2012専門分科会_new_4.pptx

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3 d dt L L = 0 q i q i d dt L L = 0 r i i r i r r + Δr Δr δl = 0 dl dt = d dt i L L q i q i + q i i q i = q d L L i + q i i dt q i i q i = i L L q i L = 0, H = q q i L = E i q i i d dt L q q i i L = L(q i, q i )

4 ρv ε + energy flux + ( F) = 0, F = momentum flux tensor ( ) = 0, ε = ρ 1 v + gz +U ( ) = ρ m r k ( ρ m v ʹ v ʹ ) ρ m v + ρ m v v + ρ m Φ δ ij ρ m v ʹ + ρ m v v ʹ + ρ m v ʹ v + ρ m Φ ʹ δ ij ( ) = ρ m ʹ ( ) r k ρ v m ʹ v ʹ ρ v ʹ v ʹ m

5 1 ρ v m + 1 ρ m N r 1 + ρ v m + 1 ρ m N r v + ρ m Φ v + ρ m v ʹ v ʹ v + ρ m N r ʹ v ʹ r = K K ʹ [ ] [ P P ʹ ] 1 ρ m v ʹ + 1 ρ m N r ʹ 1 + ρ m v ʹ + 1 ρ m N r ʹ v + ρ m Φ ʹ v ʹ = K K ʹ [ ] + [ P P ʹ ] [ K K ʹ ] = ρ m u ʹ u ʹ u x + u ʹ v ʹ u y + u ʹ w ʹ u z + v ʹ u ʹ v x + v ʹ v ʹ v y + v ʹ w ʹ v z + w ʹ u ʹ w x + w ʹ v ʹ w y + w ʹ w ʹ w z [ P P ʹ ] = ρ m N u ʹ r ʹ r x + v ʹ r ʹ r y + w ʹ r ʹ r z

6 J = pdq E ω pdq = 1 pq, α dα = E ω, J = π α : phase of oscillation suffix :,α α

7 p = ρ 0 u, q = ξ ξ A = ξ j, α u j : action j p i = ξ j, i u j : pseudomomentum j e = ξ j, t u j : pseudoenergy j ρ 0 A + F = 0

8 P + F = D + O(a3 ), : F = F y y + F z z Generalized Eliassen Palm theorem P E u c : wave activity ( pseudomomentum), A P k = Ḙ ω : action, (WKB limit) ρ m N E = 1 ρ m v ʹ + 1 F u ʹ v ʹ 1 u v N z ʹ r ʹ, r ʹ : wave energy u ʹ w ʹ ( f u y ) v ʹ r ʹ, N If D = 0 and u c 0, F = 0, Eliassen Palm theorem Eliassen Palm flux

9 u + v u * y + w u * z fv * = y u ʹ v ʹ 1 u v N z ʹ r ʹ + z u ʹ w ʹ f u 1 y N r + v r * y N w * = z r ʹ w ʹ r y r ʹ v ʹ + J, N v ʹ r ʹ + X, F u ʹ v ʹ 1 u v N z ʹ r ʹ, u ʹ w ʹ ( f u y ) v ʹ r ʹ N If D = 0 and u c 0, F = 0 ; Generalaized Charney Drazin Nonacceleration theorem

10 1 q ʹ q y + s v ʹ q ʹ = ʹ q ʹ, q y Zonal mean QG potential vorticity q = f 0 + βy u y + z f 0 ψ = f z 0 + βy u y z Disturbance of QG potential vorticity q ʹ = ψ ʹ x + ψ ʹ y + z v ʹ q ʹ = y u ʹ v ʹ A + F = D A 1 f 0 ( ) + z f 0 q ʹ = E q y u c ψ ʹ, z v ʹ r ʹ = F, f 0 N r, 0 : QG Wave activity ( pseudomomentum), and E P flux = Wave activity ( pseudomomentum) flux.

11 We consider the WKB limt ψ ʹ (x, y,z,t) = Ψ ˆ (x, y,z,t)sin χ(x, y,z,t), χ : phase function, and the local wavenumbers are defined by k = χ x, 1 q ʹ + E q y u c = 1 4 l = χ y, m = χ z, = 1 4 Ψ ˆ Ψ ˆ q (y,z) = f 0 + βy u y + z k + l + f0 ( m ) f 0 ψ z q y sin χ k + l + f0 ( m ) q y. Ψ ˆ = f 0 + βy u y f 0 z N r 0 k + l + f0 ( m ) q y cos χ u c

12 Mean zonal flow : U(y, z) A s 1 1 q ʹ q y + E U = 1 4 Ψ ˆ k l f0 ( m ) q y sin χ Ψ ˆ k + l + f0 ( m ) cos χ = 1 U 4 Ψ ˆ k + l + f0 ( m ) q y, A s + F s = A s + (A s c g ) = C s, ψ ʹ ψ ʹ ψ ʹ x x F s = 1 ψ ʹ ψ ʹ x y ψ ʹ ψ ʹ x y f 0 ψ ʹ ψ ʹ x z ψ ʹ ψ ʹ x z.

