Venkatram and Wyngaard, Lectures on Air Pollution Modeling, m km 6.2 Stull, An Introduction to Boundary Layer Meteorology,

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1 No J. C. Kaimal 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,

Air Pollution Modeling, 1988 6.3 m km 6.

2 Venkatram and Wyngaard, Lectures on Air Pollution Modeling, 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

3 Entrainment Zone Free Atmosphere Residual Lyer Stable nocturnal Boundary Layer Surface Layer 6.3 Kaimal, Stull, An Introduction to Boundary Layer Meteorology, 1988 Wangara 6.5 km 1km 100

4 Wangara Yamada and Mellor, A simulation of the Wangara atmospheric boundary data,

5 Ā t x = 1 N N A i,x, Ā t x = 1 T i=1 T 0 A t,x dt 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 σ u U, σ v U, σ w U

6 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

7 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

8 Stull, An Introduction to Boundary Layer Meteorology, 1988 covariance a b u w, w θ 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

9 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 u y 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 v y 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 w y w 2 Q Q θ t + u θ x + v θ y + w θ = Q µ ρ ν ν m 2 sec sec 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 m 2 sec 2

10 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 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

11 Closure problem Navier- Stokes 1 2 Reynolds Reynolds Reynolds Closure Problem 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

12 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

13 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

14 78 6 z 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

15 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,

16 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 A 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 closure a TKE Mellor & Yamada, 1982 level 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

17 Monin-Obukov similarity theory a Monin Obukov 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

18 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

19 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 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

20 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

21 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 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

22 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 KEYPS φ

23 φ 4 γζφ 3 1 = 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 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 ρcau c 6.11

24 d K du = 1 2 ρcau 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

25 b model 1cm mm Yeh & Brutsaert, 1971 Sellers 1985 SiB Dickinson 1984 BATS MATSIRO, MINoSGI

26 SiB

mains.dvi

mains.dvi 8 Λ MRI.COM 8.1 Mellor and Yamada (198) level.5 8. Noh and Kim (1999) 8.3 Large et al. (1994) K-profile parameterization 8.1 8.1: (MRI.COM ) Mellor and Yamada Noh and Kim KPP (avdsl) K H K B K x (avm)