An experimental and analytical study of instability of asymmetric jetstream-like currents in a rotating fluid

Abstract

Considering the barotropic instability problem of the mean westerly current in the atmosphere we have performed a series of experiments in a rotating vessel (using water and a barotropic model) to study the behaviour of a zonal asymmetric basic current with respect to small perturbations. In the centre of a rotating cylindrical vessel (of large diameter and rotation rate ω) a smaller cylinder was installed, the rotation of which relative to the vessel, at a rate Δω, generates a nearly two-dimensional field of mean relative motion within a sharply defined region. The dominant zonal velocity component $\overline{v}$ shows monotonic radial decrease within this so-called friction zone. Now what happens if the relative rotation of the inner cylinder, the source of momentum, suddenly vanishes, i.e. δΩ = 0? The main result is that the basic zonal current $\overline{v}$, which now has an asymmetric radial profile ($\overline{v} = 0 $ at the inner cylinder and the outer edge of the friction zone), breaks down into vortices, the number of which, the integer wavenumber n, is a function of the parameter ε = δω/ω alone: n = n(ε); increasing ε eff effects a decrease of n. For a theoretical discussion of the experimental results we assume this to be a problem of barotropic instability and base our analytical considerations on the two-dimensional non-divergent vorticity equation, frictional forces being neglected. By applying a perturbation method and prescribing a realistic asymmetric basic current we can derive the relation ν = {[¾π/ln (ε + 1)½]2 + 1}½, which yields the real wavenumber v as a function of the parameter ε = δω/ω. The analytical results are in good agreement with the experiments.