Chapter 5. Fundamentals of Atmospheric Modeling

Transcription

1 Overhead Slides for Chapter 5 of Fundamentals of Atmospheric Modeling by Mark Z. Jacobson Department of Civil & Environmental Engineering Stanford University Stanford, CA January 30, 2002

2 Altitude Coordinate Fig Heights of altitude-coordinate surfaces. z 6 z 5 z 5 z 4 z 3 z 2 z 1

3 Equation for Nonhydrostatic Pressure Decompose pressure into large-scale and perturbation term p a = p ˆ a + p a (5.1) Large-scale atmosphere in hydrostatic balance 1 ˆ p a ρ ˆ a z = α ˆ ˆ p a a z = g (5.2) Decompose gravitational and pressure gradient terms g + 1 ρ a z = g + α ˆ a + α a ( ) ( p z ˆ a + p a ) ˆ Substitute (5.3) into vertical momentum equation p α a a z α a g α ˆ a (5.3) dw dt = w + u w x + v w y + w w z = ˆ p α a a z + α a g (5.4) α ˆ a Take the sum of (5.4), (4.72), and (4.73) ( vˆ ρ a )+ ρ ˆ a ( v )v [ ] = ˆ ( ρ a fk v) 2 z p ˆ a 2 p a + g z (5.5) α a α ˆ 2 a

4 Equation for Nonhydrostatic Pressure Note that α a θ v α ˆ a θ ˆ c v,d p a (5.6) v c p,d p ˆ a Remove local derivative from continuity equation --> Anelastic continuity equation ( vˆ ρ a ) = 0 (5.7) Substitute (5.6) and (5.7) into (5.5) --> Diagnostic equation for nonhydrostatic pressure 2 p a g c v,d c p,d z z 2 ˆ p a + g z ρ ˆ p a a p ˆ = [ ρ ˆ a ( v )v] ρ ˆ a fk v a θ ρ ˆ p a θ ˆ + ( ρ ˆ a K m )v (5.8) p [ ]

5 Pressure Coordinate Fig Heights of pressure-coordinate surfaces. p a,1 p a,1 p a,1 p a,top p a,2 p a,3 p a,4 p a,5 p a,6

6 Intersection of Pressure and Altitude Surfaces Fig z z 2 q 1 p 1 z 1 q 3 q 2 p 2 x 1 x 2 x

7 Gradient Conversion From the Altitude to Pressure Coordinate Change in mass mixing ratio over distance q 2 q 3 = q 1 q 3 + p 2 p 1 x 2 x 1 x 2 x 1 x 2 x q 1 q 2 1 p 1 p (5.9) 2 Approximate differences as x 2 x 1 0, p 1 p 2 0 q x z = q 2 q 3 x 2 x 1 q x p = q 1 q 3 x 2 x 1 (5.10) x z = p 2 p 1 x 2 x 1 q p a x = q 1 q 2 p 1 p 2 Gradient conversion from the altitude to pressure coordinate q x z = q x p + q x z p a x (5.11) General equations x z = x p + x z p a x (5.12)

8 Gradient Conversion From the Altitude to Pressure Coordinate Substitute time for distance = + z p z p a t Horizontal gradient operator in the altitude coordinate (5.15) z = i x z + j y z (4.80) Horizontal gradient operator in the pressure coordinate p = i x p + j y p (5.14) Gradient conversion between the altitude and pressure coordinate z = p + z ( p a ) (5.13)

9 Geopotential Gradient Take gradient conversion of geopotential z Φ = p Φ + z ( p a ) Φ Note that z Φ = 0 Rearrange gradient conversion z ( p a ) = Φ pφ = g z pφ = ρ a p Φ (5.16) Component directions x y z z Φ = ρ a x p Φ = ρ a y p (5.17) (5.17)

