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1 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 : (MRI.COM ) Mellor and Yamada Noh and Kim KPP (avdsl) K H K B K x (avm) K V K K x ( : Mellor-Yamada ) - - (eb : Noh and Kim ) = E - (alo : KPP ) l l Mellor and Yamada s Turbulence Closure Model U P Θ ρ DU j ρ t = x i (ρu i ) (8.1) + ρε jkl f k U l = x k ( ρ < u k u j >) P x j g j ρ (8.) ρ DΘ = x k ( ρ < u k θ >) (8.3) D( )= U k ()= x k + ()= t g j f k ε ijk < > (8.3) Boussines (8.) Λ 85

2 8 Boussines < > closure second moment closure Kantha and Clayson (000) Mellor and Yamada (198) Rotta (1951a,b) Reynolds stress : D p ρ ui x j + u j x i E = < u i u j > δ ij 3l 1 3 +C 1 U i U j + x j x i < u i > l 1 C 1 δ ij (= 1(i = j); = 0(i 6= j)) Kolmogolov D ui u E j ν = x k x k 3 ν Λ 1 3 (8.4) Λ 1 δ ij (8.5) D p θ E = < u j θ > (8.6) ρ x j 3l D uj θ E (κ + ν) = 0 (8.7) x k x k κ l D θ θ E κ = < θ > (8.8) x k x k Λ Λ < u k u i u j > = 3 5 ls < ui u j > + < u iu k > + < u ju k > x k x j x i < uk θ > < u k u j θ > = ls uθ + < u jθ > x j x k (8.9) (8.10) < u k θ > = ls θ < θ > x k (8.11) S ;S uθ ;S θ < pθ >= 0 < pu i >= 0 86

3 8.1. Mellor and Yamada s Turbulence Closure Model Mellor-Yamada (l 1 ;Λ 1 ;l ;Λ )=(A 1 ;B 1 ;A ;B )l (8.1) l master length scale A 1 ;B 1 ;A ;B C 1 Mellor and Yamada (198) (A 1 ;B 1 ;A ;B ;C 1 )=(0:9;16:6;0:74;10:1;0:08) 8.1. The level-.5 Model level-4 level-3 ( =) (< θ >) (< θs >) (< s >) level-.5 (8.30) level- MRI.COM level-.5 ffl ffl ffl ρ DU + ρ < uw > = P + ρ fv x (8.13) ρ DV + ρ < vw > = P ρ fu y (8.14) 0 = P ρg (8.15) ρ DΘ + (ρ < wθ >) = 0 (8.16) D h i ls = P s + P b ε (8.17) P s = < wu U V > < wv > (8.18) 87

4 8 P b = g < wρ >=ρ 0 (8.19) ε = 3 =Λ 1 (8.0) < u > = l < v > = l < w > = l h 4 < wu U > h < wu U > V i + < wv > P b 4 < wv > V P b h < wu U V > + < wv > + 4P b i i (8.1) (8.) (8.3) < uv 3l h 1 > = < wu > = 3l 1 < vw > = 3l 1 < uw V > h (< w > C 1 U ) U i < vw > i g < uρ > h (< w > C 1 ) V g < vρ > i (8.4) (8.5) (8.6) < uθ 3l h > = < uw Θ > h < vθ > = 3l < wθ > = 3l U i < wθ > < vw Θ V i > < wθ > h < w Θ i > g < θρ > < θ >= Λ < wθ > Θ (8.7) (8.8) (8.9) (8.30) U < uw > = K M (8.31) V < vw > = K M (8.3) Θ < θw > = K H (8.33) K M = ls M (8.34) K H = ls H (8.35) K M ;K H 88

