Relaminarisation of turbulent stratified flow Bas van de Wiel Moene, Steeneveld, Holtslag

Overview 1)Motivation 2)A simple Couette flow analogy 3)Pressure driven flow: comparison with DNS 4)Conclusion and perspectives

(1) Motivation Why does the wind drop in the evening?

Classical picture of continuous turbulent quasi-steady SBL: z pot. T t=0t=3t=2t=1 (Nieuwstadt, 1984) Quasi-steady: Shape profiles cst. Linear heat flux profile

Central question: what happens for low pressure gradients? Continuous turbulent, quasi-steady nocturnal boundary layer only observed for strong pressure gradient conditions (high geostrophic winds)

Observational example (Cabauw, KNMI, Netherlands): Clear sky conditions Little wind near surface Collapse of turbulence→ decoupling of the surface from the atmosphere

z Temperature profiles Quasi-steady T

Rationale present work “Yet not every solution of the equations of motion, even if it is exact, can actually occur in nature. The flows that occur in nature must not only obey the equations of fluid dynamics but also be stable.” Landau and Lifschitz (1959) We hypothesize that: 1)The continuous turbulent SBL is hydrodynamically stable for high pressure gradient and are therefore observed in nature. 2)The continuous turbulent SBL is hydrodynamically unstable for low pressure gradient and are therefore not observed in nature. Instead a SBL with collapsed turbulence is observed. In fact we aim to find the transition T-L!

(2) A simple Coutte flow model Some characteristics: First order turbulence closure based on Ri No radiative divergence Rough flow using Z0=0.1 [m] BC’s: Top:Wind speed and temperature fixed Bottom:No slip and fixed surface heat flux Van de Wiel et al. (2006) Flows, Turbulence and Combustion, submitted

Turbulence closure First order closure: Two major elements controlling dominant eddie size: stratification and presence solid boundary Non-trivial in a sense that collapse of system as whole occurs way before Rc! Support locality of TKE in strongly stratified flow e.g.: Nieuwstadt ’84, Lenshow, ’88, Duynkerke ’91(Observations) Mason and Derbyshire ’90, Galmarini ’98, Basu ’05 (LES) Coleman et al (DNS); also recall presentation by Clercx

Results

Continuous turbulent case

Collapse case

Positive feedback mechanism: (following Van de Wiel et al. 2002, J. Atmos. Sc.). Increasing gradient:

Equilibrium solutions: bifurcation analysis

Linear stability analysis (i.e. on logarithmic profiles e.g. not linear!) Ansatz: (1-D!) BC’s

Criterion for instability Agreement between theory and numerical results! 0.55 Previous example: =0.52 Continuous turbulent cases Relaminarised cases

Thus: Collapse of SBL turbulence explained naturally from a linear stability analysis on the governing equations (assuming local closure) The crucial question: how close is our model in comparison with reality (here say reality~DNS)

(3) Comparison with DNS results from Nieuwstadt (2005) Pressure force Cooling BC’s Top: stress free, fixed T Bottom: no slip, prescribed heat extraction Smooth flow; Re*= 360

(3) Comparison with DNS results from Nieuwstadt (2005) We used a priori:(smooth flow) Remarkable in view of origin model

(3) Comparison with DNS results from Nieuwstadt (2005) A posteriori

DNS shows collapse at h/L~1.23 [-] Note: TKE normalised with u*^2

Our model shows collapse at h/L~1.45 [-] A priori threshold h/L~1.55

Predicting relaminarisation: Generalisation of the results Note: Continuous turbulent cases Relaminarised cases

Summary/conclusions: Relaminarization of turbulent stratified shear flows is predicted from linear stability analysis on parameterized equations In this way relaminarization critically depends on two dimensionless parameters: Re* (or Z0/h) and h/L The results seem to be confirmed by recent DNS results (at least in a qualitative sense)

z Wind speed profiles Quasi-steady U