10 Continuity Equation For Air in the Pressure Coordinate Continuity equation for air in the altitude coordinate ρ a = ρ a ( v) ( v )ρ a (3.20) Expand with horizontal operators ρ a z = ρ a z v h + w ( v z h z )ρ a w ρ a z (5.18) Gradient conversion of velocity z v h = p v h + z ( p a ) v h (5.19) Substitute gradient conversion and hydrostatic equation ρ a z ( ) = ρ a p v h + z ( p a ) v h ( v h z )ρ a + ρ a g wρ a (5.20)

11 Continuity Equation For Air in the Pressure Coordinate Define vertical scalar velocity in the pressure coordinate w p = dp a dt = z ( ) p a = + v z + ( v h z )p a + w z Substitute dz = -dp a ρ a g (5.21) w p = ρ a g z z + ( v h z ) p a wρ a g (5.22) Differentiate vertical velocity with respect to altitude w p z = g ρ a z + z ( p a ) v h z + ( v h z ) ( ) z g wρ a z (5.23) Substitute dz = -dp a ρ a g w p ρ a = ρ a z + ρ a z ( p a ) v h + v p h z a ( ) ( )ρ a ρ a g wρ a (5.24) Add (5.20) to (5.24) --> continuity equation for air p v h + w p = 0 (5.25)

12 Continuity Equation For Air in the Pressure Coordinate Expanded continuity equation u x + v y p + w p = 0 (5.26) Example 5.1. x = 5 km y = 5 km p a = -10 mb u 1 =-3 (west) u 2 =-1 m s -1 (east) v 3 = +2 (south) v 4 =-2 m s -1 (north) w p,5 =+0.02 mb s -1 (lower) ----> ( 1 + 3) m s m + ( 2 2) m s m + ( w p,6 0.02) mb s 1 10 mb = > w p,6 = mb s -1 (downward)

13 Total Derivative in the Pressure Coordinate Total derivative in Cartesian-altitude coordinate d dt = z + ( v h z ) + w z (5.27) Substitute time and horizontal gradient conversions d dt = p + z ( ) + [( v h z )p a ] + v h p + w z (5.28) Vertical velocity in altitude coordinate from (5.21) w = z + ( v h z )p a w p ρ a g (5.29) Substitute (5.29) and hydrostatic equation into (5.28) --> total derivative in Cartesian-pressure coordinates d dt = p + v h p ( ) + w p (5.30)

14 Species Continuity Equation in the Pressure Coordinate Species continuity equation in the altitude coordinate N e,t dq ( dt = ρ a K h )q + R ρ n a n=1 Apply total derivative in Cartesian-pressure coordinates dq dt = q p q + ( v h p )q + w p N e,t ( = ρ a K h )q + R ρ n a n=1 (5.31) Convert mass mixing ratio to number concentration q = Nm ρ a A (5.32)

15 Thermodynamic Energy Equation in the Pressure Coordinate Thermodynamic energy equation in the altitude coordinate dθ v dt ( = ρ a K h )θ v + ρ a θ v c p,d T v N e,h n=1 dq n dt Apply total derivative in Cartesian-pressure coordinates θ v p θ + ( v h p )θ v + w v p = ( ρ a K h )θ v + ρ a θ v c p,d T v N e,h n=1 dq n dt (5.34)

16 Horizontal Momentum Equations in the Pressure Coordinate Horizontal momentum equation in the altitude coordinate dv h dt = fk v h 1 ρ z ( ( p a )+ ρ a K m )v h a ρ a Substitute z ( p a ) = ρ a p Φ and apply total derivative in Cartesian-pressure coordinates --> v h p v + ( v h p )v h + w h ( p = fk v p h p Φ+ ρ a K m )v h a ρ a (5.35)

17 Vertical Momentum Equation in the Pressure Coordinate Assume hydrostatic equilibrium z = ρ ag Substitute g = Φ z p a = ρ a R T v T v = θ v P Hydrostatic equation in the pressure coordinate Φ = R T v = R θ v P = R θ v p a p a p a κ p a 1000 mb (5.37) Substitute κ = R c p,d p dφ = c p,d θ v d κ a = c 1000 p,d θ v dp (5.38)

