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Energy conservation and second-order statistics in stably stratified turbulent boundary layers

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Abstract

We address the dynamical and statistical description of stably stratified turbulent boundary layers with the important example of the atmospheric boundary layer with a stable temperature stratification in mind. Traditional approaches to this problem, based on the profiles of mean quantities, velocity second-order correlations, and dimensional estimates of the turbulent thermal flux run into a well-known difficulty, predicting the suppression of turbulence at a small critical value of the Richardson number, in contradiction with observations. Phenomenological attempts to overcome this problem suffer from various theoretical inconsistencies. Here we present a closure approach taking into full account all the second-order statistics, which allows us to respect the conservation of total mechanical energy. The analysis culminates in an analytic solution of the profiles of all mean quantities and all second-order correlations removing the unphysical predictions of previous theories. We propose that the approach taken here is sufficient to describe the lower parts of the atmospheric boundary layer, as long as the Richardson number does not exceed an order of unity. For much higher Richardson numbers the physics may change qualitatively, requiring careful consideration of the potential Kelvin-Helmoholtz waves and their interaction with the vortical turbulence.

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Abbreviations

\({\bf {\mathcal A}}\) :

Thermal flux production vector, (19e)

\({\bf {\mathcal B}}\) :

Pressure-temperature-gradient-vector, (19f)

\({\mathcal C_{ij}}\) :

Energy conversion tensor, (19b)

\({\mathcal D / \mathcal D t}\) :

Substantial derivative, \({\partial / \partial t + \mathcal U \cdot {\bf \nabla}}\)

D / D t :

Mean substantial derivative, \({\partial / \partial t + \varvec U \cdot {\bf \nabla}}\)

E K :

Turbulent kinetic energy per unit mass, \({\left\langle {\varvec u^2} \right\rangle/2}\)

E Θ :

“Temperature energy” per unit mass, \({\left\langle\theta^2\right\rangle/2}\)

F :

Turbulent thermal flux per unit mass, \({\left\langle {\varvec u} \theta \right\rangle}\)

F * :

Thermal flux at zero elevation z  =  0

g :

Gravity acceleration, \({{\varvec g} = -g\, \widehat{\bf{z}}}\)

L :

Monin-Obukhov length, \({u_\ast^3/\beta F_\ast}\)

ℓ(z):

Outer scale of turbulence, external parameter

\({\mathcal P_{ij}}\) :

Rate of Reynolds stress production, (19a)

\({p,\widetilde p, p_\ast}\) :

Total, fluctuating and zero level pressures

PrT :

Turbulent Prandtl number, ν T/χT

Riflux :

Flux Richardson number, β F z /τ xz S U

Rigrad :

Gradient Richardson number, \({\beta S_ \Theta / S^2_ U}\)

\({\mathcal R_{ij}}\) :

Pressure-rate-of-strain-tensor, (19c)

S U :

Mean velocity gradient, dU/dz

S Θ :

Mean potential temperature gradient, dΘ/dz

T, T * :

Molecular temperature, molecular temperature at zero level

\({\mathcal U, {\varvec U}, {\varvec u}}\) :

Total, mean, \({\left\langle \mathcal U \right\rangle}\) , and fluctuating, \({\mathcal U - {\varvec U}}\) , velocity fields

u * :

(Wall) friction velocity, \({\sqrt{\tau_\ast}}\)

\({\widehat{\bf{x}}, \widehat{\bf{z}}}\) :

Horizontal (streamwise) and vertical (wall-normal) unit vectors

β :

Buoyancy parameter, \({{\bf g} \widetilde \beta}\) (for an ideal gas g/T b)

\({\widetilde \beta}\) :

Thermal expansion coefficient, \({-\left( {\partial \rho_ {\rm b}}/{\partial T_ {\rm b}}\right)_{\rm p}/{\rho_ {\rm b}}}\)

γ RI :

Relaxation frequency of τ ij , i  =  j

\({\widetilde \gamma_{\rm RI}}\) :

Relaxation frequency of τ ij , i  ≠  j

γ RD :

Relaxation frequency of F

γ uu :

Relaxation frequency of E K

γ θθ :

Relaxation frequency of E Θ

ε ij :

Dissipation tensor of τ ij , (18)

ε :

Dissipation vector of F, (18)

ε :

Dissipation of E Θ, (18)

\({\overline{\Theta}}\) :

Potential temperature, (8)

Θ d :

Deviation of potential temperature from BRS, \({\overline{\Theta}-T_\ast}\)

Θ, θ :

Mean, \({\left\langle \Theta_d \right\rangle}\) , and fluctuating, \({\Theta_d -\left\langle \Theta_d \right\rangle}\) , parts of Θ d

θ * :

