Instabilities of the Stewartson Layer
Instabilities of the Stewartson Layer
复制标题
斯图尔森层的不稳定性
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发表时间：
2003
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影响因子：
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通讯作者：
Rainer Hollerbach
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作者：
Rainer Hollerbach
在差动旋转圆柱或球体之间流动的经典Taylor-Couette问题中，差动旋转速度通常大于平均旋转速度，因此是最简单的教科书例子。在这项工作中，我们将改为考虑整体旋转非常大而差异旋转相对较小的极限。除了是经典的Taylor-Couette问题的一个有趣的变体外，这个极限在地球物理流体动力学(例如海洋学或气象学)中也很有意义，在地球物理流体动力学中，非常快速的整体旋转通常是一个主要特征。所以，假设我们有一个球壳在快速整体旋转，另外一个差动旋转施加在内球上。用间隙宽度缩放长度！“#$，用逆旋转速度%‘&和(×)*+#$缩放长度，旋转框架中的Navier-Stokes方程变成，(，.0/21(43*56(7 98：=(？5@BA 56C(Ed GFH)，其中Ekman和Rossby数A？I&gT；H+#$C J K L/21&>分别测量总体和差异旋转速度。在这项工作中，我们将考虑(1)的数值解，在极限情况下，F和/21直到OGFH。我们从计算轴对称的基本态开始，这就产生了所谓的Stewartson层。然后我们计算了这一层的非轴对称不稳定性，发现对于P Q K R/21S T VUWF，它们是非常不同的，即对于正的和负的微分旋转。我们将这一差异与以前的实验和分析结果进行了比较和对比，并进行了大量的数值测试以阐明其来源。图1显示了角速度X如何在外边界的0和内边界的1之间变化。/21>；，对应于无限小的差异旋转，以及AY？Fz%.[]到Fz%.^，对应于越来越快的整体旋转。随着A的减小，我们非常清楚地看到，在所谓的切线柱面上出现了一个越来越薄的剪切层，Xa‘Fh在里面，而Xb在外面。为了理解为什么解应该以这种特殊的方式排列，我们回顾著名的泰勒-普罗德曼定理，该定理指出，在快速旋转的系统中，流动将倾向于使自己与旋转轴对齐。更正式地，取(1)的旋度，并使用c cZd，/21，AeMfF(实际上图1中的c cZd/212)来获得，(，根据这个结果，图1中的解非常自然地遵循：对于流体柱_，Xj是上下边界的适当边界条件，因此Xk到处都将图1：角速度的等高线，对于/212，从左到右，A？Fz%，Fz%，Fz%和Fz%。等高线间隔为1/7。3.5 4 4.5 5−1.5−1−0.5 0 Ro>；0
In the classical Taylor-Couette problem of the flow between differentially rotating cylinders or spheres, the differential rotation rate is typically greater than the average rotation rate , with (and hence ) being the simplest, textbook example. In this work we will consider instead the limit where the overall rotation is very large, and the differential rotation is relatively small. Aside from being an interesting variant on the classical Taylor-Couette problem, this limit is also of considerable interest in geophysical fluid dynamics (e.g., oceanography or meteorology), in which a very rapid overall rotation is typically a dominant feature. So, suppose we have a spherical shell in rapid overall rotation , with additionally a differential rotation imposed on the inner sphere. Scaling length by the gap width ! " # $ , time by the inverse rotation rate %'& , and ( by ) * + # $ , the Navier-Stokes equation in the rotating frame becomes , ( ,.0/21 (43*56( 7 98 : = ( ? 5 @ BA 56C (ED GFH where the Ekman and Rossby numbers A? I > H + # $ C J K L /21> measure the overall and differential rotation rates, respectively. In this work we will consider numerically computed solutions of (1), in the limit ANM F and /21 up to O GFH . We begin by computing the axisymmetric basic states, which turn out to yield this so-called Stewartson layer. We then compute the non-axisymmetric instabilities of this layer, and find them to be very different for P Q K R/21S T VUWF , that is, for positive versus negative differential rotation. We compare and contrast this difference with previous experimental and analytical results, and conduct a number of numerical tests to elucidate its origin. Figure 1 shows how the angular velocity X varies between 0 at the outer boundary and 1 at the inner. /21> , corresponding to an infinitesimal differential rotation, and AY ?FZ %.[]\ ^ to FZ %.^ , corresponding to an increasingly rapid overall rotation. As A decreases, we see very clearly the emergence of an increasingly thin shear layer on the so-called tangent cylinder _ , with Xa` FH inside _ but X b outside. In order to understand why the solutions should arrange themselves in this peculiar fashion, we recall the well-known Taylor-Proudman theorem, stating that the flow in a rapidly rotating system will tend to align itself with the axis of rotation. More formally, take the curl of (1) and use c cZd , /21 , AeMfF (in fact c cZd /212 in Fig. 1) to obtain , ( ,.g ` ih With this result, the solutions in Fig. 1 follow quite naturally: For fluid columns outside _ , X j is the appropriate boundary condition at both the upper and lower boundaries, so X k everywhere will Figure 1: Contours of the angular velocity, for /212 and, from left to right, A ?FZ %.[]\ ^ , FZ % , FZ % \ ^ and FZ %.^ . The contour interval is 1/7. 3.5 4 4.5 5 −1.5 −1 −0.5 0 Ro > 0