Implications of Nonlocal Transport and Conditionally Averaged Statistics on Monin–Obukhov Similarity Theory and Townsend’s Attached Eddy Hypothesis

Qi Li Department of Earth and Environmental Engineering, Columbia University, New York, New York

Search for other papers by Qi Li in
Current site
Google Scholar
PubMed
Close
,
Pierre Gentine Department of Earth and Environmental Engineering, Columbia University, New York, New York

Search for other papers by Pierre Gentine in
Current site
Google Scholar
PubMed
Close
,
Juan Pedro Mellado Max Planck Institute for Meteorology, Hamburg, Germany

Search for other papers by Juan Pedro Mellado in
Current site
Google Scholar
PubMed
Close
, and
Kaighin A. McColl Harvard John A. Paulson School of Engineering and Applied Sciences, and Department of Earth and Planetary Sciences, Harvard University, Cambridge, Massachusetts

Search for other papers by Kaighin A. McColl in
Current site
Google Scholar
PubMed
Close
Restricted access

We are aware of a technical issue preventing figures and tables from showing in some newly published articles in the full-text HTML view.
While we are resolving the problem, please use the online PDF version of these articles to view figures and tables.

Abstract

According to Townsend’s hypothesis, so-called wall-attached eddies are the main contributors to turbulent transport in the atmospheric surface layer (ASL). This is also one of the main assumptions of Monin–Obukhov similarity theory (MOST). However, previous evidence seems to indicate that outer-scale eddies can impact the ASL, resulting in deviations from the classic MOST scaling. We conduct large-eddy simulations and direct numerical simulations of a dry convective boundary layer to investigate the impact of coherent structures on the ASL. A height-dependent passive tracer enables coherent structure detection and conditional analysis based on updrafts and subsidence. The MOST similarity functions computed from the simulation results indicate a larger deviation of the momentum similarity function ϕm from classical scaling relationships compared to the temperature similarity function ϕh. The conditional-averaged ϕm for updrafts and subsidence are similar, indicating strong interactions between the inner and outer layers. However, ϕh conditioned on subsidence follows the mixed-layer scaling, while its updraft counterpart is well predicted by MOST. Updrafts are the dominant contributors to the transport of momentum and temperature. Subsidence, which comprises eddies that originate from the outer layer, contributes increasingly to the transport of temperature with increasing instability. However, u′ of different signs are distributed symmetrically in subsidence unlike the predominantly negative θ′ as instability increases. Thus, the spatial patterns of uw′ differ compared to θw′ in regions of subsidence. These results depict the mechanisms for departure from the MOST scaling, which is related to the stronger role of subsidence.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Qi Li, liqi1026@gmail.com

Abstract

According to Townsend’s hypothesis, so-called wall-attached eddies are the main contributors to turbulent transport in the atmospheric surface layer (ASL). This is also one of the main assumptions of Monin–Obukhov similarity theory (MOST). However, previous evidence seems to indicate that outer-scale eddies can impact the ASL, resulting in deviations from the classic MOST scaling. We conduct large-eddy simulations and direct numerical simulations of a dry convective boundary layer to investigate the impact of coherent structures on the ASL. A height-dependent passive tracer enables coherent structure detection and conditional analysis based on updrafts and subsidence. The MOST similarity functions computed from the simulation results indicate a larger deviation of the momentum similarity function ϕm from classical scaling relationships compared to the temperature similarity function ϕh. The conditional-averaged ϕm for updrafts and subsidence are similar, indicating strong interactions between the inner and outer layers. However, ϕh conditioned on subsidence follows the mixed-layer scaling, while its updraft counterpart is well predicted by MOST. Updrafts are the dominant contributors to the transport of momentum and temperature. Subsidence, which comprises eddies that originate from the outer layer, contributes increasingly to the transport of temperature with increasing instability. However, u′ of different signs are distributed symmetrically in subsidence unlike the predominantly negative θ′ as instability increases. Thus, the spatial patterns of uw′ differ compared to θw′ in regions of subsidence. These results depict the mechanisms for departure from the MOST scaling, which is related to the stronger role of subsidence.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Qi Li, liqi1026@gmail.com
Save
  • Adrian, R. J., 2007: Hairpin vortex organization in wall turbulence. Phys. Fluids, 19, 041301, https://doi.org/10.1063/1.2717527.

