Department of Applied Physics and Applied MathematicsColumbia UniversityAPPH E4210. Geophysical Fluid DynamicsSpring 2005Problem Set 4(Due Feb 24, 2005)1. Energetics of internal waves. Consider an unbounded fluid on the f-plane, with constantbuoyancy frequency N.(a) Write down the linearized governing equations in the Boussinesq approximation.(b) If the vertical velocity component w is given by a plane wave solution w = Woexp i(kx+ly + mz − ωt), find, in terms of Wo, the horizontal components of velocity (u,v), theperturbation pressure p0, and the perturbation density ρ0.(c) Use the governing equations to derive a conservation law for the energy density E(energy per unit volume). Write E as a sum of kinetic and potential energy terms.(d) Finally, for a plane wave, find the ratio of average kinetic to potential energy, wherethe average is over a complete cycle of the phase. When this ratio is 1, we say that theenergy is equipartitioned. Hint: For a plane wave solution, ψ = A exp i(k · x − ωt),it can be shown that < (Re ψ)2>=12AA∗, where <> represents an average over acomplete cycle of the phase, and A∗is the complex conjugate of A. This result provesto be rather convenient when computing averages of the kind required here.2. 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. For reasons we will discuss in class, c is known as the “equivalent phase speed.” Usingan observed profile of N from the Pacific Ocean, calculate (and plot!) the first 5 (in orderof decreasing values of c) eigenfunctions and the corresponding values of c in two differentways:(a) Numerically, and(b) Using the WKBJ approximation.Please provide a printout of the scripts or programs you use to solve and plot the solution.The N profile can be downloaded from CourseWorks. (Assume H = 4290 m.) Hints: Whensolving the problem numerically, beware that this is a boundary value problem. (Cannedroutines are generally designed to solve initial value problems.) In general, the numericalsolution of eigenvalue problems is a nontrivial business, but for present purposes a simpleminded approach in which the above ODE is discretized using finite differences to give amatrix eigen problem Ax = λBx is more than adequate. (Note that this matrix equationdefines a generalized eigenvalue problem which you can solve quite easily in software suchas MATLAB.) Also, the WKB part of the problem is not difficult at all. The solutionwe derived in class is applicable with only slight modification. The WKB solution for the“equivalent phase speed” c is very useful in