13 Time mean flow : u = { u (x, y, z), v (x, y, z) } + u x + v 1 q y ʹ + ( B) H q ( ) = ʹ s q ʹ, ψ ʹ ψ ʹ E ψ ʹ f 0 ψ ʹ ψ ʹ x y y y z ψ ʹ ψ ʹ ψ ʹ f B = E 0 ψ ʹ ψ ʹ x y x x z u ʹ v ʹ = v ʹ E E u ʹ f 0 u ʹ v ʹ f u ʹ r ʹ v ʹ r ʹ, u ʹ q ʹ = ψ ʹ ψ ʹ y x + ψ ʹ y + z v ʹ q ʹ = ψ ʹ x ψ ʹ x + ψ ʹ y + z f 0 ψ ʹ z = B xx x + B xy y + B xz z = B f 0 ψ ʹ z = B yx x + B yy y + B yz z = B ( ) x, ( ) y.

14 1 H q where n = Hq H q + u x + v 1 q y ʹ + n B. ( ) = ʹ s q ʹ H q, Assuming that the basic QG potential vorticity gradient is slowly varying, we may write, + u x + v 1 y q ʹ H q + n B + u x + v M + M y R = S M, where ( ) = ʹ s q ʹ H q, M = 1 q ʹ H q, and S s M = ʹ n x B xx + n y B q ʹ H q, and M yx R = n B = n x B xy + n y B yy. n x B xz + n y B yz

15 Approximately conservative basic state, u H q 0. ( ) / u, if the mean flow is pseudoeastward, n v, u u ( v ʹ E ) v u ʹ v ʹ M R 1 v ( u ʹ E ) u u ʹ v ʹ. u f 0 ( N v u ʹ r ʹ u v ʹ r ʹ ) 0 Utilizing u = 0, + u x + v M + M y R = S M, may be written M + M T = S M, where M T = M R + u M. H q Under the WKB limit, M T = c g M. M R = M T u M = ( c g u )M = c ˆ g M, where ˆ c g c g u : u intrinsic group velocity.

16 u + u u x + v u y fv * = Φ x + v ʹ q ʹ = Φ x + B ( ) y, v + u v x + v v y + fu * = Φ y u ʹ q ʹ = Φ y B r + u r x + v r y N 0w * = J. 1 u ʹ v ʹ + ʹ F = v ʹ u ʹ r u ʹ v ʹ 1 v ʹ u ʹ + ʹ r ( ) x, f 0 f ( F) x = ( B) y, ( F) y = ( B) x. v ʹ r ʹ u ʹ r ʹ, u + u u x + v u y fv * = Φ x + ( F), x v + u v x + v v y + fu * = Φ y + ( F), y r + u r x + v r y N 0w * = J. 3 D residual circulation u * = u + 1 f v * = v 1 f S y + u ʹ r ʹ, z S x + v ʹ r ʹ, z w * = w u ʹ r ʹ v ʹ r ʹ, x y where S = 1 u ʹ + v ʹ ʹ r.

17 streamfunction : ψ ʹ (x, y,z,t) = Ψ ˆ (x, y,z,t)sin χ(x, y,z,t), χ : phase function, sin χ + cos χ =1. H q u

18 Time mean flow : u = { u (x, y, z), v (x, y, z), w (x, y,z) } 1 u ʹ v ʹ w ʹ + ʹ F u ʹ v ʹ Du Dt Dv Dt fv * = Φ x + F x + fu * = Φ y + F y Dr Dt N w * = r ʹ w ʹ z r N u ʹ v ʹ 1 v ʹ u ʹ w ʹ + ʹ u * = u + 1 f v * = v 1 f r N w * = w x u ʹ w ʹ + f N v ʹ w ʹ f N S y + u ʹ r ʹ u + u z N S S x + v ʹ r ʹ v + v z N S v ʹ r ʹ E c ˆ gx c ˆ x u ʹ r ʹ = E c ˆ gx ˆ c y u ʹ r ʹ v ʹ r ʹ w + w N y N S E E c ˆ gy c ˆ c ˆ gz x c ˆ x E E c ˆ gy c ˆ c ˆ gz y c ˆ y 0 0 0

19 ε + E T = S E ε = p e + 1 H ψ H q H q H ψ = ( V, U ) q where e denotes energy of QG wave. ʹ

20 ( ) u ρ m = z ρ u ʹ w ʹ m 1 ρ mu = u z ρ u ʹ w ʹ m ( ) u 1 ρ m (u + u 0 ) = (u + u 0) z ρ u ʹ w ʹ m ( ) 1 ρ m (u + u 0 ) = 1 ρ u mu + ρ m u 0 ( ) u u ʹ w ʹ 0 ( ) = u z ρ u ʹ w ʹ m ( ) = (u + u 0 ) z ρ u ʹ w ʹ m 1 ρ u m + z ρ u ʹ w ʹ u m E = 1 ρ m v ʹ + 1 r ʹ N ρ m z ρ m ( ) = ρ m ʹ u w ʹ u z

21 Plane wave case A : action, e : pseudoenergy, A = E, p = ka = ω ˆ kˆ ω E, p : pseudomomentum e = ωa = ω k p = ωˆ ω E = E + u p = 1 ρ m v ʹ + 1 ρ m N r ʹ + u p p = ρ m u

1 flux flux Plumb (1985) flux F s Plumb (1985) Karoly et al. (1989) flux flux Plumb (1985) Plumb (1986) Trenberth (1986) Andrews (1983) review flux Pl

1 flux flux Plumb (1985) flux F s Plumb (1985) Karoly et al. (1989) flux flux Plumb (1985) Plumb (1986) Trenberth (1986) Andrews (1983) review flux Pl wave-activity flux flux flux ( ) E-mail: takaya@jamstec.go.jp 1 Introduction ( ) (e.g., Nakamura et al. 1997) storm track cyclogenesis downstream development (e.g., Chang 1993) wave-activity ( ) flux (