Sergej S. Zilitinkevich 1,2,3. Division of Atmospheric Sciences, University of Helsinki, Finland

Transcription

1 Atmospheric Planetary Boundary Layers (ABLs / PBLs) in stable, neural and unstable stratification: scaling, data, analytical models and surface-flux algorithms Sergej S. Zilitinkevich 1,,3 1 Division of Atmospheric Sciences,, Finland Meteorological Research, Finnish Meteorological Institute, Helsinki 3 Nansen Environmental and Remote Sensing Centre, Bergen, Norway Helsinki 7 May 1 June 007

2 Part 1 Revised theory and improved parameterization of the Stably Stratified Atmospheric Boundary Layer (SBL) in climate, NWP, AQ, and wind energy models

3 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. DOI : /s

4 Motivation

5 State of the art

6 Basic types of the SBL 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

7 1.1 Mean profiles and surface fluxes (Z and Esau, 007) Content Revision of the similarity theory for the stably stratified ABL Analytical approximations for the wind velocity and potential temperature profiles across the ABL Validation of new theory against LES and observational data Improved surface flux scheme for use in operational models

8 Turbulence in atmospheric models turbulence closure to calculate vertical fluxes: r τ and gradients: d U r / dz and d Θ / dz F θ through mean flux-profile relationships to calculate the surface fluxes: u = τ = τ z = 0, F Fθ through wind speed = U = and potential temperature = z =0 Θ 1 = z= z1 Θ at a given level z 1 U1 z z1 In NWP and climate models, the lowest computational level is z 1~30 m

9 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 )

10 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

11 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 ) is given parameter

12 Stable stratification: revised theory Zilitinkevich and Esau (005) two additional length scales besides L: 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 is the Brunt-Väisälä frequency at z>h (N ~10 - s -1 ), f is the Coriolis parameter Interpolation: 1/ 1 1 = C C N f + + L LN L f L where C N =0.1 and C f =1

13 kzτ -1/ du/dz vs. z/l (a), z/ L (b) x nocturnal; o long-lived; conventionally neutral

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

15 Vertical profiles of turbulent fluxes LES turbulent fluxes: solid lines τ / u = exp( 8 3ς ), F θ/ F θ s= exp( ς ) Approximation based on atmospheric data (e.g. Lenshow, 1988): dashed lines

16 New mean-gradient formulation (no critical Ri) βf Flux Richardson number is limited: Ri f = θ > Ri f 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 + CU1ξ ) To assure no Ri-critical, ξ -dependence of Φ should be stronger then linear. z z z Including CN and LS ABLs: Φ M = 1 + C U1, Φ H = 1+ CΘ 1 ξ + CΘ L L L H

17 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.

18 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.

19 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.

20 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

21 1/ Wind-velocity function Ψ U = kτ U ln( z / z0u ) vs. ξ = z / L, after 6 LES DATABASE64 (all types of SBL). The line: Ψ 5 / U = C U ξ, C U =3.0.

22 1/ 1 Pot.-temperature function Θ = kτ Θ Θ0 ( Fθ ) ln( z / z0u 5 (all types of SBL). The line: Ψ = C Θξ 4 / Θ with C U =3.0 and C Θ =.5. Ψ ( ) )

23 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.

24 Given τ, F θ, surface fluxes are calculated using empirical dependencies τ = exp τ 8 3 z h F, θ z = exp F h (Figures above) The equilibrium ABL height, h E, is determined diagnostically (Z. et al., 006a): 1 h = f E C R τ + N f C CN τ Algorithm fβf + C NS τ ( C R=0.6, C CN =1.36, C NS =0.51) The actual ABL height, after prognostic equation (Z. and Baklanov, 00): h r u + U h wh = K h h Ct ( h he ) ( C t = 1) t h 4 1 Θ Given h, the free-flow Brunt-Väisälä frequency is N = β h z E h h dz

25 Conclusions 1.1: mean profiles & surface fluxes Background: Generalised scaling accounting for the free-flow stability, No critical Ri (TTE closure) Stable ABL height model Verified against LES DATABASE64 (4 ABL types: TN, CN, NS and LS) Data from the field campaign SHEBA Deliverable 1: analytical wind & temperature profiles in SBLs Deliverable : surface flux scheme for use in operational models Requested: (i) roughness lengths and (ii) ABL height

