The stability of non-divergent horizontal shear flows in a homogeneous rotating fluid is investigated by means of a weakly non-linear theory. Velocity profiles of the shear flows considered in this paper are given by U(y)=tanh y and U(y)=sech² y, where y is the coordinate in the cross-stream direction. The effect of variation of the Coriolis parameter _??_(beta effect) and that of the Ekman friction at the bottom surfaces are included.
The coefficients of the generalized Landau equation, which describes temporal and spatial modulation of the amplitude of periodic waves, are calculated for various values of β=df/dy. It is found that the barotropic instability of the shear flows belongs to so-called supercritical type for all values of β used in this study. Therefore, we can expect a steady-state configuration betweenn a finite-amplitude wave and the distorted flow for a supercritical Reynolds number.
The stability of the finite-amplitude steady wave with respect to so-called side-band modes is also examined. It is found that, for a given supercritical Reynolds number, the finiteamplitude wave is stable if the wavenumber k₀ satisfies the inequality S(κ₁-κ_c)<κ₀-κ_c<S(κ₂-κ_c), where κ₁ and κ₂(κ₁<κ₂) are the wavenumbers of the neutral waves for the Reynolds number, kc is the wavenumber of the neutral wave at the critical Reynolds number and S is a constant. For U(y)=tanh y, S is 1/√3 when β=0. S becomes small as the absolute value of β is increased. Thus, the wavenumber range in which the finite-amplitude wave is stable decreases with increasing β. For U(y)=sech² y, S is equal to zero when the absolute value of β is small (in terms of non-dimensional beta, β=-1.9-0.5), so that the finite-amplitude wave is unstable. When β is outside of this range, S becomes large as the absolute value of β is increased. The value of S, however, is always smaller than 1√3.