Sergej S. Zilitinkevich - FMI

University of Helsinki

Stable Boundary Layer Parameterization

Sergej S. ZilitinkevichMeteorological Research, Finnish Meteorological Institute, Helsinki, Finland

Atmospheric Sciences and Geophysics, University of Helsinki, Finland

Nansen Environmental and Remote Sensing Centre, Bergen, Norway

Seminar: New developments in Modelling ABLs for NWP 17 June 2008


Key referencesZilitinkevich, S., and Calanca, P., 2000: An extended similarity-theory for the stably stratified

atmospheric surface layer. Quart. J. Roy. Meteorol. Soc., 126, 1913-1923.Zilitinkevich, S., 2002: Third-order transport due to internal waves and non-local turbulence in the

stably stratified surface layer. Quart, J. Roy. Met. Soc. 128, 913-925. Zilitinkevich, S.S., Perov, V.L., and King, J.C., 2002: Near-surface turbulent fluxes in stable

stratification: calculation techniques for use in general circulation models. Quart, J. Roy. Met. Soc. 128, 1571-1587.

Zilitinkevich S. S., and Esau I. N., 2005: 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, 1863-1892.

Zilitinkevich, S., Esau, I. and Baklanov, A., 2007: Further comments on the equilibrium height of neutral and stable planetary boundary layers. Quart. J. Roy. Met. Soc. 133, 265-271.

Zilitinkevich, S. S., and Esau, I. N., 2007: Similarity theory and calculation of turbulent fluxes at the surface for stably stratified atmospheric boundary layers. Boundary-Layer Meteorol. 125, 193-296.

Zilitinkevich, S.S., Elperin, T., Kleeorin, N., and Rogachevskii, I., 2007: Energy- and flux-budget (EFB) turbulence closure model for the stably stratified flows. Part I: Steady-state, homogeneous regimes. Boundary-Layer Meteorol. 125, 167-192.

Zilitinkevich, S. S., Mammarella, I., Baklanov, A. A., and Joffre, S. M., 2008: The effect of stratification on the roughness length and displacement height. Boundary-Layer Meteorol. In press


State of the art


Basic types of the stable and neutral ABLs

• Until recently ABLs were distinguished accounting only for Fbs= ∗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


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


Classical similarity theory and

current surface-flux schemes (based on the SL-concept)


Neutral stratification (no problem)

From logarithmic wall law:

kzdzdU 2/1τ

= , zk

Fdzd

T2/1τθ−

=Θ ,

uzz

kU

0

2/1

lnτ= ,

uT zz

kF

02/10 ln

τθ−

=Θ−Θ

k, Tk von Karman constants; uz0 aerodynamic roughness length for momentum;

0Θ aerodynamic surface potential temperature (at uz0 ) [ 0Θ - sΘ through Tz0 ] It follows: 2/1

1τ1

01 )/(ln −= uzzkU , =1θF 20011 )/)(ln( −Θ−Θ− uT zzUkk

1τ ∗= τ , 1θF ∗= F when 1z 30≈ m << h OK in neutral stratification


Stable stratification: current theory(i) local scaling, (ii) log-linear Θ-profile both questionable• When 1z is much above the surface layer 1τ ∗≠ τ , 1θF ∗≠ F

• Monin-Obukhov (MO) theory =Lθβ

τF−

2/3

(neglects other scales)

)(2/1 ξτ Mdz

dUkzΦ= , )(

2/1

ξτ

θH

T

dzd

Fzk

Φ=Θ , where

Lz

=ξ

• ξ11 UM C+=Φ , ξ11 Θ+=Φ CH from z-less stratification concept

⎟⎟⎠

⎞⎜⎜⎝

⎛+= ∗

sU

u LzC

zz

ku

U 10

ln ,

∗

∗−=Θ−Θ

ukF

T0 ⎟⎟

⎠

⎞⎜⎜⎝

⎛+ Θ

su LzC

zz

10

ln

• Ri 2)/)(/( −Θ≡ dzdUdzdβ Ric = 21

11

2 −−Θ UT CkCk (unacceptable)

• ~1UC 2, 1ΘC also 2~ (factually increases with z\L)


Stable stratification: current parameterizationTo avoid critical Ri modellers use empirical, heuristic correction functions to the neutral drag and heat/mass transfer coefficients

● Drag and heat transfer coefficients: CD = τ /(U1)2 , CH = –Fθs/(U1∆Θ) ● Neutral: CDn, CHn – from the logarithmic wall law

