65 6 6.1 No.4 1982 1 1981 J. C. Kaimal 1993 1994 Turbulence and Diffusion in the Atmosphere : Lectures in Environmental Sciences, by A. K. Blackadar, Springer, 1998 An Introduction to Boundary Layer Meteorology, by R. B. Stull, Kluwer Academic Publishers, 1988 Lectures on Air Pollution Modeling, Ed. by A. Venkatram and J. C. Wyngaard, AMS, 1988 The Structure of Atmospheric Turbulence, by J. L. Lumley and H. A. Panofsky, Monographs and Texts in Physics and Astronomy. Vol 12. Interscience Publ., Jon Wiley & Sons, 1964 Statistical Fluid Mechanics, Vols 1 & 2, by A. S. Monin and A. M. Yaglom, MIT Press, 1971 6.2
66 6 6.1 2 150 75 Venkatram and Wyngaard, Lectures on Air Pollution Modeling, 1988 6.3 m km 6.2 Stull, An Introduction to Boundary Layer Meteorology, 1988 Capping Inversion Convective Mixed Layer
6.3 67 Entrainment Zone Free Atmosphere Residual Lyer Stable nocturnal Boundary Layer Surface Layer 6.3 Kaimal, 1993 6.4 Stull, An Introduction to Boundary Layer Meteorology, 1988 Wangara 6.5 km 1km 100
68 6 6.5 Wangara Yamada and Mellor, A simulation of the Wangara atmospheric boundary data, 1975 6.4 6.4.1
6.4 69 6.6 1981 1. Ā t x = 1 N N A i,x, Ā t x = 1 T i=1 T 0 A t,x dt 2. 3. Ā x t = 1 N N A t,j, Ā x t = 1 L j=1 Ā e t,x = 1 N N k=1 A kt,x L 0 A t,x dt 6.4.2 1. σ u U, σ v U, σ w U
70 6 σ u 2. 1/2 1/2 1/2 u 2, σ v v 2, σ w w 2 σ u u, σ v u, σ w u u u 3. skewness : u 2, v 2, w 2 a b a Skewness
6.4 71 u 3 u 2 3/2 a b,c b c Flatness Kurtosis W q, Ū Θ, Ū W w q, u θ, u w 6.7 1981
72 6 6.8 Stull, An Introduction to Boundary Layer Meteorology, 1988 covariance a b u w, w θ 6.4.3 Bulk Richardson Number Flux Richardson Number Ri = g dθ Θ du 2 R f = gw θ Θ u w du R f = gw θ Θ u w du = g Θ K h dθ K m du du = K h K m R i Boussinesq
6.4 73 Navier-Stokes eq. u t + u u x + v u y + w u fv + 1 p ρ 0 x = µ 2 u ρ 0 x 2 + 2 u y 2 + 2 u 2 ρ 0 v t + u v x + v v y + w v + fu + 1 p ρ 0 y = µ 2 v ρ 0 x 2 + 2 v y 2 + 2 v 2 w t + u w x + v w y + w w + 1 p ρ 0 = ρ g + µ 2 w ρ 0 ρ 0 x 2 + 2 w y 2 + 2 w 2 Q Q θ t + u θ x + v θ y + w θ = Q µ ρ ν ν 1.467 10 5 m 2 sec - 1 0.5 sec 1 7 10 6 m 2 sec - 2 φ φ = φ + φ 6.1 Boussinesq 6.1 x ū t + ū ū ū ū + v + w x y f v + 1 p ρ 0 x = u ν 2 u 2 x + u v y + u w Reynolds 5 10 2 m 2 sec 2
