Consider a column of ocean water at latitude

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Fluids & Continuum Physics · Waves in two and three dimensions, dispersion and wave packets

Signal Speeds in Fluids, Ch Waves in a rotating fluid: inertial, Poincare and Kelvin waves

Problem

Consider a column of ocean water at latitude 50∘N50^\circ\mathrm N, where f=2Ωsin⁡50∘f=2\Omega\sin50^\circ. ((a)) In the well-mixed surface layer above the thermocline, treated as homogeneous, a plane inertial wave has a wave vector at φ=60∘\varphi=60^\circ to the horizontal. Derive the inertial-wave dispersion relation from the linearised rotating equations and evaluate ω\omega and the period for this wave. ((b)) Immediately below, in the thermocline, the water is stably stratified with buoyancy frequency N=0.020 s−1N=0.020\,\mathrm{s^{-1}}. Extend the dispersion relation to include buoyancy and evaluate ω\omega and the period for a wave of the same φ=60∘\varphi=60^\circ in this layer, checking that it lies between ff and NN. ((c)) A front between water masses is modelled by an active upper layer of depth h=800 mh=800\,\mathrm m over a much deeper lower layer that may be treated as motionless, with reduced gravity g′=0.050 m s−2g'=0.050\,\mathrm{m\,s^{-2}} between them. An initial interface displacement of Δh=100 m\Delta h = 100\,\mathrm m across the front is released from rest. Using conservation of potential vorticity, find the half-width of the adjusted front and its peak geostrophic current, and state the fraction of the released potential energy retained there.

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