18 Geostrophic Wind in the Pressure Coordinate Substitute x z Φ = ρ a x p y z Φ = ρ a y p (5.17) into v g = 1 fρ a x u g = 1 fρ a y (4.78) --> Geostrophic wind in the pressure coordinate v g = 1 f Φ x p u g = 1 f Φ y p (5.39) Vector form v g = iu g + jv g = i 1 f Φ y + j 1 f p Φ x p = 1 f ( k p Φ) (5.40)

19 Geostrophic Wind on a Constant Pressure Surface Fig Contour line 500 mb surface East 500 mb 5.6 km Contour line 500 mb 5.5 km North West South 510 mb 5.5 km

20 The Sigma-Pressure Coordinate Definition of a sigma level σ = p a p a,top p a,surf p a,top = p a p a,top π a (5.41) Pressure difference between column surface and top π a = p a,surf p a,top Pressure at a given sigma level p a = p a,top + σπ a (5.42) Fig Heights of sigma-pressure coordinate surfaces. σ 1 =0 σ 1, p a,1x σ 1, p a,1y p a,top σ 2 σ 3 σ 4 σ 5 σ 6 =1

21 z Intersection of Sigma-Pressure and Altitude Surfaces Fig z 2 p 1 σ 1 z 1 p 3 p 2 σ 2 x 1 x 2 x

22 Gradient Conversion From the Altitude to Sigma-Pressure Coordinate Change in pressure per unit distance p 2 p 3 = p 1 p 3 + σ 2 σ 1 p 1 p 2 x 2 x 1 x 2 x 1 x 2 x 1 σ 1 σ (5.43) 2 Approximate differences x z = p 2 p 3 x 2 x 1 x σ = p 1 p 3 x 2 x 1 (5.44) σ x z = σ 2 σ 1 x 2 x 1 σ x = p 1 p 2 σ 1 σ 2 Gradient conversion from z to σ-p coordinate x z = x σ + σ x z σ x (5.45) Conversion in gradient operator notation z ( p a ) = σ ( p a ) + ( z σ) σ (5.46) where σ = i x σ + j y σ (5.47)

23 Gradient Conversion From thealtitude to Sigma-Pressure Coordinate Generalize gradient operator z = σ + z ( σ) σ (5.48) Definition of sigma σ = ( p a p a,top ) π a Gradient of sigma ( ) 1 z σ = ( p a p a,top ) z π + z p a a π a = σ π a z π a ( ) + z ( p a ) π a (5.49) Substitute into (5.48) σ z = σ π z π a a ( ) z ( p a ) π a σ (5.50)

24 Intersection of Pressure, Altitude, and Sigma-Pressure Surfaces Fig z z 2 p q 1 1 σ 1 z 1 q 2 p 2 σ 2 q 3 x 1 x 2 x

25 Gradient Conversion From the Pressure to Sigma-Pressure Coordinate Change in mixing ratio per unit distance q 1 q 3 = q 2 q 3 + σ 1 σ 2 x 2 x 1 x 2 x 1 x 2 x 1 Gradient conversion from p to σ-p coordinate q 1 q 2 σ 1 σ (5.51) 2 q x p = q x σ + σ q x p σ x (5.52) Generalize p = σ + p ( σ) σ (5.53) Take gradient of sigma along surface of constant pressure 1 ( ) p π + p p a p top a π a p ( σ) = p a p top where Substitute (5.54) into (5.53) ( ) p ( p a ) = 0 p ( p top ) = 0 p ( π a ) = σ ( π a ) = z ( π a ) = σ π p ( π a ) a (5.54) p = σ σ π σ ( π a ) a σ (5.55)

26 Continuity Equation For Air in the Sigma-Pressure CoordinateContinuity equation for air in the pressure coordinate p v h + w p = 0 Substitute gradient conversion and σ = π a σ v h σ π σ ( π a ) v h a σ + 1 w p π a σ = 0 (5.56) Vertical velocity in the pressure coordinate w p = dp a dt where = σ dπ a dt + dσ dt π a = σ dπ a dt + σ π a (5.58) p a = p a,top + π a σ Vertical velocity in the sigma-pressure coordinate σ = dσ dt (5.57)

