Instability of flows near the onset of convection in a rotating layer with stress-free horizontal boundaries |
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| Authors: | Olga Podvigina |
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| Institution: | 1. International Institute of Earthquake Prediction Theory and Mathematical Geophysics , 84/32 Profsoyuznaya St., Moscow, 117997, Russian Federation;2. Laboratory of General Aerodynamics, Institute of Mechanics Lomonosov Moscow State University , 1, Michurinsky Ave., 119899 Moscow, Russian Federation;3. Observatoire de la C?te d'Azur , CNRS UMR 6202, BP 4229, 06304 Nice Cedex 4, France olgap@mitp.ru |
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| Abstract: | We investigate instability of convective flows of simple structure (rolls, standing and travelling waves) in a rotating layer with stress-free horizontal boundaries near the onset of convection. We show that the flows are always unstable to perturbations, which are linear combinations of large-scale modes and short-scale modes, whose wave numbers are close to those of the perturbed flows. Depending on asymptotic relations of small parameters α (the difference between the wave number of perturbed flows and the critical wave number for the onset of convection) and ε (ε2 being the overcriticality and the perturbed flow amplitude being O(ε)), either small-angle or Eckhaus instability is prevailing. In the case of small-angle instability for rolls the largest growth rate scales as ε8/5, in agreement with results of Cox and Matthews (Cox, S.M. and Matthews, P.C., Instability of rotating convection. J. Fluid. Mech., 2000, 403, 153–172) obtained for rolls with k = k c . For waves, the largest growth rate is of the order ε4/3. In the case of Eckhaus instability the growth rate is of the order of α2. |
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| Keywords: | Small-angle instability Rotation Stress-free boundaries Boussinesq convection |
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