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hw2 - AOS311 Homework 2 Spring 2008 Homework is due in...

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Unformatted text preview: AOS311 Homework 2 Spring 2008 Homework is due in class on Friday, March 14th. Please be sure to show your work and explain your answers carefully. 1. Consider a level in the tropical upper troposphere above a region of convection and latent heat release. Suppose that at this level the heating causes divergence , and that the mean advecting wind is small so that can simplify the vorticity equation to . Note that since can’t assume that the Rossby number is small, i.e. we can have . and that the heating-induced a. Suppose that the heating is switched on at time divergence is steady after that. Find as a function of , and show that as . (Hint: for any function , if then .) b. If = 10 s and c. Suppose we assume that tion affect the outcome as 2. Consider an idealized atmosphere in pressure coordinates consisting of two layers of constant potential temperature: from to and from to . a. Following the class discussion, show that each layer behaves like a shallow water layer, i.e. is independent of height (or pressure, since we’re in pressure coordinates). b. In this system, we can think of diabatic heating as a process which increases the potential temperature of parcels with to (or vice versa for cooling), thereby transferring mass from the lower to the upper layer. How will diabatic heating affect the PV of the upper and lower layers? Make a sketch to explain your answer. 3. The concept of Available Potential Energy is simple for a system consisting of two fluids with the total in the plane is . Suppose that two fluids of densities and are initially sitting next to each other, each occupying a rectangle of area . a. Show that of the system. , where , , and b. How would rotation affect the ability of the system to convert to kinetic energy? Suppose the system starts off with the two fluids next to each other but the final state is steady, geostrophically balanced motion. Has all of the been converted to kinetic energy? Explain. f” …9z f…”9z f ¨” %2pe…! r ie r“ e 6 “ “ f | Y …” ~}}&r{ f ” ¡ „ e r ƒ‚we ¡ e€ ¡ …9z “e h%f k f %q wsorih f …”  jihu…ih” gmYt ¨” ¡ …” % j f “ we r e densities and , as shown in class. for a single column is given by f Y …” yw r e ‚yxv o  pe mYl k on –—” ¡ ” r ’ ¡ ’ —” ¡ ” •¡ ” B“’ ¡ ’ –  H IE ‘ @ @ v [email protected] C# … @ ‰ ‡ vv ˆ…†…„ ƒ ‚€[email protected]# xvwC#u9 @ @ @ !   t¡  t¡  % R6hgedBcba`XWVUP Q f Y T P Q 210 ¡ P SR6  ¡ P 0 P # G6 EF    D¡  @C# @B'A @ @ # %# ! 6 '%# ! &$9 875&43  210 0 %# !  ' %# !  &$"  ) (&$"   ) at , how many days will it take before . For this situation we is small we in part so that . How does this assump? How does it affect your answer in part ?   §© ¥ £ ¡ ¨¦¤¢ d“’ ¡ ’ r ’ ¡ ’ r si pqi H IE  §© ? ˜ ™” ¡ ” , and is the initial ...
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