27 Continuity Equation For Air in the Sigma-Pressure Coordinate Material time derivative in the sigma-pressure coordinate d dt = σ + ( v h σ ) + σ σ (5.59) Substituting the total derivative of π a into (5.58) w p = σ π a σ + ( v h σ )π a + σ π a (5.60) Take partial derivative w p σ = π a σ + ( v h σ )π a + σ σ ( π a ) v h σ + π σ a σ (5.61) Substitute into (5.56) --> continuity equation for air π a σ + σ v h π a ( ) + π a σ σ = 0 (5.62) Convert to spherical-sigma-pressure coordinates R 2 e cos ϕ π a σ + uπ λ a R e e ( ) + ϕ vπ ( ar e cos ϕ) σ + π a R 2 e cos ϕ σ σ = 0 (5.63)

28 Column Pressure From the Continuity Equation Continuity equation for air π a σ + σ v h π a ( ) + π a σ σ = 0 (5.62) Rearrange and integrate 1 π a 1 0 dσ = 0 σ ( v h π a )dσ π 0 a d σ (5.64) 0 σ Prognostic equation for column pressure π a σ 1 = σ ( v h π a )dσ (5.65) 0 Analogous equation in spherical-sigma-pressure coordinates R 2 e cos ϕ π a σ = 1 0 λ e ( uπ a R e ) + ( ϕ vπ ar e cos ϕ) dσ σ (5.66)

29 Vertical Velocity From Continuity Equation Continuity equation for air π a σ + σ v h π a ( ) + π a σ σ = 0 (5.62) Rearrange and integrate σ π a d 0 σ σ = σ ( v h π a )dσ 0 σ π a dσ (5.67) 0 σ Diagnostic equation for vertical velocity σ σ π a = σ ( v h π a )dσ 0 σ π a σ (5.68) Analogous equation in spherical-sigma-pressure coordinates σ π a R 2 e cos ϕ = σ ( uπ λ a R e )+ e ϕ vπ ( ar e cos ϕ) dσ σr 0 2 e cosϕ π a σ σ (5.69)

30 Species Continuity Equation in Spherical-Sigma-Pressure Coordinates Species continuity equation in Cartesian-altitude coordinates N e,t q + ( v )q = 1 ( ρ ρ a K h )q + R n a (3.54) n=1 Apply material time derivative is sigma-pressure coordinate d dt = σ + v h σ + σ σ --> Equation in Cartesian-sigma-pressure coordinates dq dt = q σ N e,t + ( v h σ )q + σ q ( σ = ρ a K h )q + R ρ n a n=1 (5.70) Combine with continuity equation for air ( π a q ) ( σ q) N + σ ( v h π a q) ( +π a σ = π ρ a K h )q e,t a + R ρ n σ a n=1 (5.72) Apply spherical-coordinate transformations 2 R e cos ϕ ( π a q) + σ λ e ( uπ a qr e )+ ϕ vπ aqr e cos ϕ ( ) σ 2 +π a R e cosϕ σ σ q 2 ρ ( ) = π a R e cos ϕ a K h N e,t ( )q + R ρ n a n=1 (5.73)

31 Thermodynamic Energy Equation in Spherical-Sigma-Pressure Coordinates Therm. energy equation in Cartesian-altitude coordinates θ v + ( v )θ v = 1 ρ a ( ρ a K h )θ v + θ v c p,d T N e,h dq n (3.76) dt n=1 Apply the sigma-pressure coordinate material time derivative θ v σ + ( v h σ )θ v + σ θ v σ = ( ρ a K h )θ v + ρ a θ v c p,d T v N e, h n=1 dq n dt Combine with continuity equation for air ( ) ( ) π a θ v σ θ + σ ( v h π a θ v ) + π v a σ σ N ( ρ = π a K h )θ v θ e,h a + v ρ a c p,d T v n=1 Apply spherical-coordinate transformations dq n dt (5.74) (5.75) 2 R e cos ϕ ( ) π a θ v + uπ σ λ a θ v R e e ( ) + ( ) ϕ vπ aθ v R e cos ϕ 2 +π a R e cosϕ σ σ θ v ( )θ v + ρ a 2 ρ ( ) = π a R e cosϕ a K h θ v c p,d T v N e,h n=1 (5.76) dq n dt

