Agenda Monin-Obukhov Flux Richardson Prandtl
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1 2011 GFD (Sun.) :
2 Agenda Monin-Obukhov Flux Richardson Prandtl
3 0.
4 GABLS2 test case GEWEX ( 1
5 GABLS2 test case
6
7
8 Boussinesq : u i t + u iu j x j θ t + θu j x j = x j u j x j = 0. = 1 p + gθ δ ρ 0 x i θ i3 + 0 x j κ θ, x j (NB) p, θ ν u i x j, ( )
9 u i t + u iu j x j θ t + θu j x j u j x j = 0, = 1 p + gθ δ ρ 0 x i θ i3 τ ij + 0 x j x j = τ θj x j + x j κ θ x j τ ij = u i u j u i u j, τ θj = θu j θu j., ν u i x j,
10 ??? Yes.
11 The Kolmogorov length : Kolmogorov length = (ν 3 /ε) 1/4 O(10 3 m) ν = m 2 s 1 ε O(10 5 m 2 s 3 ) (Lilly et al. 1974) (1999)
12 τ ij, τ θj dissipation 1/LΔ kd
13 τ ij, τ θj τ ij, τ θj
14 τ ij, τ θj? τ ij, τ θj
15 δ O(10 3 m) Δz O(1m) O(10m) ( ) δ/δz << 1?
16 z U, Θ Θs
17 δ/δz << 1 Monin-Obukhov
18 τ ij, τ θj Kolmogorov length
19 ( )? (Mellor and Yamada ) Science and Art
20 1. Monin-Obukhov
21 Monin-Obukhov Buckingham Π Buckingham, E., 1914: On physically similar systems: illustrations of the use of dimensional equations. Phys. Rev., 4, Geoff Vallis web : ( general.html)
22 The Buckingham pi Theorem n q 1,..., q n, k f (q 1, q 2,, q n )=0 p = n k π 1,..., π p q 1,..., q n F(π 1, π 2,, π p )=0 π 1,..., π p
23 ? l[m], m[kg], g[ms -2 ] T g θ l (l/g) 1/2 T l/g. m
24 ρ σ σ U(z) ρ, σ, z u * κ (0.4 ) du dz = u κz, u := σ ρ.
25 U(z) = u κ log z z 0 z 0 ( ). z ρu w = σ U(z) = u κ log z z 0 : σ u w = σ ρ = u2. σ
26 U [m/s] u = u z κ log, z 0 κ = 0.4, u = [m/s], z 0 = [m] Z [m]
27 m m
28 Monin-Obukhov? Monin-Obukhov( ) ρ σ Q0 g/θ0, z Monin-Obukhov u := σ/ρ θ := Q 0 /u L := u 3 Θ 0 /(κgq 0 )
29 Monin-Obukhov Monin-Obukhov Monin-Obukhov Q0 > 0: Q0 < 0: 0 1/L
30 Monin-Obukhov F F F * F F * ς = z/l F F = g F (ζ) g F (ς)
31 Monin-Obukhov Monin-Obukhov U/ z u /L = g m(ζ), Θ/ z θ /L = g h(ζ). : U z = u κz φ m(ζ), φ m (ζ) := κζg m (ζ), Θ z = θ κz φ h(ζ), φ h (ζ) := κζg h (ζ). φ m (ς), φ h (ς) φ m (ς) 1
32 ( ) ς < 1 ς
33 U(z), Θ(z) U(z) = u κ Θ(z) = θ κ ψ m (ς), ψ h (ς) ψ m (ζ) := ψ h (ζ) := z log z log ζ 1 φ m (ζ ) ζ 0 ζ dζ, ζ z 0m z 0h ζ 0 1 φ h (ζ ) ζ dζ. u *, θ * U(z), Θ(z) ψ m (ζ) ψ h (ζ), + Θ s.
