| description abstract | When a turbulence closure model is used for cloud simulations, the effect of turbulence-scale saturation must be taken into account in determining the turbulent buoyancy flux. Sommeria and Deardorff derived statistical relations through which the fractional saturation or cloudiness and the mean liquid water specific humidity can be calculated from the moments of conservative thermodynamic quantities, such as total water specific humidity and liquid water potential temperature. In deriving these relations, however, they had to rely on assumptions that were only empirically justified. Mellor showed, by a direct integration of the probability density function, that the assumptions which Sommeria and Deardorff used were not necessary for deriving those relations. But Mellor's direct integration method had to be carried out for each different relation, and can not be generalized to other derivations. In the present paper, the same relations are derived from theorems on Gaussian distributions by introducing a new quantity, qx, which converts the original two-parameter turbulence-scale saturation parameterization problem to a single-parameter problem. A systematic solution method is established based on these theorems. Quantities involving liquid water specific humidity can be obtained through this parameterization from those conservative variables for which predictive equations are available and for which Gaussian distributions are presumed. | |