● To account for stratification, correction functions (dependent only of Ri):

fD (Ri1) = CD / CDn and fH (Ri1) = CH / CHn Ri1= β(∆Θ)z1/(U1)2 (surface-layer “Richardson number”) - given parameter


Revised similarity theory and

ABL flux-profile relationships


Revised similarity theory Zilitinkevich, Esau (2005) besides Obukhov’s L = – τ3/2(βFθ)-1

two additional length scales:

NL =N

2/1τ non-local effect of the free flow static stability

fL =||

2/1

fτ the effect of the Earth’s rotation

N ~10-2 s-1 – Brunt-Väisälä frequency at z>h, f – Coriolis parameter τ= τ(z)

Interpolation: ∗L

1 = 2/12221

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛

f

f

N

N

LC

LC

L where NC = 0.1 and fC =1


ΦM = kzτ--1/2dU/dz vs. z/L (a), z/ ∗L (b) x nocturnal; o long-lived; □ conv. neutral

Velocity gradient (conventionally neutral ABLs!)


HΦ = (kTτ1/2z/Fθ)dΘ/dz vs. z/L (a), z/ ∗L (b) x nocturnal; o long-lived


Turbulent fluxes: data points – LES; dashed lines – atmospheric data, Lenshow, 1988); solid lines 2/ ∗uτ = )exp( 2

38 ς− , θF / sFθ = )2exp( 2ς− ; ζ = z/h. ABL height h = ?

Vertical profiles of turbulent fluxes


New mean-gradient formulation (no critical Ri)

Flux Richardson number is limited: =fRidzdU

F/τ

β θ− > ∞fRi 2.0≈

Hence asymptotically Ldz

dU

f∞→

Ri

2/1τ , and interpolating ξ11 UM C+=Φ

Gradient Richardson number becomes Ri 2)/(/dzdUdzdΘ

≡β

21

2

)1()(ξξξ

U

H

T Ckk

+Φ

=

To assure no Ri-critical, ξ -dependence of HΦ should be stronger then linear.

Including CN and LS ABLs: MΦ =∗

+LzCU11 , HΦ =

2

211 ⎟⎠

⎞⎜⎝

⎛++

∗Θ

∗Θ L

zCLzC


MΦ vs. ∗= Lz /ξ , after LES DATABASE64 (all types of SBL). Dark grey points for z<h; light grey points for z>h; the line corresponds to .21 =UC


HΦ vs. ∗= Lz /ξ (all SBLs). Bold curve is our approximation: 8.11 =ΘC , 2.02 =ΘC ; thin lines are =ΦH 0.2 2ξ and traditional =ΦH 1+2ξ .


Ri vs. Lz /=ξ , after LES and field data (SHEBA - green points). Bold curve is our model with 1UC =2, 1ΘC =1.6, 2ΘC =0.2. Thin curve is HΦ =1+2ξ .


Mean profiles and flux-profile relationships

We consider wind/velocity and potential/temperature functions

uU z

zzkU

02/1 ln)(

−=Ψτ

and [ ]u

T

zz

Fzk

0

02/1

ln)(−

−Θ−Θ

=ΨΘθ

τ

Our analyses show that UΨ and ΘΨ are universal functions of ∗= Lz /ξ

6/5ξUU C=Ψ , 5/4ξΘΘ =Ψ C , with UC =3.0 and ΘC =2.5

The problem is solved given (i) z0u and (ii) τ and L as functions of z/h


Wind-velocity function UΨ )/ln( 02/1

uzzUk −= −τ vs. ∗= Lz /ξ , after LES DATABASE64 (all types of SBL). The line: 6/5ξUU C=Ψ , UC =3.0.


Pot.-temperature function ΘΨ ( ) )/ln()( 01

02/1

uzzFk −−Θ−Θ= −−θτ

(all types of SBL). The line: 5/4ξΘΘ =Ψ C with UC =3.0 and ΘC =2.5.


Analytical wind and temperature profiles (SBL)

12/5

2226/5

02/1

)()(1ln ⎥

⎦

⎤⎢⎣

⎡ ++⎟

⎠⎞

⎜⎝⎛+= L

fCNCLzC

zzkU fN

Uu ττ

5/2

2225/4

0

02/1 )()(

1ln)(

⎥⎦

⎤⎢⎣

⎡ ++⎟

⎠⎞

⎜⎝⎛+=

−Θ−Θ

Θ LfCNC

LzC

zz

Fk fN

u

T

ττ

θ

where NC =0.1 and fC =1. Given U(z), Θ(z) and N, these equations allowdetermining τ , θF , and 12/3 )( −−= θβτ FL , at the computational level z.


