Stable Boundary Layer Parameterization

Transcription

1 Stable Boundary Layer Parameterization Sergej S. Zilitinkevich Meteorological Research, Finnish Meteorological Institute, Helsinki, Finland Atmospheric Sciences and Geophysics,, Finland Nansen Environmental and Remote Sensing Centre, Bergen, Norway Seminar: New developments in Modelling ABLs for NWP 17 June 008

2 Key references Zilitinkevich, S., and Calanca, P., 000: An extended similarity-theory for the stably stratified atmospheric surface layer. Quart. J. Roy. Meteorol. Soc., 16, Zilitinkevich, S., 00: Third-order transport due to internal waves and non-local turbulence in the stably stratified surface layer. Quart, J. Roy. Met. Soc. 18, Zilitinkevich, S.S., Perov, V.L., and King, J.C., 00: Near-surface turbulent fluxes in stable stratification: calculation techniques for use in general circulation models. Quart, J. Roy. Met. Soc. 18, Zilitinkevich S. S., and Esau I. N., 005: Resistance and heat/mass transfer laws for neutral and stable planetary boundary layers: old theory advanced and re-evaluated. Quart. J. Roy. Met. Soc. 131, Zilitinkevich, S., Esau, I. and Baklanov, A., 007: Further comments on the equilibrium height of neutral and stable planetary boundary layers. Quart. J. Roy. Met. Soc. 133, Zilitinkevich, S. S., and Esau, I. N., 007: Similarity theory and calculation of turbulent fluxes at the surface for stably stratified atmospheric boundary layers. Boundary-Layer Meteorol. 15, Zilitinkevich, S.S., Elperin, T., Kleeorin, N., and Rogachevskii, I., 007: Energy- and flux-budget (EFB) turbulence closure model for the stably stratified flows. Part I: Steady-state, homogeneous regimes. Boundary-Layer Meteorol. 15, Zilitinkevich, S. S., Mammarella, I., Baklanov, A. A., and Joffre, S. M., 008: The effect of stratification on the roughness length and displacement height. Boundary-Layer Meteorol. In press

3 State of the art

4 Basic types of the stable and neutral ABLs Until recently ABLs were distinguished accounting only for F bs = F : neutral at F =0 stable at F <0 Now more detailed classification: truly neutral (TN) ABL: F =0, N=0 conventionally neutral (CN) ABL: F =0, N>0 nocturnal stable (NS) ABL: F <0, N=0 long-lived stable (LS) ABL: F <0, N>0 Realistic surface flux calculation scheme should be based on a model applicable to all these types of the ABL

5 Content Revision of the similarity theory for stable and neutral ABLs Analytical approximation of mean profiles across the ABL Validation against LES and observational data (to be proceeded) Diagnostic & prognostic ABL height equations (to be included in operational routines) Surface-flux & ABL-height schemes for use in operational models

6 Classical similarity theory and current surface-flux schemes (based on the SL-concept)

7 Neutral stratification (no problem) From logarithmic wall law: du dz 1/ τ =, kz dθ dz = k T τ F θ 1/, z U 1/ τ z = ln, k z 0u Θ Fθ Θ0 = ln 1/ k τ T z z 0u k, k T von Karman constants; z 0 u aerodynamic roughness length for momentum; Θ 0 aerodynamic surface potential temperature (at z 0 u ) [ Θ0-Θ s through z 0 T ] It follows: 1/ 1 τ1 = ku 1 (ln z / z 0u ), F θ1 = kk TU ( Θ1 Θ 0 )(ln z / z 0u τ1 = τ, Fθ 1= F when z1 30 m << h OK in neutral stratification 1 )

8 Stable stratification: current theory (i) local scaling, (ii) log-linear Θ-profile both questionable When z 1 is much above the surface layer τ 1 F τ, θ1 F 3/ τ Monin-Obukhov (MO) theory L = (neglects other scales) βfθ 1/ kz du ktτ z dθ z = Φ ( ξ ) 1/ M, = Φ H ( ξ ), where ξ = τ dz F dz L θ Φ M = + C ξ, Φ H = + C ξ from z-less stratification concept Ri 1 U1 u 1 Θ1 z z ln + CU, zu0 L U = 1 k s Θ Θ β ( dθ / dz)( du / dz) Ric = F 0 = ln + C Θ 1 kt u zu0 Ls k C k T C (unacceptable) 1 Θ1 U1 CU1 ~, C Θ1 also ~ (factually increases with z\l) z z