5 8.1. Mellor and Yamada s Turbulence Closure Model S M S H S M [6A 1 A G M ]+S H [1 3A B G H 1A 1 A G H ]=A S M [1 +6A 1G M 9A 1 A G H ] S H [1A 1G H + 9A 1 A G H ]=A 1 (1 3C 1 ) (8.36) G M l h U V i + G H l g ρ ρ 0 (8.37) (8.38) ρ= S M S H l (8.34) (8.35) K M ;K H (8.17) D h K i h U V i = K g ρ M + + K H ρ 0 ε (8.39) K = ls MRI.COM S S M (G H = 0) S = 0: S = S c S M =S Mn (S c = 0:;S Mn = S M (G H = 0) =0:397) = l = 0 (8.40) ( ρ s ) (τ s ) u τ (τ s =ρ s ) 1= ρ s 3 =Λ 1 = τ s u τ =l (8.1) = B =3 1 u τ (8.41) master length scale MRI.COM : Z 0 Z 0 l = γ jz 0 jdz 0 = z b dz 0 z b (8.4) γ = 0: z b Mellor and Yamada (198) n MYSL5 n n+1 l (8.34) (8.36) n 89

6 8 n+1 l (8.39) (8.1) (8.0) (8.41) K 8.4 master length scale (8.4) 8. Noh and Kim (1999) Mellor and Yamada Noh and Kim (1999) Mellor and Yamada second moment closure 8..1 U V B = g ρ=ρ o E DU DV DB DE = < uw > 1 P + fv ρ x (8.43) = < vw > 1 P fu ρ y (8.44) = < bw > R = p < w ρ + uu + vv + ww > < uw > U (8.45) V < vw > < bw > ε (8.46) R R= DU DV DB DE = K U = K V B = K B E = K E 1 P ρ x P 1 ρ R + K U + fv (8.47) fu (8.48) y U + K V V + B K B K;K B ;K E (ε) ( =(E) 1= ) (l) (8.49) ε (8.50) K = Sl (8.51) K B = S B l (8.5) K E = S E l (8.53) ε = C 3 l 1 (8.54) 90

7 8.. Noh and Kim (1999) (S;S B ;S E ;C) S = S 0 = 0:39 Pr = S=S B = 0:8 σ = S=S E = 1:95 C = C 0 = 0:06 l b = =N (N = B=) 1= K ο l b ο lri t (8.55) Ri t Ri t =(Nl=) (8.56) N Ri t K ) Ri t (8.55) S S=S 0 =(1 + αri t ) 1= (8.57) α Noh and Kim (1999) α ο 10:0 C C=C 0 =(1 + αri t ) 1= (8.58) l = κ(z + z 0 ) (1 + κ(z + z 0 )=h) (8.59) z 0 (z 0 = 1[m]) z h K U B K B E K E = τ ρ 0 (8.60) = Q 0 (8.61) = mu 3 Λ (8.6) m Noh and Kim (1999) m = 100 N < 0 K = K B = 1:0[m s 1 ] K E (K) 8.. nkoblm.f90 E 8.50 E N E 91

8 8 8.3 K Profile Parameterization (KPP) K profile parameterization (KPP) Monin-Obukhov ν x MRI.COM Mellor and Yamada (198) KPP nonlocal K profile model(troen and Mahrt 1986) Large et al.(1994) MRI.COM KPP NCEP (NCOM) X < wx > X U V T S B x u v T s b w ( ) m s t X = z < wx > (8.63) KPP X nonlocal < wx >= K x ( z X γ x ) (8.64) MRI.COM KPP K x nonlocal γ x ffl < wx 0 > ffl L ffl h ffl φ x ffl w x ffl K x ffl ν x ffl nonlocal γ x 8.3. Monin-Obukhov Monin-Obukhov d(= z) < wx 0 > nonlocal X= ( ) x= ( ) 9

