Leprovost Nicolas, Kim Eun-Jin
Department of Applied Mathematics, University of Sheffield, Sheffield S3 7RH, United Kingdom.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Jul;78(1 Pt 2):016301. doi: 10.1103/PhysRevE.78.016301. Epub 2008 Jul 7.
Rotation and shear flows are ubiquitous features of many astrophysical and geophysical bodies. To understand their origin and effect on turbulent transport in these systems, we consider a forced turbulence and investigate the combined effect of rotation and shear flow on the turbulence properties. Specifically, we study how rotation and flow shear influence the generation of shear flow (e.g., the direction of energy cascade), turbulence level, transport of particles and momentum, and the anisotropy in these quantities. In all the cases considered, turbulence amplitude is always quenched due to strong shear (xi=nuk_y2/A<<1 , where A is the shearing rate, nu is the molecular viscosity, and ky is a characteristic wave number of small-scale turbulence), with stronger reduction in the direction of the shear than those in the perpendicular directions. Specifically, in the large rotation limit (OmegaA) , they scale as A-1 and A-1|ln xi| , respectively, while in the weak rotation limit (Omega<<A) , they scale as A-1 and A-2/3 , respectively. Thus, flow shear always leads to weak turbulence with an effectively stronger turbulence in the plane perpendicular to shear than in the shear direction, regardless of rotation rate. The anisotropy in turbulence amplitude is, however, weaker by a factor of xi1/3|ln xi| ( proportional, variantA;-1/3|ln xi|) in the rapid rotation limit (OmegaA) than that in the weak rotation limit (OmegaA) since rotation favors almost-isotropic turbulence. Compared to turbulence amplitude, particle transport is found to crucially depend on whether rotation is stronger or weaker than flow shear. When rotation is stronger than flow shear (OmegaA) , the transport is inhibited by inertial waves, being quenched inversely proportional to the rotation rate (i.e., proportional, variantOmega;-1 ) while in the opposite case, it is reduced by shearing as A-1 . Furthermore, the anisotropy is found to be very weak in the strong rotation limit (by a factor of 2) while significant in the strong shear limit. The turbulent viscosity is found to be negative with inverse cascade of energy as long as rotation is sufficiently strong compared to flow shear (OmegaA) while positive in the opposite limit of weak rotation (OmegaA) . Even if the eddy viscosity is negative for strong rotation (OmegaA) , flow shear, which transfers energy to small scales, has an interesting effect by slowing down the rate of inverse cascade with the value of negative eddy viscosity decreasing as |nuT| proportional, variantA-2 for strong shear. Furthermore, the interaction between the shear and the rotation is shown to give rise to a nondiffusive flux of angular momentum ( Lambda effect), even in the absence of external sources of anisotropy. This effect provides a mechanism for the existence of shearing structures in astrophysical and geophysical systems.
旋转流和剪切流是许多天体物理和地球物理天体普遍存在的特征。为了理解它们的起源以及对这些系统中湍流输运的影响，我们考虑一种受迫湍流，并研究旋转和剪切流对湍流特性的综合影响。具体而言，我们研究旋转和流剪切如何影响剪切流的产生（例如能量串级的方向）、湍流水平、粒子和动量的输运，以及这些量中的各向异性。在所有考虑的情况下，由于强剪切（$\xi=\frac{\nu k_y^2}{A}\ll1$，其中$A$是剪切率，$\nu$是分子粘性，$k_y$是小尺度湍流的特征波数），湍流振幅总是被抑制，在剪切方向上的减小比垂直方向上更强。具体来说，在大旋转极限（$\Omega\gg A$）下，它们分别按$A^{-1}$和$A^{-1}|\ln\xi|$缩放，而在弱旋转极限（$\Omega\ll A$）下，它们分别按$A^{-1}$和$A^{-2/3}$缩放。因此，无论旋转速率如何，流剪切总是导致弱湍流，在垂直于剪切的平面内的湍流比在剪切方向上的湍流更有效。然而，在快速旋转极限（$\Omega\gg A$）下，湍流振幅的各向异性比在弱旋转极限（$\Omega\ll A$）下弱$\xi^{1/3}|\ln\xi|$倍（与$A^{-1/3}|\ln\xi|$成比例），因为旋转有利于几乎各向同性的湍流。与湍流振幅相比，发现粒子输运关键取决于旋转比流剪切强还是弱。当旋转比流剪切强（$\Omega\gg A$）时，输运受到惯性波的抑制，与旋转速率成反比地被抑制（即与$\Omega^{-1}$成比例），而在相反的情况下，它因剪切而按$A^{-1}$减小。此外，发现在强旋转极限下各向异性非常弱（相差一个因子2），而在强剪切极限下很显著。只要旋转相对于流剪切足够强（$\Omega\gg A$），湍流粘性就被发现为负，能量出现反向串级，而在弱旋转的相反极限（$\Omega\ll A$）下为正。即使在强旋转（$\Omega\gg A$）时涡粘性为负，将能量转移到小尺度的流剪切也有一个有趣的效果，即通过减缓反向串级速率，使得负涡粘性的值在强剪切时按$|\nu_T|\propto A^{-2}$减小。此外，即使在没有外部各向异性源的情况下，剪切和旋转之间的相互作用也被证明会产生角动量的非扩散通量（$\Lambda$效应）。这种效应为天体物理和地球物理系统中剪切结构的存在提供了一种机制。
