论文标题
压力驱动的球形壳中的粘性流体旋转
Stress-driven spin-down of a viscous fluid within a spherical shell
论文作者
论文摘要
我们研究了球形壳内部的粘性流体的稳定和轴对称应力驱动的旋转流的线性特性，无论是在不可压缩的和无弹性的近似值内，以及小粘度的渐近极限。从边界层分析中，我们为3D不可压缩的稳定流提供了一个分析地质解，在圆柱体内外，与内壳相切。位于$ \ Mathcal {C} $上的Stewartson图层由两个厚度$ O（E^{2/7}）$的嵌套剪切层组成，$（E^{1/3}）$。我们得出了$ e^{2/7} $ - layer的最低顺序解决方案。对沿切线圆柱体放置的$ e^{1/3} $ - 层的简单分析表明，它是振幅上升流$ o（e^{1/3}）$的上升流的位置。尽管它狭窄，但该剪切层集中了旋转流的大部分全球子午下动能。此外，只要prandtl数量足够小，稳定的分层就不会扰动旋转流动。如果不是这种情况，Stewartson层消失，子午循环被局限于热层。已经发现，在最低阶的所有三个区域中，无弹性二级流的幅度的尺度与不可压缩的流量相同。但是，由于速度不再符合Taylor-Proudman定理，因此其形状在切线缸$ \ Mathcal {C} $之外，即发生差异旋转的地方。最后，我们发现在粘性的时间到达稳态的沉降，以解决虚弱，强烈和热未分层的不可压缩流。与星体和地球物理系统相关的大密度变化往往会略微缩短瞬态。
We investigate the linear properties of the steady and axisymmetric stress-driven spin-down flow of a viscous fluid inside a spherical shell, both within the incompressible and anelastic approximations, and in the asymptotic limit of small viscosities. From boundary layer analysis, we derive an analytical geostrophic solution for the 3D incompressible steady flow, inside and outside the cylinder $\mathcal{C}$ that is tangent to the inner shell. The Stewartson layer that lies on $\mathcal{C}$ is composed of two nested shear layers of thickness $O(E^{2/7})$ and $O(E^{1/3})$. We derive the lowest order solution for the $E^{2/7}$-layer. A simple analysis of the $E^{1/3}$-layer laying along the tangent cylinder, reveals it to be the site of an upwelling flow of amplitude $O(E^{1/3})$. Despite its narrowness, this shear layer concentrates most of the global meridional kinetic energy of the spin-down flow. Furthermore, a stable stratification does not perturb the spin-down flow provided the Prandtl number is small enough. If this is not the case, the Stewartson layer disappears and meridional circulation is confined within the thermal layers. The scalings for the amplitude of the anelastic secondary flow have been found to be the same as for the incompressible flow in all three regions, at the lowest order. However, because the velocity no longer conforms the Taylor-Proudman theorem, its shape differs outside the tangent cylinder $\mathcal{C}$, that is, where differential rotation takes place. Finally, we find the settling of the steady-state to be reached on a viscous time for the weakly, strongly and thermally unstratified incompressible flows. Large density variations relevant to astro- and geophysical systems, tend to slightly shorten the transient.
