Mechanisms and Parameterizations of Geostrophic Adjustment and a Variational Approach to Balanced Flow

Abstract

Geostrophic balance is shown to be the minimum energy state, for a given linear potential vorticity field, for small deviations of the height field around a resting state, in the shallow-water equations. This includes (but is not limited to) the linearized shallow-water equations. Quasigeostrophic motion is evolution on the slow manifold defined by advection of linear potential vorticity by the velocity field that minimizes that energy. Other linear and nonlinear arguments suggest that geostrophic adjustment is a process whereby the energy of a flow is minimized consistent with the maintenance of the potential vorticity field. A variational calculation that minimizes energy for a given potential vorticity field leads to a balance relationship that for the unapproximated shallow-water equations is similar but not identical to geostrophic balance. Preliminary numerical evidence, involving the inversion of potential vorticity for a simple model, indicates that this balance is a somewhat better approximation to the primitive equations than geostrophy. It is also shown how the process of geostrophic adjustment may be significantly accelerated, or parameterized, in the primitive equations by the addition of certain terms to the equations of motion. Application of the parameterization to an unbalanced state in a primitive equation model is very effective in achieving a balanced state and in continuously filtering gravity waves. It is more accurate and less sensitive to tunable parameters than pure divergence damping, and may also be a useful and much simpler alternative to nonlinear normal-mode schemes whenever those may be inappropriate.