74 6 E S t + ū E S x + v E S y = ρ wg ρ 0 ū ρ 0 w u 2 + w E S ū p x + v p y + w p x + u v y + u w u w x + v w + w 2 y ρ 0 v u v x + v 2 y + v w 6.2 ES E S ρ 0 2 6.2 E S t ū2 + v 2 + w 2 + {E S ū + ρ 0 ūu x 2 + vu v + wu w ū = ρ wg + ρ 0 u 2 x + u v ū y + u w ū + } + ū p + y {} + {} u t + ū u x u + v y = fv 1 ρ 0 p x + + w u + u ū x ū + v y u 2 x + u v y + u w E T t + } {E T ū + ρ 0 u x 3 + u v 2 + u w 2 + u p = gρ w ū ρ 0 u 2 x + u v ū y + u w ū + ū u u u + w + u + v + w x y + y {} + {} E T E T ρ 0 2 u 2 + v 2 + w 2 E S, E T i ii iii Reynolds
6.5 75 6.4.4 Closure problem Navier- Stokes 1 2 Reynolds Reynolds Reynolds Closure Problem 6.5 6.5.1 closure K-theory closure. Dū Dt = fv 1 p ρ 0 D θ Dt = u 2 x + u v y + u w x u θ x + v θ y + w θ Dū Dt = t + ū y + v + w 6.3 u w w θ - q u q = K q x, v q = K q y, w q = K q K u w = K m U, θ w = K h Θ K h 1.35K m
76 6 K h 2.5K m a 1925 S z z Sz Sz δz z S S = Sz δz Sz { = Sz + S } δz + Sz = S δz 6.4 w du/ > 0 w = cu du/ < 0 w = cu u u = du/δz w = c du δz w S S w = c du S δz2 = K S E 6.5. l K E l 2 du, l 2 cδz 2 b K examples 6.9 l = kz c i Log profile l = kz K E = κ 2 z 2 du 0 = d du K E = d κ 2 z 2 du du
6.5 77 6.9 Stull, An Introduction to Boundary Layer Meteorology, 1988 du/ > 0 0 = d κ 2 z 2 du 2 = 2κ 2 z 2 du + 2κ 2 z 2 du d 2 U 2 du + z d2 U 2 = 0 U = γ ln z z 0
78 6 z 0 6.10 6.10 1994 ii Ekman spiral / t = 0 / x = / y = 0 θ/ = 0 U g, V g w = 0 x fv = du w K m - fu U g = dv w fv = K m d 2 U 2 fv = K m d 2 fu U g = K m d 2 V 2 2 d 2 V fu g + K m 2 = Km 2 d 4 V 4
6.5 79 z U U g, V 0 V = A expλz f 2 = Kmλ 2 4 λ = ± ±i f K m z z = 0 V = U = 0 z U U g f λ = 1 ± i 2K m U = U g [ 1 e γz cosγz ] V = U g e γz sinγz f γ = 2K m h E = π/γ K m 10 m 2 /sec 2, f 10 4 sec 1 h E 1400 m 6.11 Elman Stull, An Introduction to Boundary Layer Meteorology, 1988 d Refine K-theory Mellor & Yamada level 2 Mellor, G.L. and T. Yamada, 1974: A hierarchy of turbulence closure models for planetary boundary layers. J. Atmos. Sci., 31, 1791-1807.
