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1 wave-activity flux flux flux ( ) takaya@jamstec.go.jp 1 Introduction ( ) (e.g., Nakamura et al. 1997) storm track cyclogenesis downstream development (e.g., Chang 1993) wave-activity ( ) flux ( ) A F flux A t + F = D, (1) D 0 C g F = C g A F flux sources and sinks, A F ( ) 1 flux Eliassen-Palm (E-P) flux (e.g., Andrews and McIntyre 1976; Edmon et al. 1980; McIntyre 198; Andrews et al. 1987) flux ( ) flux Hoskins et al. (1983) Trenberth (1986) Plumb (1986) flux flux 1

2 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 Plumb (1985) F s Plumb (1985) Plumb (1985) straightforward flux Takaya and Nakamura (1997) Takaya and Nakamura (000) 1

3 wave-activity flux wave-activity flux feedback ( TEM ).1 wave-activity flux: EP flux (x, y, z) β [A] t + 1 ρ 0 F = D () 0 F = ρ 0 [u v ] (3) f[v θ ]/θ 0z [U] t + f 0 v = 1 ρ 0 F (4) v [A] A. Trenberth (1986) flux F = ρ 0 (v u )/ u v f[v θ ]/θ 0z (5) du dt + f 0v = 1 ρ 0 F (6) A A flux.3 Plumb (1986) flux A t + 1 ρ 0 M T = D (7) M T = UA + M R (8) {U(v e) V u v } M R = ρ 0 { Uu v + V (u e)} (9) u f {Uv θ θ + V u θ } 0z TEM V 0 du dt + f 0v = 1 ρ 0 M R (10) 3

4 flux flux Trenberth (1986) flux.4 Plumb (1985) flux F s = ρ 0 A s t + 1 ρ 0 F s = C s (11) ( ψ ) ψ ψ ψ ( ψ ) z ψ ( f N ( ψ z ψ )) 3 wave-activity flux flux Plumb (1985) flux Plumb (1985) feedback ( ) (1) 4

5 3 p- β potential vorticity equation u = ( ψ, ψ, 0) q t + u q = 0 (13) q = f 0 + βy + ψ + ψ + p ( f S ψ p ) (14) (S ) ψ ψ = Ψ + ψ U potential vorticity eq. q t + U q + v = 0 (15) Q potential vorticity q potential vorticity q = ψ + ψ + p ( f S p ) (16) ψ = ψ 0 exp{i(kx + ly + mp ωt)} (15) (16) ω = ku x C x C x = U k k + l + (f/s) m (17) k + l + (f/s) m (18) C gx = U + k l (f/s) m {k + l + (f/s) m } ( ) (19) C gy = C gp = kl {k + l + (f/s) m } ( ) (0) ( f S )km {k + l + (f/s) m } ( ) (1) (18) (C x = 0) K = (k, l, ( f S )m) U = ( )/ K 5

6 C g = U K k kl ( f S )km = ( ) K 4 k kl ( f S )km () U = (U, V ) potential vorticity eq. q t + U q + V q + u + v = 0 (3) ψ = ψ 0 exp{i(kx + ly + mp ωt)} ω = ku + lv + ( l k)/ K (4) C gx = U + (k l (f/s) m )( ) kl( ) K 4 (5) C gy = V + (k l + (f/s) m )( ) + kl( ) K 4 (6) C gp = km( ) lm( ) K 4 (7) C g = 1 K (Appendix ) k U + klv klu + l V (kmu + mlv ) f S (8) 4 flux U flux vorticity eq. = β ζ t + U ζ + βv = 0 6

