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Instability of flows near the onset of convection in a rotating layer with stress-free horizontal boundaries

机译：在无应力水平边界的旋转层中，对流开始附近的流动不稳定

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摘要

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 a (the difference between the wave number of perturbed flows and the critical wave number for the onset of convection) and epsilon (epsilon(2) being the overcriticality and the perturbed flow amplitude being O(epsilon)), either small-angle or Eckhaus instability is prevailing. In the case of small-angle instability for rolls the largest growth rate scales as epsilon(8/5), 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 epsilon(4/3). In the case of Eckhaus instability the growth rate is of the order of alpha(2).
机译：我们研究在对流开始附近具有无应力水平边界的旋转层中简单结构（对流，驻波和行波）对流的不稳定性。我们显示出，气流总是不稳定地受到扰动的影响，扰动是大尺度模式和短尺度模式的线性组合，它们的波数与被摄动流的波数接近。根据小参数的渐近关系，a（对流发生时的扰动波数与临界波数之差）和ε（ε（2）为过临界度，扰动波幅为O（ε）） ，无论是小角度还是Eckhaus不稳定性都盛行。在小角度轧辊不稳定性的情况下，最大增长率标度为epsilon（8/5），这与Cox和Matthews（Cox，SM和Matthews，PC，旋转对流的不稳定性。J. Fluid。Mech）的结果一致。 （例如2000），403、153-172）所获得的值，其中k = k（c）。对于波浪，最大的增长率为epsilon（4/3）。在Eckhaus不稳定的情况下，增长率约为alpha（2）。

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