Day18-slides - Fundamental Eqns Momentum Continuity: r r "#...

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1 Atms 502 Numerical Fluid Dynamics Tue., Oct. 24, 2006 Wave equations in Geophysical Fluid Dynamics References : Durran chaps. 1,7 Haltiner & Williams chap. 1,2 Ferziger and Peric (2002, 3rd edition) 10/25/06 Atms 502 - Fal 2006 - Jewett 2 Continuity: Equation of state: Note definition of total derivative! Fundamental Eqns Momentum 1 st law - thermodynamics No heat exchange Q du dt = " 1 # dp dx + f + u tan $ a % & ( ) * v + F + dv dt = " 1 dp dy " f + u tan a % & ( ) * u + F dw dt = " 1 dp dz " g " F z "# dt = $ r % r V P = " RT d dt = 0; = T p 0 p # $ % & ( R / c p da dt = a t + V # a 10/25/06 Atms 502 - Fal 2006 - Jewett 3 Fundamental Eqns Exner form Eliminates p and ρ from the motion and continuity equations π is the nondimensional Exner function. If use ideal gas equation + def’n of π , θ + thermodynamics, continuity equations: = p p 0 # $ % & ( R / C p ) 1 * + p = c p , + d dt = # R c v $% r v 10/25/06 Atms 502 - Fal 2006 - Jewett 4 Euler equations, minus Coriolis: Inviscid flow: far from solid surfaces; neglect all viscous effects Euler Eqns d r V dt = " c p $ % " g ˆ k d dt = 0 d dt = " R c v $& r v
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2 10/25/06 Atms 502 - Fal 2006 - Jewett 5 Euler equations in Durran: Part of full compressible equations Includes sound waves; restricts time step Klemp and Wilhelmson (1978): The term f π is often neglected . Pressure field: prognostic d " dt = # R c v $% r v # $ dt + c 2 % C p & v 2 x v u ( ) + y v v ( ) + z %& v w ( ) ( ) * + , = f , where f = - u # x - v # y - w # z - R d # C v u x + v y + w z . / 0 1 2 3 + c 2 C p v 2 d v dt 10/25/06 Atms 502 - Fal 2006 - Jewett 6 In, e.g., a Boussinesq system, you solve a diagnostic equation for pressure at every step. Take divergence of momentum equations p’ is that which keeps evolving velocity field nondivergent. Pressure field: diagnostic r v t + 1 # 0 $ % p = F ( r v , % ) where F ( r v , % ) = & r v r $ v & g % 0 r k d % dt + w d dz = 0 r $ r v = 0 ( ) * * * * + * * * * , $ 2 % p = 0 r $ r F Durran chap. 7 10/25/06 Atms 502 - Fal 2006 - Jewett 7 u dt = # r V $ r % u # 1 p x + % 2 u v dt = # r V $ r % v # 1 p y + % 2 v w dt = # r V $ r % w # 1 p z + g ( + % 2 w "( t = # r V $ r % + Q ( x , y , z , t ) + % 2 p t = # c s 2 u x + v y ) * + , - .
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This note was uploaded on 10/06/2009 for the course ATMOSPHERI 502 taught by Professor Jewett during the Spring '09 term at University of Illinois at Urbana–Champaign.

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Day18-slides - Fundamental Eqns Momentum Continuity: r r "#...

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