7 ψ ψ t ζ + Uψ ζ + βψ ψ = 0 ζ = ψ + ψ ψ t ψ = (ψ t ψ ) t {1 ( ψ ) } ψ ψ = (ψ ψ ) {1 ( ψ ) } e t + U e ( ψ + ( t ψ ( t + U + U ) 1 βψ ) e flux F ) = 0 (9) e = 1 {( ψ ) + ( ψ ) } = 1 (u + v ) (30) F = ( Ue ψ ( t v g + U v g) 1 βψ ψ ( t u g + U u g) u g = ψ, v g = ψ ) (31) t e + F = 0 (3) ψ = ψ 0 sin(kx + ly ωt) (17) ω = kc x U C x = β/ K 1 ( K = (k, l)) (3) ( 1 1 t 4 K ψ0 + 4 U K ψ0 1 (k β/ K )ψ0 1 4 βψ 0 1 (klβ/ K )ψ0 ) = 0 (33) ( 1 4 U K ψ0 1 (k β/ K )ψ0 1 ) ( 1 4 βψ 0 1 (klβ/ K )ψ0 = 4 U K ψ0 1 4 (k l ( f S m) )β/ K )ψ0 ) 1 (klβ/ K )ψ0 ( U + (k l ( f = S m) )β/ K 4 ) 1 klβ/ K 4 4 K ψ0 = C g ē overbar 1 7

8 t ē + F = 0, F = Cg ē (34) vorticity eq. u t + U u fv = Φ v t + U v + fu = Φ e ( t ē + F 0 = 0, F 0 Ue + ua Φ = v a Φ ) (35) (36) (37) (u a, v a ) ageostrophic flux F 0 (3) flux F (31) (35) (36) u = u g + u a u g t + U u g (f 0 + βy)(v g + v a ) = Φ v g t + U v g + (f 0 + βy)(u g + u a ) = Φ (38) (39) (31) F = ( Ue + ua Φ ( 1 βyψ ) v a Φ + ( 1 βyψ ) ) (38) (39) (40) F F 0 F = F 0 + ( ( 1 βyψ ) + ( 1 βyψ ) F = F 0 F = F 0 F 0 C g C g F flux C g C g ) 8

9 5 wave activity flux U U wave activity density potential vorticity eq. q t + U q + v = s (41) s Sources and/or sinks q enstrophy Eq. 1 t q + 1 U q + v q = s q (4) v q = v { v u + p ( f S p )} = v v = = B (R) u u u v 1 (v u ( f S p ) ) u v v e u v f S f S p p u v + p ( ψ f S p ) 1 { f S p } (43) y t (4) A t + U A + B(R) = C (44) A = 1 q /( ) (45) wave activity Density U = (U, 0, 0) B (T ) = B (R) + UA (46) A t + B(T ) = C (47) B (T ) 9

10 ( ) 5.1 : Plumb (1986) V = 0 B (T ) ψ = ψ 0 sin(kx + ly + mp ωt) overbar K = (k, l, f S m) Ā = q B (T ) = = K 4 4 ψ0, Ue + v e u v f S p ē = 1 {u + v + ( f S p ) } = K 4 ψ 0 = U + {k l ( f S m) } / K 4 kl /K 4 K 4 ( f S ) km 4 ψ0 / K 4 = C g Ā (48) Ā t + B(T ) = C, B (T ) = C g Ā (49) 5. : Plumb (1985) B (T ) Plumb(1985) B (T ) flux G(i.e. G = 0) F s = B (T ) + G (50) F s A t + F s = C s (51) C s = C + G 10

11 G = 1 4 ψ + p ( f ) S p ψ r q / ψ f S p ψ r r = s s = 0 G = 0 F s F s = 1 ( ψ ) ψ ψ f ( ψ S + ψ q + [(Uq r q )/ p ψ ( ψ ) ψ ( p ψ )) (5) (41) t = 0 U q + ψ = s (Uq + ψ r ) = 0 Uq + ψ q r q = 0 F s F s = 1 f ( ψ S ( ψ p ) ψ ψ ψ ( ψ ) ψ ( p ψ )) ] (53) (54) 1 order ψ = ψ 0 sin(kx + ly + mp ωt) F s k F s ψ 0 kl (55) f km/s ( C g = ) k K 4 kl (56) ( f )km S 11