32 Momentum Equation in the Sigma- Pressure Coordinate Horizontal momentum equation in Cartesian-altitude coordinates dv dt = fk v Φ 1 ρ a p a + η a ρ a 2 v + 1 ρ a ( ρ a K m )v (4.70) Material time derivative of velocity dv h = v h + ( v dt h σ )v h + σ v h σ σ Apply to horizontal momentum equation v h σ + ( v h σ )v h + σ v h σ + fk v h = 1 ρ z ( ( p a )+ ρ a K m )v h a ρ a (5.77) Pressure gradient term 1 ρ z ( p a ) = p Φ = σ Φ σ a π σ ( π a ) Φ a σ (5.78) Substitute into momentum equation v h σ + ( v h σ )v h + σ v h σ + fk v h (5.79) = σ Φ + σ π σ ( π a ) Φ a ( σ + ρ a K m )v h ρ a

33 Coupling Horizontal and Vertical Momentum Equations Hydrostatic equation in the pressure coordinate Φ σ = π a R T v p a = π a ρ a = α a π a (5.80) Re-derive specific density α a = R T v p a = κc p,d θ v P p a = c p,d θ v P = c p,d θ v π a P σ (5.82) Combine terms above with momentum and continuity equations ( ) v h π a σ + v h σ ( v h π a )+ π a ( v h σ )v h +π a ( σ σ v h ) P = π a fk v h π a σ Φ σc p,d θ v σ σ( π a ) ( ρ +π a K m )v h a (5.83) ρ a Expand advection terms v h σ v h π a ( ) = iu ( uπ a ) x ( ) + vπ a y ( ) + jv uπa x ( ) + vπ a y (5.84) π a ( v h σ )v h = iπ a u u x + v u + jπ y a u v x + v v (5.85) y

34 Momentum Equation in Spherical- Sigma-Pressure Coordinates 2 R e cos ϕ ( π a u ) + σ λ e ( π a u 2 R e ) + ϕ π auvr e cosϕ ( ) σ ( ) 2 + π a R e cosϕ σ σ u = π a uvr e sinϕ + π a fvr 2 Φ P π e cos ϕ R e π a + σc λ p,d θ a v e σ λ e σ + R e 2 cos ϕ π a ρ a ( ρ a K m )u (5.86) 2 R e cos ϕ ( π a v) + σ π λ a uvr e e ( ) + ( ) ϕ v2 π a R e cos ϕ σ + π a R e 2 cos ϕ σ ( σ v) = π a u 2 R e sin ϕ π a fur 2 Φ e cosϕ R e cosϕ π a ϕ + σc P p,dθ v σ + R e 2 cos ϕ π a ρ a ( ρ a K m )v (5.87) π a ϕ σ

35 The Sigma-Altitude Coordinate Sigma-altitude value s = z top z = z top z (5.89) z top z surf Z t Altitude difference between column top and surface Z t = z top z surf Altitude of a sigma surface z = z top Z t s (5.90)

36 Gradient Conversion From the Altitude to Sigma-Altitude Coordinate Gradient conversion between z and s-z coordinate z = s + z ( s) s (5.91) Horiz. gradient of sigma along const. altitude surface z ( s) = z top z Z2 z ( Z t ) = s t Z z ( Z t ) (5.92) t Substitute into gradient conversion z = s s Z z ( Z t ) t s (5.93)

