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

摘要

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.