12 F s = ( 1 4 K 4 ψ0 )C g (57) (57) F s = C g A WKB limit A K = /U A s 1 (A + e U ) = 1 K 4 ψ0 4 (58) WKB limit A s t + F s = C s (59) F s = C g A s (60) 6 U,V wave activity flux: Plumb (1986) Plumb(1986) U = (U, V, 0) wave activity flux β potential vorticity eq. q t + U q + V q + u + v = s (61) q enstrophy Eq. q t + U q + V q + u q + v q = s q (6) u q = u { v u + p ( f S p } = u v + v u + v v u u ψ p ( f S p } 1

13 = u v + (1 (v u )) p ( f S u v = 1 (v u + f ( ψ S p ) ) = f S u v e u f S p p p } + ( f S ( ψ p ) ) enstrophy Eq. q t + U q + V q + ( ) u v e u f S p + ( ) v e u v f S p = s q (63) wave activity density A = q H Q ( H = (,, 0) ) 1 A t + 1 H Q Uq u v v e 1 V q + H Q e u + H Q u v = D (64) 0 1 H Q H Q H Q 1 Uq 1 V q 0 u v e u f S p v e u v f S p f S p 1 { H Q { { H Q H Q 1 Uq 1 V q 0 u v e u f S p v e u v f S p f S p } (65) } (66) } (67) (65) enstrophy potential vorticity (66) (67) potential vorticity flux U + V 0 H Q u U 13 u V

14 ( u = (U, V, 0)) (64) M T = A t + M T = D (68) UA + 1 u {U(v e) V u v } V A + 1 u { Uu v + V (u e)} f ψ {U S u p + V ψ p } (69) ψ = ψ 0 sin(kx + ly + mp ωt) A = K 4 ψ 0 4 H Q M T = C g A (70) 7 pseudoenergy : Andrews (1983) pseudoenergy Andrews(1983) wave activity flux potential vorticity eq. U + V = 0 (71) q t + U q + V q + u + v = s (71) U = (U, V, 0) = ( ψ q t + U q + u 0 Q = 0 (7), ψ, 0) Q Q = Q( ψ; p) Λ( ψ; p) = ψ (73) U Λ = U( ψ Λ ψ Λ ) + V ( ψ ψ ) = 0 (74) 14

15 (7) u 0 Q = Λu 0 ψ = Λ( ψ ψ + ψ ψ ) = ΛU ψ f = ɛ S = U (Λψ ) (75) q t = U (q Λψ ) (76) q = ψ + ψ + p (ɛ ψ p ) (77) ψ q t = 1 t {( ψ ) + ( ψ ) + ɛ( ψ p ) } + (ψ t ) + (ψ t ) + p (ɛψ t p ) (78) (78) Λ 1 q q t = 1 t (q /Λ) 1 t {( ψ ) + ( ψ ) + ɛ( ψ p ) + q /Λ} + ( ψ t ) + ( ψ t ) + p ( ɛψ t p ) 1 q = Λ t (q Λψ ) (79) (76) 1 q Λ t (q Λψ ) = (Λ) 1 U (q Λψ) = U [(q Λψ ) /Λ] = [U(q Λψ ) /Λ] (80) α t + I = 0 (81) 15

16 α = 1 t {( ψ ) + ( ψ ) + ɛ( ψ p ) + q /Λ} (8) ψ ( t ) + U Λ (q Λψ ) I = ψ ( t ) + V Λ (q Λψ ) (83) ɛψ ( t p ) Pseudoenergy overbar 1 I = C g ᾱ (84) V = 0 Λ = /U (83) ψ ( t I = ) U ψ ( t ɛψ ( t (q + Q ) p ) Q U ψ ) α = e UA (86) (85) 8 flux (ψ ) (PV) A E A (pseudomomentum) ( Andrews and McIntyre 1976) E Uryu (1974) ( ) ( ) M (A + E)/ 16

