Spin up of a rapidly rotating gas in thermally insulated annulus

Authors: Lindblad, I., Bark, F.H., Zahrai, S.
Document Type: Article
Pubstate: Published
Journal: J. Fluid Mech.
Volume: 274   383-404
Year: 1994


The linear spin-up problem for a rapidly rotating viscous diffusive ideal gas is considered in the limit of vanishing Ekman number E. Particular attention is given to gases having a large molecular weight. The gas is enclosed in a cylindrical annulus, with flat top and bottom walls, which is rotating around its axis of symmetry with rotation rate Omega . The walls of the container are adiabatic. In a rotating gas (of any molecular weight), the Ekman layers on adiabatic walls are weak, which implies that there is no distinct non-diffusive response of the gas outside the Ekman and Stewartson boundary layers on the timescale E/sup -1/2/ Omega /sup -1/ for spin-up of a homogeneous fluid. For the case of adiabatic walls, it is shown that the spin-up mechanisms due to viscous diffusion and Ekman suction are, from a formal point of view, equally strong. Therefore, the gas will adjust to the increased rotation rate of the container on the diffusive timescale E/sup -1/ Omega /sup -1/. However, if E/sup 1/3/<< gamma -1<<1 and M approximately 1, which characterizes rapidly rotating heavy gases (where gamma is the ratio of specific heats of the gas and M the Mach number), it is shown that the gas spins up mainly by Ekman suction on the shorter timescale ( gamma -1)/sup 2/E/sup -1/ Omega /sup -1/. In such cases, the interior motion splits up into a non-diffusive part of geostrophic character and diffusive boundary layers of thickness ( gamma -1) outside the Ekman and Stewartson layers. The motion approaches the steady state of rigid rotation algebraically instead of exponentially as is usually the case for spin-up.