A layer of water of uniform density obeys the rotating shallow-water equations on an…

PHYSICS · Multi-part · Fermi University Physics · Earth, Climate & Energy Physics · Atmosphere and ocean dynamics · Atmosphere and ocean dynamics

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

Problem

A layer of water of uniform density obeys the rotating shallow-water equations on an ff-plane, DuDt+fz^×u=−g∇hη,DhDt=−h ∇h⋅u,\frac{D\mathbf u}{Dt} + f\hat{\mathbf z}\times\mathbf u = -g\nabla_h\eta, \qquad \frac{Dh}{Dt} = -h\,\nabla_h\cdot\mathbf u, where hh is the local depth of a fluid column, η\eta the free-surface height, ff is constant, and ωz\omega_z is the vertical component of ∇×u\nabla\times\mathbf u. ((a)) Take the curl of the momentum equation to obtain the relative-vorticity equation Dωz/Dt=−(ωz+f)∇h⋅uD\omega_z/Dt = -(\omega_z+f)\nabla_h\cdot\mathbf u, then eliminate ∇h⋅u\nabla_h\cdot\mathbf u using the depth equation to derive DDt(ωz+fh)=0.\frac{D}{Dt}\left(\frac{\omega_z+f}{h}\right) = 0. ((b)) A current at latitude 30∘N30^\circ\mathrm N flows with zero relative vorticity over a sea floor where the water column is 2000 m2000\,\mathrm m deep, then crosses a seamount where the column depth is reduced to 1200 m1200\,\mathrm m. Using Ω=7.292×10−5 s−1\Omega = 7.292\times10^{-5}\,\mathrm{s^{-1}} for the Earth's rotation, find the relative vorticity the column acquires over the seamount and state whether it turns cyclonically or anticyclonically.

Browse the problem library