17 log-p β- PV (q) q t + u q + v q = s; q = f 0 + βy + ψ + ψ + f 0 p ( ) p ψ z N. (87) z (u, v) T = u = ( ψ y, ψ x ) T s f = f 0 + βy z = H ln p p =(pressure/1000hpa) H N = R ap κ θ H z θ R a κ R a U = (U, V, 0) T ɛ u = U(x, y, z) + u, v = V (x, y, z) + v, ψ = Ψ(x, y, z) + ψ, q = Q(x, y, z) + q (88) O(ɛ ) PV q q t + U Hq + u H Q = s ; q = ψ + ψ + f 0 p ( p ) z N, (89) z H gradient operator ( U C P energy e = ψ x + ψ y + (f 0 ψ z/s) ) / A E A pq /( H Q ) E pe/( U C P ) C U U x- y- C U = C P U U = ( U U C P, V U C P, 0) T (90) U C P zonal U ( H Q) 0 n PV n H Q/ H Q γ( V, U, 0) T / U γ = 1 pseudoeastward (Andrews 1984, Plumb 1986) WKB ( ) H Q, U n C P U (89) pq / H Q A [ A U t + p H q ] + n u 0 q = D 1, (91) H Q D 1 = ps q / H Q WKB A A t + (E + C UA) + N (1) = D 1. (9) 17

18 N (1) E N (1) (U C U )A, pn u 0 q E = p U U(ψ x e) + V ψ xψ y Uψ xψ y + V (ψ y e) f 0 N {Uψ xψ z + V ψ yψ z} (93) (V = 0) WKB (9) A (9) Plumb (1986) (.19) (89) pψ /( U C P ) E pe/( U C P ) E t p (k Hψ ) (ψ H Q) U C P ( U pψ H q U + C U U C P ) + R(1) U C P = D (94) D = ps ψ /( U C P ) k ( R (1) = p ψ ψ xt, ψ ψ yt, f ) 0 T ψ N ψ zt A E H t E + (H + C UE) + N () R = D +, (95) U C P p U U (ψ H q ) pnψ (k H q ) H = p U U(e ψ ψ xx) V ψ ψ xy V (e Uψ ψ xy) V ψ ψ yy f 0 N {Uψ ψ xz + V ψ ψ yz} (96) (94) p (k Hψ ) (ψ H Q) N () = (k HQ)ψ U C P ( U C P ) R R = p ( e t + U U C e P + V U C P U H Q ψ = U ( U C P ) ) ( e q + pψ t + U U C q P + V U C P C P R = 0 (95) M 1 (A + E) = p q ) (97) (98) ( ) q H Q + e. (99) U C P M ψ = ψ 0 exp(z/h)sin(kx+ ly + mz ωt) M M K 4 ψ 0 4 H Q (100) 18

19 K = (k, l, (f 0 S 1 )m) T H 1 m M (pseudomomentum) M M (9) (95) M t + W = D N + R ( U C P ), (101) D = (D 1 + D )/ D 0 s, N = (N (1) + N () )/, W = (E + H)/ + C U M W N N (1) N () N = p U 4 U ( q H Q ψ ) [ ( U C P )q + H Q ψ ]. (10) U C P (89) r 0 U [ ( U CP )q + H Q ψ ] s, (103) U (10) N = 1 r0 D 0 D 0 (101) ( ) N (s = 0) N = 0 D T = D N = (D 0 r 0 r 0 D 0 )/ C p R = 0 M M t + W = D T. (104) W W s (E + H)/ W = W s + C U M W = p U U(ψ x ψ ψ xx) + V (ψ xψ y ψ ψ xy) U(ψ xψ y ψ ψ xy) + V (ψ y ψ ψ yy) f 0 N {U(ψ xψ z ψ ψ xz) + V (ψ yψ z ψ ψ yz)} + C UM. (105) (104) (105) W W C P ( C P = 0 ) C U M W s Takaya and Nakamura (1997) wave-activity flux ( W ) V = 0 D T = (D 0 r 0 r 0 D 0 )/ W D T M 19