Remain to be determined:

• ABL height and

• Roughmess length


ABL height


ReferencesZilitinkevich, S., Baklanov, A., Rost, J., Smedman, A.-S., Lykosov, V.,

and Calanca, P., 2002: Diagnostic and prognostic equations for the depth of the stably stratified Ekman boundary layer. Quart, J. Roy. Met. Soc., 128, 25-46.

Zilitinkevich, S.S., and Baklanov, A., 2002: Calculation of the height of stable boundary layers in practical applications. Boundary-Layer Meteorol. 105, 389-409.

Zilitinkevich S. S., and Esau, I. N., 2002: On integral measures of the neutral, barotropic planetary boundary layers. Boundary-Layer Meteorol. 104, 371-379.

Zilitinkevich S. S. and Esau I. N., 2003: The effect of baroclinicity on the depth of neutral and stable planetary boundary layers. Quart, J. Roy. Met. Soc. 129, 3339-3356.

Zilitinkevich, S., Esau, I. and Baklanov, A., 2007: Further comments on the equilibrium height of neutral and stable planetary boundary layers. Quart. J. Roy. Met. Soc., 133, 265-271.


Factors controlling ABL height


Neutral and stable ABL height

The dependence of the equilibrium conventionally neutral PPL height, hE, on the free-flow Brunt-Väisälä frequency N after new theory (Z et al., 2007a, shown by the curve), LES (red) and field data (blue). Until recently the effect of N on hE was disregarded


General case


Modelling the stable ABL height

● Equilibrium

∗

=τ22

21

RE Cf

h +

∗τ2

||

CNCfN + 22

||

∗

∗

τβ

NSCFf ( RC =0.6, CNC =1.36, NSC =0.51)

● Baroclinic: substitute Tu = *u (1+C0Γ/N)1/2 for *u in the 2nd term on the r.h.s. ● Vertical motions: corr−Eh = Eh + hw Tt , where Tt = tC Eh / *u ● Generally prognostic equation (Z. and Baklanov, 2002):

)(2E

Ethh hh

hu

ChKwhUth

−−∇=−∇⋅+∂∂ ∗

r ( tC = 1)

Given h, the free-flow Brunt-Väisälä frequency is ∫ ⎟⎠⎞

⎜⎝⎛

∂Θ∂

=h

h

dzzh

N2 2

4 1 β


Conclusions (surface fluxes and ABL height) Background: Generalised scaling accounting for the free-flow stability,

No critical Ri (ΦH 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 2: surface flux and ABL height schemes

for use in operational models


Roughmess length


Stability dependences of the roughness length and displacement height

S. S. Zilitinkevich1,2,3, I. Mammarella1,2,A. Baklanov4, and S. M. Joffre2

1. Atmospheric Sciences, University of Helsinki, Finland2. Finnish Meteorological Institute, Helsinki, Finland3. Nansen Environmental and Remote Sensing Centre /

Bjerknes Centre for Climate Research, Bergen, Norway4. Danish Meteorological Institute, Copenhagen, Denmark

New developments in modelling ABLs for NWP 17 June 2008


Reference

S. S. Zilitinkevich, I. Mammarella, A. A. Baklanov, and S. M. Joffre, 2007: The roughness length in environmental fluid mechanics: the classical concept and the effect of stratification. Boundary-Layer Meteorology. In press.


Content

• Roughness length and displacement height:

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛Ψ+

−=

Lz

zdz

kuzu u

u

u

0

0* ln)(

• No stability dependence of uz0 (and ud0 ) in engineering fluid mechanics: neutral-stability 0z = level, at which )(zu plotted vs. zln approaches zero;

0z 251~ of typical height of roughness elements, 0h

• Meteorology / oceanography: 0h comparable with MO length sF

uLθβ−

= ∗3

• Stability dependence of the actual roughness length, uz0 : uz0 < 0z in stable stratification; uz0 > 0z in unstable stratification


Surface layer and roughness length

Self similarity in the surface layer (SL) 1.6 0h <z< 110− h Height-constant fluxes: τ

05| hz=≈ τ 2∗≡ u

∗u and z serve as turbulent scales: ∗uuT ~ , zlT ~ Eddy viscosity ( 4.0≈k ) MK (~ TT lu )= zku∗ Velocity gradient kzuKzU M /// ∗==∂∂ τ Integration constant: constantln1 += ∗

− zukU )/ln( 01

uzzuk ∗−=

uz0 (redefined constant of integration) is “roughness length” “Displacement height” ud0 [ ]00

