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 valu es that would be obtained if subgrid-scale variability were taken into acco unt. The biases are important because they are systematic and hence have cu mulative effects. Several types of biases are discussed in this paper. Name ly, numerical models that employ convex autoconversion formulas underpredic t (or, more precisely, never overpredict) autoconversion rates, and numeric al models that use convex functions to diagnose specific liquid water conte nt and temperature underpredict these latter quantities. One may call these biases the "grid box average autoconversion bias,'' "grid box average liqu id water content bias, '' and "grid box average temperature bias, '' respec tively, because the biases arise when grid box average values are substitut ed into formulas valid at a point, not over an extended volume. The existen ce 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 dem onstrate that the biases can be large in important cases. In addition, the authors prove that the average liquid water content and te mperature of an isolated, partly cloudy, constant-pressure volume of air ca nnot increase, even temporarily. The proof assumes that liquid water conten t can be written as a convex function of conserved variables with equal dif fusivities. The temperature decrease is due to evaporative cooling as cloud y and clear air mix. More generally, the authors prove that if an isolated volume of fluid contains conserved scalars with equal diffusivities, then t he average of any convex, twice-differentiable function of the conserved sc alars cannot increase.