  • Albertson, J. D., 1996: Large eddy simulation of land–atmosphere interaction. Ph.D. dissertation, University of California, Davis, 185 pp.

  • Albertson, J. D., and M. B. Parlange, 1999: Surface length scales and shear stress: Implications for land-atmosphere interaction over complex terrain. Water Resour. Res., 35, 21212132, https://doi.org/10.1029/1999WR900094.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Andreas, E. L, and B. B. Hicks, 2002: Comments on “Critical test of the validity of Monin–Obukhov similarity during convective conditions.” J. Atmos. Sci., 59, 26052607, https://doi.org/10.1175/1520-0469(2002)059<2605:COCTOT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Antonia, R. A., and A. J. Chambers, 1978: Note on the temperature ramp structure in the marine surface layer. Bound.-Layer Meteor., 15, 347355, https://doi.org/10.1007/BF02652606.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Antonia, R. A., A. J. Chambers, C. A. Friehe, and C. W. Van Atta, 1979: Temperature ramps in the atmospheric surface layer. J. Atmos. Sci., 36, 99108, https://doi.org/10.1175/1520-0469(1979)036<0099:TRITAS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Baars, W. J., N. Hutchins, and I. Marusic, 2017: Reynolds number trend of hierarchies and scale interactions in turbulent boundary layers. Philos. Trans. Roy. Soc. London, 375A, 20160077, https://doi.org/10.1098/rsta.2016.0077.

    • Search Google Scholar
    • Export Citation
  • Banerjee, T., D. Li, J.-Y. Juang, and G. Katul, 2016: A spectral budget model for the longitudinal turbulent velocity in the stable atmospheric surface layer. J. Atmos. Sci., 73, 145166, https://doi.org/10.1175/JAS-D-15-0066.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Beljaars, A. C. M., and A. A. M. Holtslag, 1991: Flux parameterization over land surfaces for atmospheric models. J. Appl. Meteor., 30, 327341, https://doi.org/10.1175/1520-0450(1991)030<0327:FPOLSF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Berg, L. K., and R. B. Stull, 2004: Parameterization of joint frequency distributions of potential temperature and water vapor mixing ratio in the daytime convective boundary layer. J. Atmos. Sci., 61, 813828, https://doi.org/10.1175/1520-0469(2004)061<0813:POJFDO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Berkooz, G., P. Holmes, and J. L. Lumley, 1993: The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech., 25, 539575, https://doi.org/10.1146/annurev.fl.25.010193.002543.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Betts, A. K., and C. Jakob, 2002: Evaluation of the diurnal cycle of precipitation, surface thermodynamics, and surface fluxes in the ECMWF model using LBA data. J. Geophys. Res., 107, 8045, https://doi.org/10.1029/2001JD000427.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Betts, R. A., 2000: Offset of the potential carbon sink from boreal forestation by decreases in surface albedo. Nature, 408, 187190, https://doi.org/10.1038/35041545.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bose, S. T., P. Moin, and D. You, 2010: Grid-independent large-eddy simulation using explicit filtering. Phys. Fluids, 22, 105103, https://doi.org/10.1063/1.3485774.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bou-Zeid, E., C. Meneveau, and M. Parlange, 2005: A scale-dependent Lagrangian dynamic model for large eddy simulation of complex turbulent flows. Phys. Fluids, 17, 025105, https://doi.org/10.1063/1.1839152.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Brasseur, J. G., and T. Wei, 2010: Designing large-eddy simulation of the turbulent boundary layer to capture law-of-the-wall scaling. Phys. Fluids, 22, 021303, https://doi.org/10.1063/1.3319073.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Brunet, Y., and M. R. Irvine, 2000: The control of coherent eddies in vegetation canopies: Streamwise structure spacing, canopy shear scale and atmospheric stability. Bound.-Layer Meteor., 94, 139163, https://doi.org/10.1023/A:1002406616227.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Businger, J. A., 1973: A note on free convection. Bound.-Layer Meteor., 4, 323326, https://doi.org/10.1007/BF02265241.

