The collapse of atmospheric turbulence

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1 The collapse of atmospheric turbulence Bas van de Wiel, Arnold Moene, Harm Jonker, Peter Baas, Bosveld, Jielun Sun,Sukanta Basu, Bert Holtslag, Judith Donda, Herman Clercx

3 Terre Incognita? GABLS I: GABLS II: GABLS III: Ugeo = 8 m/s Ugeo = 9.5 m/s Ugeo = m/s (Source: Svensson & Holtslag) But what when 0< Geo < 6 m/s?

4 Cessation of evening turbulence & temperature extremes

5 Ugeo: Cabauw (200m), The Netherlands (KNMI, Bosveld, Baas) AVG Stress After sunset? Ugeo

6 Cabauw 397 clear nights 4-hour evening averages stress and wind speed Cabauw (200m), The Netherlands (KNMI, Bosveld, Baas)

7 Hockey-stick Goal: find mechanism & predict threshold..a minimum wind speed needed for sustaining turbulence

8 Today 1: naïve model based on MO under assumption fixed bulk shear 2: fixed shear assumption must be wrong on long term! 3:.but on short-term it appears to be realistic! 4: application model formerly known as naïve

9 (1) MO-answer Simplified surface energy balance: Qn-G H Collapse possible when: Qn-G > H.. surface inversion intensifies, H to decreases further! (pos. feedback) Mechanism likely during clear skies, weak winds

10 (1) MO-answer e.g. from Businger-dyer: H c p c D UT ( 1Rb ) 2 parabolic graph known since Taylor (1971), Later e.g. : Malhi, Mahrt, Delage, Derbyshire, Van de Wiel, Basu, Here: consequences explicitly interpreted in term of surface energy balance: The maximum sustainable heat flux

11 Turbulent stress [N m -2 ] Our naïve model For each fixed value of U_40, diagnose if solution is possible in: Q n T c p c D UT f ( R ) b 0 T Is found, and stress diagnosed via: c U f ( 2 D R b ) Qn=25,l=6 Qn=25,l=4 Qn=25,l=8 Qn=15,l=6 Qn=35,l=6 Obs Finally: Stress Wind speed at level 40 m [m s -1 ]..is it that simple? U_40

12 (2) Unfortunately, Atmospheric bulk shear (or wind) is not fixed! Pressure driven flows recover after collapse: shear increases. naïve MO-approach appears good for wrong reason. Step 3: deeper analysis needed on shear evolution: theoretical channel flow

13 Stratified channel flow Pressure force free slip h No heat flux at z=h no slip H_0 Extraction heat normalized as: h L EXT gh T c ref p u H 0 3 * EXT u EXT (1/ )( P / x) h *

14 Multilayer model z x P t U 1 1 z H c t T p 1 z U K m z T K c H H p Ri f z U l K n m H ) ( 2, Ri/Rc f(ri/rc) local scaling (N84) (log-linear similarity functions) 100 vertical levels

15 Surface stress [-] Collapse of turbulence h/l=0.31 h/l=0.61 h/l=1.01 h/l=1.14 h/l=1.15 h/l =surface heat extraction Maximum sustainable flux 1.00 Inevitable recovery! Time [-] (scaled) Our challenge in transient part!

16 Wind profile development More shear than in neutral conditions Solution, next slide:

17 Long-term analytical solution L h q q q L h q q q u q U EXT * 1 2atanh atanh 1 2 ) ( z x P t U 1 1 z H c t T p 1 U 0 z 0 z h z 0 H 0 H H 0 In steady-state : ) ( 0 z t z U ) ( 0 z H t z T flux height,), (, z T z U f H Solution: with q=z/h Assymptotically converges to MO close to surface

18 Wind profile development Can we make quantitative prediction on transient shape?

19 z/h [-] Pseudo-steady state concept uw Pseudo Eq. wt Pseudo Eq resembling long-term steady state gamma instead of h! 0.4 g Flux [-] 0.0 Long-term steady state: Here: surface stress/h = pressure gradient surface stress (and hence gamma) is unknown! Alternative momentum constraint needed

20 Profile development 1) Momentum tends to be conserved over depth gamma: g q0 U ˆ (ˆ t ) Uˆ (0) dq 0 PS 2) A velocity crossing point appears Due to rapid redistribution of momentum in lower domain

