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. 2025 Jul 15;15(1):25632.
doi: 10.1038/s41598-025-10815-3.

Towards definitive functional forms for Monin-Obukhov similarity functions in stable and very stable surface layers

G Casasanta  1 , M Conte  2 , R Sozzi  3 , A Cecilia  1 , I Petenko  1 , S Argentini  1
Affiliations

Towards definitive functional forms for Monin-Obukhov similarity functions in stable and very stable surface layers

G Casasanta et al. Sci Rep. .

Abstract

Monin-Obukhov similarity functions are key components in all numerical models of atmospheric flows, yet their exact functional forms remain a matter of debate. Existing formulations, typically derived through empirical curve fitting, often result in inconsistencies and physically questionable behaviour, particularly under stable and very stable conditions. This paper bridges the well-established Monin-Obukhov Similarity Theory (MOST) with the more recent Energy and Flux Budget (EFB) second-order closure to analytically derive the functional forms of all MOST similarity functions under stable conditions. In addition, it identifies and formalises a set of constrain relationships that characterise the physical connection among the universal functions, highlighting their interdependences. Our results aim to advance the theoretical understanding of the stable surface layer and offer a pathway toward more physically grounded turbulence parameterizations, with implications for improving the performance of numerical weather prediction, air quality, ocean, and climate models.

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Conflict of interest statement

Competing interests: The authors declare no competing interests.

Figures

Fig. 1
Fig. 1
Evolution of the four formula image presented in section “Universal function for the standard deviation of potential temperature” as a function of the stability parameter formula image; discrepancies exceed two orders of magnitude at high stability.
Fig. 2
Fig. 2
Evolution of formula image as a function of formula image for the four formula image formulations reported in Table 1; only the Businger–Dyer formulation converges to the expected value.
Fig. 3
Fig. 3
Behaviour of the three normalized vertical fluxes of momentum (left panel) and potential temperature (right panel) as a function of formula image; the contrasting trend of formula image compared to both formula image and formula image reflects the TKE exchange among components.
Fig. 4
Fig. 4
Behaviour of the universal functions for standard deviations (left panel) and potential temperature (right panel) as a function of ζ; according to the intercomponent exchange of TKE, wind standard deviations tend towards horizontal isotropy as stability increases.
Fig. 5
Fig. 5
Evolution of formula image and formula image as a function of formula image; linear and two-thirds-power scalings support the functional forms proposed in previous literature.
See this image and copyright information in PMC

References

    1. Monin, A. & Obukhov, A. M. Basic laws of turbulent mixing in the surface layer of the atmosphere. Contrib. Geophys. Inst. Acad. Sci. USSR1954, 163–187 (1954).
    1. Edwards, J. M., Beljaars, A. C. M., Holtslag, A. A. M. & Lock, A. P. Representation of Boundary-Layer processes in numerical weather prediction and climate models. Boundary-Layer Meteorol.177, 511–539. 10.1007/s10546-020-00530-z (2020).
    1. Falasca, S. et al. Sensitivity of near-surface meteorology to PBL schemes in WRF simulations in a port-industrial area with complex terrain. Atmos. Res.264, 105824. 10.1016/j.atmosres.2021.105824 (2021).
    1. Tennekes, H. Similarity relations, scaling laws and spectral dynamics. Atmos. Turbul. Air Pollut. Model A1981, 37–68. 10.1007/978-94-010-9112-1_2 (1982).
    1. Nieuwstadt, F. T. M. The turbulent structure of the stable, nocturnal boundary layer. J. Atmos. Sci.41, 2202–2216. (1984).

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