20 (U = U(y, p)) (104) C P = 0 W F s (Plumb, 1985) (104) C g (100) W = C g M W ( ) 9 W W C g W Plumb (1985) F s ( ) transformed Eulerian-mean (TEM) (V = 0) W s Plumb (1985) F s v t + U v + f(u + u (a) ) = φ, (106) u (a) ( ) β- c (U c) v + f 0u (a) + βyu = 0 (107) ψ xx = v x f 0 u (a) /(U c) ψ xy = u x f 0 v (a) /(U c) f 0 Θ z ψ xz/n = θ x Θ z w (a) /(U c) v (a) w (a) U > c ( ψ ψ xx, ψ ψ xy, f 0 N ψ ψ xz) T (Φ u (a), Φ v (a), Φ w (a) ) T /(U c) Φ F s ( W s ) C g (Figs. 1a 1b) ( ψ ψ xx, ψ ψ xy, f 0 ψ N ψ xz) T C g F s ( W s ) v (Fig. 1c) ψ - 0

21 (v ) ψ - F s (W s ) u v (Fig. 1c) F s (W s ) v θ (Fig. 1b) F s ψ - flux W ( W s ) ψ = ψ 0 sin(kx + ly + mz ωt) ψ - F s W Plumb (1986) O(ɛ ) D (0) U () Dt fv () + Φ() = u u v = ψ ψ yy ψ ψ xy, (108) () D (0) U () /Dt U () / t + (U )U () + (U () )U (108) D (0) U () Dt fv () + Φ() = 1 ( u + ψ ψ yy) 1 (u v + ψ ψ xy) (109) (109) ( ) (109) D (0) U () Dt fv () + Φ() = 1 (ψ xq ψ q x) 1

22 1 [ ] (ψ x ψ ψ xx) + (ψ y ψ ψ yy) 1 [ ] f p z p 0 N (ψ xψ z ψ ψ xz). D (0) U () fk U Dt (a) = G + X, (110) D (0) Θ () + W dθ (a) Dt dz = Q, (111) X Q TEM (110) G G = p k (u0 q ψ k H q ) p k P (11) (110) Ua U (a) = (U (a), V (a), W (a) )T = U () (a) + 1 pr, (113) p () (a) = (U(a), V () (a), W () (a) )T Euler (113) R R = 1 (v θ ψ θ x)/θ z (u θ + ψ θ [ y)/θ z ]. (114) (v ψ ψ xx) + (u ψ ψ yy) /f U () (110), (111), (11), (113), (114) Plumb (1986) (4.3), (4.4), (4.5) (4.6) Plumb (1986) PV ( H Q) s s k n G G = p [(P n)s (P s)n]. (115) (93), (96) W s (E + H)/ β- W s p n (u0 q ψ k H q ) = p P n (116) (115) W s Plumb (1986) W s PV

23 10 W Enomoto and Matsuda (1999) [40 N, 90 E] (Fig. a) Fig. b 14 ψ W C P = 0 (Fig. c) W flux W ψ W 3 Plumb (1985) F s W F s F s Figs. 1c 1d W W W NCEP/NCAR 8 W p Fig.3 flux plot W 300-hPa, 600-hPa W W U C P 3 Enomoto and Matsuda (1999) 3

24 11 flux W flux flux A E M M W ψ - flux W W flux W W flux W Takaya and Nakamura (000) W W W 4

25 Appendix: Rossby potential vorticity eq. q t + U q + V q + u + v = 0 (A1) ψ = ψ 0 exp{i(kx + ly + mp ωt)} K = k + l + f 0 N m ω = ku + lv + l K k K = k(u / K ) + l(v + / K ) (A) U + V = 0 (A3) (A) (A3) ω = k(u / K ) + l(v V U / K ) = k(u / K ) + l V (U U / K ) = (k + l V )(U U / K ) (A4) ω = ( U V k + l)(v + / K ) ω = 0 (U / K ) = 0, k l = V U (V + / K ) = 0 (A6) ( 1) (A7) ( ) (A5) (A6) (A7) 5

