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PaperIII_70 (2) - MATHEMATICAL TRIPOS Part III Thursday 29...

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MATHEMATICAL TRIPOS Part III Thursday, 29 May, 2014 1:30 pm to 4:30 pm PAPER 70 FLUID DYNAMICS OF THE ENVIRONMENT Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS Cover sheet None Treasury Tag Script paper You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator. 2 1 Consider small-amplitude, two-dimensional monochromatic internal waves in a Boussinesq fluid with constant buoyancy frequency, N 2, where N 2 = − g ρ0 dz , and ρ0 is a characteristic value of the background density distribution ρ(z). In a stationary ambient fluid, derive the dispersion relation for waves associated with a two- dimensional vertical displacement perturbation ζ = Aζ cos(kx + mz − Ωt), where Aζ , k, m and Ω are all real. Show that statically unstable regions are predicted to develop in the flow if |Aζ | > |1/m|. Now consider a situation where the ambient fluid velocity varies with height, such that U (z) = sz for z > 0 where s is a positive real constant. A two- dimensional wave packet propagates upwards into the upper half-plane, with horizontal wavenumber k > 0 and vertical wavenumber m0 < 0 at z = 0. Using the ray-tracing equations (which you may quote without proof ), show that the intrinsic frequency ω and the horizontal phase speed cx = ω/k measured by a stationary observer remain constant for all time. Hence derive an expression for the height zc of the critical level where cx = U (zc ). Furthermore, show that the ray followed by this wave packet is defined implicitly as the solution to the differential equation dx ksz N 2 dz = tan |Θ| + N sin |Θ| cos2 |Θ| , tan2 Θ(z) = (ω − ksz)2 − 1.
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