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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.

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