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Columbia APPH E4210 - Problem Set 8

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Department of Applied Physics and Applied MathematicsColumbia UniversityAPPH E4210. Geophysical Fluid DynamicsSpring 2004Problem Set 8(Due April 8, 2004)1. Poincare waves on a β-plane. Consider the shallow water equations on a β-plane. Forsimplicity assume a flat bottom. In class we showed that the gravity wave solution, namelythe Poincare waves, are only slightly modified by the presence of β. Quantify this by derivingthe lowest order correction to the dispersion relation. Give a numerical estimate of thiscorrection for a mid-latitude (45◦N) baroclinic plane wave of zonal wavenumber kλd= 3,and meridional wavenumber lλd= 0. Assume a first baroclinic phase speed of 3 ms−1.Hint: You may find it convenient to work with the cubic equation for ω that we derived inclass. In particular, observe (graphically) what happens to the gravity wave root as β → 0,and then approximate accordingly.2. Rossby wave dispersion. Derive expressions for the group velocity and zonal phase speed(ω/k) for shallow water Rossby waves. For l = 0, make a plot of ω as a function of k.(Nondimensionalize axes in a sensible manner.) Indicate on the figure the direction in whichphase and energy propagate. Assuming a mean stratification of N = 2 × 10−3and a depth of4300 m, provide numerical estimates of the zonal group and phase velocity for mid-latitudefirst baroclinic mode Rossby waves in the ocean. Based on these calculations, what can yousay about the linearity assumption made in deriving the governing equations.3. In the presence of a mean zonal current (or wind) U, the governing equation (linearizedabout the background flow) for low frequency motions is given by:(∂∂t+ U∂∂x)(∇2−1λ2d)η + β∂η∂x= 0.(a) Derive and physically interpret the dispersion relation for small amplitude waves gov-erned by this equation.(b) The figure below shows a longitude-time plot of 500 mbar geopotential height anoma-lies averaged over 45-50◦N. The data are for a 25 day period beginning on January 1,1997. The mean zonal winds at this latitude and for that period were roughly 20 ms−2.Suppose we were to interpret the pattern of height anomalies as that associated withbarotropic Rossby waves. (These are all gross approximations, but adequate for thepurpose at hand.) Based on the results of problem 2 above, what is the most strikingfeature of this plot? Use part (a) above to interpret these observations. A qualitativeexplanation will do, and probably the best one can do given the not atypically noisydata. But this should not stop you from suspending disbelief and plugging in somenumbers to see if your qualitative argument make quantitative sense. (You can makeyour own plots and animations here:


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