Holton 9.2.2 semigeostrophic 1 9.2.2 semigeostrophic,.., Φ(x, y, z, t) = (p p 0 )/ρ 0, Θ = θ θ 0,,., p 0 (z), θ 0 (z).,,,, Du Dt fv + Φ x Dv Φ + fu + Dt DΘ Dt + w dθ 0 dz = 0, (9.2) = 0, (9.3) = 0, (9.4) b gθ θ 00 = Φ z, (9.5) u x + v + w z D Dt = t + u x + v + w z = 0, (9.6) *1., b, θ 00.,., 2 2.,, y., L x, L y :, U, V :, L x L y, U V 9.4. U 10m s 1, V 1m s 1, L x 1000km, L y 100km *1,, 1 ρ 0 Hp = HΦ., H., θ = Θ(x, y, z, t) + θ 0(z). 25 1 28 ()
Holton 9.2.2 semigeostrophic 2 9.4: x., x y. D/Dt V/L y, Ro V/fL y 1, x, y., ( ) U Du/Dt fv Dv/Dt fu UV/L y fv V 2 /L y fu Ro ( V Ro U 1, V ) 10 2., Du Dt }{{} O(1) fv }{{} O(1) / Dv + fu }{{} Dt O(10 2 ) }{{} O(1) + Φ = 0, (9.2 ) }{{} x O(1) + Φ = 0. (9.3 ) }{{} O(1) (9.3 ) u 1 % Φ. (9.2 ) v., fu g = Φ/, fv g = Φ/ x,, u = u g, v = v g + v a., v g, v a., (9.2), (9.3), (9.4), (9.6) Du g Dt fv a = 0, (9.7) 25 1 28 ()
Holton 9.2.3 3 fu g = Φ, Db Dt + wn 2 = 0, (9.8) v a + w z = 0 (9.9)., (9.7) (9.2) u = u g,. (9.9) (9.6),. (9.8) (9.5)., N N 2 g θ 0 θ 00 z. u g, f u g z = b (9.10)., (9.7) (9.8) *2.,., (v a, w)., D Dt = ( t + u g x + v g + v a + w ) z }{{} ( ) Dg Dt., (6.8),. (9.7),, semigeostrophic *3. 9.2.3 (9.7) (9.10), u g b v a, w. 9.3,.. 6.4.2. *2 semi-geostrophic. *3, Hoskins 1975. 25 1 28 ()
Holton 9.2.3 4 (9.8) y, (9.7) z,, ( ) D b = Q 2 v a b Dt w ( N 2 + b ) z ( D f u ) g = Q 2 + v ( a Dt z z f f u ) g + w b z (9.11) (9.13)., Q 2 = u g b x v g b (9.12), 6.4.2 Q y.,. Q 2,., Q 2 2.. 25 1 28 ()
Ù É ¾ Ù Ù Ú Holton 9.2.3 5 Æz9S U!t lo =^ om Æz9S U = 9SéÌåïµ Ù Þ fbéwú³ž U = Þ 0vMÍÚp Ù UÿC Ú tpo æ æ Ø Ú çée xrw Ot`opV t É ¾ Ú UC\ ËÍe º½½µ Ø 9SéÌåïµx Ë É ¾ qëíe pìt ËÍe º½ µ Ø Þ., Q 2,,. 2 y, z 2, v a = ψ M / z, w = ψ M / (9.14) ψ M., (9.9). (9.11) (9.13), (9.10), (9.14), N 2 s 2 ψ M 2 + F 2 2 ψ M z 2 + 2S 2 2 ψ M z = 2Q 2 (9.15) 25 1 28 ()
Holton 9.2.3 6., N 2 s N 2 + b z, F 2 f ( f u ) g = f M, S2 b (9.16), M fy u g. (9.15) (9.7) (9.8) N 2 2 ψ M 2 + f 2 2 ψ M z 2 = 2Q 2 (9.17)., N f., semigeostrophic N s, S F. N 2 f 2 > 0 (9.17), *4. Q 2 ψ M. 9.1b v g / b/, Q 2., y. semigeostrophic, (9.15), N 2 s F 2 S 4 > 0,. *5, 9.1 *6.. (9.16), 9.5 2 *7.,., semigeostrophic. semigeostrophic,., Q 2, 2, T/ T/.,, semigeostrophic.. *4. *5 9.1, P P F 2 N 2 s S 4, (9.15), P > 0. *6. *7. 25 1 28 ()
¾ Û Æ ¾ ¼ É ¾ ¾» É Þ ¾É ¾ Æ ¾ É ¾ Ú ¾ Ú ¾É ¾ ¾Ë Holton 9.2.3 7 9.5: 2,.,.,. É% Ë% Ø Ø É ¾ U ::pÿc Û Ú ¾ Ú Þ ¾ ¼ Ú Þ UÿC ¾ Ú Þ ¾Ë¾ Ú UÿC É x ¾ pÿc É x Ú ¾ pÿc Ø Ø!=px ÑŸ ÅÌ «s`!=pxƒ tÿc YwÑŸ ÅÌ «QG SG.., w = 0. F 2 v a z. 25 1 28 ()
Holton 9.3 8,. QG,. SG,. QG, SG 1., ( ). 2.,. Q 2. Q 2 (9.15). SG, QG Q 2, Q 2. 9.3,.,,. 2. 9.1.,,,.,,,.,,.,,., 25 1 28 ()
Holton 9.3 9 9.1:. (2.7.3) (7.5.1) Ns 2 > 0 F 2 > 0 x 2.7.3, 7.5.1, θ = θ 0 + Θ, M fy u g.,,.,,, *8., N 2 s /(f M/) *9. 10 4 *10., M θ *11.,, *12. θ y, 7.5.1., z θ., θ, M., M θ 9.5, *8,. *9 δz, δy, D 2 Dt 2 (δz) = N 2 s δz F vδz, (2.52) D 2 δy Dt 2 M = f δy F hδy (7.53)., F v, F h.,., / F v = Ns 2 f M F h.. *10 N 2 s O(10 2 ), f M f 2 O(10 8 ), 10 6. *11,. *12. 25 1 28 ()
Holton 9.3 10 *13., θ M, f( M/) θ < 0 (9.19), θ.,. (9.19) M θ (7.54) *14. (9.19) g( θ/ p),, (4.12) f P < 0 (9.20)., P.,,.,,. (9.19), 9.6 1, 2. y 1, y 2 = y 1 + δy, x, 2 *15.,., δ(ke),.,,,. x,, M 1 = fy 1 u 1 = fy 1 u g (y 1, z), M 2 = fy 2 u 2 = fy 1 + fδy u g (y 1 + δy, z + δz) (9.21)., M 1 = fy 1 + fδy u 1 = M 1, M 2 = fy 1 u 2 = M 2 (9.22) *13. *14 f M ( ) = f f ug > 0 = 0 < 0 (7.54) *15, x,. x,. 25 1 28 ()
Holton 9.3 11 9.6:. 1, 2., M... (9.21), (9.22) M 1, M 2,, u 1 = fδy + u 1, u 2 = fδy + u 2.., δ(ke) = 1 ( u 2 2 1 + u 2 ) 1 ( 2 u 2 2 1 + u 2 2) = fδy(u 1 u 2 + fδy) = fδy(m 2 M 1 ) (9.23) fδy(m 2 M 1 ) < 0, δ(ke),. θ, (δy) 2 > 0 (δy) 2, 1 δy f(m 2 M 1 ) = f δm ( ) M δy = f < 0 (9.19).,., δm = M M δy + z δz θ 25 1 28 ()
Holton 9.3 12. M M, δm = 0 δz, M δy., δm = 0 ( ) ( δz = M )/ ( ) ( M = f u )/ ( ) g ug (9.24) δy z z M., ( ) ( δz = θ )/ ( ) ( θ = f u )/ ( ) g g θ δy z z θ 00 z θ (9.25).,,. (9.25) (9.24) ( ) δz δy M / ( ) ( δz = f f u ) ( g g θ δy θ θ 00 z., (9.16). )/ [ ( ) ] 2 f 2 ug = F 2 Ns 2 z S 4 M θ, ( ) δz δy M / ( ) ( δz = f f u ) / g Ri f 2 = F 2 Ns 2 δy θ S 4 < 1 (9.26) *16., Ri, Ri ( ) / ( ) g θ 2 ug θ 00 z z., u g / = 0,, Ri< 1. (9.26), F 2 N 2 s S 4 < 0. (9.20). 9.1, F 2 N 2 s S 4 = (ρfg/θ 00 ) P (9.27). P,, (9.27),.,.,,,, 9.5 *17. *16, M θ. *17 (Emanuel, 1986). 25 1 28 ()
