Prediction of gradient‐based similarity functions from the Mellor–Yamada model

Gradient‐based implicit similarity functions are derived theoretically from a few variants of the Mellor–Yamada algebraic turbulence‐closure model under assumptions of local equilibrium conditions in a stably stratified shear flow. The solutions are compared with empirical functions presented by Sorbjan. Good agreement is found in the range of gradient Richardson number Ri extending up to around 0.2. The gradient‐based scaling framework offers better accuracy and reliability than the traditional Monin–Obukhov framework, as it circumvents the problems of small values of fluxes and cross‐correlations. It is also possible to separate the specification of the master length‐scale from the calculation of similarity functions describing second moments and dissipation rate. The related discussion of model performance at higher Ri is included. It is unclear whether the current countermeasures against spurious flow decoupling reflect the observed features of turbulence in the very stable regime, due to the apparent breakdown of the Richardson–Kolmogorov energy cascade, as found by Grachev et al. We present a theoretical derivation of gradient‐based similarity functions from a turbulence‐closure model within the frameworks of three scaling systems and compare the results. This article shows an alternative, self‐consistent way to establish the form of similarity functions (in contrast to polynomial fitting to experimental data, which invites certain inconsistencies) and presents a way to achieve adjustment of closure constants in turbulence models. It also addresses the issue of critical Richardson number and master length‐scale specifications.