Department of Applied Physics and Applied MathematicsColumbia UniversityAPPH E4210. Geophysical Fluid DynamicsSpring 2008Problem Set 2(Due Feb 7, 2008)1. Energetics of internal waves. Consider an unbounded fluid on the f-plane, with constantbuoyancy frequency N.(a) Making the Boussinesq approximation, derive an equation governing the vertical veloc-ity component w. Do not make the hydrostatic approximation. Note that, the Boussi-nesq approximation we studied in class can also be stated as follows: in the momentumequations, variations in density may be neglected except when computing buoyancyforces, i.e., wherever density is coupled to gravity. Thus, in the x-momentum equationρo∂u∂t− fv= −∂p0∂x,we may replace ρowith ρ, where ρ is a mean density. The z-momentum equationbecomes:ρ∂w∂t= −∂p0∂z− ρ0g.Use this more conventional statement of the Boussinesq approximation in your deriva-tion of the equation for w.(b) Substitute a plane wave solution w = Woexp i(kx+ly+mz− ωt) to find the dispersionrelation for internal gravity waves.(c) For a plane wave, find, in terms of Wo, the horizontal components of velocity (u,v), theperturbation pressure p0, and the perturbation density ρ0.(d) Use the governing equations to derive a conservation law for the energy density E(energy per unit volume) of the wave. Write E as a sum of kinetic and potential energyterms.(e) Finally, find the ratio of average kinetic to potential energy, where the average is overa complete cycle of the phase. When the ratio is 1, we say that the energy is equiparti-tioned. You may find the following relationship useful: For ψ = A exp i(k · x − ωt),< (Re ψ)2>=12AA∗,where, <> represents an average over a complete cycle of the phase, and A∗is thecomplex conjugate of A.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. 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.) The WKB solution for the“equivalent phase speed” c is very useful in