Department of Applied Physics and Applied MathematicsColumbia UniversityAPPH E4210. Geophysical Fluid DynamicsSpring 2004Problem Set 6(Due March 4, 2004)1. Warm up and review. Consider a homogeneous layer of depth H. Assuming no rotation,show that the perturbation pressure p0satisfies Laplace’s equation. Derive the linearizedboundary conditions p0is subject to at the bottom, z = −H, and the top, z = 0. Solvethe resulting problem, and find the gravity wave dispersion relation. (Do not make the hy-drostatic approximation.) Assuming that the wave vector is directed in the x direction, i.e.,k = (k, 0), make a plot of the ratio of group velocity to phase speed as a function of kH.2. Energetics of internal waves. Consider an unbounded fluid on the f-plane, with constantbuoyancy frequency N. Making the Boussinesq approximation, derive an equation govern-ing the vertical velocity component w. Substitute a plane wave solution w = Woexp i(kx +ly + mz − ωt) to find the dispersion relation for internal gravity waves. Also find, in termsof Wo, the horizontal components of velocity (u,v), the perturbation pressure p0, and the per-turbation density ρ0. Use the governing equations to derive a conservation law for the energydensity E (energy per unit volume). Write E as a sum of kinetic and potential energy terms.Finally, for a plane wave, find the ratio of average kinetic to potential energy, where theaverage is over a complete cycle of the phase. (You may find it useful to review problem 5from last weeks’ assignment.) When the ratio is 1, we say that the energy is equipartitioned.3. Normal modes for the ocean. The separation of variables procedure we applied in classresults in the following equation for the vertical structure functionˆh:d2ˆhdz2+N2(z)c2ˆh = 0,subject to the following (linearized) boundary conditions:ˆh(z = −H) = 0 andˆh(z = 0) = 0.(Here, we have made the Boussinesq and rigid lid approximations.) Given N(z), these equa-tions define a Sturm-Liouville eigen problem for the eigenfunctionsˆh(z) and the eigenvalues1/c2. Using an observed profile of N, calculate (and plot!) the first 5 (in order of decreasingvalues of c) eigenfunctions and the corresponding values of c in two different ways:(a) Numerically, and(b) Using the WKBJ approximation.The N profile can be downloaded from CourseWorks or http://www.ldeo.columbia.edu/˜spk/Classes/APPH4210_GFD/N_profile. (Assume H = 4290 m.) Hints:When solving the problem numerically, beware that this is a boundary value problem. (Cannedroutines are generally designed to solve initial value problems.) Also, the WKB part of theproblem is not difficult at all. The solution we derived in class is applicable with only slightmodification. The WKB solution for the “equivalent phase speed” c is very useful in