atmospheric planetary boundary layer (abl / pbl): theory...

University of Helsinki

Atmospheric Planetary Boundary Layer (ABL / PBL): theory, modelling

and applicationsSergej S. Zilitinkevich

Division of Atmospheric Sciences, University of Helsinki, FinlandMeteorological Research, Finnish Meteorological Institute, Helsinki

Nansen Environmental and Remote Sensing Centre, Bergen, Norway


Part 1Stably Stratified Atmospheric

Boundary Layer (SBL)


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. Submitted to Boundary-Layer Meteorol.


Motivation


State of the art


Basic types of the SBL

• 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


MEAN PROFILES & SURFACE FLUXESContent

• 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_______________________________________________________________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.


Turbulence in atmospheric models

• turbulence closure – to calculate vertical fluxes: τr and θF through mean

gradients: dzUd /r

and dzd /Θ • flux-profile relationships – to calculate the surface fluxes: 0

2 | =∗∗ == zu ττ ,0| =∗ = zFF θ through wind speed =1U

1| zzU = and potential temperature

=Θ1 1| zz=Θ at a given level 1z

• Warning: In NWP and climate models, the lowest computational level

is 1z ~30 m


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


Stable stratification: revised theory Zilitinkevich and Esau (2005) two additional length scales besides L:

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

Interpolation: ∗L

1 = 2/12221

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛

f

f

N

N

LC

LC

L where NC =0.1 and fC =1


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


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


LES turbulent fluxes: solid lines 2/ ∗uτ = )exp( 23

8 ς− , θF / sFθ = )2exp( 2ς−

Approximation based on atmospheric data (e.g. Lenshow, 1988): dashed lines

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


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 allow determining τ , θF , and 12/3 )( −−= θβτ FL , at the computational level z.


Given τ , θF , surface fluxes are calculated using empirical dependencies

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−=

∗

2

38exp

hz

ττ , ⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−=

∗

2

2exphz

FFθ (Figures above)

The equilibrium ABL height, Eh , is determined diagnostically (Z. et al., 2006a):

∗

=τ22

2

1

RE Cf

h +

∗τ2

||

CNCfN + 22

||

∗

∗

τβ

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

The actual ABL height, after 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

Algorithm


Conclusions (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 2: surface flux scheme for use in operational models Requested: (i) roughness lengths and (ii) ABL height


STRATIFICATION EFFECT ON THE ROUGHNESS LENGTH

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


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. Submitted to Boundary-Layer Meteorology.


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) 5 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 Monin-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 ( 050 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


ExperimentalExperimental datasetsdatasets

Sodankyla MeteorologicalObservatory, 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 length)

• Traditional concept: roughness length and displacement height 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

• Especially: urban air pollution episodes in very stable stratification


NEUTRAL and STABLE ABL HEIGHT

Sergej Zilitinkevich 1,2,3, Igor Esau3 and Alexander Baklanov4

1 Division of Atmospheric Sciences, University of Helsinki, Finland2 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


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 PBL height


Scaling analysis


Dominant role of the smallest scale


How to verify h-equations?


Stage I: Truly neutral ABL


Stage I: Transition TN CN ABL


Stage I: Transition TN NS ABL


Stage II: General case


The height of the conventionally neutral (CN) ABL

Z & Esau, 2002, 2007: the effect of free-flow stability (N) on CN ABL height, hE,, (LES – red; field data – blue; theory – curve). Traditional theory overlooks this dependence and overestimates hE up to an order of magnitude.


Conclusions (SBL height)● Eh , depends on many factors multi-limit analysis / complex formulation ● difficult to measure: baroclinic shear (Γ), vertical velocity ( hw ), Eh itself ● hence use LES, DNS and lab experiments

● baroclinic ABL: substitute Tu = *u (1+C0Γ/N)1/2 for *u in the 2nd term of

∗

=τ22

21

RE Cf

h +

∗τ2

||

CNCfN + 22

||

∗

∗

τβ

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

● account for vertical motions: corr−Eh = Eh + hw Tt , where Tt = tC Eh / *u ● generally prognostic (relaxation) equation (Z. and Baklanov, 2002):

)(2E

Ethh hh

huChKwhU

th

−−∇=−∇⋅+∂∂ ∗

r ( tC = 1)

atmospheric planetary boundary layer (abl / pbl): theory...

Documents