The Linear Stability and Structure of Convection in a Circular Mean Shear

Abstract

An investigation in made of the linear stability and structure of convection embedded in a mean shear flow with a circular hodograph. This can he considered an extension of Asai's work, but with emphasis on the rotational and helicity features of the disturbances. It also examines the relevance of the Beltrami flow solutions presented previously by Lilly and Davics-Jones, which could not be directly extended to consider the effects of buoyancy. The Boussinesq equations we applied to neutrally and unstably stratified fluids, with emphasis placed on the inviscid solutions. Upper and lower boundary conditions are free slip and rigid. Lateral conditions are periodic, which allows casting the disturbance equations into a horizontally periodic normal-mode structure. The growth rates and disturbance forms are generally fairly similar to the results presented by Asai, except that the most unstable modes are nearly always oriented transverse to the shear component at the channel center. The most rapidly growing modes at small Richardson number are found to be highly helical, with the helicity obtained from the Beltrami mean state. The helicity transfer efficiency and disturbance relative helicity decrease rapidly, however, for negative, Richardson numbers greater than about 1.