THE EFFECT OF STRATIFICATION ON THE ROUGHNESS LENGTH AN DISPLACEMENT HEIGHT

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1 THE EFFECT OF STRATIFICATION ON THE ROUGHNESS LENGTH AN DISPLACEMENT HEIGHT S. S. Zilitinkevich 1,2,3, I. Mammarella 1,2, A. Baklanov 4, and S. M. Joffre 2 1. Atmospheric Sciences,, Finland 2. 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

2 Reference S. S. Zilitinkevich, I. Mammarella, A. A. Baklanov, and S. M. Joffre, 28: The effect of stratification on the roughness length and displacement height. Boundary-Layer Meteorology. In press.

3 Content Roughness length and displacement height: u* z du u( z) = ln + Ψ k z No stability dependence of z u (and d u) in engineering fluid mechanics: neutral-stability z = level, at which u (z) plotted vs. ln z approaches zero; z ~ 1 of typical height of roughness elements, h 25 Meteorology / oceanography: h comparable with MO length Stability dependence of the actual roughness length, z u : z u < z in stable stratification; z u > z in unstable stratification u u z L L u 3 = βfθ s

4 Surface layer and roughness length 1 Self similarity in the surface layer (SL) 5h <z< 1 h 2 Height-constant fluxes: τ τ z= 5h u u and z serve as turbulent scales: u T ~ u, l T ~ z Eddy viscosity ( k. 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 / zu ) z u (redefined constant of integration) is roughness length 1 Displacement height u U = k u ln ( z d u ) / zu Not applied to the roughness layer (RL) <z<5h d [ ]

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

comparable with the Monin-Obukhov turbulent length scale, L,

6 Stability Dependence of Roughness Length For urban and vegetation canopies with roughness-element heights (2-5 m) comparable with the Monin-Obukhov turbulent length scale, L, the surface resistance and roughness length depend on stratification

7 Background physics and effect of stratification Physically z u = depth of a sub-layer within RL ( < z < 5h ) with 9% of the velocity drop from U R ~ u (approached at z ~ h ) 2 From τ = K M ( RL ) U / z, τ ~ u and U / z ~ R / z u~ / z u z u ~ KM ( RL) / u U u K (RL) M = K M ( h + ) from matching the RL and the surface-layer Neutral: Stable: K Unstable: K M ~ h u classical formula z u ~ h 1 = ku z(1 + Cu z / L) ~ u L z 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/ 3 )

8 Recommended formulation Neutral stable z z u 1 = 1+ C h SS / L Neutral unstable z z 1/3 u h = 1+ CUS L Constants: C =8. 13±.21, C = 1.24±.5 SS US

9 Experimental datasets Sodankyla Meteorological Observatory, Boreal forest (FMI) BUBBLE urban BL experiment, Basel, Sperrstrasse (Rotach et al., 24) 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

10 Stable stratification z zu Bin-average values of / (neutral- over actual-roughness lengths) versus h /L in stable stratification for Boreal forest (h=13.5 m; z =1.1±.3 m). Bars are standard errors; the curve is z / z = h / L u +.

11 Stable stratification Bin-average values of z / z u (actual- over neutral-roughness lengths) versus h /L in stable stratification for boreal forest (h=13.5 m; z =1.1±.3 m). Bars are standard errors; the curve is z / z 1 u = ( h / L).

12 Stable stratification Displacement height over its neutral-stability value in stable stratification. Boreal forest (h = 15 m, d = 9.8 m). ( ) The curve is 1 d u / d 1+.5( h / L) h / = L

13 Unstable stratification Convective eddies extend in the vertical causing z > zu 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

14 Unstable stratification Unstable stratification, Basel, z / z u vs. Ri = (gh /Θ 32 )(Θ 18 Θ 32 )/(U 32 )2 Building height =14.6 m, neutral roughness z =1.2 m; BUBBLE, Rotach et al., 25). h /L through empirical dependence on Ri on (next figure) The curve (z / z u =1+5.31Ri 6/13 ) confirms theoretical z u /z = (h /-L) 1/3

15 Unstable stratification Empirical Ri =.365 (h / L) 13/18

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

17 STABILITY DEPENDENCE OF THE ROUGHNESS LENGTH in the meteorological interval -1 < h /L <1 after new theory and experimental data Solid line: z u /z versus h /L Thin line: traditional formulation z u = z

18 STABILITY DEPENDENCE OF THE DISPLACEMENT HEIGHT in the meteorological interval -1 < h /L <1 after new theory and experimental data Solid line: d u /d versus h /L Dashed line: the upper limit: d = h

19 Conclusions (Roughness length) Traditional: roughness length and displacement height fully characterised by geometric features of the surface New: 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

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

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