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Department of Applied Physics and Applied MathematicsColumbia UniversityAPPH E4210. Geophysical Fluid DynamicsSpring 2005Problem Set 5(Due March 10, 2005)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. Geostrophic adjustment. Consider the shallow water equations on an f-plane. Suppose thatat t = 0, the velocity field is zero and the surface elevation is given byη = ηo, −a ≤ y ≤ a,and zero elsewhere. (Make sure to attach plots of all solutions and printouts of any script-s/programs.)(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.There, and here too, we integrate the differential equation over a small interval centeredabout some point yo, and then let the interval go to zero. (The choice of yodepends onthe problem at hand.) This establishes the continuity (or lack thereof) of η and dη/dyacross yo. The change in η or its derivative across yois known as a “jump condition”and provides us with the necessary constraints.(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 a/λd, where λdisthe deformation radius.(e) Transient solution. Now that you have found the steady (particular) solution, lets nowcalculate the time-dependent (homogeneous) solution. While this transient solution canbe found analytically by means of Fourier or Laplace transforms, the inverse transformsare difficult to work out. (Of course, we could just look these up in tables!) Here, Iwalk you through the steps necessary to obtain the solution numerically. The basic ideais to use the discrete Fourier transform (implemented as the FFT in most math softwareincluding matlab) to do the inverse transform.i. Write down the PDE for the homogeneous part, ηh, and the initial conditions it issubject to.ii. Take the Fourier transform of the equation (in the spatial direction) and solve theresulting ODE (for the Fourier transform). Equivalently, you could simply writedown the solution as a sum of left- and right-going plane wave solutions using theknown (Poincare) dispersion relation for this equation.iii. Use the DFT (FFT) to invert the transform solution back into physical space.iv. Make plots of the full time dependent solution η = ηh+ηsat several times showingthe approach to a steady state. If you nondimensionalize time and space appropri-ately, your life will be greatly


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Columbia APPH E4210 - Problem Set

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