Environmental Fluid Mechanics

, Volume 9, Issue 3, pp 267–295 | Cite as

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

  • Victor S. L’vov
  • Itamar Procaccia
  • Oleksii Rudenko
Original Article


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.


Atmospheric boundary layer Stable stratification Richardson number Transport equations Second order closure Algebraic model 


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

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 / Dt

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


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


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


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


Thermal flux at zero elevation z  =  0


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


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


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


Turbulent Prandtl number, ν TT


Flux Richardson number, β F z xz S U


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

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

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


Mean velocity gradient, dU/dz


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


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

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

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


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

\({\widetilde \beta}\)

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


Relaxation frequency of τ ij , i  =  j

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

Relaxation frequency of τ ij , i  ≠  j


Relaxation frequency of F


Relaxation frequency of E K


Relaxation frequency of E Θ


Dissipation tensor of τ ij , (18)


Dissipation vector of F, (18)


Dissipation of E Θ, (18)


Potential temperature, (8)


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


Potential temperature at zero elevation, F */u *


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


Viscous lengthscale, ν/u *


Kinematic and turbulent viscosity

ρ, ρb

Total and BRS density of the fluid


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


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

χ, χT

Kinematic and turbulent thermal conductivity


Basic reference state


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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  • Victor S. L’vov
    • 1
  • Itamar Procaccia
    • 1
  • Oleksii Rudenko
    • 1
  1. 1.Department of Chemical PhysicsThe Weizmann Institute of ScienceRehovotIsrael

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