Department of Applied Physics and Applied MathematicsColumbia UniversityAPPH E4210. Geophysical Fluid DynamicsSpring 2004Problem Set 7(Due March 11, 2004)1. Consider the shallow water equations on an f-plane, and a plane wave solution of the formη = Re ηoexp i(kx + l y − ωt).(a) Find the velocity field, (u,v ), in terms of η.(b) Write the flow field in terms of a component parallel to the wave vector (uk) and acomponent perpendicular to the wave vector (u⊥). Show that the horizontal velocityvector traces out an ellipse. In which direction (clockwise or counter clockwise) doesthe velocity vector rotate?2. Poincare Waves. Show that the perturbation potential vorticity (PV), q0, of a Poincare waveis exactly zero. What does this imply about the relation between the vorticity ξ and thesurface elevation η? (What this shows is that these waves “carry” no PV, and that the finalgeostrophic steady state can be determined from the initial PV distribution.)3. Geostrophic adjustment. Consider the shallow water equations on an f-plane. Suppose thatat t = 0, the surface elevation and meridional velocity v are both zero and the zonal velocityis given byu = Uo, −L ≤ y ≤ L,and zero elsewhere.(a) Write down the appropriate Klein-Gordon equation governing the time evolution of η.(b) Write the solution as the sum of a time-dependent homogeneous solution (ηh) and asteady particular solution (ηs). Find the steady, geostrophic solution ηs. Hint: Youwill find that the problem to be solved is a 2d order, inhomogeneous ODE, whichrequires the specification of 2 boundary conditions (or constraints). Apparently, theonly boundary conditions available are that η not blow up as y → ±∞. What to do?Recall that a similar situation is encountered when solving for the Green’s function.(If this sounds unfamiliar, look it up in any standard ODE or PDE book, Haberman,say.) There, and here too, we integrate the differential equation over a small intervalcentered about some point yo, and then let the interval go to zero. (The choice of yodepends on the problem at hand.) This establishes the continuity (or lack thereof) of ηand dη/dy across yo. The change in η or its derivative across yois known as a “jumpcondition” and provides us with the necessary constraints. It is easier than it sounds.(c) Use the momentum equations to find the geostrophic velocity field.(d) Compute the ratio R of the total energy in the final geostrophic state to that in theinitial state. Express, and make a plot of, this ratio as a function of L/λd, where λdisthe deformation radius.(e) What are the initial conditions satisfied by ηh? Solve the homogeneous problem subjectto these initial conditions. You may do it numerically or analytically. (I have notattempted the latter, but would be very happy if one of you can do it.) Make plots ofthe full time dependent solution η = ηh+ ηsat several times showing the approach to asteady state. Hint: If you do the problem numerically, it may be easiest to write ηhas asuperposition of plane wave solutions (making use of the dispersion relation) and thenevaluate the Fourier integral using the discrete Fourier transform. (A few lines of codein