  • Businger, J. A., and S. P. Oncley, 1990: Flux measurement with conditional sampling. J. Atmos. Oceanic Technol., 7, 349352, https://doi.org/10.1175/1520-0426(1990)007<0349:FMWCS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Businger, J. A., J. C. Wyngaard, Y. Izumi, and E. F. Bradley, 1971: Flux-profile relationships in the atmospheric surface layer. J. Atmos. Sci., 28, 181189, https://doi.org/10.1175/1520-0469(1971)028<0181:FPRITA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Calaf, M., M. Hultmark, H. J. Oldroyd, V. Simeonov, and M. B. Parlange, 2013: Coherent structures and the k−1 spectral behaviour. Phys. Fluids, 25, 125107, https://doi.org/10.1063/1.4834436.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cancelli, D. M., M. Chamecki, and N. L. Dias, 2014: A large-eddy simulation study of scalar dissimilarity in the convective atmospheric boundary layer. J. Atmos. Sci., 71, 315, https://doi.org/10.1175/JAS-D-13-0113.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chung, D., and G. Matheou, 2012: Direct numerical simulation of stationary homogeneous stratified sheared turbulence. J. Fluid Mech., 696, 434467, https://doi.org/10.1017/jfm.2012.59.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Coleman, G. N., J. H. Ferziger, and P. R. Spalart, 1994: A numerical study of the convective boundary layer. Bound.-Layer Meteor., 70, 247272, https://doi.org/10.1007/BF00709121.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Couvreux, F., F. Hourdin, and C. Rio, 2010: Resolved versus parametrized boundary-layer plumes. Part I: A parametrization-oriented conditional sampling in large-eddy simulations. Bound.-Layer Meteor., 134, 441458, https://doi.org/10.1007/s10546-009-9456-5.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Delire, C., J. A. Foley, and S. Thompson, 2004: Long-tern variability in a coupled atmosphere–biosphere model. J. Climate, 17, 39473959, https://doi.org/10.1175/1520-0442(2004)017<3947:LVIACA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Dyer, A. J., 1974: A review of flux-profile relationships. Bound.-Layer Meteor., 7, 363372, https://doi.org/10.1007/BF00240838.

