Energetics of Nonlinear Geostrophic Adjustment

Abstract

Solutions and energetics for nonlinear geostrophic adjustment with an initial height perturbation of the order of the total fluid depth are computed and compared to solutions derived assuming linear dynamics. Both axisymmetric and zonally uniform profiles in 1½- and 2-layer shallow-water models are considered. Nonlinearities are present due to the finite perturbation in the initial depth and the nonzero centripetal acceleration. The comparison yields differences in both the magnitude and the partition of energy. In the adjusted state of a zonally uniform step, the total energy of the linear solution is a very good approximation to, and is slightly less than, the total energy of the nonlinear solution. Less resemblance is found for a horizontally bounded perturbation where for a positive (negative) perturbation in initial depth, the nonlinear final state has more (less) energy than the linear one. Addition of a second layer increases the contrast between the linear and nonlinear solutions, especially when one of the layers is shallow. In all the adjustment problems considered, the ratio of the adjusted state kinetic energy to the potential energy released during the adjustment is smaller than or equal to 1/3. A simple model describing the adjustment of a convective chimney illustrates the dependence of its energetics on initial radius and depth. The available potential energy of its adjusted state is important because it determines the growth rate of baroclinic instability.