5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1

4 1 1.1 ( )

5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1 da n i n da n i n + 3 A ni n n=1 3 n=1 di A n n, di n/ = Ω i n, d a A = da + Ω A (1.2.3), d a /, d/,, Ω A (1.2.3) V a, V a (1.2.2) V, d a V a d(v + Ω r) = + Ω (V + Ω r).

6 Ω, d a V a = dv + 2Ω V + Ω (Ω r). (1.2.4),, g a, F (1.2.4) (1.2.1), dv = α p 2Ω V + g + F (1.2.5), α, g g a Ω (Ω r) 2 2Ω V, (coriolis force) 1.3 2,, δxδyδz, (x, y, z) x, [ρu ( ρu/ x) δx 2 ] δyδz, [ρu+( ρu/ x)δx 2 ] δyδz, x,, (ρu)/ x, ( ) x 2, ρ = (ρv ) (1.3.6) t, (continuity equation) d/ = / t+v, (ρv ) = ρ V + V ρ (1.3.6), dρ = ρ V. (1.3.7) 1.4 3 p, α, T, f(p, α, T ) = (1.4.8)

7,,,, (ideal gas),, n pv = nr T (1.4.9), R (= 8.31447 JK 1 mol 1 ), (1.4.9),,, p p k, p = k p k V, k M k, m k,, p k = M k m k V R T p = R T V Mk m k = R T ( Mk V ) M k m k Mk m = M k / M k /m k α = V/ M k, pα = RT (1.4.1), R = R / m, m = 28.97 g mol 1, R = 287 J kg 1 K 1 1.5,,

8,,,,,,,,, Q = di + W, Q, W, di/, c v dt/, c v, pdα/,, dt Q = c v + pdα (1.5.11) (1.5.11),, (1.5.11) dt = c v + pdα (1.4.1), α dlnt = c p R dlnp, c p = c v + R, T = C p R/c p. (1.5.12) C p θ, C θ = T (p /p) κ, κ = R/c p (1.5.13)

9, p 1hPa. θ (potential temperature) θ, (1.5.11), d(lnθ) = Q c p T (1.5.14) (1.5.14) (Q = ) dθ/ =, θ, (1.5.11) Q L dq c ll dt = c dt p αdp (1.5.15), q, l, L, c l, Q 1.6, (1.2.5), (1.3.6), (1.4.1), (1.5.11), 6 6 p, α, T, u, v, w F Q,,,,,,, (1.2.5) dv H = α p f V + F H (1.6.16) = α p z g (1.6.17)

1, V H, 2, F H f = fk, f(= 2Ω sin φ) ( ) 1.7,,, (1.2.5), *1. (q 1, q 2, q 3 ) q 1 = λ, q 2 = φ, q 3 = r = z + a, λ, φ, r, a, z ds i h i ds 1 = r cos dλ, ds 2 = rdφ, ds 3 = dr. h 1 = r cos φ, h 2 = r, h 3 = 1. (1.7.18), î, ĵ, ˆk, V = uî + vĵ + wˆk u = ds 1 = r cos φdλ, v = ds 2 = r dφ, w = ds 3 = dr = dz. (1.7.19) *2, dv = V ( t + (V ) V + uv r tan φ + uw ) î ( r u 2 + r tan φ + uw ) ) ĵ + ( u2 + v 2 ˆk. r r *1, *2 dv = [ ( Vi ) t + V V i + i l ( Vi V l h i h l h i q l V lv l h l h i ) ] h l ẽ i (1.7.2) q i l, l = i

11, 2Ω V =2Ω(ĵ cos φ + ˆk sin φ) (uî + vĵ + wˆk) =(2Ω cos φ w fv)î + fuĵ 2Ω cos φ uˆk, (1.2.5) du dv dw = 1 p ρ x + (2Ω + u r cos φ )(v sin φ w cos φ) + F λ, = 1 p ρ y (2Ω + u r cos φ )u sin φ vw r + F φ, (1.7.21) = 1 p ρ z + (2Ω + u v2 )u cos φ + r cos φ r + F z., f df = f t + f dλ λ + f dφ φ + f dz z = f t + u f x + v f y + w f z, (x, y, z),,,, (Phillips, 1966)., d [r cos φ (u + Ωr cos φ)] = r cos φ F λ (1.7.22) *3., F λ, r a,, (1.7.21) w (1.7.22), r a, (1.6.16) *4. (1.7.21) ( du p f + u tan φ ) v + F λ, (1.7.23) dv dw = 1 ρ x + a = 1 ( p ρ y f + u tan φ ) u + F φ, (1.7.24) a = 1 p ρ z g + F z. (1.7.25) *3 (1.7.21), r cos φ *4 (1.7.21) w,, w 2Ωw cos φ, uw/r, vw/r, 2Ωu cos φ, u 2 /r, v 2 /r