37 Conversions in the Sigma-Altitude Coordinate Time-derivative conversion between z and s-z coordinate z = s (5.94) Scalar velocity in the sigma-altitude coordinate s = ds dt = ( v h z )s + w s z = ( v h z )s w (5.95) Z t where s z = 1 Z t Material time derivative in the sigma-altitude coordinate d dt = s + ( v h s ) + s s (5.96)

38 Continuity Equation For Air in the Sigma-Altitude CoordinateContinuity equation for air in the z coordinate ρ a z = ρ a z v h + w ( v z h z )ρ a w ρ a z Apply gradient conversion to horizontal velocity z v h = s v h + z ( s) v h s Apply gradient conversion to dry air density z ( ρ a ) = s ( ρ a ) + z ( s) ρ a s Substitute these two terms into continuity equation above ρ a s = ρ a s v h + z s v h s ρ a ( ) + z s ( ) v h ( ) ρ a s + w z s w ρ a z (5.97)

39 Continuity Equation For Air in the Sigma-Altitude Coordinate Rewrite vertical velocity equation [ ( ) s ] w = Z t v h z s Differentiate with respect to altitude w z = Z t z s ( ) v h ( ) s z + v h z z s z (5.98) Substitute s z = 1 Z t w z = s s z s ( ) v h Substitute w = Z t v h z s s + 1 ( v Z h z )Z t (5.99) t [ ( ) s ], (5.99) and s z = 1 Z t into (5.97) ρ a s = ρ a s v h + s s + 1 Z t ( v h z )Z t ( v h s )ρ a s ρ a s (5.100)

40 Continuity Equation For Air in the Sigma-Altitude Coordinate ( ) = s ( Z t ) and compress --> Substitute z Z t Nonhydrostatic continuity equation for air in s-z coordinate ρ a s = 1 Z t s v h ρ a Z t ( ) ( s ) s ρ a ( ) = 1 uρ a Z t Z t x ( ) + vρ a Z t y s ( s ρ a ) s (5.101) Hydrostatic equation in the sigma-altitude coordinate ρ a = 1 g p a z = 1 Z t g p a s (5.102) Substitute into (5.101) --> Hydrostatic continuity equation p a s = s v p a h s s s p a s (5.103)

41 Species Continuity Equation in the Sigma-Altitude Coordinate Apply material derivative in the s-z coordinate to the continuity equation for a trace species in the z coordinate dq dt s = q s + ( v h s )q + s q N e,t ( s = ρ a K h )q + R ρ n a n=1 (5.104)

42 Thermodynamic Energy Equation in the Sigma-Altitude Coordinate Apply material derivative in the s-z coordinate to the thermodynamic energy equation in the z coordinate θ v s + ( v h s )θ v + s θ v ( s = ρ a K h )θ v + ρ a θ v c p,d T v N e,h n=1 dq n dt (5.106)

43 Horizontal Momentum Equation in the Sigma-Altitude Coordinate Horizontal equation in the z coordinate dv h dt = fk v h 1 ρ z ( ( p a )+ ρ a K m )v h a ρ a Apply material time derivative of velocity v h s + ( v h s )v h + s v h s + fk v h = 1 ρ z p a a ( ( )+ z ρ a K m z)v h ρ a (5.107) Gradient conversion of pressure z ( p a ) = s ( p a ) s Z z ( Z t ) t s (5.108) Substitute gradient conversion v h s + ( v h s )v h + s v h s = fk v h 1 ρ a s ( p a ) s Z z Z t t ( ) (5.109) s ρ ( ak m )v h

44 Vertical Momentum Equation in the Sigma-Altitude Coordinate Substitute s z = 1 Z t into vertical momentum eq. in z coordinate w + u w x + v w y s + s w s = g + 1 Z t ρ a ( s + ρ a K m )w ρ a (5.113) Substitute [ ( ) s ] w = Z t v h z s Another form of vertical momentum equation + u s x s + v y s + s s Z tu s x z + Z t v s y z Z t s = g + 1 Z t ρ a s ( ) Z t u s + 1 ρ a ρ a K m x z + Z t v s Z y t s z (5.114)