Energetics of linear geostrophic adjustment in stratified rotating fluids

Abstract

The energy conversion ratio, g , is shown to be bounded below by 0 and above by 1/2 in the two-dimensional linear geostrophic adjustment of a continuously stably stratie ed, incompressible, inviscid non-Boussinesq e uid. ‘ ‘ Two-dimensional’ ’ refers to problems in which the initial isopycnal displacement e eld is an arbitrary function of the vertical (parallel to the rotation axis) and a single horizontal coordinate. By using Fourier analysis techniques, the paper also identie ed classes of initial isopycnal displacement proe les for which the adjustment process leads to g . 1/3. Finally, an expression for g is derived when the initial isopycnal displacement proe le is three dimensional. The problem of how a e uid, initially not in geostrophic balance, adjusts to that balance is a fundamental problem in the theory of rotating e uids. Blumen (1972) gave the e rst review in the area, and discussed many concepts which have been used since, including potential vorticity conservation and minimum energy principles. In this paper, upper and lower bounds are derived for the energy conversion ratio in the geostrophic adjustment problem for a uniformly rotating stratie ed e uid. A feature of all geostrophic adjustment problems is that only a fraction of the potential energy released, D PE, is converted into kinetic energy, D KE, of the e nal geostrophically adjusted state. The energetics of the geostrophic adjustment problem are usually expressed in terms of the energy conversion ratio g 5 D KE/D PE. In the literature, following Blumen’ s work, the ‘ ‘ classical’ ’ lineargeostrophic adjustment problem refers to the adjustment of a horizontally unbounded, uniformly rotating, barotropic e uid which initially is at rest (with respect to the rotating frame of reference) with a step in the e uid surface which is maintained by a vertical barrier. Upon removal of the ›