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

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.