The linear spin-up of a strongly stratified fluid of small Prandtl number

Abstract

The spin-up of a thermally stratified Boussinesq fluid in a circular cylinder with insulated side walls is analysed under the conditions of strong stratification (Brunt-Väisälä frequency N [Gt ] rotation frequency Ω) and small Prandtl number. An earlier paper (Sakurai, Clark & Clark 1971) showed that complete spin-up is achieved in the Eddington-Sweet time. The present work considers in detail the spin-up transients corresponding to shorter time scales.The analysis reveals a complicated system of merging and bifurcating horizontal layers in the interior flow. Following the spin-up of the cylindrical container, a rotational shear layer, of the kind discovered by Holton (1965), forms near each horizontal boundary. At the same time, a thermal boundary layer begins diffusing outward from each boundary. When the thermal layer reaches the shear layer, the two merge and form a higher order layer, which diffuses at a rate proportional to t¼. At a later time, the layer splits into a steady layer and another diffusing layer, this time following a t½ law. One important conclusion from the analysis is that the lifetime of the rotational shear layer is not great: it is of order (Ω/N)2 (R2/χ), where R is the radius of the cylinder and χ is the thermal diffusivity.The problem of computing the angular velocity from the poorly converging series is dealt with in some detail, and graphs are given of representative values. The results show that, for spin-up, there are appreciable adverse gradients of angular momentum near the side wall, and thus there is some question about the stability of the spin-up configuration.Finally, a discussion is given of continuous spin-up of the container and the results are applied qualitatively to the solar spin-down problem. The principal conclusion is that the Ekman time scale is unimportant in the solar case.