12, f = 2Ω sin φ (1.7.22),, dw/, (1.7.24) = 1 p ρ z g (1.7.26), (1.5.14), d = t + u r cos φ λ + v r φ + w z

13 1.8 z( r), p, ln(p/p ), σ = p/p s, θ 1.8.1: A ζ ζ z z ζ (x, y, z, t), ζ z (x, y, ζ, t) ( ) A, A (x, y, z, t) A (x, y, ζ, t) A, A (x, y, ζ, t) A (x, y, z(x, y, ζ, t), t) 1.8.1 xz 3 A, B, C, z A, ζ A, B z A x,,, A (B) A (A) x = A (C) A (A) x A (C) A (B) ζ z ζ z x (1.8.27), A (A), A A. (1.8.27), x, ( ) ( ) A A = x x z ζ A z x y t, s = x, y, t (, A A ) ( ) ( ) A A = + A ( ) z (1.8.28) s s z s ζ z z x ζ

14 A ζ = A z z ζ, A z = A ζ ζ z. (1.8.29) (1.8.29) (1.8.28), ( ) A = s ζ ( ) A s z + A ζ ζ z ( ) z s ζ (1.8.3) s = x, y, A B 2 z ζ ζ A = z A + A ζ ζ ζ B = z B + B ζ z ζ z, (1.8.31) ζ z ζ z. (1.8.32) s = t, ( ) A = t ζ ( ) A t z + A ζ ζ z, ζ, ( ) da A = + V ζ A + t ζ ( ) z t ζ ζ A ζ (1.8.33) (1.8.34), V, ζ = dζ/ ζ. A,, z ζ (1.8.31), α z p = α ζ p + α p z ζz = α ζ p ζ Φ (1.8.35), Φ = gz, ζ, dv = α ζ p ζ Φ fk V. (1.8.36) (1.8.29),, α p ζ ζ z + g = or α p ζ + Φ ζ =. (1.8.37)

15 z d(lnρ) + z V + w z = (1.8.38) ζ (1.8.29) (1.8.32), z V + w z = ζ V ζ V z ζ ζ z + w ζ ζ z (1.8.39), w = ż = ( ) z + V ζ z + t ζ ζ z ζ, w ζ = ( ) z z + V ζ t ζ ζ + V ζ ζ z + ζ z ζ ζ + ζ ζ ( ) z ζ (1.8.39), z V + w z = ζ V + ζ ( z t + V ζ + ζ ) z ζ ζ + ζ ζ. (1.8.38), ζ ( ) d ln p ζ + ζ V + ζ ζ = (1.8.4) θ, z., c p T dlnθ = Q (1.8.41), c p dt αdp = Q (1.8.42), ζ

16 1.8.1 ζ = p, ζ p = p p, p/ p = 1, (1.8.36), (1.8.37) dv Φ p = p Φ fk V + F (1.8.43) = α (1.8.44) (1.8.4), ζ = p p V + ω p = (1.8.45), ω = dp/ = ṗ p p ω = p = p, ω ω = p p V dp (1.8.46), p = ω =,, ω s = dp s = p s t + V s p s. (1.8.47) (1.8.46) p s (x, y, t), p s t + V s p s = ps p V dp (1.8.48), dp/ ω. z p, (1.8.45), (1.8.43), p V g = 1 f k Φ, V g p = 1 f k Φ p

