Monin–Obukhov similarity theory

The Monin-Obukhov similarity theory describes the vertical characteristics of non-dimensionalized mean flow in the atmospheric boundary layer, named after Russian scientists A. M. Obukhov and A. S. Monin. The similarity theory originated from the discovery of the Obukhov length, a characteristic length scale of boundary layer turbulences derived in Obukhov's notable 1946 paper.

The similarity theory becomes a classical theory in boundary layer meteorology, leading to various important applications.

The Obukhov length

The Obukhov length is the characteristic length of the dynamic sublayer of the boundary layer. It was developed based on Prandtl's semi-empirical work on turbulence and Richardson's criterion for dynamic stability .

$L={\dfrac {-u_{*}^{3}}{\kappa {\dfrac {g}{T}}{\dfrac {Q}{c_{p}\rho }}}}$

The Obukhov length can also be defined more precisely as ,

$L={\dfrac {-u_{*}^{3}}{\kappa {\dfrac {g}{\theta _{v}}}{\overline {w'\theta _{v}'}}}}$

However this is not often used.

Governing formulae

The similarity theory parameterizes fluxes in the dynamic sublayer as a function of the dimensionless length parameter $\zeta =z/L$, based on Buckingham's Π-theorem. The functions of $\zeta$ used to determine the shape $u_{*}$ and vertical flux profiles are called universal functions. The universal functions are not uniquely determined, but approximated empirically.

$u_{*}={\dfrac {\kappa }{\psi _{m}(\zeta ){\dfrac {\partial u}{\partial \ln z}}}}$

Universal functions of the Monin-Obukhov similarity theory

Several forms were proposed to represent the universal functions of the similarity theory. Based on the results of the 1968 KANSAS experiment, the following universal functions are determined for momentum transport and heat transport,

$\psi _{m}(\zeta )=(1-15\zeta )^{-1/4}\quad -2<\zeta <0$

$\psi _{m}(\zeta )=1+4.7\zeta \quad 0<\zeta <1$

$\psi _{h}(\zeta )=0.74(1-9\zeta )^{-1/2}\quad -2<\zeta <0$

$\psi _{h}(\zeta )=0.74+4.7\zeta \quad 0<\zeta <1$

Validity and limitations

Works for near-neutral to moderately convective ABLs.

Works for dry conditions because universal functions under moist conditions were not well studied.

10%~20% errors

References

1. Obukhov, A. M. (1971). Turbulence in an atmosphere with a non-uniform temperature. Boundary-Layer Meteorology 2: 7-29.

2. Monin, A. S., and Obukhov, A. M. (1954). Basic laws of turbulent mixing in the surface layer of the atmosphere (English translation). Tr. Akad. Nauk SSSR Geophiz. Inst. 24(151): 163-187.

3. Foken, T. (2006). 50 Years of the Monin-Obukhov similarity theory. Boundary-Layer Meteorology 119: 431-447.

4. Johansson, C., Smedman, A., Högström, U., Brasseur, J.G., and Khanna, S. (2001). Critical test of the validity of Monin–Obukhov similarity during convective conditions. J. Atmos. Sci. 58: 1549-1566.

5. Khanna, S., and Brasseur, J. G. (1997). Analysis of Monin-Obukhov similarity from large-eddy simulation. J. Fluid Mech. 345: 251–286.

6. McNaughton, K. G., and Brunet, Y. (2002). Townsend's hypothesis, coherent structures and Monin–Obukhov similarity. Boundary-layer meteorology 102(2): 161-175.

7. Pahlow, M., Parlange, M. B., and Porté-Agel, F. (2001). On Monin–Obukhov similarity in the stable atmospheric boundary layer. Boundary-layer meteorology 99(2): 225-248.

8. Panofsky, H. A., Tennekes, H., Lenschow, D. H., and Wyngaard, J. C. (1977). The characteristics of turbulent velocity components in the surface layer under convective conditions. Boundary-Layer Meteorology 11(3): 355-361.

9. Wilson, J. D. (2008). Monin-Obukhov functions for standard deviations of velocity. Boundary-layer meteorology 129(3): 353-369.

10. Cheng, Y., Parlange, M. B., and Brutsaert, W. (2005). Pathology of Monin‐Obukhov similarity in the stable boundary layer. J. Geophys. Res. 110: D06101.