Spin-up of a strongly stratified fluid in a sphere

Abstract

A linear theory is developed for the spin-up of a compressible fluid, stratified by a spherical gravity field. Numerical results are obtained for the case of strong stratification (Brunt–Väisälä frequency N much greater than the rotation frequency ω₀). The interior flow is solved in terms of a set of angular eigenfunctions which have been obtained numerically. The principal result is that the spin-up is limited to a layer adjacent to the spherical boundary, the thickness δ of the layer being of the order of L(ω₀/N), where L is the radius of the boundary. The solution is qualitatively similar to that found by Holton (1965), Walin (1969), and Sakurai (1969a, b) for a stratified fluid in a cylinder. The thickness of the spin-up layer diminishes with latitude ϕ, the variation being described roughly by the formula δ ∼ LΩ₀| sin ϕ|/N. For the case of slow continuous spin-up, the Ekman suction velocity has been calculated, and the results show that |ϕ| = 24° is the dividing angle between suction (|ϕ| > 24°) and blowing (|ϕ| < 24°).