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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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##### References:
 [1] Legras, B.; Santangelo, P.; Benzi, R., High-resolution numerical experiment for forced two-dimensional turbulence, Europhys. lett., 5, 37-42, (1988) [2] Maltrud, M.E.; Vallis, G.K., Energy and enstrophy transfer in numerical simulations of two-dimensional turbulence, Phys. fluids A, 5, 1760-1775, (1993) [3] Chekhlov, A.; Orszag, S.A.; Sukoriansky, S.; Galperin, B.; Staroselsky, I., Direct numerical simulation tests of eddy viscosity in two dimensions, Phys. fluids, 6, 2548-2550, (1994) [4] Sivashinsky, G.; Yakhot, V., Negative viscosity effect in large-scale flows, Phys. fluids, 28, 1040-1042, (1985) · Zbl 0584.76045 [5] Gama, S.; Vergassola, M.; Frisch, U., Negative eddy viscosity in isotropically forced two-dimensional flow: linear and nonlinear dynamics, J. fluid mech., 260, 95-126, (1994) [6] Gotoh, K.; Yamada, M.; Mizushima, Y., The theory of stability of spatially periodic flows, J. fluid mech., 127, 45-58, (1983) · Zbl 0523.76037 [7] Gotoh, K.; Yamada, M., The instability of rhombic cell flows, Fluid dyn. res., 1, 165-176, (1986) [8] Meshalkin, L.D.; Sinai, I.G., Investigation of the stability of a stationary solution of the system of equations for the plane movement of an incompressible viscous liquid, J. appl. math. mech. (prikl. mat. mekh.), 25, 1700-1705, (1961) · Zbl 0108.39501 [9] Beaumont, D.N., The stability of spatially periodic flows, J. fluid mech., 108, 461-474, (1981) · Zbl 0486.76065 [10] Lorenz, E.N., Barotropic instability of Rossby wave motion, J. atmos. sci., 29, 258-269, (1972) [11] Gill, A.E., The stability of planetary waves on an infinite beta-plane, Geophys. fluid dyn., 6, 29-47, (1974) [12] Frisch, U.; Legras, B.; Villone, B., Large-scale Kolmogorov flow on the beta-plane and resonant wave interactions, Physica D, 94, 36-56, (1996) · Zbl 0899.76230 [13] Legras, B.; Villone, B.; Frisch, U., Dispersive stabilization of the inverse cascade for the Kolmogorov flow, Phys. rev. lett., 82, 4440-4443, (1999) [14] Manfroi, A.J.; Young, W.R., Slow evolution of zonal jets on the beta plane, J. atmos. sci., 56, 784-800, (1999) [15] Dolzhanskiy, F.V., Effect of Ekman layer on the stability of planetary waves, Izvestiya atmos. Ocean phys., 21, 292-297, (1985) [16] Stuhne, G.R., One-dimensional dynamics of zonal jets on rapidly rotating spherical shells, Physica D, 149, 43-79, (2001) · Zbl 1050.76057 [17] Panetta, R.L., Zonal jets in wide baroclinically unstable regions: persistence and scale selection, J. atmos. sci., 50, 2073-2106, (1993) [18] Salmon, R., Baroclinic instability and geostrophic turbulence, Geophys. astrophys. fluid dyn., 15, 167-211, (1980) [19] Simmons, A.J.; Hoskins, B.J., The life cycles of some nonlinear baroclinic waves, J. atmos. sci., 35, 414-432, (1978) [20] James, I.N.; Gray, L.J., Concerning the effect of surface drag on the circulation of a baroclinic planetary atmosphere, Quart. J. R. meteorol. soc., 112, 1231-1250, (1986) [21] J. Pedlosky, Geophysical Fluid Dynamics, Springer, Berlin, 1987, 710+xiv pp. · Zbl 0713.76005 [22] Rhines, P.B., Waves and turbulence on a beta-plane, J. fluid mech., 69, 417-443, (1975) · Zbl 0366.76043 [23] Sukoriansky, S.; Galperin, B.; Chekhlov, A., Large scale drag representation in simulations of two-dimensional turbulence, Phys. fluids, 11, 3043-3053, (1999) · Zbl 1149.76554 [24] Huang, H.-P.; Galperin, B.; Sukoriansky, S., Anisotropic spectra in two-dimensional turbulence on the surface of a rotating sphere, Phys. fluids, 13, 225-240, (2001) · Zbl 1184.76235
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