Stress-driven spin-down of a viscous fluid within a spherical shell. (arXiv:2003.10403v2 [physics.flu-dyn] UPDATED)
<a href="http://arxiv.org/find/physics/1/au:+Gagnier_D/0/1/0/all/0/1">Damien Gagnier</a>, <a href="http://arxiv.org/find/physics/1/au:+Rieutord_M/0/1/0/all/0/1">Michel Rieutord</a>
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.
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.
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