34 Businger (1971) ψ m (ζ) = 4.7ζ, (ζ 0), log[(1 + ξ(ζ)) 2 (1 + ξ(ζ) 2 )/8] 2 tan 1 ξ(ζ)+π/2, (ζ < 0), ψ h (ζ) = 4.7ζ, (ζ 0), 2 log[(1 + η(ζ))/2], (ζ < 0), ξ(ζ) =(1 15ζ) 1/4, η(ζ) =(1 9ζ) 1/2. Beljaas and Holtslag (1991) ψ m (ζ) = b(ζ c/d) exp( dζ) aζ bc/d (ζ 0), log[(1 + ξ(ζ)) 2 (1 + ξ(ζ) 2 )/8] 2 tan 1 ξ(ζ)+π/2, (ζ < 0), ψ h (ζ) = b(ζ c/d) exp( dζ) (1 + 2aζ/3) 3/2 bc/d + 1, (ζ 0), 2 log[(1 + ξ(ζ) 2 )/2], (ζ < 0), a = 1, b = 2/3, c = 5, d = 0.35, ξ(ζ) =(1 16ζ) 1/4.
35 z 0 = 0.1 [m], u* = 0.3 [m/s], Θ 0 = 288 [K] L = -100, -200, 100, 200 [m] ( )?( ) Altitude [m] 10 1 Altitude [m] U [m/s] Θ - Θ 0 [K]
36 :? Monin-Obukhov log-profile :
37 (u *2 ) (u * θ * ) U(z), Θ(z) u *, θ * σ/ρ = u 2 = C m U 2, Q 0 = u θ = C h U (Θ Θ s ), C m = C h =? log z z 0m κ 2 ψ m zl 2, κ 2 log z z 0m ψ m zl log z z0h ψ h zl.
38 No C m, C h L L u *, θ * ( ) L C m, C h u *, θ *
39 Richardson Rb := gzθ Θ 0 U 2 = z L Richardson log z z0h ψ zl h log z z 0m ψ zl 2. m Rb L C m, C h u *, θ *
40 Richardson (Louis 1979, Louis et al. 1982) Rb C m, C h u *, θ *
41 Louis (1979) Businger(1971) u 2 = a 2 mu 2 F m (z/z 0m, Rb), a m := κ/[log(z/z 0m )], a h := κ/[log(z/z 0h )], z 1 brb Rb < 0 1+c F m, Rb = m Rb 1/2 z 1 0m (1+b Rb 0 Rb) 2 z z 1 brb Rb < 0 1+c F h,, Rb = h Rb 1/2 z 0m z 1 0h (1+b Rb 0 Rb) 2 u θ = a m a h U Θ F h (z/z 0m, z/z 0h, Rb), b = 2b = 9.4, c m = 7.4a 2 mb z z 0m 1/2, c h = 5.3a m a h b z z 0h 1/2.
42 Louis et al. (1982) F m F h u 2 = a 2 mu 2 F m (z/z 0m, Rb), u θ = a m a h U Θ F h (z/z 0m, z/z 0h, Rb), a m := κ/[log(z/z 0m )], a h := κ/[log(z/z 0h )], 2bRb z 1 Rb < 0 1+c m Rb 1/2 F m, Rb = 1 z 0m 1 + 2bRb Rb 0 1+ f Rb F h z z 0m, z, Rb z 0h b = c = f = 5, = 1 3bRb Rb < 0 1+c h Rb 1/2 (1 + 3bRb 1 + f Rb) 1 Rb 0 z 1/2, c h = 3ba m a h c c m = 3ba 2 mc z 0m z z 0h 1/2.
43 z/z 0 = 10 2 z/z 0 = z/z 0 = z/z 0 = C m C h z/z 0 = z/z 0 = z/z 0 = 10 4 z/z 0 = Rb Rb Businger (1971) Louis (1979)
44 z/z 0 = 10 2 z/z 0 = 10 2 z/z 0 = z/z 0 = C m C h z/z 0 = z/z 0 = z/z 0 = 10 4 z/z 0 = Rb Rb Louis et al. (1982) Louis (1979)
45 z/z 0 = 10 2 z/z 0 = 10 2 z/z 0 = z/z 0 = C m z/z 0 = z/z 0 = z/z 0 = 10 4 C h z/z 0 = Rb Rb Louis et al. (1982) Beljaas and Holtslag (1991)
46 2 Richardson??