17 1.8.2 σ σ p s = p s (x, y, t), σ = p/p s., α z p = α σ p + α p z σz = α σ (σp s ) σ Φ α σ (σp s ) = ασ p s, σ (1.8.36), V t + V σv + σ V σ = σφ RT p s p s fk V + F (1.8.49) (1.8.37), p/ σ = (σp s )/ σ = p s, Φ σ + αp s = (1.8.5), (1.8.4) 1 d(ln p/ σ)/, p/ σ = p s, σ d(lnp s ) + σ V + σ σ = (1.8.51) σ, 1 p s /p s p s t + V p s + σ p s σ + p s σ V + p s σ σ =, p s σ p s t = σ (p s V ) (p s σ) σ (1.8.52) σ = dσ/, σ = p s /p s = 1 σ = /p s = σ =, (1.8.52), p 1 s t = (p s V )dσ (1.8.53)

18, σ = σ, σ σ σ p σ s t + (p s V )dσ = p s σ (1.8.54) p s / t (1.8.53), (1.8.54) σ. σ, (1.8.49) σ V / σ, σ V, T σ =, σ = 1 ( ) p σ, σ (1.8.53) 1.8.3 θ, 193-194, 197 θ θ,,.,,,, θ θ,, α z p = α θ p + α p z θz = α θ p θ Φ (1.8.55) = θ lnθ = θ lnt R c p θ lnp, θ p = (pc p /RT ) θ T, (1.8.55) α z p = θ (c p T ) θ Φ, θ V t + V θv + θ V θ = θ(c p T + Φ) fk V (1.8.56)

19, c p T + Φ = M (Montgomery streamfunction) θ,, z, 1 θ = 1 T T θ R p pc p θ. p θ = p z z θ = ρ Φ θ θ θ (c pt + Φ) = c pt θ (1.8.57) θ (1.8.4), ( d ln p ) + θ V + θ θ θ = (1.8.58) σ d/ θ, p s / t (1.8.41) θ, θ = θ top θ =, θ = θ s θ = θ s t + V s θ s

2 1.9,.,,, 1.9.1, V,, dk = K t + V pk + ω K p = V pφ + V F (1.9.59), K = V 2 /2, V F, *5 K, (1.9.59) K t + p (KV ) + (Kω) p, 3, = p (ΦV ) + Φ p V + V F. (1.9.6) Φ p V = Φ ω p = (ωφ) + ω Φ p p, φ/ p = α = RT/p, K t + [(K + Φ)ω] p [(K + Φ)V ] + p = RT p ω + V F (1.9.61) 1 2, 1 2 *5 p, p V + ω p =

21 1.9.2 P, ps P = gzρdz = Φ dp [ ] ps Φp g = 1 ps p Φ g g p dp, g,, P = Φ sp s g + 1 g ps RT dp (1.9.62) (internal energy)i, I = c v T ρdz = ps c v T dp g (1.9.63) c v + R = c p, I P I + P (total potential energy), ps P + I = g 1 Edp + g 1 Φ s p s (1.9.64), E = c p T (enthalpy),,,, (1.8.42) E t + V pe + ω E p = RT p ω + Q (1.9.65) p E, (1.9.65), E t + p (EV ) + ω E p = RT p ω + Q (1.9.66)

22 (1.9.61) (1.9.66), RT ω/p,,, K + E (1.9.61) (1.9.66), (K + E) t + p [(K + E + Φ)]V ] + [(K + E + Φ)ω] = Q + V F (1.9.67) p, ρ dxdydz = dxdydp/g, ps ps (K + E)dpdxdy + [(K + E + Φ)V ]dpdxdy t (1.9.68) + [(K + E + Φ)ω] s dxdy = Q + V F., ps ( ) = ( )ρdxdydz = ( )dxdydp. Q, V F 3, / t *6 p p s, { t ps (K + E)dp p s t (K + E) s + ps [(K + E + Φ)V ]dp p s [(K + E + Φ)V ] s + [(K + E + Φ)ω] s }dxdy = Q + V F. (1.9.69),, t ( ), 3, { ps t ( K + Ē) g 1 t (K + E) s + p s [(K + E + Φ)V ] s } (1.9.7) [(K + E + Φ) s ω s ] dxdy = Q + V F. *6 f(x, α) a(α) b(α) x, α, d b(α) f(x, α)dx = db(α) dα a(α) dα f(b(α), α) da(α) b(α) f(a(α), α) + f(x, α)dx dα a(α) α