9 Stable stratification: current parameterization To avoid critical Ri modellers use empirical, heuristic correction functions to the neutral drag and heat/mass transfer coefficients Drag and heat transfer coefficients: C D = τ /(U 1 ), C H = F θs /(U 1 Θ) Neutral: C Dn, C Hn from the logarithmic wall law To account for stratification, correction functions (dependent only of Ri): f D (Ri 1 ) = C D / C Dn and f H (Ri 1 ) = C H / C Hn Ri 1 = β( Θ)z 1 /(U 1 ) (surface-layer Richardson number ) - given parameter

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12 Revised similarity theory and ABL flux-profile relationships

13 Revised similarity theory Zilitinkevich, Esau (005) besides Obukhov s L = τ 3/ (βf θ ) -1 two additional length scales: 1/ τ L = non-local effect of the free flow static stability N N τ L f = f 1/ the effect of the Earth s rotation N ~10 - s -1 Brunt-Väisälä frequency at z>h, f Coriolis parameter τ= τ(z) Interpolation: 1/ 1 1 = C C N f + + L LN Lf L where C N= 0.1 and C f =1

14 Velocity gradient (conventionally neutral ABLs!) Φ M = kzτ --1/ du/dz vs. z/l (a), z/ L (b) x nocturnal; o long-lived; conv. neutral

15 Φ H = (k T τ 1/ z/f θ )dθ/dz vs. z/l (a), z/ L (b) x nocturnal; o long-lived

16 Vertical profiles of turbulent fluxes Turbulent fluxes: data points LES; dashed lines atmospheric data, Lenshow, 1988); solid lines τ = exp( 8 ς ) / u 3, θ F / F θ = exp( ς ) s ; ζ = z/h. ABL height h =?

17 New mean-gradient formulation (no critical Ri) βf Flux Richardson number is limited: Ri f = θ > Rif 0. τdu / dz 1/ du τ Hence asymptotically, and interpolating Φ M = 1+ CU1 ξ dz Ri f L dθ/ dz Gradient Richardson number becomes Ri β k ξφ H ( ξ) = ( du / dz) k T (1 + CU 1ξ ) To assure no Ri-critical, ξ -dependence of Φ should be stronger then linear. H Including CN and LS ABLs: z z z Φ M = 1 + C U1, Φ H= 1+ C Θ1 + CΘ L L L

18 Φ M vs. ξ = z / L, after LES DATABASE64 (all types of SBL). Dark grey points for z<h; light grey points for z>h; the line corresponds to C = U1.

19 Φ H vs. ξ = z / L (all SBLs). Bold curve is our approximation: C Θ1 = 1. 8, C Θ = 0.; thin lines are Φ H = 0.ξ and traditional Φ H = 1+ξ.

20 Ri vs. ξ = z / L, after LES and field data (SHEBA - green points). Bold curve is our model with C U1=, C Θ1=1.6, C Θ=0.. Thin curve is Φ H =1+ξ.

21 Mean profiles and flux-profile relationships We consider wind/velocity and potential/temperature functions Ψ U = ku ( z) ln 1/ τ z z 0u and Ψ Θ = k T τ 1/ [ Θ( z) Θ ] F θ 0 ln z z 0u Our analyses show that Ψ U and Ψ Θ are universal functions of ξ = z / L 5 / 6 Ψ =, U C U ξ 4 / 5 ΨΘ = C, with U Θξ C =3.0 and C Θ =.5 The problem is solved given (i) z 0u and (ii) τ and L as functions of z/h

22 1/ Wind-velocity function Ψ U = kτ U ln( z / z0u ) vs. ξ = z / L, after 5 / 6 LES DATABASE64 (all types of SBL). The line: Ψ U = C U ξ, C U =3.0.

23 Ψ ( ) ln( z / z ) 1/ 1 Pot.-temperature function Θ = kτ Θ Θ0 Fθ ) 5 (all types of SBL). The line: Ψ Θξ 4 / Θ = C with C U =3.0 and ( 0u C =.5. Θ

24 Analytical wind and temperature profiles (SBL) 5 /1 6 5 / 0 1/ ) ( ) ( 1 ln = L f C N C L z C z z ku f N U u τ τ 5 / 5 4 / 0 0 / 1 ) ( ) ( 1 ln ) ( = Θ Θ Θ L f C N C L z C z z F k f N u T τ τ θ where N C =0.1 and f C =1. Given U(z), Θ(z) and N, these equations allow determining τ, θ F, and 1 3/ ) ( = θ β τ F L, at the computational level z.