26 ( ) (A1) ( ) ( Q contour ) ( 1) ( 1) (U / K ) = 0, (A) (V + / K ) = 0 (A8) C gx = U + (k l (f/s) m )( ) kl( ) K 4 C gy = V + (k l + (f/s) m )( ) + kl( ) K 4 (A9) (A10) C gp = km( ) lm( ) K 4 (A11) (A8) C g = 1 K k U + klv klu + l V (kmu + mlv ) f S (A1) 6

27 ( wave-activity flux ) Andrews, D. G., 1983: A conservation law for small-amplitude quasigeostrophic disturbances on a zonally asymmetric basic flow. J.Atmos.Sci., 40, , 1984: On the existence of nonzonal flows satisfying sufficient conditions for instability. Geophys. Astrophys. Fluid Dyn., 8, , and M. E. McIntyre, 1976: Planetary waves in horizontal and vertical shear: The generalized Eliassen-Palm relation and the mean zonal acceleration. J. Atmos. Sci., 33, , J.R. Holton, and C.B. Leovy, 1987: Middle Atmosphere Dynamics., Academic press, 489pp. Chang, E. -K. M., 1993: Downstream development of baroclinic waves as inferred from regression analysis. J. Atmos. Sci., 50, , I. Orlanski, 1994: On energy flux and group velocity of waves in baroclinic flows. J.Atmos.Sci., 51, Edmon, H. J., B. J. Hoskins and M. E. McIntyre, 1980: Eliassen-Palm cross sections for the troposphere. J. Atmos. Sci., 37, [See also Corrigendum, 1981: J. Atmos. Sci., 38, 1115, especially nd last item]. Enomoto, T., and Y. Matsuda, 1999: Rossby wavepacket propagation in a zonally-varying basic flow. Tellus in press. Hoskins, B. J., I. N. James, and G. H. White, 1983: The shape, propagation and mean-flow interaction of large-scale weather system. J. Atmos. Sci., 40, Karoly, D.J., R.A. Plumb, and M. Ting, 1989: Examples of the horizontal propagation of quasi-stationary waves, J. Atmos. Sci., 46, Longuet-Higgins, M. S., 1964: On group velocity and energy flux in planetary wave motions. Deep-Sea Res., 11, McIntyre, M. E., 198: How well do we understand the dynamics of stratospheric warmings? J. Meteor. Soc. Japan., 60, Nakamura, H., M. Nakamura, and J.L. Anderson, 1997: The role of high- and low-frequency dynamics in the blocking formation, Mon. Wea. Rev., 15, Orlanski, I., J. Sheldon, 1993: A case of downstream baroclinic development over western north america. Mon.Wea.Rev., 11, Pedlosky, J., 1987: Geophysical Fluid Dynamics. nd ed. Springer-Verlag, 710pp. Plumb, R. A., 1985: On the three-dimensional propagation of stationary waves. J. Atmos. Sci., 4, 17-9., 1985: An alternative form of Andrews conservation law for quasi-geostrophic waves on a steady,nonuniform flow. J.Atmos.Sci., 4, , 1986: Three-dimensional propagation of transient quasi-geostrophic eddies and its relationship with the eddy forcing of the time-mean flow, J. Atmos. Sci., 43, Takaya, K., and H. Nakamura, 1997: A formulation of a wave-activity flux for stationary Rossby waves on a zonally varying basic flow. Geophys. Res. Lett., 4,

28 and, 000: A formulation of a phase-independent wave-activity flux for stationary and migratory quasi-geostrophic eddies on a zonally-varying basic flow. J. Atmos. Sci., submitted. Trenberth, K. E., 1986: An assessment of the impact of transient eddies on the zonal flow during a blocking episode using localized Eliassen-Palm flux diagnostics. J. Atmos. Sci., 43, Uryu, M., 1974: Mean zonal flows induced by a vertically propagating Rossby wave packet. J. Meteor. Soc. Japan., 56,

2012専門分科会_new_4.pptx

2012専門分科会_new_4.pptx 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