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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
University of Helsinki
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
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State of the art
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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
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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
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Classical similarity theory and
current surface-flux schemes (based on the SL-concept)
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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
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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)
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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
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Revised similarity theory and
ABL flux-profile relationships
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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
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ΦM = kzτ--1/2dU/dz vs. z/L (a), z/ ∗L (b) x nocturnal; o long-lived; □ conv. neutral
Velocity gradient (conventionally neutral ABLs!)
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HΦ = (kTτ1/2z/Fθ)dΘ/dz vs. z/L (a), z/ ∗L (b) x nocturnal; o long-lived
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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
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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
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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
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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 .
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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 .
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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
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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.
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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.
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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.
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Remain to be determined:
• ABL height and
• Roughmess length
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ABL height
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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.
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Factors controlling ABL height
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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
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General case
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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
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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
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Roughmess length
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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
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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.
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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
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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
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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
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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
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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 −
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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
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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
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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+ .
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Stable stratification
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 .
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Stable stratification
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
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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
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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
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Unstable stratification
Empirical Ri = 0.0365 (h0/–L)13/18
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Unstable stratification
Displacement height in unstable stratification (Basel):
The line confirms theoretical dependence:
Ri versus1/0 −oudd
3/10
00 )/(1 LhC
ddDC
u −+=
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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
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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
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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
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Thank you
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