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PS02_solutions

# PS02_solutions - AOS 311 Problem Set 2 Solutions 1 Consider...

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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 0 = V D , and that the mean advecting wind is small so that ( 29 ( 29 f t f dt d + + ζ ζ . For this situation we can simplify the vorticity equation to ( 29 ( 29 D f f t + - + ζ ζ . Note that since f is small we can’t assume that the Rossby number is small, i.e. we can have f ζ . a) Suppose that the heating is switched on at time t = 0, and that the heating-induced divergence D is steady after that. Find ζ as a function of t , and show that f - ζ as t . (Hint: for any function η , if λη η - = t then ( 29 t o t λ η η - = = exp ) ( .) For ( 29 ( 29 D f f t + - + ζ ζ : let ( 29 f + = ζ η let λ = D Therefore As t , ( 29 0 + f ζ , b) If D = 10 -6 s -1 , and 0 = ζ at 0 = t , how many days will it take before 2 / f f < + ζ ? Given above, the equation becomes ( 29 ( 29 Dt f f t t - - = + = exp | 0 ) ( ζ 2 / f f < + ζ when ( 29 2 1 exp < - Dt : < - 2 1 ln Dt ( 29 ( 29 ( 29 Dt f f t t - + - = + = exp | 0 ) ( ζ ζ so f - ζ as t

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c) Suppose we assume that 1 << o R in part (a) so that f << ζ . How does this assumption affect the outcome as t ? Equation becomes ( 29 fD f t - + ζ , which can be solved by integrating through time: ( 29 dt fD dt f t t t t t t t = = = = - + 0 0 ζ ( 29 ( 29 fDt f f t t - = + - + = 0 | | ζ ζ Therefore 02 . 8 10 931 . 6 2 1 ln 1 5 = - s x D t days ( 29 ( 29 fDt f f t t - + = + = 0 | | ζ ζ , and vorticity decreases linearly through time.
How does this affect your answer in part (b)?

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PS02_solutions - AOS 311 Problem Set 2 Solutions 1 Consider...

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