1 /)(ln uu zdzukU −= ∗−

Not applied to the roughness layer (RL) 0<z<5h0


Parameters controlling z 0u

Smooth surfaces: viscous layer uz0 ~ ∗u/ν

Very rough surfaces: pressure forces depend on: obstacle height 0h velocity in the roughness layer RU ~ ∗u

uz0 = uz0 ( 0h , ∗u )~ 0h (in sand roughness experiments uz0 0301 h≈ )

No dependence on ∗u ; surfaces characterised by uz0 = constant

Generally uz0 = 0h )(Re00f where Re0 = ν/0hu∗

Stratification at M-O length 13 −∗−= bFuL comparable with 0h


Stability Dependence of Roughness Length

For urban and vegetation canopies with roughness-element heights (20-50 m) comparable with the Obukhov turbulent length scale, L, the surface resistance and roughness length depend on stratification


Background physics and effect of stratification

Physically =uz0 depth of a sub-layer within RL ( 06.10 hz << ) with 90% of the velocity drop from ~RU ∗u (approached at 0~ hz )

From zUK RLM ∂∂= /)(τ , 2~ ∗uτ and zU ∂∂ / ~ uR zU 0/ ~ uzu 0/∗

∗uKz RLMu /~ )(0

)RL(MK = )0( 0 +hKM from matching the RL and the surface-layer

Neutral: MK 0~ hu∗ ⇒ classical formula 00 ~ hz u Stable:

1)/1( −∗ += LzCzkuK uM Lu∗~ ⇒ Lz u ~0

Unstable: 3/43/11 zFCzkuK bUM−

∗ += 3/43/1~ zFb ⇒ 3/1000 )/(~ Lhhz u −


Recommended formulation

Neutral ⇔ stable LhCz

z

SS

u

/11

00

0

+=

Neutral ⇔ unstable 3/1

0

0

0 1 ⎟⎠⎞

⎜⎝⎛−

+=L

hCzz

USu

Constants: 13.8=SSC ±0.21, =USC 1.24±0.05


Experimental datasets Experimental datasets

Sodankyla Meteorological Observatory, Boreal forest (FMI)

BUBBLE urban BL experiment, Basel, Sperrstrasse (Rotach et al., 2004)

h ≈ 13 m, measurement levels 23, 25, 47 m h ≈ 14.6 m, measurement levels 3.6, 11.3, 14.7, 17.9, 22.4, 31.7 m


Stable stratification

Bin-average values of uzz 00 / (neutral- over actual-roughness lengths) versus h0/L in stable stratification for Boreal forest (h0=13.5 m; 0z =1.1±0.3 m). Bars are standard errors; the curve is uzz 00 / = Lh /13.81 0+ .



Bin-average values of 00 / zz u (actual- over neutral-roughness lengths) versus h0/L in stable stratification for boreal forest (h0=13.5 m; 0z =1.1±0.3 m). Bars are standard errors; the curve is 00 / zz u = 1

0 )/13.81( −+ Lh .



Displacement height over its neutral-stability value in stable stratification. Boreal forest (h0 = 15 m, d0= 9.8 m).

The curve is ( ) 10000 /05.1)/(5.01/ −++= LhLhdd u


Unstable stratificationConvective eddies extend in the vertical causing uzz 00 >

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


Unstable stratification

Unstable stratification, Basel, z0/ z0u vs. Ri = (gh0/Θ32)(Θ18–Θ32)/(U32)2

Building height =14.6 m, neutral roughness z0 =1.2 m; BUBBLE, Rotach et al., 2005). h0/L through empirical dependence on Ri on (next figure)The curve (z0/ z0u =1+5.31Ri6/13) confirms theoretical z0u/z0 = 1 + 1.15(h0/-L)1/3



Empirical Ri = 0.0365 (h0/–L)13/18



Displacement height in unstable stratification (Basel):

The line confirms theoretical dependence:

Ri versus1/0 −oudd

3/10

00 )/(1 LhC

ddDC

u −+=


STABILITY DEPENDENCE OF THE ROUGHNESS LENGTHin the “meteorological interval” -10 < h0/L <10 after new theory and experimental dataSolid line: z0u/z0 versus h0/L Thin line: traditional formulation z0u = z0


STABILITY DEPENDENCE OF THE DISPLACEMENT HEIGHTin the “meteorological interval” -10 < h0/L <10 after new theory and experimental dataSolid line: d0u/d0 versus h0/L Dashed line: the upper limit: d0 = h0


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


Thank you

Sergej S. Zilitinkevich - FMI

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