A cylindrical tank of water rotates about its vertical axis at constant angular velocity

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Convection, rotation and geophysical flows · Convection, rotation and geophysical flows

Fluid Dynamics, Ch Rotating fluids: geostrophy, Taylor columns and Ekman layers

Problem

A cylindrical tank of water rotates about its vertical axis at constant angular velocity Ω=2.0 rad s−1\Omega = 2.0\,\mathrm{rad\,s^{-1}}. The water is H=25 cmH = 25\,\mathrm{cm} deep, bounded below by a rigid floor and above by a rigid lid, and a small bump of horizontal scale L=6.0 cmL = 6.0\,\mathrm{cm} sits on the floor. Once solid rotation is established, a slow steady flow of speed U=1.5 cm s−1U = 1.5\,\mathrm{cm\,s^{-1}} is set up past the bump. Take the kinematic viscosity of water as ν=1.00×10−6 m2 s−1\nu = 1.00\times10^{-6}\,\mathrm{m^2\,s^{-1}} and treat the tank as an ff-plane with f=2Ωf = 2\Omega (a container spinning about a vertical axis), the density of the water uniform. ((a)) Starting from the geostrophic balance 2Ω×u=−ρ−1∇P2\boldsymbol{\Omega}\times\mathbf{u} = -\rho^{-1}\nabla P for this uniform-density fluid, take its curl and use incompressibility to derive the Taylor-Proudman theorem (Ω⋅∇)u=0(\boldsymbol{\Omega}\cdot\nabla)\mathbf{u} = 0. ((b)) Explain why the rigid floor and lid, together with the theorem, force w=0w = 0 everywhere in the tank, and hence why the fluid above the bump behaves as if a solid column stood on the bump and reached the lid. ((c)) Compute the Rossby and Ekman numbers for this flow, using LL as the horizontal length scale and HH as the vertical one, and confirm that both are small enough for the theorem to hold away from the side walls.

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