On forced oscillations in a rotating stratified fluid

Abstract

An attempt is made to study the axisymmetric forced oscillations of an inviscid rotating stratified fluid confined in a finite circular cylinder. An unsteady flow is generated in the fluid by harmonic oscillations of the plane ends of the cylinder. The principal features of the flow phenomena are determined by the relative magnitudes of the angular velocity Ω of rotation, the forcing frequency Ω of the imposed oscillations and the Brunt Vaisala frequency N. It is shown that the solution for the disturbance pressure (and hence the velocity field) consists of two components–the periodic solution oscillating harmonically with frequency Ω and the normal inertial modes each of which is spatially periodic in the direction of the axis of rotation. The latter contains a doubly infinite set of modes whose frequencies form a discrete spectrum dense in [min (2Ω, N), max ((2Ω, N)]. Both of these solutions are independently modified by the density stratification. The solution is found to have a resonant behaviour when the forcing frequency Ω is equal to one of the resonant frequencies Ωmn. In contrast to the singular behaviour of the solution at Ω = 2Ω which usually occurs in the forced motion of a rotating liquid, the present solutions for this critical case Ω = 2Ω and the case Ω = N represent the inertial modes only. A qualitative assessment of the possible effects of viscosity on the flow has been made. Several interesting features of the rotating stratified flow phenomena have been analyzed. The initial value problem is solved by using the joint Laplace and the Hankel transforms of the second kind.