  • Farge, M., G. Pellegrino, and K. Schneider, 2001: Coherent vortex extraction in 3D turbulent flows using orthogonal wavelets. Phys. Rev. Lett., 87, 054501, https://doi.org/10.1103/PhysRevLett.87.054501.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fedorovich, E., and A. Shapiro, 2009: Turbulent natural convection along a vertical plate immersed in a stably stratified fluid. J. Fluid Mech., 636, 4157, https://doi.org/10.1017/S0022112009007757.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Findell, K. L., P. Gentine, B. R. Lintner, and C. Kerr, 2011: Probability of afternoon precipitation in eastern United States and Mexico enhanced by high evaporation. Nat. Geosci., 4, 434439, https://doi.org/10.1038/NGEO1174.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Foken, T., 2006: 50 years of the Monin–Obukhov similarity theory. Bound.-Layer Meteor., 119, 431447, https://doi.org/10.1007/s10546-006-9048-6.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gao, Z., H. Liu, E. S. Russell, J. Huang, T. Foken, and S. P. Oncley, 2016: Large eddies modulating flux convergence and divergence in a disturbed unstable atmospheric surface layer. J. Geophys. Res. Atmos., 121, 14751492, https://doi.org/10.1002/2015JD024529.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Garcia, J. R., and J. P. Mellado, 2014: The two-layer structure of the entrainment zone in the convective boundary layer. J. Atmos. Sci., 71, 19351955, https://doi.org/10.1175/JAS-D-13-0148.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gentine, P., D. Entekhabi, A. Chehbouni, G. Boulet, and B. Duchemin, 2007: Analysis of evaporative fraction diurnal behaviour. Agric. For. Meteor., 143, 1329, https://doi.org/10.1016/j.agrformet.2006.11.002.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gentine, P., D. Entekhabi, and J. Polcher, 2011a: The diurnal behavior of evaporative fraction in the soil–vegetation–atmospheric boundary layer continuum. J. Hydrometeor., 12, 15301546, https://doi.org/10.1175/2011JHM1261.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gentine, P., J. Polcher, and D. Entekhabi, 2011b: Harmonic propagation of variability in surface energy balance within a coupled soil-vegetation-atmosphere system. Water Resour. Res., 47, W05525, https://doi.org/10.1029/2010WR009268.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gentine, P., A. A. M. Holtslag, F. D’Andrea, and M. Ek, 2013: Surface and atmospheric controls on the onset of moist convection over land. J. Hydrometeor., 14, 14431462, https://doi.org/10.1175/JHM-D-12-0137.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ghannam, K., T. Duman, S. T. Salesky, M. Chamecki, and G. Katul, 2017: The non-local character of turbulence asymmetry in the convective atmospheric boundary layer. Quart. J. Roy. Meteor. Soc., 143, 494507, https://doi.org/10.1002/qj.2937.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Green, M. A., C. W. Rowley, and G. Haller, 2007: Detection of Lagrangian coherent structures in three-dimensional turbulence. J. Fluid Mech., 572, 111120, https://doi.org/10.1017/S0022112006003648.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Haller, G., 2015: Lagrangian coherent structures. Annu. Rev. Fluid Mech., 47, 137162, https://doi.org/10.1146/annurev-fluid-010313-141322.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Haller, G., and G. Yuan, 2000: Lagrangian coherent structures and mixing in two-dimensional turbulence. Physica D, 147, 352370, https://doi.org/10.1016/S0167-2789(00)00142-1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Head, M. R., and P. Bandyopadhyay, 1981: New aspects of turbulent boundary-layer structure. J. Fluid Mech., 107, 297338, https://doi.org/10.1017/S0022112081001791.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Högström, U., 1988: Non-dimensional wind and temperature profiles in the atmospheric surface layer: A re-evaluation. Bound.-Layer Meteor., 42, 5578, https://doi.org/10.1007/BF00119875.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hommema, S. E., and R. J. Adrian, 2003: Packet structure of surface eddies in the atmospheric boundary layer. Bound.-Layer Meteor., 106, 147170, https://doi.org/10.1023/A:1020868132429.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Huang, J., M. Cassiani, and J. D. Albertson, 2009: Analysis of coherent structures within the atmospheric boundary layer. Bound.-Layer Meteor., 131, 147171, https://doi.org/10.1007/s10546-009-9357-7.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hunt, J. C. R., and J. F. Morrison, 2000: Eddy structure in turbulent boundary layers. Eur. J. Mech., 19B, 673694, https://doi.org/10.1016/S0997-7546(00)00129-1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Johansson, C., A.-S. Smedman, U. Högström, J. G. Brasseur, and S. Khanna, 2001: Critical test of the validity of Monin–Obukhov similarity during convective conditions. J. Atmos. Sci., 58, 15491566, https://doi.org/10.1175/1520-0469(2001)058<1549:CTOTVO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Jonker, H. J. J., P. G. Duynkerke, and J. W. M. Cuijpers, 1999: Mesoscale fluctuations in scalars generated by boundary layer convection. J. Atmos. Sci., 56, 801808, https://doi.org/10.1175/1520-0469(1999)056<0801:MFISGB>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kader, B. A., and A. M. Yaglom, 1990: Mean fields and fluctuation moments in unstably stratified turbulent boundary layers. J. Fluid Mech., 212, 637662, https://doi.org/10.1017/S0022112090002129.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kaimal, J. C., and J. J. Finnigan, 1994: Atmospheric Boundary Layer Flows: Their Structure and Measurement. Oxford University Press, 289 pp.