26 1. STRATIFICATION EFFECT ON THE ROUGHNESS LENGTH 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

27 Reference (1.) 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. Submitted to Boundary-Layer Meteorology.

28 Content (1.) 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 u < z 0 in stable stratification; z 0 u > z 0 in unstable stratification 0u u z L L u 3 = βfθ s

29 Surface layer and roughness length 1 Self similarity in the surface layer (SL) 5h 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

30 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

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

32 Background physics and effect of stratification Physically z 0 u = depth of a sub-layer within RL ( 0 < z < 5h0 ) with 90% of the velocity drop from U R ~ u (approached at z ~ h0 ) From τ = K M ( RL ) U / z, τ ~ u and U / z ~ R / z u~ / z u z 0 u ~ KM ( RL) / u U 0 u 0 K (RL) M = K M ( h 0 + 0) from matching the RL and the surface-layer Neutral: Stable: K Unstable: K M ~ h0 u classical formula z0 u ~ h0 1 = ku z(1 + Cu z / 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 M M + U b F b 1/ )

33 Recommended formulation Neutral stable z z 0u 0 1 = 1+ C h SS 0 / L Neutral unstable z z 1/3 0u h0 = 1+ CUS 0 L Constants: C =8. 13±0.1, C = 1.4±0.05 SS US

34 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

35 Stable stratification z z0u Bin-average values of / (neutral- over actual-roughness lengths) versus h 0 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.

36 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 0 / L).

37 Stable stratification Bin-average values of the ratio 0 / versus parameter h 0 / L h c /L. d d0, neutral

38 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

39 Unstable stratification Bin-average values of z0 u / z0 vs. Ri=(g/Θ 31 )(Θ 31 Θ 18 )h 0 / U 31, for the city of Basel (h 0 ~14.6 m; z0 1.±0.4) in unstable stratification. Bars are standard errors; the curve is z0 u / z0 =1+1.3 Ri 3/14.

40 Unstable stratification Actual over neutral displacement height, d d0, neutral 0 /, versus Ri=[(g/T ref )(θ 31m - θ 18m )h c /U 31m ].

41 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 Dashed line: traditional formulation z 0u = z 0

42 Conclusions: 1. Roughness length Traditional concept: roughness length fully characterised by geometric features of the surface New theory and data: essential dependence on hydrostatic stability especially strong in stable stratification Applications: to urban and terrestrial-ecosystem meteorology Practically sound: urban air pollution episodes in very stable stratification

43 1.3 NEUTRAL and STABLE ABL HEIGHT Sergej Zilitinkevich 1,,3, Igor Esau 3 and Alexander Baklanov 4 1 Division of 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

44 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., 00: Further comments on the equilibrium height of neutral and stable planetary boundary layers. Quart. J. Roy. Met. Soc., 133,

45 Factors controlling PBL height

46 Scaling analysis

47 Dominant role of the smallest scale

48 How to verify h-equations?

49 Stage I: Truly neutral ABL

50 Stage I: Transition TN CN ABL

51 Stage I: Transition TN NS ABL

52 Stage II: General case

53 The height of the conventionally neutral (CN) ABL Z & Esau, 00, 007: the effect of free-flow stability (N) on CN ABL height, h E,, (LES red, field data blue, theory curve). Classical theory overlooks it and overestimates h E up to an order of magnitude.

54 Conclusions: 1.3 SBL height h E, depends on many factors multi-limit analysis / complex formulation difficult to measure: baroclinic shear (Γ), vertical velocity ( w h), h E itself hence necessity to use LES, DNS and lab experiments baroclinic ABL: substitute u T = u * (1+C 0 Γ/N) 1/ for u * in the nd term of 1 h = f E C R τ + N f C CN τ account for vertical motions: fβf + C NS τ h E corr = h E + ( C R=0.6, w h T C CN =1.36, C NS =0.51) t, where t T = Ct h E / u * generally prognostic (relaxation) equation (Z. and Baklanov, 00): h r u + U h wh = K h h Ct ( h he ) ( C t = 1) t h E

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