The Stability Functions and Realizability of the Turbulent Kinetic Energy\u2013Scalar Variance Closure for Moist Atmospheric Boundary-Layer Turbulence

Abstract

The problem of realizability of the second-order turbulence closure models (parametrization schemes) is addressed through the consideration of the so-called “stability functions”. The emphasis is on the turbulence kinetic energy–scalar variance (TKESV) closure scheme that carries prognostic transport equations for the turbulence kinetic energy (TKE) and for the variances and covariance of two quasi-conservative scalars suitable for describing moist atmospheric boundary-layer turbulence. Stability functions appear within the framework of truncated closure schemes, where (i) the Reynolds-stress and scalar-flux equations (and, within the framework of one-equation TKE schemes, also equations for scalar variances and covariance) are reduced to the diagnostic algebraic formulations by neglecting the substantial derivatives and the third-order transport terms, and (ii) simplified linear parametrizations of the pressure-scrambling terms are used. The stability functions are ill-behaved (tend to infinity or become negative) over a certain range of governing parameters, e.g., mean velocity shear and buoyancy gradient. Using the approach of Helfand and Labraga (J Atmos Sci 45:113–132, 1988), we develop regularized stability functions for the TKESV scheme that reveal no pathological behaviour over their entire parameter space. The physical meaning of the regularization procedure and its relation to non-linear parametrizations of the pressure-scrambling terms and to weak non-equilibrium hypothesis are discussed. Finally, realizability of turbulence closures is considered within a more general framework of the moments problem of the probability theory.