Holton 9.3 13, F 2 N 2 s S 4 > 0, (9.15).,, Q 2, (9.15).,. y., 2 t 2 ( 2 ψ M z 2 ) + Ns 2 2 ψ M 2 + F 2 2 ψ M z 2 + 2S 2 2 ψ M z = 0 (9.28) F *18., (9.15). F 2 N 2 s S 4 > 0, (9.28)., F 2 N 2 s S 4 < 0, (9.28), *19.. *18. *19. 25 1 28 ()
Holton 9.3 14.,,.,. (9.19), M θ. (9.20),. (9.27). (9.28),. F 2 N 2 s S 4 > 0,, Q 2., Q 2. F 2 N 2 s S 4 < 0, Q 2... 25 1 28 ()
Holton 9.3 13-1 semi-geostrophic, 6 QG 9 semi-geostrophic SG. 6, 9, *20. z, w. f. D g Dt u a x + v a QG SG D g u g Du g = f 0 v a (6.11) Dt Dt = f 0v a (9.7) D g v g p = f 0 u a (6.11) Dt = f 0u g + w z = 0 (6.12) v a + w z = 0 (9.9) D g b Dt + N 2 Db w = 0 (6.13b) Dt + N 2 w = 0 (9.8) u g f 0 z = b D g Dt ( ) b = Q 2 N 2 w ( ) D g u g f 0 = Q 2 + f 2 v a 0 Dt z z u g (3.29) f 0 z = b (6.48) (6.47) D g (9.10) Dt + v a + w z ( ) D b = Q 2 w Dt N 2 v a b w b (9.11) z ( ) D u g f 0 = Q 2 + f 2 v a 0 Dt z z f 0 v a z u g + w b z V g b (6.45a) Q 2 V g b (9.12) v a = ψ M z, w = ψ M v a = ψ M z, N 2 2 ψ M 2 + f0 2 2 ψ M z 2 = 2Q 2 (9.17) SE N 2 2 ψ M 2 + f0 2 + b z 2 ψ M 2 (9.13) w = ψ M (9.14) 2 ψ M z 2 u g 2 ψ M z 2 +2S 2 2 ψ M z = 2Q 2 (9.15) *20 b, b. 25 1 28 ()
Holton 9.3 13-2 9.1 θ, (4.12) ( P (ζ θ + f) g θ ) p., u g, ( ) ug ζ θ =., { ( ) } ( ug P = f g θ ) p θ θ (4.12) (ex9.1.1)., (1.27), θ ( ) ( ) ug ug = + u ( ) g θ θ z z., (ex9.1.1) { ( ) ug P = f + u ( ) } ( g θ g θ ) θ p., *21, (ex9.1.2) { ( ) ug P = f z g θ p = 1 θ ρ z = θ 00 gρ N s 2 z } { θ00 gρ N s 2 ug + θ, 2, { ( ) } ug θ00 P = f gρ N s 2 1 fρ., 2 u g θ z θ θ z = u g z = 1 b f z z ( ) } ( θ 1 z ρ b ( ) θ z ) θ z (ex9.1.2) (ex9.1.3). (9.10). (9.16) F, S, (ex9.1.3). *21 (9.16),. P = θ 00 fgρ F 2 Ns 2 θ 00 fgρ S4 = θ 00 [ F 2 Ns 2 S 4] (ex9.1.4) fgρ N 2 s = N 2 + b z = g θ 00 [θ0 + Θ] = g θ z θ 00 z 25 1 28 ()
