Department of Applied Physics and Applied MathematicsColumbia UniversityAPPH E4210. Geophysical Fluid DynamicsSpring 2008Problem Set 6(Due Apr 10, 2008)1. In this problem you will complete the solution of the Eady baroclinic instability problem.Recall that we are considering quasigeostrophic motions of a uniformly rotating (f-plane),uniformly stratified (N constant) fluid bounded by two horizontal surfaces at z = 0 andz = H. The instability problem considers small perturbations about a background (basic)state consisting of a steady, uniformly sheared, zonal flow u(z) = (Uo/H)z. This flow is inthermal wind balance with the basic state density (temperature) structure.(a) Make a contour plot of the growth rate (Im ω) as a function of k and l. (Nondimen-sionalize the growth rate by the advective timescale in the problem, λd/Uo.) Note thatthe fastest growth occurs when l = 0. Wave motion is then purely in the meridionaldirection (a consequence of the O(1) horizontal velocity field being nondivergent), i.e.,down the mean temperature gradient, and the release of available potential energy ismaximized.(b) In the Eady problem, long waves, i.e., waves with µH less than a critical value, areunstable (c is complex).i. Find the full solution ψ0(x, y, z, t) = Reˆψ(z) exp i(kx + ly − ωt) for the unstablewaves. It is convenient to writeˆψ(z) = |ˆψ(z)| exp iα(z).ii. For µH corresponding to the most unstable wave, make plots of |ˆψ(z)| and α(z)for both the growing and decaying solutions. Indicate in which direction the phaselines tilt with height for the two solutions. Note that |ˆψ(z)| takes on a minimumvalue at zc= H/2. At this height, known as the steering level, the phase speed isequal to the local mean flow, i.e., cr= u(zc).iii. Calculate the meridional heat flux, ρoCpv0θ0, associated with the disturbance. (Cpis the specific heat capacity at constant pressure of air.) Show that a wave withphase lines tilting westward with height is associated with poleward heat flux, thatis, a growing disturbance transports heat poleward (as it must if it is to draw uponthe potential energy stored in the mean flow).iv. Using values of various parameters typical of the mid-latitude atmosphere, com-pute the poleward heat flux associated with the fastest growing wave.v. For the most unstable wave, make contour plots (in the x-z plane) of the pressurefield (ψ0) and the temperature perturbation (θ0). Note the characteristic tilt of thephase lines. Also note that at the surface (z = 0), there is a phase shift betweenthe temperature and pressure perturbations associated with the disturbance, withwarm air just ahead (westward) of the pressure trough.vi. For the most unstable wave, make a vector plot (in the y-z plane) of the velocityfield (v0,w0). (You will need to pick a particular value of x.)(c) For a disturbance with k = l (a so called “square Eady wave”), find the maximumgrowth rate and wavelength of the most unstable perturbation. Assuming a buoyancyfrequency of N = 10−2s−1, what is the e-folding time (in days) for growth for