An Experimental and Theoretical Study of Barotropic Instability

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Abstract
The barotropic instability of horizontal shear flows is investigated by means of a laboratory experiment. Two kinds of basic flows with different velocity profiles am examined, one a free-shear layer and the other a jet. It is found that for both flows the stability is described by a single nondimensional parameter, a Reynolds number R=VL/v where V is the characteristic velocity of the basic flow, L=(E/4)¼H is the characteristic length of the basic flow, v the kinematic viscosity, H the depth of the fluid layer, and E=v(ΩH2)−1 the Ekman number, with Ω the angular velocity of the basic rotation. The experimentally-determined critical Reynolds number Rc and critical wavenumber kc show excellent agreement with those predicted by a linear stability theory in which both Ekman friction and internal viscosity are incorporated. It is found that the internal viscosity plays an important role in explaining the observed values of Rc and kc. When R is larger than Rc, several organized eddies develop along the shear zone of the basic, flows. Them eddies are quite steady and stable. In general, the number of eddies decreases as R is increased. This tendency is opposite to that shown by the results of linear stability theory in which the wavenumber k of the most unstable wave increases with R. The number of eddies realized for a certain Reynolds number is not unique, i.e., several different configurations are stable for a fixed value of R. Domains on the R-k plane in which finite-amplitude waves are stable are determined both for the shear layer and the jet. Having determined the domains, we are able to simulate the hysteresis phenomena in wavenumber selection. The result that stable eddies are realized for the jet contradicts the prediction of the weakly nonlinear theory (Niino, 1982a) in which only Ekman friction is considered. It is found, however, that this contradiction can be removed if both Ekman friction and internal viscosity are incorporated in the weakly nonlinear theory.

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