VORTICITY IN STRATIFIED FLUIDS .2. FINITE CROSS-SECTION TUBES AND RINGS

A recent paper (Arendt, 1992a) began an investigation of vorticity in a polytropic or isothermal fluid stratified by gravity, and derived the velocity fields of an infinite straight vortex tube and a vortex ring, each having an infinitesimal cross-section. It was found that the density stratification induces motion of the vortices in a direction perpendicular to gravity: vortex tubes translate while vortex rings expand or contract, depending on their direction. The present paper extends these calculations to vortex tubes and rings of finite cross-section whose vorticity distributions are constructed so as to be steady state. Expressions for the velocity field, propagation velocity, and kinetic energy are presented. The calculated corrections to the infinitesimal cross-section results are found to be significant for intermediate values (10(-5)-10(-1)) of the ratios b/lambda or b/a where b is the vortex (ring or tube) cross-sectional radius, lambda is the density scale height at the location of the vortex, and a is the vortex ring radius. The implications for sunspot formation and for exploding solar granules are discussed.