Systematic Biases in the Microphysics and Thermodynamics of Numerical Models That Ignore Subgrid-Scale Variability

A grid box in a numerical model that ignores subgrid variability has biases in certain microphysical and thermodynamic quantities relative to the values that would be obtained if subgrid-scale variability were taken into account. The biases are important because they are systematic and hence have cumulative effects. Several types of biases are discussed in this paper. Namely, numerical models that employ convex autoconversion formulas underpredict (or, more precisely, never overpredict) autoconversion rates, and numerical models that use convex functions to diagnose specific liquid water content and temperature underpredict these latter quantities. One may call these biases the ‘‘grid box average autoconversion bias,’’ ‘‘grid box average liquid water content bias,’’ and ‘‘grid box average temperature bias,’’ respectively, because the biases arise when grid box average values are substituted into formulas valid at a point, not over an extended volume. The existence of these biases can be derived from Jensen’s inequality. To assess the magnitude of the biases, the authors analyze observations of boundary layer clouds. Often the biases are small, but the observations demonstrate that the biases can be large in important cases. In addition, the authors prove that the average liquid water content and temperature of an isolated, partly cloudy, constant-pressure volume of air cannot increase, even temporarily. The proof assumes that liquid water content can be written as a convex function of conserved variables with equal diffusivities. The temperature decrease is due to evaporative cooling as cloudy and clear air mix. More generally, the authors prove that if an isolated volume of fluid contains conserved scalars with equal diffusivities, then the average of any convex, twice-differentiable function of the conserved scalars cannot increase.