Potential temperature at zero elevation, F */u *

Λ(z):

Local Monin-Obukhov length, \({\tau_{xz}\sqrt{-\tau_{xz}}\left/\beta F_z\right.}\)

λ* :

Viscous lengthscale, ν/u *

ν,ν T :

Kinematic and turbulent viscosity

ρ, ρ b :

Total and BRS density of the fluid

τ ij :

Reynolds stress tensor per unit mass, \({\left\langle u_i u_j \right\rangle}\)

τ * :

Mechanical momentum flux at zero elevation (at the ground), νS U(0)

χ, χT :

Kinematic and turbulent thermal conductivity

BRS:

Basic reference state

References

  1. Umlauf L, Burchard H (2005) Second-order turbulence closure models for geophysical boundary layers. A review of recent work. Continent Shelf Res 25: 725–827. doi:10.1016/j.csr.2006.02.008

    Google Scholar 

  2. Weng W, Taylor P (2003) On modelling the one-dimensional atmospheric boundary layer. Bound Layer Meteorol 107: 371–400. doi:10.1023/A:1022126511654

    Article  Google Scholar 

  3. Zeman O (1981) Progress in the modeling of planetary boundary layers. Ann Rev Fluid Mech 13: 253–272. doi:10.1146/annurev.fl.13.010181.001345

    Article  Google Scholar 

  4. Canuto VM (2002) Critical Richardson numbers and gravity waves. Astron Astrophys 384: 1119–1123. doi:10.1051/0004-6361:20011773

    Article  Google Scholar 

  5. Galperin B, Sukoriansky S, Anderson PS (2007) On the critical Richardson number in stably stratified turbulence. Atm Sci Lett 8(3): 65–69. doi:10.1002/asl.153

    Article  Google Scholar 

  6. Mellor GL, Yamada T (1974) A hierarchy of turbulence closure models for planetary boundary layer. J Atmos Sci 31: 1791–1806 110.1175/1520-0469(1974)031<1791:AHOTCM>2.0.CO;2

    Article  Google Scholar 

  7. Richardson LF (1920) The supply of energy from and to atmospheric eddies. Proc. Roy. Soc. London A 97: 354–373

    Article  Google Scholar 

  8. Richardson LF (1922) Weather prediction by numerical process. Cambridge University Press, xii+236 pp

  9. Zilitinkevich SS, Elperin T, Kleeorin N, Rogachevskii I (2007) Energy- and flux-budget (EFB) turbulence closure model for stably stratified flows. Part I: steady-state, homogeneous regimes. Bound Layer Meteorol 125(2): 167–191. doi:10.1007/s10546-007-9189-2

    Google Scholar 

  10. Nieuwstadt FTM (1984) The turbulent structure of the stable, nocturnal boundary layer. J Atmos Sci 41: 2202–2216 10.1175/1520-0469(1984)041<2202:TTSOTS>2.0.CO;2

    Article  Google Scholar 

  11. Zilitinkevich SS (2002) Third-order transport due to internal waves and non-local turbulence in the stably stratified surface layer. Q J Royal Meteorol Soc 128: 913–925. doi:10.1256/0035900021643746

    Article  Google Scholar 

  12. Boussinesq J (1903) The’orique Analytique de la Chaleur, vol 2. Gauthier-Villars, Paris

    Google Scholar 

  13. Oberbeck A (1879) Über die Wärmeleitung der Flüssigkeiten bei Berücksichtigung der Strömung infolge Temperaturdifferenzen. Ann Phys Chem (Leipzig) 7: 271–292.

    Google Scholar 

  14. Landau LD, Lifshitz EM (1987) Course of theoretical physics: fluid mechanics. Pergamon, New York, p 552

    Google Scholar 

  15. Zilitinkevich SS (1972) On the determination of the hight of the Ekman boundary layer. Bound Layer Meteorol 3: 141–145. doi:10.1007/BF02033914

    Article  Google Scholar 

  16. Hauf T, Höller H (1987) Entropy and potential Temperature. J. Atm Sci 44: 2887–2901 10.1175/1520-0469(1987)044<2887:EAPT>2.0.CO;2

    Article  Google Scholar 

  17. L’vov VS, Rudenko O (2008) Equations of motion and conservation laws in a theory of stably stratified turbulence. Physica Scripta T132, to be published; also: arXiv:0803.2627v1 [physics.flu-dyn]

  18. Hunt JCR, Stretch DD, Britter RE (1988) Length scales in stably stratified turbulent flows and their use in turbulence models. In: Puttock JS (ed) Proceedings of I.M.A. conference on “stably stratified flow and dense gas dispersion”. Clarendon Press, pp 285–322

