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Unformatted text preview: AOS 311 Problem Set 2 Solutions 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 = V D , and that the mean advecting wind is small so that ( 29 ( 29 f t f dt d + + . For this situation we can simplify the vorticity equation to ( 29 ( 29 D f f t +- + . Note that since f is small we cant assume that the Rossby number is small, i.e. we can have f . a) Suppose that the heating is switched on at time t = 0, and that the heating-induced divergence D is steady after that. Find as a function of t , and show that f- as t . (Hint: for any function , if - = t then ( 29 t o t - = = exp ) ( .) For ( 29 ( 29 D f f t +- + : let ( 29 f + = let = D Therefore As t , ( 29 + f , b) If D = 10-6 s-1 , and = at = t , how many days will it take before 2 / f f < + ? Given above, the equation becomes ( 29 ( 29 Dt f f t t-- = + = exp | ) ( 2 / f f < + when ( 29 2 1 exp <- Dt : <- 2 1 ln Dt ( 29 ( 29 ( 29 Dt f f t t- +- = + = exp | ) ( so f- as t c) Suppose we assume that 1 << o R in part (a) so that f << . How does this assumption affect the outcome as t ? Equation becomes ( 29 fD f t- + , which can be solved by integrating through time: ( 29 dt fD dt f t t t t t t t = = = =- + ( 29 ( 29 fDt f f t t- = +- + = | | Therefore 02 . 8 10 931 . 6 2 1 ln 1 5 = - s x D t days ( 29 ( 29...
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This homework help was uploaded on 03/27/2008 for the course ATM OCN 311 taught by Professor Deweaver during the Spring '08 term at Wisconsin.
- Spring '08