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ComputerProb5 - Oct 31 2006 ATMS 502 CS 505 CSE 566 Jewett...

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ATMS 502 Computer problem 5, page 1 Fall 2006 - Jewett Oct. 31, 2006 ATMS 502 - CS 505 - CSE 566 Jewett Computer Problem 5 2D nonlinear quasi-compressible flow Due : Thursday, Nov. 16 (recommended!) / Thursday, Nov. 30 (last date accepted) Problem : colliding dry outflows – nonlinear, quasi-compressible flow Methods : directional splitting; non/monotonic piecewise linear advection, Lax-W. A. Equations The set of equations includes four unknowns: horizontal (u) and vertical (w) wind, potential temperature ( θ ) , and pressure (p) . There are advection, diffusion, and pressure/buoyancy terms, with sound waves present. The continuous equation form: u-momentum: u t = " uu x " wu z " 1 # $ p x + K $ u xx + $ u zz ( ) ; $ u = u " U ( z ) w-momentum: w t = " uw x " ww z " 1 $ p z + g $ % + K m w xx + w zz ( ) ; $ = " ( z ) (thermodynamic): " t = # u ( ) x # w ( ) z + u x + w z ( ) Perturbation pressure: " p t = # c s 2 $ u x + z w ( ) ( ) * + Equations of state and potential temperature: p = R d T and = T p 0 p $ % ( ) R d / C p Dependent variables: u(x,z,t), w(x,z,t), θ (x,z,t), p’(x,z,t), with units of m s -1 , m s -1 , ˚K and Pa (pascals) for u,w, θ , and p’, and g kg -1 for . The discrete equation form: u: 2 t u = # u x x u ( ) ( n ) x # w x z u ( ) ( n ) z # 1 x % p ( n # 1) + K xx u + zz % u ( ) ( n # 1) w: 2 t w = # u z x w ( ) ( n ) x # w z z w ( ) ( n ) z # 1 ( ) z z % p ( n # 1) + g # ( ) * + , ( n ) z + K xx w + zz w ( ) ( n # 1) : t = $ 1 % t F i + 1/2 u , ( ) $ F i $ 1/2 u , ( ) [ ] ( n ) + ( n ) x u [ ] ( n ) $ 1 % t F k + 1/2 w , ( ) $ F k $ 1/2 w , ( ) [ ] ( n ) + ( n ) z w [ ] ( n ) ( ( ) * + + + K xx + zz , ( ) ( n ) or t = $ u 2 x + u 2 % t 2 xx $ w 2 z + w 2 % t 2 zz - . / 0 1 2 ( n ) + K xx + zz , ( ) ( n ) " p : 2 t # p = $ c s 2 x u ( n + 1) + z ( ) z w ( n + 1) { } ( ) * +
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ATMS 502 Computer problem 5, page 2 Fall 2006 - Jewett U, W advection follows the “box method,” not to be confused with an implicit scheme of the same name. Pressure and diffusion terms are lagged (at time n-1 ). θ is advected with Lax-Wendroff or the MPL scheme (see below). Note carefully: time levels, averaging! B. Grid layout and boundary conditions
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