25 Remain to be determined: ABL height and Roughmess length

26 ABL height

27 References Zilitinkevich, S., Baklanov, A., Rost, J., Smedman, A.-S., Lykosov, V., and Calanca, P., 00: Diagnostic and prognostic equations for the depth of the stably stratified Ekman boundary layer. Quart, J. Roy. Met. Soc., 18, Zilitinkevich, S.S., and Baklanov, A., 00: Calculation of the height of stable boundary layers in practical applications. Boundary-Layer Meteorol. 105, Zilitinkevich S. S., and Esau, I. N., 00: On integral measures of the neutral, barotropic planetary boundary layers. Boundary-Layer Meteorol. 104, Zilitinkevich S. S. and Esau I. N., 003: The effect of baroclinicity on the depth of neutral and stable planetary boundary layers. Quart, J. Roy. Met. Soc. 19, Zilitinkevich, S., Esau, I. and Baklanov, A., 007: Further comments on the equilibrium height of neutral and stable planetary boundary layers. Quart. J. Roy. Met. Soc., 133,

28 Factors controlling ABL height

29 Neutral and stable ABL height The dependence of the equilibrium conventionally neutral PPL height, h E, on the freeflow Brunt-Väisälä frequency N after new theory (Z et al., 007a, shown by the curve), LES (red) and field data (blue). Until recently the effect of N on h E was disregarded

30 General case

31 Equilibrium 1 = h E C R Modelling the stable ABL height f τ + N f C CN τ fβf + C NS τ ( C R=0.6, C CN =1.36, C NS =0.51) Baroclinic: substitute u T = u * (1+C 0 Γ/N) 1/ for u * in the nd term on the r.h.s. Vertical motions: h E corr = E h + w h T t, where t T = Ct h E / u * Generally prognostic equation (Z. and Baklanov, 00): h r u + U h wh = K h h Ct ( h he ) t h E ( C t = 1) h 4 1 Θ Given h, the free-flow Brunt-Väisälä frequency is N = β h z h dz

32 Conclusions (surface fluxes and ABL height) Background: Generalised scaling accounting for the free-flow stability, No critical Ri (Φ H consistent with TTE closure) Stable ABL height model Verified against LES DATABASE64 (4 ABL types: TN, CN, NS and LS) Data from the field campaign SHEBA More detailed validation needed Deliverable 1: analytical wind & temperature profiles in SBLs Deliverable : surface flux and ABL height schemes for use in operational models

33 Roughmess length

34 Stability dependences of the roughness length and displacement height S. S. Zilitinkevich 1,,3, I. Mammarella 1,, A. Baklanov 4, and S. M. Joffre 1. Atmospheric Sciences,, Finland. Finnish Meteorological Institute, Helsinki, Finland 3. Nansen Environmental and Remote Sensing Centre / Bjerknes Centre for Climate Research, Bergen, Norway 4. Danish Meteorological Institute, Copenhagen, Denmark New developments in modelling ABLs for NWP 17 June 008

35 Reference S. S. Zilitinkevich, I. Mammarella, A. A. Baklanov, and S. M. Joffre, 007: The roughness length in environmental fluid mechanics: the classical concept and the effect of stratification. Boundary-Layer Meteorology. In press.

36 Content Roughness length and displacement height: u* z d0u u( z) = ln + Ψ k z No stability dependence of z 0 u (and d 0 u ) in engineering fluid mechanics: neutral-stability z 0 = level, at which u (z) plotted vs. ln z approaches zero; z 0 ~ 1 of typical height of roughness elements, h 5 0 Meteorology / oceanography: h 0 comparable with MO length Stability dependence of the actual roughness length, z 0 u : z 0 < z 0 in stable stratification; z 0 > z 0 in unstable stratification u 0u u u z L L u 3 = βfθ s

37 Surface layer and roughness length 1 Self similarity in the surface layer (SL) 1.6h 0 <z< 10 h Height-constant fluxes: τ τ z= 5h u 0 u and z serve as turbulent scales: u T ~ u, l T ~ z Eddy viscosity ( k 0. 4) K M (~ u l )= ku z T T Velocity gradient U / z = τ / K M = u / kz 1 1 Integration constant: U = k u ln z + constant = k u ln( z / z0u ) z 0 u (redefined constant of integration) is roughness length 1 Displacement height u U = k u ln ( z d u 0 ) / zu Not applied to the roughness layer (RL) 0<z<5h 0 d 0 [ ] 0

38 Parameters controlling z 0u Smooth surfaces: viscous layer z 0 u ~ ν /u Very rough surfaces: pressure forces depend on: obstacle height h 0 velocity in the roughness layer U R~ u z 0 u = z 0 u ( h 0, u )~ h 0 (in sand roughness experiments No dependence on u ; surfaces characterised by Generally z 0 u= h 0 f 0 (Re 0 ) where Re0 = u h 0 / ν Stratification at M-O length ) z 1 0 u 30 h 0 z 0 u= constant 3 1 L = u Fb comparable with h 0

