On vertical boundary layers in a rapidly rotating gas

Abstract

The effect of compressibility on vertical boundary layers, of thickness E^¼ and $E^{\frac{1}{3}}$ respectively (see Stewartson 1957), in a rapidly rotating gas of constant temperature is studied. Such layers have been considered by Nakayama & Usui (1974). However, these authors used a power-series expansion of the basic density field in their calculations, which means that their results are not uniformly valid in the boundary-layer co-ordinate. In this paper, a uniformly valid asymptotic solution and a numerical solution for the velocity field in each layer are computed. The asymptotic solutions are valid for scale heights of the basic density field large in terms of the reference length for the respective boundary layer. The agreement between the asymptotic and numerical solutions is shown to be good. Although the limits considered have no meaning for density scale heights of the order of the respective boundary-layer thickness, one can formally find an exact solution for the E^¼-layer for any density scale height. In the limit of large density scale height this solution agrees to third order with the corresponding asymptotic solution. It is shown that for both layers the boundary-layer thickness increases (decreases) if the basic density field decreases (increases) in the direction of the respective boundary-layer co-ordinate.