Holton 9.3 13-3, δ,., a 2 2 ψ x 2 + c2 2 ψ = δ(x, y) 2 (ap9.15.1)., a, c, δ.,., 2 G, 2 ψ x 2 + 2 ψ = δ(x, y) 2 ψ(x, y) = G(x, y), G(x, y) = 1 ln r, r (x, y) (ap9.15.2) 2π., (ap9.15.1), x = ax, y = cy, 2 ψ X 2 + 2 ψ = δ(ax, cy ) Y 2 (ap9.15.3)., (ap9.15.2) ψ(x, Y ) = 1 2π ln X 2 + Y 2. (ap9.15.4), ψ(x, y) ψ(x, y) = 1 (x ) 2 ( y ) 2 2π ln + (ap9.15.5) a c., x 1/a, y 1/c 1., (1/).,, a 2 2 ψ x 2 + 2b2 2 ψ x + c2 2 ψ = δ(x, y) 2 (ap9.15.6). ( ) / x ( / x, /) A ψ = δ(x, y), / A ( a 2 b 2 b 2 c 2 ) (ap9.15.7) 25 1 28 ()
Holton 9.3 13-4. 2 *22., ( ) λ 2 0 Λ 0 µ 2 A, Λ = R 1 AR., R A, Λ. RΛR 1 = A., (ap9.15.7) ( / x, /) RΛR 1 ( / x / ) ψ = δ(x, y) (ap9.15.8) (ap9.15.9)., ( / X, / Y ) ( / x, /) R, ( / X / Y ) R 1 ( / x / ) *23, (ap9.15.9) ( / X, / Y ) Λ ( / X / Y ) ψ = δ(x, y) (ap9.15.10). λ 2 2 ψ X 2 + µ2 2 ψ = δ(x, y), Y 2 { x = R 11 X + R 12 Y, y = R 21 X + R 22 Y (ap9.15.11)., (ap9.15.11) (X, Y ), X 1/λ, Y 1/µ. (X, Y ) (x, y) R,, ψ., 1.,. 2,.,.,.,, x, y,.,,,. *22, (a 2 c 2 ) 2 + 4b 4 = 0,. *23 R,., 2 1,. 25 1 28 ()
Holton 9.3 13-5 óõ s8 óõ Ü 1:,. 2:,. 25 1 28 ()
Holton 9.3 13-6 3.,,.,.,,. 3:. 4.,,,.,,.,.,,. F (9.15), *24. *24, (9.15). 25 1 28 ()
Holton 9.3 13-7 4:.,.,,., u g = u g (y, z), b = (y, z). 25 1 28 ()
Holton 9.3 13-8 v a = ψ/ z, w a = ψ/ ψ(y, z)., (9.12) Q 2 = 0,,., (9.10) y *25,. z, z Dv a Dt + fu g + Φ = 0 ( ) Dva + f u g Dt z + b = 0. (F.1), (9.11), (9.13) D [ { }] Dva + N 2 2 ψ M s Dt z Dt 2 + F 2 2 ψ M z 2 + 2S 2 2 ψ M z = 0 (9.15 )., Q 2., x, 2, D Dt = t + v a + w a z t., 2 ( 2 ) ψ t 2 z 2 + Ns 2 2 ψ 2 + F 2 2 ψ z 2 + 2S2 2 ψ z = 0. (F.3) (9.28) 2 ( 2 ) ψ M t 2 z 2 + Ns 2 2 ψ M 2 + F 2 2 ψ M z 2 + 2S 2 2 ψ M z = 0 (9.28)., ψ M = A(t)e i(ly+mz)., l, m,, A(t). (9.28), m 2 d2 dt 2 A(t) = [ N 2 s l 2 + F 2 m 2 + 2S 2 lm ] A(t) (ap9.28.1) *25, Q 2,,. 25 1 28 ()
Holton 9.3 13-9., A(t),.., A(t) = e at a 2 [ N 2 s l 2 + F 2 m 2 + 2S 2 lm ] (ap9.28.2) (ap9.28.3)., { a 2 > 0 a 2 < 0,., a 2 a 2 = 1 N 2 s N 4 s {l + S2 N 2 s } 2 m + { Ns 2 F 2 S 4} }{{} ( 1) m 2 (ap9.28.4)., ( 1)., ( 1), a 2 < 0 N 2 s F 2 S 4 > 0 (ap9.28.5) a 2 > 0 N 2 s F 2 S 4 < 0 (ap9.28.6) *26., { F 2 Ns 2 S 4 > 0 F 2 Ns 2 S 4 < 0. *26, (ap9.28.6)., a 2, (ap9.28.4) 1., (ap9.28.6)., Ns 2,,, Ns 2 > 0 F 2 > 0, Ns 2. 25 1 28 ()