Turbulence closure and self-similarity of atmospheric turbulence: Stable stratification

Since Richardson (1920), it has been generally agreed that the effect of stratification on shear-generated turbulence is determined by the gradient Richardson number Ri. The concept of Ri-similarity postulates that dimensionless characteristics of turbulence are universal functions of Ri. Monin and Obukhov (1954) proposed a widely recognised Monin-Obukhov similarity theory (MOST) for the atmospheric surface layer with approximately height-constant vertical turbulent fluxes of buoyancy and momentum. MOST postulates that dimensionless characteristics of the surface-layer turbulence are fully determined by z/L, but the Ri-similarity is equally applicable to any stably stratified, shear-generated, locally balanced turbulence. The resistance and heat/mass transfer laws resulting from these analyses have never been questioned, and still form the bases for parameterization of the surface-layer turbulence in modelling applications. We demonstrate that both the above assumptions are inconsistent with observational evidence and contradict modern theory. We also employ the new Energy- & Flux-Budget (EFB) turbulence closure theory (Zilitinkevich et al, 2007, 2008, 2009, 2013) to examine applicability of MOST and conventional closure models to very stable stratification. We show that Ri-similarity is inherent to any locally balanced, stably stratified sheared flow. MOST is justified in both the surface and boundary layers, provided that generation of TKE is locally balanced by its dissipation and conversion into turbulent potential energy. Besides these quite expected conclusions, EFB theory has given insight into the general nature of self-similarity. That the dimensionless combinations of turbulent parameters (e.g., energies, fluxes, eddy viscosity, eddy conductivity) are universal functions of Rif, Ri, or z/L is explained by the equivalence (with the accuracy of dimensionless constant factors) of the dissipation time scales for all second moments under consideration. Furthermore, it is understood that the concept of self-similarity is generally irrelevant to the third and higher moments. Widely recognised conclusions from MOST concerning the so-called z-less stratification regime are shown to be generally misleading. In particular, MOST prescribes that at high values of z/L, the turbulent Prandtl number should tend to a universal constant. However, extensive experimental data, as well as the EFB turbulence-closure theory, yield a very strong asymptotic Ri-dependence of Pr ≈ 4Ri, where the factor 4 is determined empirically. EFB theory has allowed an understanding of a long-term confusion in use of the phrase “strongly stable stratification”. In boundary-layer meteorology, it implies that the strongest stratifications achievable in the atmospheric turbulent boundary layer is z/L ~ 3-4 or Ri ~ 0.2-0.25. However, such values of Ri factually correspond to the weakly-stable stratification inherent to the strong-turbulence regime with turbulent Prandtl number PrT ~ 0.8, practically independent of Ri or z/L. The genuine strongly-stable stratification associated with the weak-turbulence regime (with Pr ≈ 4Ri) is observed only outside boundary layers, e.g., in the free atmosphere, where Ri varies typically from 1 to 103. It was just the above terminological confusion that has led to erroneous treatment of the widely recognised z-less stratification regime (factually associated with maximal z/L achievable in the surface layer) as the ultimate strongly-stable stratification regime. In imbalanced turbulence, no grounds exist to expect exact self-similarity. Moreover, traditional similarity criteria, such as, Rif, Ri or z/L, based on essentially local parameters, become strongly variable and uncertainly determined. To overcome this difficulty, we propose a concept of approximate similarity based on the “energy stratification criterion” defined as ∏, the ratio of potential to kinetic energy. In contrast to the traditional criteria, the new one is applicable to any stably stratified turbulence, in particular to the turbulence generated by orbital motions in long internal gravity waves (Zilitinkevich et al., 2009). In the steady state, the EFB theory is equivalent to traditional Rif - or Ri-similarity (and therefore consistent with MOST). However, in imbalanced turbulence, local similarity concepts fail, whereas ∏-similarity, based on the trustworthy prognostic values of turbulent energies, remains a reasonable approximation. This idea underlies a hierarchy of EFB turbulence closure models for research and operational modelling applications (Zilitinkevich et al., 2013).