NOTE ON THE NONLINEAR STABILITY OF STRATIFIED FLUID EQUILIBRIA

We study the nonlinear stability of hydrostatic equilibria of an ideal incompressible stratified fluid. We obtain a new a priori estimate fo r finite-amplitude perturbation of the basic equilibrium state. The ma in idea of our approach is based on a special decomposition of the den sity perturbation, namely, we split the density perturbation in two pa rts, the first part depends on time but has zero initial value, the se cond one is in some sense time-independent (its L-2-norm is time-indep endent). This decomposition allows us to obtain the a priori estimate for the time-dependent part of the perturbation and hence for the tota l perturbation. In our approach we avoid the problem of a smooth exten sion of a locally convex function beyond its initial domian of definit ion that arises in applications of Arnold's method. Taking advantage o f this fact, we consider the nonlinear stability of equilibrium states of stratified fluid endowed with two densities. Such a kind of proble m appears, e.g., in atmospheric physics when symmetric flows of a stra tified fluid are considered. As a result, we obtain a sufficient condi tion for nonlinear stability of a general equilibrium state of such a doubly stratified fluid as well as an a priori estimate for perturbati ons of arbitrary amplitude. Copyright (C) 1998 Elsevier Science B.V.