Spin-up from rest in a stratified fluid: core flows
Résumé
We investigate the spin-up from rest of a stratified fluid with initial Brunt-Väisälä fre- quency N bound within a cylindrical container of height H and radius R which is set to rotate impulsively with angular speed . The initial phase of motion is marked by the establishment of axisymmetric corner vortices fed by radial Ekman transport, a process detailed in Flor, Ungarish &Bush (2001). The subsequent evolution of the central vortex depends critically on N/f. For N/f > 1 and H/R > 1, the axisymme- try of the system is retained throughout the spin-up process: the central vortex attains a state of near solid body rotation by the diffusion of vorticity from the sidewalls. For N/f > 1 and H/R < 1, the central vortex breaks up into a series of vertical vortices that enhance transfer of angular momentum from the boundaries and so expedite the spin-up process. For N/f < 1, the central vortex becomes unstable through a tilting instability. In a short tank, this is marked by a simple tipping of the central stratified vortex. In a tall tank, the centerline of the central vortex is twisted from its vertical po- sition, and the resulting instability gives rise to a stack of vortices with approximately constant Burger ratio Nh/fR.