80 6 Full u w, v w = lq S U M, V w θ = lq S Θ H S M 3A 1 γ 1 C 1 6A 1 + 3A 2 Γ/B 1 γ 1 γ 2 Γ + 3A 1 Γ/B 1 γ 1 γ 2 Γ S H 3A 2 γ 1 γ 2 Γ γ 1 1 3 2A 1, γ 2 B 2 + 6A 1 B 1 B 1 w θ R f Γ, R f βg 1 R f u w U + v w V = γ 1 γ 1 + γ 2 6.5.2 1.5 closure a TKE Mellor & Yamada, 1982 level 2.25 2.5 q 2 t + u q2 + = Prod + Dissip + x x Prod Buoy + Shear = βg w θ θ K m = S M l q, K m = S H l q, K q = S q q K q q 2 x + u ū i i u j = gk θ H x j + K m ūi + ū j 2 x j x i 3 δ ijq 2 S M b TKE Yamada, 1983 q 2 l model Level 2.5 l a q 2 l level 2 c k ϵ Rodi, 1985; Detering & Etling, 1985; Kitada, 1987 k ϵ σ Yamada & Mellor
6.5 81 6.5.3 Monin-Obukov similarity theory a Monin Obukov 1946 1954 z m 2 /sec 2 τ 0 ρ u w K m/sec m/sec 2 /K H 0 c p ρ w θ Q 0 g Θ Monin-Obukov u 1/2 τ0 = u ρ w 1/2 Monin-Obukov T Q 0 u = w θ u L u3 Θ κgq 0 = u3 Θ κgw θ κ 0.4 Monin-Obukov L F F F F = F * G F z/l G F z/l ζ z/l
82 6 Θ > 0 w θ < 0, T > 0, L > 0 Θ = 0 w θ = 0, T = 0, L = Θ < 0 w θ > 0, T < 0, L < 0 b i Monin-Obukov 6.6 ζ du = u κl G mz/l 6.6 dθ = T κl G hz/l 6.7 Uz = u κl f mz/l f m z 0 /L f m ζ ζ G m ζ z 0 roughness κ u [Uz U L /2] κ u [Uz U L /2] = f m ζ f m ±0.5 ζ 6.12 Monin-Obukov
6.5 83 6.12 1981 ii L >> 1, ζ << 1 ζ = 0 logarithmic law or log law U z = u z κ ln z 0 du = u 1 κ z = u L κl z = u κl 1 ζ G m = 1 ζ z 0 6.6 G m z = 0 φ m ζ = ζg m ζ 6.8 φ h ζ = ζg h ζ 6.9 φ m 0 = 1 ζ << 1 φ m
84 6 φ m ζ = 1 + βζ + du = u κl G m = u κlζ φ mζ U z u κl ln z z 0 + β z z 0 L log+linear law u 1 + βζ κlζ ζ >> 1 L << 1 Q 0 u z z Q 0 u z zgq0 u f Θ 0 1/3 z T f Q 0 u f u f z 1/3 u L T f z 1/3 T L z/l >> 1 φ m φ h = κz dθ T κz T f T z T f z 1/3 T L ζ >> 1 z z L
6.5 85 du u L z φ m = κz du u φ h = κz dθ T κz u u L z L κz T T L z L 6.13 1981 iii u 2 = τ 0 ρ = u w du = K m K m = u2 du = κu L G m ζ = κu z φ m ζ K m = u T dθ = κu L G h ζ = κu z φ h ζ
86 6 Richardson Gradient Richardson number Pr = K m K h = G m G h = φ m φ h Ri = g dθ Θ du 2 = g T Θ κl G h u 2 κl 2 G 2 m = G h G 2 = Pr m G m Flux Richardson number R f gw θ /Θ u w U Z de T dt = u w U + g w θ Θ = u w U 1 R f flux Richardson 1 K R f = gw θ /Θ u w U = K H g K m Θ θ/ U/ 2 = K H R i K m 6.14 1981 KEYPS φ
6.5 87 φ 4 γζφ 3 1 = 0 6.10 Kazanski & Monin 1956, Ellison 1957, Yamamoto 1959, Panofsky 1961 KEYPS Obukov 1946 O KEYPS KEYPS 6.10 K n K n = κzu φ m κzu ζ 1/3 R f >> 1, or ζ >> 1 K f K f = κzu φ m κzu R f 1/4 K = κzu 1 γr f 1/4 K = κzu φ, R f = ζ φ 6.10 γ 10 6.5.4 Canopy z = h z = d u = u z d κ ln, forz h z 0 d zero-plane displacement d h d 0.7h a a F = 1 2 ρcau2 6.11 c 6.11
88 6 6.15 1981 d K du = 1 2 ρcau2 6.12 a K l K = l 2 du/ l 6.12 z = h u h u = u h exp [γz h], γ = ρ ac 4l 2 1/3 z < h z = h 6.16 1981
6.5 89 b model 1cm 2 10 4 100mm 0.3 3 Yeh & Brutsaert, 1971 Sellers 1985 SiB Dickinson 1984 BATS MATSIRO, MINoSGI
90 6 6.17 SiB