    • Crossref
    • Export Citation
  • Karrasch, D., and G. Haller, 2013: Do finite-size Lyapunov exponents detect coherent structures? Chaos, 23, 043126, https://doi.org/10.1063/1.4837075.

  • Katul, G. G., 1994: A model for sensible heat flux probability density function for near-neutral and slightly-stable atmospheric flows. Bound.-Layer Meteor., 71, 120, https://doi.org/10.1007/BF00709217.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Katul, G. G., J. D. Albertson, M. B. Parlange, C.-I. Hsieh, P. S. Conklin, J. T. Sigmon, and K. R. Knoerr, 1996: The “inactive” eddy motion and the large-scale turbulent pressure fluctuations in the dynamic sublayer. J. Atmos. Sci., 53, 25122524, https://doi.org/10.1175/1520-0469(1996)053<2512:TEMATL>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Katul, G. G., A. Porporato, E. Daly, A. C. Oishi, H. S. Kim, P. C. Stoy, J. Y. Juang, and M. B. Siqueira, 2007: On the spectrum of soil moisture from hourly to interannual scales. Water Resour. Res., 43, W05428, https://doi.org/10.1029/2006WR005356.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Katul, G. G., A. G. Konings, and A. Porporato, 2011: Mean velocity profile in a sheared and thermally stratified atmospheric boundary layer. Phys. Rev. Lett., 107, 268502, https://doi.org/10.1103/PhysRevLett.107.268502.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Katul, G. G., D. Li, M. Chameki, and E. Bou-Zeid, 2013: Mean scalar concentration profile in a sheared and thermally stratified atmospheric surface layer. Phys. Rev., 87E, 023004, https://doi.org/10.1103/PhysRevE.87.023004.

    • Search Google Scholar
    • Export Citation
  • Katul, G. G., A. Porporato, S. Shah, and E. Bou-Zeid, 2014: Two phenomenological constants explain similarity laws in stably stratified turbulence. Phys. Rev., 89E, 023007, https://doi.org/10.1103/PhysRevE.89.023007.

