problem_set6

problem_set6 - Problem Set #6 Due: Friday 18 Nov. 2011...

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Unformatted text preview: Problem Set #6 Due: Friday 18 Nov. 2011 UNDERGRAD: Also hand in HOLTON 3.10, 3.11 GRAD students do only b.), c.), and f.) (As usual you may use Excel for your plotting/programming and/or software/language [C, For- tran, Java, Grace, etc.] of your choice to calculate the profiles.) Assume that there are no large-scale troughs or ridges such that the temperature gradient varies in the meridional direction only and can be represented as a linear function of pressure via the fol- lowing 5_T =_( 1 _3) * ay(p) 0‘ 700hPap 7 where oc = 30 K/1000 km is a ‘representative’ surface temperature gradient. a.) Plot the temperature gradient profile from p = psfc = 1000 mb to p = 50 mb. b.) Using the thermal wind equation (i.e. Holton 3.30) and * above, derive an analytic relationship for the geostrophic wind as a function of pressure, i.e. ago?) . c.) Using your answer to part b) above, plot the geostrophic wind, ago?) , profile from p = 1000 mb to p = 50 mb (use 10 mb increments), where f 2 f0 = I 0‘4 s'] . Your y-axis (pressure) should be logarithmic. Assume ugwsfc) = 0 . d.) Describe/show how you could graphically use your results in 0.) above to estimate the hori- zontal temperature gradient at any given level. e.) Using your results in a)-c) above, answer the following questions: i. At what level are your winds a maximum? Why is this (i.e., why don’t the winds continue to increase all the way up to 50 mb)? ii. What do we call this wind maximum? 1‘.) GRADUATE STUDENTS ONLY : Note that you can arrive at an approximation to the exact wind profile (0 above) by applying the trapezoidal rule (yikes it’s all coming back to haunt you now!), i.e., N If(x)dx ~ 2 [fix/6+ 1)2+f(xk) k=l (xk+1_xk) For this problem we have, and applying the trapezoidal rule to both sides for N = l we have pl RplaTa p] ——Pk+1 laugi‘rlaf laug~ug<k+1>—ug<k> f<x>+x Z p0 pa pa —— Pk and, Pl 5 alilulil} 0C (L _§)_E(L 3)] _ fiayp f2 pk+1700pk+1 7 pk700P/c 7 (mm 19;) pl) Combining and rearranging the two we have: Rl 0c 1 3 0c 1 3 [(+1 ~ k +—— — (_ __)__(_ __):| _ ** “g( ) “gm f2[pk+1700pk+1 7 pk 700” 7 (pkfl pk) You can see here that the geostrophic wind at a given level is a function of the geostrophic wind at the level below + the average meridional temperature gradient over the layer (1/2 X bracket term) multiplied by the layer depth (pk+1-pk). Code the approximation given above (**) using a pro- gramming language of your choice and Ap of 200, 100, 50, 10 hPa. Plot the 4 wind profiles along with your analytic wind profile in part c.) above. ...
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problem_set6 - Problem Set #6 Due: Friday 18 Nov. 2011...

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