39 Stability Dependence of Roughness Length For urban and vegetation canopies with roughness-element heights (0-50 m) comparable with the Obukhov turbulent length scale, L, the surface resistance and roughness length depend on stratification

40 Background physics and effect of stratification Physically z 0 u = depth of a sub-layer within RL ( 0 < z < 1.6h0 ) with 90% of the velocity drop from U R ~ u (approached at z ~ h0 ) From τ = K U / z, τ u and U / z ~ U R / z0u~ u / z0u M ( RL ) z ~ 0 u ~ KM ( RL) / u K M (RL) = K M ( h 0 + 0) from matching the RL and the surface-layer Neutral: Stable: K Unstable: ~ u h classical formula z0 u ~ h0 1 = ku z(1 + Cuz / L) ~ u L z0 u ~ L 1 1/ 3 4 / 3 1/ 3 4 / 3 K = ku z C F z ~ z z u ~ h ( h / L K M 0 M M + U b F b 1/ )

41 Recommended formulation Neutral stable z 0u = 1 z 0 1+ C SS h 0 / L Neutral unstable 1/3 z h 0u = 1+ C 0 US z L 0 Constants: C =8. 13±0.1, C = 1.4±0.05 SS US

42 Experimental datasets Sodankyla Meteorological Observatory, Boreal forest (FMI) BUBBLE urban BL experiment, Basel, Sperrstrasse (Rotach et al., 004) h 13 m, measurement levels 3, 5, 47 m h 14.6 m, measurement levels 3.6, 11.3, 14.7, 17.9,.4, 31.7 m

43 Stable stratification Bin-average values of z / z (neutral- over actual-roughness lengths) versus h 0 0u 0/L in stable stratification for Boreal forest (h0=13.5 m; z 0 =1.1±0.3 m). Bars are standard errors; the curve is z / z = h / L 0 0u + 0.

44 Stable stratification Bin-average values of z / z 0 u 0 (actual- over neutral-roughness lengths) versus h 0 /L in stable stratification for boreal forest (h0=13.5 m; z 0 =1.1±0.3 m). Bars are standard errors; the curve is z / z 1 0 u 0 = ( h / L. 0 )

45 Stable stratification Displacement height over its neutral-stability value in stable stratification. Boreal forest (h 0 = 15 m, d 0 = 9.8 m). ( ) The curve is 1 d0 u / d ( h0 / L) h0 / = L

46 Unstable stratification Convective eddies extend in the vertical causing z > 0 z0u VOLUME 81, NUMBER 5 PHYSICAL REVIEW LETTERS 3 AUGUST 1998 Y.-B. Du and P. Tong, Enhanced Heat Transport in Turbulent Convection over a Rough Surface

47 Unstable stratification Unstable stratification, Basel, z 0 / z 0u vs. Ri = (gh 0 /Θ 3 )(Θ 18 Θ 3 )/(U 3 ) Building height =14.6 m, neutral roughness z 0 =1. m; BUBBLE, Rotach et al., 005). h 0 /L through empirical dependence on Ri on (next figure) The curve (z 0 / z 0u =1+5.31Ri 6/13 ) confirms theoretical z 0u /z 0 = (h 0 /-L) 1/3

48 Unstable stratification Empirical Ri = (h 0 / L) 13/18

49 Unstable stratification Displacement height in unstable stratification (Basel): The line confirms theoretical dependence: d 0u d0 / d ou 1 versus d0 = 1/3 1+ C ( h / L) DC 0 Ri

50 STABILITY DEPENDENCE OF THE ROUGHNESS LENGTH in the meteorological interval -10 < h 0 /L <10 after new theory and experimental data Solid line: z 0u /z 0 versus h 0 /L Thin line: traditional formulation z 0u = z 0

51 STABILITY DEPENDENCE OF THE DISPLACEMENT HEIGHT in the meteorological interval -10 < h 0 /L <10 after new theory and experimental data Solid line: d 0u /d 0 versus h 0 /L Dashed line: the upper limit: d 0 = h 0

52 Conclusions (roughness & displacement) Traditional: roughness length and displacement height fully characterised by geometric features of the surface New: essential dependence on hydrostatic stability especially in stable stratification Logarithmic intervals in the velocity profiles diminish over very rough surfaces in both very stable and very unstable stratification Applications: to urban and terrestrial-ecosystem meteorology Especially: urban air pollution episodes in very stable stratification

53 Thank you