Vertical length scale selection for pancake vortices in strongly stratified viscous fluids
Vertical length scale selection for pancake vortices in strongly stratified viscous fluids
复制标题
强分层粘性流体中薄饼涡的垂直长度尺度选择
DOI：
10.1017/s0022112004008067
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发表时间：
2004
影响因子：
3.7
通讯作者：
P. Billant
中科院分区：
文献类型：
--
作者：
R. Godoy;J. Chomaz;P. Billant
在分层槽实验中研究了不同长径比的煎饼偶极子的演化。这里报告了两种情况，即$\alpha_0 = 0.4$（情况I）和$\alpha_0 = 1.2$（情况II）的偶极子初始纵横比$\alpha_0 = L_v/L_h$（其中$L_v$和$L_h$分别是垂直和水平长度尺度）的值。在第一种情况下，观察到通常的衰减情景，偶极子随着厚度的增加和衰减环流缓慢扩散。在情形II中，我们观察到偶极子的厚度减小，而涡旋水平中层的环流保持不变。我们表明，这种垂直长度尺度减小的状态可以用偶极子顶部和底部两个边界层的脱落来解释，这些边界层实际上剥离了涡度层。水平平流和垂直扩散在这一状态下协同作用，在一个时间尺度$\alpha_0 \hbox{\it Re}^{1/2}T_A$上向粘性垂直长度尺度$\delta = L_h\hbox{\it Re}^{-1/2}$减小，$T_A$为平流时间$L_h/U$。通过对Boussinesq近似中分层粘性流体方程的标度分析，讨论了依赖于参数$R = \hbox{\it Re}F_h^2$的两个主要平衡，其中$F_h = U/NL_h$为水平弗鲁德数，$\hbox{\it Re} = UL_h/\nu$为雷诺数，$U$、$N$和$\nu$分别为偶极子的平移速度、Brunt-Väisälä频率和运动粘度。当$R\gg 1$垂直长度尺度由浮力效应决定为$L_b = U/N$级。本文所介绍的实验是在$R$小的情况下进行的，在这种情况下，粘性效应决定了垂直长度尺度的选择。我们表明，如果最初$L_v \leq \delta$，流动在垂直方向上扩散（情况I），而如果$L_v \gg \delta$（情况II），垂直剪切的水平平流将垂直长度缩小到$\delta$。这种粘性状态可以解释实验和数值模拟的结果，在分层流动的后期演化中，观察到衰减与浮力频率无关$N$。
The evolution of pancake dipoles of different aspect ratio is studied in a stratified tank experiment. Two cases are reported here for values of the dipole initial aspect ratio $\alpha_0 = L_v/L_h$ (where $L_v$ and $L_h$ are vertical and horizontal length scales, respectively) of $\alpha_0 = 0.4$ (case I) and $\alpha_0 = 1.2$ (case II). In the first case, the usual decay scenario is observed where the dipole diffuses slowly with a growing thickness and a decaying circulation. In case II, we observed a regime where the thickness of the dipole decreases and the circulation in the horizontal mid-plane of the vortices remains constant. We show that this regime where the vertical length scale decreases can be explained by the shedding of two boundary layers at the top and bottom of the dipole that literally peel off vorticity layers. Horizontal advection and vertical diffusion cooperate in this regime and the decrease towards the viscous vertical length scale $\delta = L_h\hbox{\it Re}^{-1/2}$ occurs on a time scale $\alpha_0 \hbox{\it Re}^{1/2}T_A$, $T_A$ being the advection time $L_h/U$. From a scaling analysis of the equations for a stratified viscous fluid in the Boussinesq approximation, two dominant balances depending on the parameter $R = \hbox{\it Re}F_h^2$ are discussed, where $F_h = U/NL_h$ is the horizontal Froude number and $\hbox{\it Re} = UL_h/\nu$ is the Reynolds number, $U$, $N$ and $\nu$ being, respectively, the translation speed of the dipole, the Brunt–Väisälä frequency and the kinematic viscosity. When $R\gg 1$ the vertical length scale is determined by buoyancy effects to be of order $L_b = U/N$. The experiments presented in this paper pertain to the case of small $R$, where viscous effects govern the selection of the vertical length scale. We show that if initially $L_v \leq \delta$, the flow diffuses on the vertical (case I), while if $L_v \gg \delta$ (case II), vertically sheared horizontal advection decreases the vertical length scale down to $\delta$. This viscous regime may explain results from experiments and numerical simulations on the late evolution of stratified flows where the decay is observed to be independent of the buoyancy frequency $N$.