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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 - Fall 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 - Fall 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 - Fall 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 - Fall 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 - Fall 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 - Fall 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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