An Explicit Algorithm for Calculating Surface-Layer Parameters in Convective Conditions Derived from a Turbulence Closure Model

Lech Łobocki Department of Environmental Engineering, Warsaw University of Technology, Warsaw, Poland

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Abstract

Equations relating surface-layer parameters (or, surface fluxes) to differences of wind speed, temperature, and humidity between two levels can be obtained either via integrating empirical flux–gradient relationships or via integrating turbulence closure models. These equations form a nonlinear set, which is usually solved with iterative techniques. Under some restrictions imposed on the choice of computational heights, it is possible to apply a precomputed, explicit approximated solution of this equation system instead. This approach is desired in certain situations, for example, for massive computations in numerical meteorological models implemented on vector or parallel computer architecture. This paper presents a systematic derivation of such an explicit algorithm, valid for the unstable range, from the popular Mellor–Yamada Level 2 model. Special attention is paid to the consistency with asymptotic properties of the model and to providing smooth transition to the free-convection regime throughout the entire range of light-wind, convective conditions. The proposed solution is mainly intended for use in modeling. The validity of the algorithm is restricted to a specified range of the level-heights ratio, corresponding to typical values of roughness parameters and lowest grid levels used in atmospheric models. Because the relationship between the bulk Richardson number and the Monin–Obukhov stability parameter is not universal, the paper focuses on the method of deriving the practical approximations rather than on the particular results.

Corresponding author address: Dr. Lech Łobocki, Institute of Environmental Engineering Systems, Warsaw University of Technology, Nowowiejska 20, 00-653 Warsaw, Poland.

lech.lobocki@is.pw.edu.pl

Abstract

Equations relating surface-layer parameters (or, surface fluxes) to differences of wind speed, temperature, and humidity between two levels can be obtained either via integrating empirical flux–gradient relationships or via integrating turbulence closure models. These equations form a nonlinear set, which is usually solved with iterative techniques. Under some restrictions imposed on the choice of computational heights, it is possible to apply a precomputed, explicit approximated solution of this equation system instead. This approach is desired in certain situations, for example, for massive computations in numerical meteorological models implemented on vector or parallel computer architecture. This paper presents a systematic derivation of such an explicit algorithm, valid for the unstable range, from the popular Mellor–Yamada Level 2 model. Special attention is paid to the consistency with asymptotic properties of the model and to providing smooth transition to the free-convection regime throughout the entire range of light-wind, convective conditions. The proposed solution is mainly intended for use in modeling. The validity of the algorithm is restricted to a specified range of the level-heights ratio, corresponding to typical values of roughness parameters and lowest grid levels used in atmospheric models. Because the relationship between the bulk Richardson number and the Monin–Obukhov stability parameter is not universal, the paper focuses on the method of deriving the practical approximations rather than on the particular results.

Corresponding author address: Dr. Lech Łobocki, Institute of Environmental Engineering Systems, Warsaw University of Technology, Nowowiejska 20, 00-653 Warsaw, Poland.

lech.lobocki@is.pw.edu.pl

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