con_angmom_hw - Constant Angular Momentum Oscillations...

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Unformatted text preview: Constant Angular Momentum Oscillations: Graduate Student HW: Due: 02 December 2011 ALL GRAPHS SHOULD BE DONE IN AN EXCEL SPREADSHEET AND SHOULD BE SUBMITTED VIA EMAIL AS AN ATTACHMENT UNLESS OTHERWISE SPECIFIED. DO NOT BURN PAPER UNECESSARILY! Assume that a parcel accelerates from rest under the influence of a N/S pressure gradient (note which direction the PGF is). Assuming that the flow is horizontal, inviscid, and on an f-plane, show that the solution to the coupled differ- ential equations 02: Jag dt pax 1v; nag *"‘ dt p632 is given by the following I70) = §(cosft—l)i—1J—isinft} where K = (l /p)5p/ay. ALSO, answer the following questions: 1. Derive an expression for the parcel displacement x(t), y(t). 2. Plot two curves, 1.) zonal wind u(t) and 2.) meridional wind v(t) on the intervalft = [0, 27:]. Assume that f210'4 s'], and K' 2 40-3 ms'Z. 3. Using your results in 2.) above, indicate whether the quantity listed in the first column is >0, < 0, or = 0 over the interval given. ft [0, m2] [7r/2, 7:] [753m] [3 7r/2,27r] W) Va) du/dt dv/dt 4. Calculate the geostrophic wind (be sure to express in terms of a vector) at (l) = 45 N and 90 N. ,2.) W1, 5. On a separate graph from #3 above, plot the following curves (forft = [0,27r], and ¢ = 90 N) l.) IVg and 3.) — . Indicate (by lightly shading) regions of subgeostrophic and supergeostrophic flow. 6. On your graph in 5.) above, place an asterisk (*) at the point(s) in which the wind speed is geostrophic, and a plus (+) where the wind direction is geostrophic. 7. Using your results in #3-#6 above, briefly discuss why the wind never attains geostrophic balance. 8. Use your TRACKS account to run the code provided (coriolis.f and the input file c0riolis.input at ~slazarus/met4305/201 l/hw/). You must first compile the code (directions for doing this are embedded in the code). Once compiled, you will have an ‘executable’ called “coriolis” (without the .f appendage). Directions on how to run the executable are also included in the code. You will also need the file c0riolis.input (it contains the latitude input). You can edit the code using the command ‘pico’ (or “vi”) and the filename (i.e. type pico coriolis.f or pico corio- lis.input). The use of pico is pretty straightforward as you can move around in the code with the arrow keys and the ‘edit’ commands are listed at the bottom ofyour window. When you run the code, it will generate a file called corio- lis.txt. This file has column output with header information describing each column. If you remove the header, you can import this data directly into an EXCEL spreadsheet. The output file has 7 columns of data. Columns 4 and 5 are the transformed (i.e., x',y’) locations of the parcel defined by the following: x’(t) x(t) — ugt y'(t) = y(t) —ug/f This transformation is similar to that in Holton equations 1.15a,b (page 19), where x0 = ugt and ya = 245/)”. A. Graph the trajectories (2) defined by the coordinate pairs (x, y) in #2 above and (x', y’ ). Use can use the output of the code for both ofthese curves (columns 2 and 3 are x and y, and columns 4 and 5 are x', y’ ). Do this for both 30 N and 90 N (PLOT ALL OF THESE ON THE SAME FIGURE). B. What kind of trajectory (i.e., shape) does the transformed coordinate sweep out? Show that what you see ‘visually’ makes sense by subtracting ugt from x(t) and ué/f from y(t) using the x(t) and y(t) you obtained in #1 above (HINT: ug = -K/f). C. Calculate the radius of curvature (ROC) for the transformed trajectory for (1) = 30 N and 90 N. Show that your cal- culations match the ROC from the graphs in 8A. D. m the oscillation period, I (the time required for a Foucault pendulum to turn through an angle of 180°), versus sin¢ for ¢ = 10 N to 90 N (use a Ad) = 100 and normalize by dividing by the length of a sidereal day - hence your y axis should be in terms of a fraction of a day). How does this compare with your results in 8A above (i.e., compare the period of oscillation from your trajectory in 8A with that predicted by Holton Eq. 1.16). ...
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con_angmom_hw - Constant Angular Momentum Oscillations...

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