47 Richardson Richardson Richardson ζ (1/L ) Rb
48 Richardson Businger (1971): ψ m (ζ) =ψ h (ζ) = 4.7ζ, Rb = ζ [log(z/z 0h) ψ h (ζ)] [log(z/z 0m ) ψ m (ζ)] 2 4.7ζ ζ 2 = 1 4.7, (ζ ) Beljaas and Holtslag (1991): Rb 2 2ζ 5/2 =, (ζ ) 3 3aζ2
49 Businger (1971), Louis (1979) Beljaas and Holtslag (1991), Louis et al. (1982) ψ m, ψ h C h = log(z/z 0m) ψ m (ζ) C m log(z/z 0h ) ψ h (ζ)
50 Ri, Rf, Pr Richardson flux Richardson Prandtl Gradient Richardson number Flux Richardson number Turbulent Prandtl number Ri := Rf := Pr := K m K h g dθ θ 0 dz du dz 2 g θ 0 θ w u w du dz = Ri Rf
51 Ri, Rf, Pr Ri, Rf, Pr φ m (ς), φ h (ς) : u w = u 2, θ w = u θ, L = U z = u κz φ m(ζ), Θ z = θ κz φ h(ζ), Ri = ζφ h(ζ) φ 2 m(ζ), Rf = ζ φ m (ζ), Richardson Ri c? u 2 κg/θ 0 θ, Pr 1 = φ m(ζ) φ h (ζ). = Pr Rf
52 Rf(Ri) and Pr 1 (Ri) Rf 1/Pr Ri Ri Beljaas and Holtslag (1991) Businger (1971)
53 Businger (1971), Beljaas and Holtslag (1991), Louis (1979), Louis et al. (1982) Businger (1971) Richardson Beljaas and Holtslag (1991), Louis et al. (1982) Ri, Rf, Pr
54 2.
55 ( ): RANS ( ): LES(Large-eddy Simulation): ( ) u i t + u j u i x i = 0. u i x j = p x i + ν u i,
56 RANS LES RANS F(x, t) := P(F; x, t) F(x, t) LES G(x) F(x, t) := F(x, t)p(f; x, t)df. F(x, t)g(x x )dx. (GS) RANS
57 RANS LES RANS: LES: ( km km) ( m)
58 : ( u i t + u iu j x j u i x i = 0. u i u j u i = u i + u i ) = p x i u i u j x j + ν u i,
59 2 u i u j u i u j u k u i u j u k u i u j u k u l ( )
60 2 3 u i u j t + x k U k u i u j + u k u i u j ν u x i u j k U j U = u i u k u x j u i k β(g k x j u i θ + g i u j θ) + p k u j θ + U t x k u j θ + u j u k θ νθ u j θ αu k x j k x k + pu x i + pu j x j + f k ( ikl u i u l + jkl u j u l ) i u i u j + u j u i + x j pθ + jkl f k u l θ Θ = u j u k u x k θ U j βg k x j θ 2 + p θ (α + ν) u j θ, k x j x k x k θ 2 t + U x k θ 2 + u k θ 2 α θ2 = 2u k x k θ Θ 2α θ θ. k x k x k x k 2ν u i u k u j u k,
61 ( ) ( )
62 ( ) (Boussinesq 1877): ui u j = ν T S ij δ iju k u k, S ij = 1 u i + u j. 2 x j x i ν T?