23 (1.8.47), p s / t + V p s = ω s, (K + E) s, Φ s (ω s V p s ) Φ s p s / t, (1.9.7) t [K + E + 1 g Φ sp s ] = Q + V F (1.9.71), Φ s p s 2 (1.9.71) (1.9.64) ( d/), (1.9.71), 1.1,,, Lorenz(1955) (available potential energy(ape)), p, T ( θ ) (1.9.61),, (R/g) (T ω/p)dxdydp, T ω T ω(= T ω + T ω) ω =, T ω = T ω Lorenz, T ω =, (1.8.45), p ω dxdy = ωdxdy = p V dxdydp (1.1.72),,, p V dxdy = ω =, T ω = T ω =, A p s = 1hPa, θ = T (p s /p) κ

24, (1.9.64) T P E = c p g ps T dp + Φ sp s g = c p g 1 p κ s ps θp κ dp + Φ sp s g, κ = R/c p, T P E = c { pp κ s [θp ] θt } 1+κ p s g(1 + κ) + p 1+κ dθ + Φ sp s θ S g,, T P E = c pp κ s p g(1 + κ) 1+κ dθ (1.1.73), p Lorenz, APE (1.1.73) TPE, Ā = c pp κ s p g(1 + κ) 1+κ p 1+κ dθ. (1.1.74), p = p + p p 1+κ p 1+κ = ( p + p ) 1+κ = p 1+κ + (1 + κ) p κ p + κ(1 + κ) pκ 1 p 2 + 2! (1.1.74), Ā = 1 θt 2 κc pg 1 p κ s p 1+κ (p / p) 2 dθ. (1.1.75) θ S, APE, θ( T ) θ T p, p = p[θ(p)] and p = p(θ θ ) p(θ) = θ p/ θ., (p /p) 2 = (1/ p 2 )(θ p/ θ) 2 = (1/ p 2 )(θ ) 2 ( p/ θ) 2, (1.1.75) Ā = 1 θt 2 κc pg 1 p κ p κ p s θ 2 θ S = κc p 2gp κ s p κ 1 θ2 p s ( θ θ ( ) 2 p dθ θ ) 1 (1.1.76) ) 2 ( θ p

25, p θ κ θ γ d p = 1 γ d γ, γ = T z, γ d = g/c p θ = T (p s /p) κ and θ /θ = T /T (1.1.76) Ā Ā = 1 2 ps T ( T γ d γ T ) 2 dp (1.1.77) γ = 2γ d /3, T 2 = (15 K) 2, Ā/T P E 1/2 (1.1.78), 1% K TPE K = 1 ps V 2 dp, 2g, c 2 = c p RT/c v 2 V/c 1/2, T P E = c ps v c 2 dp (1.1.79) gr K/T P E 1/2 and K/ Ā 1/1 (1.1.8),, 1, APE,.1% TPE, 2% (1.9.66) (1.9.71) (E A ),,,.

26 1.11,, ( 2 ) 2 2, (vorticity equation) (divergence equation) 1.11.1,., 2,,, ζ(= k V ), ζ = 1 v a cos φ λ 1 u a φ + u tan φ. (1.11.81) a ζ (1.7.23) (1.7.24), (V )V = (V 2 /2) + ( V ) V, dv = V t + (V 2 /2) + ζk V + w V z = α p fk V + F, f = 2Ω sin φ η = f + ζ, ζk V fk V, ηk V, k,, ηk V, k [ (ηk V )] = k [ηk V V ηk (ηk )V + (V )ηk] V 2, k V = k =, 2, ζ ζ +V η +w t z = η V +k w V z +k p α+k F (1.11.82)

27, f/ t = f/ z =, (1.11.82) (ζ + f) t (ζ + f) + V (ζ + f) + w z = (ζ + f) V + k w V z + k p α + k F (1.11.83), d(ζ + f)/, 1.11.2, 3 3 V 3, D = V, V = 1 u a cos φ λ + 1 v a φ v a tan φ D,, D t V + [(V )V ] + (fk V ) + w z + w D z = (α p) + F (1.11.84),,, D t D V + V D + w + w z z + D2 2J(u, v) +(k V ) f fζ = (α p) + F (1.11.85), J(u, v) = ( u/ x)( v/ y) ( v/ x)( u/ y)., 2,,,

28 (1) (2) (3), (1.11.85) 5, 3 (1.11.83), ζ V,, z.,, p, Φ, k Φ =, (1.11.83) k p α, (1.11.85) 2 Φ