Pollicott S L, Matthews P C, Cox S M
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom.
Phys Rev E Stat Nonlin Soft Matter Phys. 2003 Jan;67(1 Pt 2):016301. doi: 10.1103/PhysRevE.67.016301. Epub 2003 Jan 9.
We analyze thermal convection in a fluid layer confined between isothermal horizontal boundaries at which the tangential component of the fluid stress vanishes. The layer rotates about an oblique, nearly vertical axis. Using a model set of equations for w, the horizontal planform of the vertical velocity component, and psi, a stream function related to a large-scale vertical vorticity field, we describe the instabilities of convection rolls. We show how the usual Küppers-Lortz instability, which leads to a continual precession of the roll pattern, can be suppressed by the oblique rotation vector. Of particular interest is the small-angle instability of rolls, to perturbations in the form of rolls that are almost aligned with the primary rolls; at finite Prandtl number, this instability is not prevented by the horizontal component of the rotation vector, unless this component is sufficiently strong, in which case stability is confined to small-amplitude rolls near the marginal stability boundary. A one-dimensional instability leading to amplitude-modulated rolls is unaffected by the oblique rotation. Numerical simulations of the model equations are presented, which illustrate the instabilities analyzed.
我们分析了夹在等温水平边界之间的流体层中的热对流，在这些边界处流体应力的切向分量消失。该流体层绕着一个倾斜的、近乎垂直的轴旋转。使用一组关于垂直速度分量的水平平面形式(w)和与大规模垂直涡度场相关的流函数(\psi)的方程，我们描述了对流卷的不稳定性。我们展示了通常会导致卷模式持续进动的库珀斯 - 洛茨不稳定性如何被倾斜的旋转向量抑制。特别令人感兴趣的是卷的小角度不稳定性，即对于几乎与主卷对齐的卷形式的扰动；在有限普朗特数下，除非旋转向量的水平分量足够强，否则这种不稳定性不会被其阻止，在这种情况下稳定性仅限于接近边际稳定性边界的小振幅卷。导致振幅调制卷的一维不稳定性不受倾斜旋转的影响。给出了模型方程的数值模拟，其说明了所分析的不稳定性。