63 Prandtl (1925) U z l u l U z. u w U(z + l) U(z) + l U z U(z)
64 Prandtl ( ) w u u w u w = l 2 U l Prandtl( ) z U z. ν T = l 2 U z. ( )?
65 Prandtl ν T = lv l v ν l v
66 Mellor-Yamada K m = lqs m, K h = lqs h. q S m, S h l, q, S m, S h
67 Anderson, P. S., 2009: Measurement of Prandtl Number as a Function of Richardson Number Avoiding Self-Correlation, Bound.-Layer Meteorol., 131, Balarac G., H. Pitsch, and V. Raman, 2008: Modeling of the subfilter scalar dissipation rate using the concept Modeling of the subfilter scalar dissipation rate using the concept of optimal estimators, Phys. Fluids, 29, Beljaas, A. C. M., and A. A. M. Holtslag, 1991: Flux parameterization over land surfaces for atmospheric models, J. Appl. Meteor., 30, Boussinesq, 1877: Essai sur la théorie des eaux courantes, Mém. prés. par div. savants à l Acad. Sci., 23, Buckingham, E., 1914: On physically similar systems: illustrations of the use of dimensional equations, Phys. Rev., 4, Businger, J. A., J. C. Wungaard, Y. Izumi, and E. F. Bradley, 1971: Flux-profile relationships in the atmospheric surface layer, J. Atmos. Sci., 28, Canuto, V. M., Y. Cheng, A. M. Howard, and I. N. Esau, 2008: Stably Stratified Flows: A Model with No Ri(cr), J. Atmos. Sci., 65, Cheng, Y., V. M. Canuto, and A. M. Howard, 2002: An Improved Model for the Turbulent PBL, J. Atmos. Sci., 59,
68 Galperin, B., S. Sukoriansky, and P. S. Anderson, 2007: On the critical Richardson number in stably stratified turbulence. Atmos. Sci. Lett., 8, Honnert, R., V. Masson, and F. Couvreux, 2011: A Diagnostic for evaluating the Representation of Turbulence in Atmospheric Models at the Kilometric Scale, J. Atmos. Sci., in press.,, 1999:,. Kitamura, Y., 2010: Modifications to the Mellor-Yamada-Nakanishi-Niino (MYNN) Model for the Stable Stratification Case, J. Meteor. Soc. Japan, 88, Langford, J. A., and Robert D. Moser, 1999: Optimal LES formulations for isotropic turbulence, J. Fluid Mech., 398, Lilly, D., D. E. Waco, and S. I. Adelfang, 1974: Stratospheric mixing estimated from highaltitude turbulence measurements., J. Appl. Meteor., 13, Louis, J. F., 1979: A parametric model of vertical eddy fluxes in the atmosphere, Bound.- Layer Meteorol., 17, Louis, J. F., M. Tiedtke, and J. F. Geleyn, 1982: A short history of the Operational PBL - parameterization at ECMWF, Proc. Workshop on Planetary Boundary Layer Parameterization,
69 Mauristen, T., G. Svensson, S. S. Zilitinkevich, I. Esau, L. Enger, and B. Grisogono, 2007: A total turbulent energy closure model for neutrally and stably stratified atmospheric boundary layers, J. Atmos. Sci., 64, Mellor, G. L., and T. Yamada, 1982: Development of a turbulence closure model for geophysical fluid problems, Rev. Geophys. and Space Phys., 20, Moreau A., O. Teytaud, and J. P. Bertoglio, 2006: Optimal estimation for large-eddy simulation of turbulence and application to the analysis of subgrid models, Phys. Fluids, 18, Nakanishi, M., 2001: Improvement of Mellor-Yamada turbulence closure model based on large-eddy simulation data, Bound.-Layer Meteor., 99, Nakanishi, M., and H. Niino, 2004: An improved Mellor-Yamada level-3 model with condensation physics: its design and verification, Bound.-Layer Meteor., 112, Nakanishi, M., and H. Niino, 2006: An improved Mellor-Yamada level-3 model: its numerical stability and application to a regional prediction of advection fog, Bound.-Layer Meteor., 119, Nakanishi, M., and H. Niino, 2009: Development of an improved turbulence closure model for the atmospheric boundary layer, J. Meteor. Soc. Japan, 87,
70 Prandtl, L., 1925: Bericht über die ausgebildete Turbulenz, Zs. angew. Math. Mech., 5, Sukoriansky, S., B. Galperin, and I. Staroselsky, 2005: A quasinormal scale elimination model of turbulent flows with stable stratification, Phys. Fluids, 17, ,, 1981: 1.,,,. Wyngaard, J. C., 2004: Toward Numerical Modeling in the ``Terra Incognita'', J. Atmos. Sci., 61, Zilitinkevich, S. S., T. Elperin, N. Kleeorin, and I. Rogachevskii, 2007: Energy- and flux-budget (EFB) turbulence closure model for stably stratified flow. Part I: steady-state, homogeneous regimes, Bound.-Layer Meteor., 125, Blackadar, A., 1962: A., 1962: The vertical The vertical distribution distribution of wind and turbulent of wind and exchange turbulent in exchange a neutral in a atmosphere. neutral in a neutral atmosphere. J. Geophys. atmosphere. J. Geophys. Res., 67, Res., J.Geophys , Res., 67, Frisch, U., U., 1995: Turbulence - The legacy - The of legacy A. N. Kolmogorov, of A. N. Kolmogorov, Canbridge University Canbridge Press. Blackadar, A., 1962: The vertical distribution of wind and turbulent exchange University Press.
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