A unified linear theory of homogeneous and stratified rotating fluids

Abstract

A unified picture of the linear dynamics of rotating fluids with given arbitrary stratification is presented. The range of stratification which lies outside the region of validity of both the theories of homogeneous fluids, $\sigma S < E^{\frac{2}{3}}$ and the strongly stratified fluids, σ S > E ½ , is studied, where σ S = v α g Δ T /κΩ 2 L and E = v /Ω L 2 . The transition from one dynamics to the other is elucidated by a detailed study of the intermediate region E 2/3 < σ S < E ½ . It is shown that, within this intermediate stratification range, the dynamics differs from that of either extreme case, except in the neighbourhood of horizontal boundaries where Ekman layers are present. In particular the side wall boundary layer exhibits a triple structure and is made up of (i) a buoyancy sublayer of thickness (σ S ) −1/4 E ½ in which the viscous and buoyancy forces balance, (ii) an intermediate hydrostatic, baroclinic layer of thickness (σ S ) ½ and (iii) an outer E ¼ -layer which is analogous to the one occurring in a homogeneous fluid. In the interior, the dynamics is mainly controlled by Ekman-layer suction, but displays hybrid features; in particular the dynamical fields can be decomposed into a ‘homogeneous component’ which satisfies the Taylor-Proudman theorem, and into a ‘stratified component’ which is baroclinic and which satisfies a thermal wind relation. In all regions the structure of the flow is displayed in detail.