  19. Schumann U, Gerz T (1995) Turbulent mixing in stably stratified shear flows. J Appl Meteorol 34: 33–48 10.1175/1520-0450(1995)034<0033:TMISSS>2.0.CO;2

    Google Scholar 

  20. Hanazaki H, Hunt JCR (2004) Structure of unsteady stably stratified turbulence with mean shear. J Fluid Mech 507: 1–42. doi:10.1017/S0022112004007888

    Article  Google Scholar 

  21. Keller K, Van Atta CW (2000) An experimental investigation of the vertical temperature structure of homogeneous stratified shear turbulence. J Fluid Mech 425: 1–29

    Article  Google Scholar 

  22. Stretch DD, Rottman JW, Nomura KK, Venayagamoorthy SK (2001) Transient mixing events in stably stratified turbulence. In: 14th Australasian fluid mechanics conference, Adelaide, Australia, 10–14 December 2001

  23. Elperin T, Kleeorin N, Rogachevskii I, Zilitinkevich S (2002) Formation of large-scale semiorganized structures in turbulent convection. Phys Rev E 66: 066305. doi:10.1103/PhysRevE.66.066305

    Article  Google Scholar 

  24. Cheng Y, Canuto VM, Howard AM (2002) An improved model for the turbulent PBL. J Atm Sci 59: 1550–1565 10.1175/1520-0469(2002)059<1550:AIMFTT>2.0.CO;2

    Article  Google Scholar 

  25. Rehmann CR, Hwang JH (2005) Small-scale structure of strongly stratified turbulence. J Phys Oceanogr 32: 154–164. doi:10.1175/JPO-2676.1

    Google Scholar 

  26. Pope SB (2001) Turbulent flows. Cambridge University Press, 771 pp

  27. Rotta JC (1951) Statistische theorie nichthomogener turbulenz. Z Physik 129: 547–572. doi:10.1007/BF01330059

    Article  Google Scholar 

  28. L’vov VS, Pomyalov A, Procaccia I, Zilitinkevich SS (2006) Phenomenology of wall bounded Newtonian turbulence. Phys Rev E 73: 016303. doi:10.1103/PhysRevE.73.016303

    Article  Google Scholar 

  29. L’vov VS, Procaccia I, Rudenko O (2006) Analytic model of the universal structure of turbulent boundary layers. JETP Lett 84: 67–73. doi:10.1134/S0021364006140049

    Google Scholar 

  30. Kolmogorov AN (1941) Energy dissipation in locally isotropic turbulence. Doklady AN SSSR 32(1): 19–21

    Google Scholar 

  31. Monin AS, Obukhov AM (1954) Main characteristics of the turbulent mixing in the atmospheric surface layer. Trudy Geophys Inst AN SSSR 24(151): 153–187

    Google Scholar 

  32. L’vov VS, Procaccia I, Rudenko O (2008) Universal model of finite reynolds number turbulent flow in channels and pipes. Phys Rev Lett 100: 054504. doi:10.1103/PhysRevLett.100.054504

    Article  Google Scholar 

  33. Beare RJ, Macvean MK, Holtslag AAM, Cuxart J, Esau I, Golaz J-C, Jimenez MA, Khairoutdinov M, Kosovic B, Lewellen D, Lund TS, Lundquist JK, Mccabe A, Moene AF, Noh Y, Raasch S, Sullivan P (2006) An intercomparison of large-eddy simulations of the stable boundary layer. Bound Layer Meteorol 118: 247–272. doi:10.1007/s10546-004-2820-6

    Article  Google Scholar 

  34. Basu S, Porté-agel F, Foufoula-Georgiou E, Vinuesa J-F, Pahlow M (2006) Revising the local scaling hypothesis in stably stratified atmospheric boundary-layer turbulence: an integration of field and laboratory measurements with large-eddy simulations. Bound Layer Meteorol 119: 473–500. doi:10.1007/s10546-005-9036-2

    Article  Google Scholar 

  35. Zilitinkevich SS, Esau I (2007) Similarity theory and calculation of turbulent fluxes at the surface for the stably stratified atmospheric boundary layer. Bound Layer Meteorol 125: 193–205. doi:10.1007/s10546-007-9187-4

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Chemical Physics, The Weizmann Institute of Science, Rehovot, 76100, Israel

    Victor S. L’vov, Itamar Procaccia & Oleksii Rudenko

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  1. Victor S. L’vov
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  2. Itamar Procaccia
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Correspondence to Oleksii Rudenko.

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L’vov, V.S., Procaccia, I. & Rudenko, O. Energy conservation and second-order statistics in stably stratified turbulent boundary layers. Environ Fluid Mech 9, 267–295 (2009). https://doi.org/10.1007/s10652-008-9117-0

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