9 8.3. K Profile Parameterization (KPP) 8.1: KPP ffl u Λ =(< wu 0 > + < wv 0 > ) 1= = j~τ 0 j=ρ 0 (8.65) ffl S Λ = < ws 0 >=u Λ (8.66) ffl Monin-Obukhov L = u Λ3 =(κb f ) (8.67) ~τ 0 ρ 0 κ = 0:4 von Karman B f ; ( ) d < εh [ε fi 1 ε ο 0:1]) < wx 0 > u Λ, S Λ, L ζ = d=l φ m (ζ ) = κd u Λ z (U +V ) 1= φ s (ζ ) = κd (8.68) S Λ z S K x w x G(σ ) K x h K x (σ )=hw x (σ )G(σ ) (8.69) σ = d=h( / ) G(σ ) (O Brien 1970) G(σ )=a 0 + a 1 σ + a σ + a 3 σ 3 (8.70) 93

10 8 8.: ( ) G(1) = σ G(1) =0 G(σ ) ( ) h=l = 1;0:1;0; 1; 5 w x (σ )=(κu Λ ) (h=l < 0) w s (σ ) ( ) w m (σ ) ( ) (h=l 0) ( ) Large et al.(1994) 8. G(σ ) w x (8.70) (σ = 0) K x = 0 a 0 = 0 (σ < ε[= 0:1]) Monin-Obukhov (8.64[γ x = 0]) (8.68) (8.69) w x (σ )(a 1 + a σ )= κuλ < wx(d) > (8.71) φ x (ζ ) < wx 0 > < wx > (Lumley and Panofski 1964; Tennekes 1973) (8.71) κuλ w x (σ )= φ x (ζ ) (8.7) (ζ (= d=l) < 0) σ = ε(ο 0:1) w x (σ )= φ κuλ ε < σ < 1 ζ < 0 x (εh=l) (8.73) w x (σ )= φ κuλ otherwise x (σh=l) w x ( 8.) φ x ζ (= d=l) (h=l = 0) κu Λ (h=l < 0) (h=l 0) Large et al.(1994) ( 8.3) 94

11 8.3. K Profile Parameterization (KPP) 8.3: ζ φ x Large et al.(1994) φ m = φ s = 1 + 5ζ 0» ζ φ m = (1 16ζ ) 1=4 ζ m» ζ < 0 φ m = (a m c m ζ ) 1=3 ζ < ζ m φ s = (1 16ζ ) 1= ζ s» ζ < 0 (8.74) φ s = (a s c s ζ ) 1=3 ζ < ζ s (ζ s ;c s ;a s ;ζ m ;c m ;a m )=( 1:0;98:96; 8:86; 0:;8:38;1:6) (h=l < 0) w s (σ ) w m (σ ) (h=l 0) (ζ < ζ x ) w s w Λ φ x φ x =(a x c x ζ ) 1=3 ζ < ζ x < 0 (8.75) (8.67) (8.73) w Λ =( B f h) 1=3 (8.76) w x = κ(a x u Λ3 + c x κσw Λ3 ) 1=3! κ(c x κσ) 1=3 w Λ σ < ε w x = κ(a x u Λ3 + c x κεw Λ3 ) 1=3! κ(c x κε) 1=3 w Λ ε» σ < 1 (8.77)! < wx > (8.71) < wx(σ ) >=<wx 0 >= 1 β r σ =ε = a 1 + a σ (8.78) σ = 0 (8.70) a 1 = 1;a = β r =ε σ = 1 G(1) = σ G(1) =0 ε = 0:1 a = ;a 3 = 1;β r = 0: 95

12 8 8.4: ( h) h d k 1 < h < d k d k 1 Large et al.(1994) KPP ν x (MRI.COM Tsujino et al.(000) ) ( d k 1 ) 8.4 δ = (h d k 1 )=(d k d k 1 ) K Λ x = (1 δ) K x (d k 1 )+δ K x (d k 1 ) (8.79) Λ x = (1 δ)ν x (d k 1 )+δk Λ x 8.4 ν x K x (d) (h) d k 1 Λ x K Λ x h d k 1 d k 1 < h < d 1 k d 1 k (8.79) K x (d k 1 ) ν x (d k 1 ) h B(d) ~ V (d) (B r B(d))d Ri b (d) = j ~ V r ~ V (d)j +Vt (8.80) (d) 96

