Spin-up from rest in a stratified fluid: boundary flows
Résumé
We present the results of an integrated experimental, numerical and theoretical examination of spin-up from rest of a stratified fluid. A vertical cylindrical container of radius R and height 2H containing fluid of viscosity ν and characterized by a constant buoyancy frequency N is set impulsively to rotate about its symmetry axis with angular speed Ω = f/2. The characteristic Ekman number E = ν/ΩR2 is small and the Schmidt number S = ν/Ds (where Ds is the diffusivity of salt) is large. The investigation is focused on elucidating the initial stage of spin-up, which is characterized by an axisymmetric circulation driven by nonlinear Ekman layers adjoining the horizontal boundaries. Fluid is drawn by the boundary layers from the stationary, stratified interior and transported into corner regions. It is shown that the corner regions are restricted to a height of approximately 0.3Rf/N from the horizontal boundaries, above which the fluid remains unperturbed apart from that spun up by diffusion of momentum from the sidewall boundary. Two distinct regions thus emerge: rotating corner regions, and a quiescent stratified core. After a time 1.3/(E1/2N), the corner regions cover the bulk of the horizontal boundaries and the boundary layer suction is suppressed. Our study provides a framework for understanding the subsequent evolution of the spin-up process, which may be characterized by axisymmetry-breaking instabilities of the stratified core.