    • Search Google Scholar
    • Export Citation
  • Khanna, S., and J. G. Brasseur, 1997: Analysis of Monin–Obukhov similarity from large-eddy simulation. J. Fluid Mech., 345, 251286, https://doi.org/10.1017/S0022112097006277.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Khanna, S., and J. G. Brasseur, 1998: Three-dimensional buoyancy- and shear-induced local structure of the atmospheric boundary layer. J. Atmos. Sci., 55, 710743, https://doi.org/10.1175/1520-0469(1998)055<0710:TDBASI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kumar, V., J. Kleissl, C. Meneveau, and M. B. Parlange, 2006: Large-eddy simulation of a diurnal cycle of the atmospheric boundary layer: Atmospheric stability and scaling issues. Water Resour. Res., 42, W06D09, https://doi.org/10.1029/2005WR004651.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Laubach, J., and K. G. McNaughton, 2009: Scaling properties of temperature spectra and heat-flux cospectra in the surface friction layer beneath an unstable outer layer. Bound.-Layer Meteor., 133, 219252, https://doi.org/10.1007/s10546-009-9422-2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lenschow, D. H., and P. L. Stephens, 1980: The role of thermals in the convective boundary layer. Bound.-Layer Meteor., 19, 509532, https://doi.org/10.1007/BF00122351.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Li, D., and E. Bou-Zeid, 2011: Coherent structures and the dissimilarity of turbulent transport of momentum and scalars in the unstable atmospheric surface layer. Bound.-Layer Meteor., 140, 243262, https://doi.org/10.1007/s10546-011-9613-5.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Li, D., G. G. Katul, and E. Bou-Zeid, 2012: Mean velocity and temperature profiles in a sheared diabatic turbulent boundary layer. Phys. Fluids, 24, 105105, https://doi.org/10.1063/1.4757660.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Li, Q., E. Bou-Zeid, W. Anderson, S. Grimmond, and M. Hultmark, 2016: Quality and reliability of LES of convective scalar transfer at high Reynolds numbers. Int. J. Heat Mass Transfer, 102, 959970, https://doi.org/10.1016/j.ijheatmasstransfer.2016.06.093.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mahrt, L., 1991: Eddy asymmetry in the sheared heated boundary layer. J. Atmos. Sci., 48, 472492, https://doi.org/10.1175/1520-0469(1991)048<0472:EAITSH>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Maronga, B., and J. Reuder, 2017: On the formulation and universality of Monin–Obukhov similarity functions for mean gradients and standard deviations in the unstable surface layer: Results from surface-layer-resolving large-eddy simulations. J. Atmos. Sci., 74, 9891010, https://doi.org/10.1175/JAS-D-16-0186.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Marusic, I., R. Mathis, and N. Hutchins: 2010: Predictive model for wall-bounded turbulent flow. Science, 329, 193196, https://doi.org/10.1126/science.1188765.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mathis, R., N. Hutchins, and I. Marusic, 2011: A predictive inner–outer model for streamwise turbulence statistics in wall-bounded flows. J. Fluid Mech., 681, 537566, https://doi.org/10.1017/jfm.2011.216.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • McColl, K. A., G. G. Katul, P. Gentine, and D. Entekhabi, 2016: Mean-velocity profile of smooth channel flow explained by a cospectral budget model with wall-blockage. Phys. Fluids, 28, 035107, https://doi.org/10.1063/1.4943599.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • McColl, K. A., C. C. van Heerwaarden, G. G. Katul, P. Gentine, and D. Entekhabi, 2017: Role of large eddies in the breakdown of the Reynolds analogy in an idealized mildly unstable atmospheric surface layer. Quart. J. Roy. Meteor. Soc., 143, 21822197, https://doi.org/10.1002/qj.3077.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • McNaughton, K. G., 2004: Attached eddies and production spectra in the atmospheric logarithmic layer. Bound.-Layer Meteor., 111, 118, https://doi.org/10.1023/B:BOUN.0000010997.51745.0f.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • McNaughton, K. G., and Y. Brunet, 2002: Townsend’s hypothesis, coherent structures and Monin–Obukhov similarity. Bound.-Layer Meteor., 102, 161175, https://doi.org/10.1023/A:1013171312407.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • McNaughton, K. G., R. J. Clement, and J. B. Moncrieff, 2007: Scaling properties of velocity and temperature spectra above the surface friction layer in a convective atmospheric boundary layer. Nonlinear Processes Geophys., 14, 257271, https://doi.org/10.5194/npg-14-257-2007.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • McPhee, M. G., 1994: On the turbulent mixing length in the oceanic boundary layer. J. Phys. Oceanogr., 24, 20142031, https://doi.org/10.1175/1520-0485(1994)024<2014:OTTMLI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mellado, J. P., C. C. van Heerwaarden, and J. R. Garcia, 2016: Near-surface effects of free atmosphere stratification in free convection. Bound.-Layer Meteor., 159, 6995, https://doi.org/10.1007/s10546-015-0105-x.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mellado, J. P., C. S. Bretherton, B. Stevens, and M. C. Wyant, 2018: DNS and LES for simulating stratocumulus: Better together. J. Adv. Model. Earth Syst., 10, 14211438, https://doi.org/10.1029/2018MS001312.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mestayer, P. G., C. H. Gibson, M. F. Coantic, and A. S. Patel, 1976: Local anisotropy in heated and cooled turbulent boundary layers. Phys. Fluids, 19, 12791287, https://doi.org/10.1063/1.861649.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Moeng, C.-H., 1984: A large-eddy-simulation model for the study of planetary boundary-layer turbulence. J. Atmos. Sci., 41, 20522062, https://doi.org/10.1175/1520-0469(1984)041<2052:ALESMF>2.0.CO;2.

    • Crossref