Characteristic updrafts for computing distribution-averaged cloud droplet number and stratocumulus cloud properties

A computationally effective framework is presented that addresses the contribution of subgrid-scale vertical velocity variations in predictions of cloud droplet number concentration (CDNC) in large-scale models. Central to the framework is the concept of a characteristic updraft velocity w(star), which yields CDNC value representative of integration over a probability density function (PDF) of updraft (i.e., positive vertical) velocity. Analytical formulations for w(star) are developed for computation of average CDNC over a Gaussian PDF using the Twomey droplet parameterization. The analytical relationship also agrees with numerical integrations using a state-of-the-art droplet activation parameterization. For situations where the variabilities of vertical velocity and liquid water content can be decoupled, the concept of w(star) is extended to the calculation of cloud properties and process rates that complements existing treatments for subgrid variability of liquid water content. It is shown that using the average updraft velocity (w) over bar (instead of w(star)) for calculations of N-d, r(e), and A (a common practice in atmospheric models) can overestimate PDF-averaged N-d by 10%, underestimate r(e) by 10%-15%, and significantly underpredict autoconversion rate between a factor of 2-10. The simple expressions of w(star) presented here can account for an important source of parameterization tuning in a physically based manner.