internal-gravity waves.

This mathematical result is well known but, in practise, little exploited. Using potential vorticity explicitly poses two major problems: (1) it forces one to solve a nonlinear diagnostic equation for one of the primitive variables (velocity, density, pressure or a combination thereof); and (2) numerical methods are traditionally not suited for conservative advection. In this talk, a new approach is presented which overcomes these two problems. Theoretically, the equations are reformulated in a mathematically convenient way, revealing the existence of an underlying Monge-Ampere equation, a nonlinear diagnostic equation for one of the primitive variables. The reformulation uses the ageostophic horizontal vorticity, a first-order estimate for the imbalanced (wave-part) of the flow. Numerically, explicit potential vorticity conservation is handled by contour advection, which tracks potential vorticity contours in a grid-free way on density surfaces. These contours are converted to gridded values for the purpose of solving the Monge-Ampere equation, and the remaining part of the numerical algorithm uses conventional methods (e.g. pseudo-spectral).

An example of a strongly anticyclonic vortex is presented. We focus on the behaviour of a notoriously difficult field, the vertical velocity, which is typically 10,000 times weaker than the horizontal velocity. Our solutions are shown to be highly accurate, as judged indirectly by comparison with the vertical velocity diagnosed from an approximate balance relation (the quasi-geostrophic omega equation). The method also appears able to accurately quantify the radiation of internal-gravity waves.