A numerical model that ignores subgrid-scale variability has biases in any quantity that is diagnosed from a convex or concave function. The biases are important because they are systematic and hence have cumulative effects. We will discuss several examples of such biases drawn from the study of atmospheric clouds. Namely, numerical models that employ convex autoconversion formulas underpredict drizzle formation rates, and numerical models that diagnose liquid water content and temperature underpredict these latter quantities. 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. The biases could be largely eliminated by accounting for subgrid variability using simplified probability density functions (PDFs).

Finally, we will prove that turbulent mixing cannot cause the average of a convex function of conserved scalars to increase with time, even temporarily. An example is that mixing of cloudy and clear air cannot cause average liquid water to increase.