13 8.3. K Profile Parameterization (KPP) Ri c (MRI.COM 0.3) B r ~ V r Vt =d (8.80) d = h (B r B(h))h Ri c = j ~ V r ~ V (h)j +Vt (8.81) (h) (j ~ V r ~ V (h)j = 0) B r N (N = B=) 8.5 (B r B(h)) = (h h e )N (h h e ) (8.64) (8.69) (8.76) (8.77) γ b fi N G(h e =h)=(h h e ) =h N(h e )=N=C v ; C v = 1:8 ( 8.5) d = h e < wb e >=<wb 0 >= β T (= 0:); <wb 0 > B f V t (d) =C v( β T ) 1= Ri c κ =3 (c s ε) 1=6 hnw Λ (8.8) w Λ (8.77) (w s ) V t (d) =C v( β T ) 1= Ri c κ (c s ε) 1= dnw s (8.83) (8.80) Vt N h K x N Vt (8.83) N h K x Nonlocal 8.5 local (0:35 < d=h < 0:8) (8.64 [γ x = 0]) < wb > (< 0) 8.5 (nonlocal ) < wb 0 > (> 0) < wb > local nonlocal counter gradient nonlocal local local < wb > h Nonlocal (Deardroff 197)Mailhôt and Benoit(198) nonlocal γ s γ s = C Λ < ws 0 > w Λ h C Λ = 10 γ x = 0 ζ 0 γ m = 0 ζ < 0 < γ s = C ws 0 > s ζ < 0 w s (σ )h < γ θ = C wθ 0 > + < wθ R > s ζ < 0 w s (σ )h (8.84) (8.85) C s = C Λ κ(c s κε) 1=3 (8.86) 97

14 8 8.5: (< wb 0 >) Large et al.(1994) < wθ R > nonlocal I 9 < wθ R >=[(I=ρC p ) 0 (I=ρC p ) hγ ] (8.87) C p h γ nonlocal References Deardroff, J. W., 197: Theoretical expression for the countergradient vertical heat flux, J. Geophys. Res., 77, Kantha, L. H., and C. A. Clayson, 000: Small Scale Processes in geophysical Fluid Flows, Academic Press, 888pp. Large, W. G., J. C. McWilliams, and S. C. Doney, 1994: Oceanic vertical mixing: a review and a model with a nonlocal boundary layer parameterization, Rev. Geophys., 3, Lumley, J. A., and H. A. Panofsky, 1964: The structure of the atmospheric turbulence, 39pp., John Wiley, New York. Mailhôt, J., and R. Benoit, 198: A finite-element model of the atmospheric boundary layer suitable for use with numerical weather prediction models, J. Atmos. Sci., 39, Mellor, G. L., and T. Yamada, 198: Development of a turbulence closure model for geophysical fluid problems, Rev. Geophys. Space Phys., 0,

15 8.3. K Profile Parameterization (KPP) Noh, Y., and H.-J. Kim, 1999: Simulations of temperature and turbulence structure of the oceanic boundary layer with the improved near-surface process, J. Geophys. Res., 104, 15,61-15,634. O Brien, J. J., 1970: A note on the vertical structure of the eddy exchange coefficient in the planetary boundary layer, J. Atmos. Sci., 7, Rotta, J. C., 1951a: Statistische Theotie nichthomogener Turbulenz, Z. Phys., 19, Rotta, J. C., 1951b: Statistische Theotie nichthomogener Turbulenz, Z. Phys., 131, Tennekes, H., 1973: A model for the dynamixcs of the inversion above a convective boundary layer, J. Atmos. Sci., 30, Treon, I. B., and L. Mahrt, 1986: A simple model of the atmospheric boundary layer; sensitivity to surface evaporation, Boundary Layer Meteorol., 37, Tsujino, H., H. Hasumi, and N. Suginohara, 000: Deep pacific circulation controlled by vertical diffusivity at the lower thermocline depth, J. Phys. Oceanogr., 30,

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