Dmitrii Wironow, German Weather Service, Offenbach am Main, Germany


The (So-Called) Stability Functions of Truncated Second-Order Turbulence Closure Models:
Origin of Ill Behaviour and Remedial Measures


The problem of realizability of the second-order turbulence closure models (parameterization schemes) is addressed through the consideration of the so-called "stability functions" (Mellor and Yamada 1974, 1982). Stability functions appear within the framework of truncated turbulence closure schemes, where (i) the Reynolds-stress and scalar-flux (and often also scalar-variance) equations are reduced to the diagnostic algebraic formulations by neglecting the substantial derivatives and the third-order transport (diffusion) terms, and (ii) the linear parameterizations (in the second-order moments) of the pressure-scrambling terms are used. The stability functions are known to be ill-behaved over a part of their parameter space. They become infinite in the case of growing turbulence, where the actual value of the turbulence kinetic energy (TKE) is smaller than the equilibrium TKE corresponding to the steady-state production-dissipation balance. Helfand and Labraga (1988) analyzed the stability functions of the one-equation TKE scheme (the level 2.5 scheme in the nomenclature of Mellor and Yamada) that is very popular in geophysical applications. Using rather plausible physical arguments, they developed "regularized'' stability functions that reveal no pathological behaviour over their entire parameter space.

We extend the approach of Helfand and Labraga to the TKE - Scalar Variance (TKESV) scheme (the level 3 scheme in the nomenclature of Mellor and Yamada) that carries prognostic equations for the variances and covariance of scalar quantities (temperature and humidity in the atmosphere, temperature and salinity in the ocean) with due regard for the third-order transport. The cause of pathological behaviour of the stability functions of the TKESV schemes is analysed. The regularized stability functions for the TKESV scheme are developed that are well-behaved at any values of their governing parameters (gradient Richardson number, dimensionless velocity shear, and dimensionless scalar variances characteristic of the turbulence potential energy). The Helfand and Labraga regularization procedure is compared to some other realizability constraint techniques, and physical interpretation of the various techniques is offered. Those techniques are placed into a more general context of the problem of moments.


Monday November 19, 11:15 am


Rossbysalen C609, Arrhenius laboratory, Svante Arrhenius väg 16C, 6th floor