Variational principles and stability of stationary flows of barotropic ideal fluid

Abstract

Some well-known results of Arnold are generalized and it is proved that the stationary three-dimensional flow of an ideal (inviscid) barotropic fluid yields an extremum of the total mechanical energy with respect to variations of the hydrodynamic fields that possess the same vorticity. In the two-dimensional case some functionals are constructed using integrals of the energy and vortex conservation laws. These functionals are also integrals of the equations of motion and their extremum corresponds to the stationary flows under consideration. The formulae for the second variations of these functionals are then derived, and in this way some sufficient conditions for the stability of the corresponding stationary flows (in the linear approximation or in an exact non-linear sense) are found. In particular, it is proved that plane stationary flows of a barotropic ideal fluid are stable in the Lyapunov sense if two conditions hold: the first one is a natural generalization of the Rayleigh-Arnold criterion, the latter a requirement of the subsonic character of the flow.