Jensen's inequality and systematic biases in numerical simulations

Dr. Vincent E. Larson,

Atmospheric Science Group, Department of Mathematical Sciences

University of Wisconsin --- Milwaukee

A numerical model that ignores subgrid variability has biases in certain microphysical and thermodynamic quantities. The biases are important because they are systematic and hence have cumulative effects. Several types of biases are discussed in this talk. 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).