21 Hint One conclusion can already be made: Existence point of fixed velocity suggest Rehabilitation naïve collapse mechanism! Next, assume explicitly: g q0 U ˆ (ˆ t ) Uˆ (0) dq 0 PS Now Initial momentum & surface cooling determine the pseudo-steady solution

22 Validation (I) z/h Initial profile Short-term Theory: pseudo steady Long-term Theory: steady state Depth pseudosteady layer Velocity crossing point model predicts level crossing point, all cases! U/u *

23 Validation (II) Solution curve for given initial momentum and surface heat extraction x=u * /u *N h/l EXT Theoretical maximum sustainable heat flux: Numerical simulations indicate: h/l_max=1.29 h/l_max=1.14

24 pseudo-steady state real thing? With pseudo-steady initial condition system should be happy for a while: system indeed locked in this initial condition

25 Back to Atmosperic BL Wind speed composite from cases with 5<Ugeo<10 200m U 40 m 10m Appears constant around 40m Time after sunset [hr] i.e. rehabilitation fixed wind assumption!

26 The naïve model valid again: For each fixed value of U_40 one can diagnose if a solution is possible in: Q n T c p c D UT f ( R ) b 0 T Is found, and stress diagnosed via: c U f ( 2 D R b ) Finally: Stress U_40

27 Turbulent stress [N m -2 ] Prediction critical speed Qn=25,l=6 Qn=25,l=4 Qn=25,l=8 Qn=15,l=6 Qn=35,l=6 Obs Hockey-stick robust feature Wind speed at level 40 m [m s -1 ]

28 Level of fixed speed: z_ref Why 40m? What if fixed wind occurs at other level? Sun et al, JAS (2011) Theory predicts: U 2 z (ln( z z 1/ 3 min ref ref 0)) i.e. approximately logarithmic increase of Umin with z_ref

29 Conclusion Collapse understood from maximum sustainable heat flux mechanism Fixed shear approach useful (~few hours after sunset) (to be submitted Quarterly J. Roy. Met. Soc.

30 Poster Judith Donda... Back to classical continuous turbulent SBL (Ugeo > 5m/s), exciting scaling with external forcings

31 GABLS IV GABLS I: GABLS II: GABLS III: Ugeo = 8 m/s Ugeo = 9.5 m/s Ugeo = m/s GABLS IV: low forcing: 0< Geo < 6 m/s?

32 Thank you for your attention

33 z/h [-] z/h [-] Extra Normalized cooling h/l_ext = t= 0+ t* t= 10 t* t= 20 t* t= 25 t* t= 0+ t* t= 10 t* t= 20 t* t= 25 t* U/u* [-] U/u* [-] DNS 1-D Eddy diffusivity Model

Surface stress [-] EXTRA 2.50 2.00 1.50 h/l=0.31 h/l=0.61 h/l=1.01 h/l=1.14 h/l=1.15 1.

34 Surface stress [-] EXTRA h/l=0.31 h/l=0.61 h/l=1.01 h/l=1.14 h/l= Time [-] t_ps t_ps/2 Rapid evolution to pseudo-steady state

35 TKE_scaled [-] EXTRA 4 3 h/l=0.4 h/l= t/t* [-]

36 Kinematic stress m 2 s -2 ] Application to atmosphere Atmospheric velocity crossing point typically at 30-60m. Here say ~ 40m. limiting case Wind speed at level 40 m [m s -1 ] minimum velocity for surviving turbulence ~5 m/s

37 Positive feedback Radiative cooling (demand) > turbulent heat flux (supply): More Surface cooling Increase density stratification Demand ~fixed Decrease turbulent heat supply Decrease turbulent mixing until all turbulent activity ceases...

38 Transformation H 1 (1 R b ) u* ( U ) h ln z0 Ri or DT 0.8 u* N U h ln z x=u * /u *N Later this result extended to nonconstant flux layer picture explains hockey-stick..unfortunately Van de 0.2 Wiel et al, h/l Flow, Turbulence and N Combustion

The collapse of turbulence in the evening

The collapse of turbulence in the evening B.J.H. Van de Wiel 1, A. F. Moene 2, H.J.J. Jonker 3, P. Baas 4, S. Basu 5, J. Sun 6, and A.A.M. Holtslag 2 1 Fluid Dynamics Lab., Eindhoven, Technical University,