Skip to main content Accessibility help

Multi-point Monin–Obukhov similarity in the convective atmospheric surface layer using matched asymptotic expansions

  • Chenning Tong (a1) and Mengjie Ding (a1)


The multi-point Monin–Obukhov similarity (MMO) was recently proposed (Tong & Nguyen, J. Atmos. Sci., vol. 72, 2015, pp. 4337–4348) to address the issue of incomplete similarity in the framework of the original Monin–Obukhov similarity theory (MOST). MMO hypothesizes the following: (1) The surface-layer turbulence, defined to consist of eddies that are entirely inside the surface layer, has complete similarity, which however can only be represented by multi-point statistics, requiring a horizontal characteristic length scale (absent in MOST). (2) The Obukhov length $L$ is also the characteristic horizontal length scale; therefore, all surface-layer multi-point statistics, non-dimensionalized using the surface-layer parameters, depend only on the height and separations between the points, non-dimensionalized using $L$ . However, similar to MOST, MMO was also proposed as a hypothesis based on phenomenology. In this work we derive MMO analytically for the case of the horizontal Fourier transforms of the velocity and potential temperature fluctuations, which are equivalent to the two-point horizontal differences of these variables, using the spectral forms of the Navier–Stokes and the potential temperature equations. We show that, for the large-scale motions (wavenumber $k<1/z$ ) in a convective surface layer, the solution is uniformly valid with respect to $z$ (i.e. as $z$ decreases from $z>-L$ to $z<-L$ ), where $z$ is the height from the surface. However, for $z<-L$ the solution is not uniformly valid with respective to $k$ as it increases from $k<-1/L$ to $k>-1/L$ , resulting in a singular perturbation problem, which we analyse using the method of matched asymptotic expansions. We show that (1)  $-L$ is the characteristic horizontal length scale, and (2) the Fourier transforms satisfy MMO with the non-dimensional wavenumber $-kL$ as the independent similarity variable. Two scaling ranges, the convective range and the dynamic range, discovered for $z\ll -L$ in Tong & Nguyen (2015) are obtained. We derive the leading-order spectral scaling exponents for the two scaling ranges and the corrections to the scaling ranges for finite ratios of the length scales. The analysis also reveals the dominant dynamics in each scaling range. The analytical derivations of the characteristic horizontal length scale ( $L$ ) and the validity of MMO for the case of two-point horizontal separations provide strong support to MMO for general multi-point velocity and temperature differences.


Corresponding author

Email address for correspondence:


Hide All
Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill.
Betchov, R. & Yaglom, A. M. 1971 Comments on the theory of similarity as applied to turbulence in an unstable stratified fluid. Izv. Akad. Nauk. Ser. Fiz. Atmos. Okeana 7, 829832; English translation.
Businger, J. A. 1973 A note on free convection. Boundary-Layer Meteorol. 4, 323326.10.1007/BF02265241
Businger, J. A., Wyngaard, J. C., Izumi, Y. & Bradley, E. F. 1971 Flux-profile relationships in the atmospheric surface layer. J. Atmos. Sci. 28, 181189.10.1175/1520-0469(1971)028<0181:FPRITA>2.0.CO;2
Caughey, S. J. & Palmer, S. G. 1979 Some aspects of turbulence structure through the depth of the convective boundary layer. Q. J. R. Meteorol. Soc. 105, 811827.10.1002/qj.49710544606
Cousteix, J. & Mauss, J. 2007 Asymptotic Analysis and Boundary Layers. Springer.10.1007/978-3-540-46489-1
Ding, M., Nguyen, K. X., Liu, S., Otte, M. J. & Tong, C. 2018 Investigation of the pressure–strain-rate correlation and pressure fluctuations in convective and near neutral atmospheric surface layers. J. Fluid Mech. 854, 88120.10.1017/jfm.2018.576
Grachev, A. A. W., Fairall, C. & Zilitinkevich, S. S. 1997 Surface-layer scaling for the convection induced stress regime. Boundary-Layer Meteorol. 83, 423439.10.1023/A:1000281625985
Kader, B. A. 1988 Three-layer structure of an unstably stratified atmospheric surface layer. Izv. Akad. Nauk. Ser. Fiz. Atmos. Okeana 24, 907918; English translation.
Kaimal, J. C. 1978 Horizontal velocity spectra in an unstable surface layer. J. Atmos. Sci. 35, 1824.10.1175/1520-0469(1978)035<0018:HVSIAU>2.0.CO;2
Kaimal, J. C., Wyngaard, J. C., Izumi, Y. & Coté, O. R. 1972 Spectral characteristic of surface-layer turbulence. Q. J. R. Meteorol. Soc. 98, 563589.10.1002/qj.49709841707
Kosović, B. 1997 Subgrid-scale modelling for the large-eddy simulation of high-Reynolds-number boundary layer.. J. Fluid Mech. 336, 151182.10.1017/S0022112096004697
Lumley, J. L. & Panofsky, H. A. 1964 The Structure of Atmospheric Turbulence, Interscience Monographs and Texts in Physics and Astronomy, vol. 12. Interscience.
Lundgren, T. S. 2003 Kolmogorov turbulence by matched asymptotic expansions. Phys. Fluids 15, 10741081.10.1063/1.1558332
Moeng, C.-H. 1984 A large-eddy simulation model for the study of planetary boundary-layer turbulence. J. Atmos. Sci. 41, 20522062.10.1175/1520-0469(1984)041<2052:ALESMF>2.0.CO;2
Moeng, C. H. & Wyngaard, J. C. 1988 Spectral analysis of large-eddy simulations of the convective boundary layer. J. Atmos. Sci. 45, 35733587.10.1175/1520-0469(1988)045<3573:SAOLES>2.0.CO;2
Monin, A. S. & Obukhov, A. M. 1954 Basic laws of turbulent mixing in the ground layer of the atmosphere. Trans. Inst. Teoret. Geofiz. Akad. Nauk SSSR 151, 163187.
Monin, A. S. & Yaglom, A. M. 1975 Statistical Fluid Mechanics. MIT Press.
Nguyen, K. X., Horst, T. W., Oncley, S. P. & Tong, C. 2013 Measurements of the budgets of the subgrid-scale stress and temperature flux in a convective atmospheric surface layer. J. Fluid Mech. 729, 388422.10.1017/jfm.2013.302
Nguyen, K. X. & Tong, C. 2015 Investigation of subgrid-scale physics in the convective atmospheric surface layer using the budgets of the conditional mean subgrid-scale stress and temperature flux. J. Fluid Mech. 772, 295329.10.1017/jfm.2015.171
Obukhov, A. M. 1946 Turbulence in the atmosphere with inhomogeneous temperature. Trans. Inst. Teoret. Geofiz. Akad. Nauk SSSR 1, 95115.
Otte, M. J. & Wyngaard, J. C. 2001 Stably stratified interfacial-layer turbulence from large-eddy simulation. J. Atmos. Sci. 58, 34243442.10.1175/1520-0469(2001)058<3424:SSILTF>2.0.CO;2
Sullivan, P. P., McWilliams, J. C. & Moeng, C.-H. 1994 A subgrid-scale model for large-eddy simulation of planetary boundary-layer flows. Boundary-Layer Meteorol. 71, 247276.10.1007/BF00713741
Sullivan, P. P., Mcwilliams, J. C. & Moeng, C.-H. 1996 A grid nesting method for large-eddy simulation of planetary boundary-layer flows. Boundary-Layer Meteorol. 80, 167202.10.1007/BF00119016
Sykes, R. I., Henn, D. S. & Lewellen, W. S. 1993 Surface-layer description under free-convection conditions. Q. J. R. Meteorol. Soc. 119, 409421.10.1002/qj.49711951103
Tong, C. & Ding, M. 2018 Monin-Obukhov similarity and local-free-convection scaling in the atmospheric boundary layer using matched asymptotic expansions. J. Atmos. Sci. 75, 36913701.10.1175/JAS-D-18-0016.1
Tong, C. & Nguyen, K. X. 2015 Multipoint Monin–Obukhov similarity and its application to turbulence spectra in the convective atmospheric surface layer. J. Atmos. Sci. 72, 43374348.10.1175/JAS-D-15-0134.1
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics. The Parabolic Press.
Wyngaard, J. C. & Coté, O. R. 1971 The budgets of turbulent kinetic energy and temperature variance in the atmospheric surface layer. J. Atmos. Sci. 28, 190201.10.1175/1520-0469(1971)028<0190:TBOTKE>2.0.CO;2
Wyngaard, J. C., Coté, O. R. & Izumi, Y. 1971 Local free convection, similarity, and the budgets of shear stress and heat flux. J. Atmos. Sci. 28, 11711182.10.1175/1520-0469(1971)028<1171:LFCSAT>2.0.CO;2
Yaglom, A. M. 1994 Fluctuation spectra and variances in convective turbulent boundary layers: a reevaluation of old models. Phys. Fluids 6, 962972.10.1063/1.868328
Zilitinkevich, S. S. 1971 On the turbulence and diffusion under free convection conditions. Izv. Akad. Nauk. Ser. Fiz. Atmos. Okeana 7, 12631269.
Zilitinkevich, S. S., Hunt, J. C. R., Esau, I. N., Grachev, A. A., Lalas, D. P., Akylas, E., Tombrou, M., Fairall, C. W., Fernando, H. J. S., Baklanov, A. A. & Joffre, S. M. 2006 The influence of large convective eddies on the surface-layer turbulence. Q. J. R. Meteorol. Soc. 132, 14231456.10.1256/qj.05.79
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Multi-point Monin–Obukhov similarity in the convective atmospheric surface layer using matched asymptotic expansions

  • Chenning Tong (a1) and Mengjie Ding (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed