Jensen's inequality and systematic biases in numerical simulations

Numerical models that ignore subgrid-scale variability have 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 autoconversion rates, and numerical models that use a convex function to diagnose 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 local formulas, that is, formulas valid at a point, not over an extended volume. The biases are a consequence of Jensen's inequality.

To assess the magnitude of the biases, observations of boundary layer clouds are analyzed. Many times the biases are small, but the observations demonstrate that the biases can be large in important cases.