An ocean at latitude has buoyancy frequency

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

An ocean at latitude 45∘N45^\circ\mathrm{N} has buoyancy frequency N=2.4×10−3 rad s−1N = 2.4\times10^{-3}\,\mathrm{rad\,s^{-1}}, taken as uniform over the depth of interest. The rotation rate of the Earth is Ω=7.292×10−5 rad s−1\Omega = 7.292\times10^{-5}\,\mathrm{rad\,s^{-1}}. Someone argues as follows: "An inertial wave oscillates at ff and an internal wave oscillates at NN. In a fluid that is both stratified and rotating, the restoring effects add, so the frequency of the combined inertia-gravity wave is simply ω=N+f,\omega = N + f, whatever the direction of the wave vector." Using this reasoning they compute f=2Ωsin⁡45∘=1.03×10−4 rad s−1f = 2\Omega\sin45^\circ = 1.03\times10^{-4}\,\mathrm{rad\,s^{-1}} and quote ω=2.50×10−3 rad s−1\omega = 2.50\times10^{-3}\,\mathrm{rad\,s^{-1}} for every wave in this ocean. ((a)) Find the error in this reasoning, stating which physical bound it violates. ((b)) Compute the correct frequency, in rad s−1\mathrm{rad\,s^{-1}}, for a wave whose wave vector makes an angle φ=30∘\varphi = 30^\circ with the horizontal.

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