A cylindrical tank of water deep is in solid rotation at

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 H=15 cmH = 15\,\mathrm{cm} deep is in solid rotation at Ω=1.5 rad s−1\Omega = 1.5\,\mathrm{rad\,s^{-1}}. Its angular velocity is then increased by a small increment ΔΩ≪Ω\Delta\Omega\ll\Omega, so that the flow relative to the new rotation rate is a weak swirl with Rossby number much less than one. Take the water's kinematic viscosity as ν=1.00×10−6 m2 s−1\nu = 1.00\times10^{-6}\,\mathrm{m^2\,s^{-1}} and f=2Ωf = 2\Omega for the tank. ((a)) The bottom Ekman layer pumps fluid out of (or into) the interior at rate wE=12δE ωzw_E = \tfrac12\delta_E\,\omega_z, where ωz\omega_z is the vertical relative vorticity of the geostrophic interior flow and δE=2ν/f\delta_E = \sqrt{2\nu/f}. This pumping stretches or squashes the interior column of height HH, so that dH/dt=−wE\mathrm{d}H/\mathrm{d}t = -w_E. Using also the shallow-water result that (ωz+f)/H(\omega_z+f)/H is conserved by a fluid column, and that ωz≪f\omega_z\ll f for this weak swirl, derive a first-order equation dωz/dt=−ωz/τE\mathrm{d}\omega_z/\mathrm{d}t = -\omega_z/\tau_E and give τE\tau_E in terms of HH, δE\delta_E and ff. ((b)) Show that τE\tau_E reduces to H/νΩH/\sqrt{\nu\Omega}. ((c)) Evaluate τE\tau_E for the tank described and compare it with the diffusive time H2/νH^2/\nu that would apply if rotation played no part.

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