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Stability of \(\beta\)-plane Kolmogorov flow. (English) Zbl 0983.86002
Summary: We show that the geophysical \(\beta\)-effect strongly affects the linear stability of a sinusoidal Kolmogorov flow. If \(\alpha\) denotes the angle between the flow direction and the planetary vorticity gradient then the critical Reynolds’ number, \(R_c(\alpha,\beta)\), is zero for \(\beta\neq 0\), provided that \(\sin 2\alpha\neq 0\). In particular, the small \(\beta\) limit is discontinuous: \(\lim_{\beta\rightarrow 0} R_c(\alpha,\beta)=0\), rather than the classical value \(R_c(\alpha,0)=\sqrt{2}\). Moreover, though the Kolmogorov flow is non-zonal, the most unstable modes are large-scale quasizonal flows. These results are obtained using asymptotic analysis and confirmed by numerical solution. The simulations show the saturating effects of nonlinearities.

86A10 Meteorology and atmospheric physics
76E20 Stability and instability of geophysical and astrophysical flows
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