On determining characteristic velocities for calculation of PDF-averaged cloud droplet number concentration, effective radius and autoconversion rate

Description of cloud processes (such as droplet activation and autoconversion of cloudwater to rain) in regional and global models is often done by applying parameterizations that require information of sub-grid scale vertical velocity. The nonlinear link between updraft velocity, CDNC and all subsequent cloud processes necessitates the integration of parameterizations over the distribution of updrafts implied by the turbulent kinetic energy spectrum; although rigorous, this method is computationally expensive. Instead, it is common practice to calculate cloud droplet number concentration (CDNC) using the average updraft velocity “diagnosed” from grid-scale turbulent kinetic energy. This CDNC is then applied to calculate cloud effective radius, optical depth and autoconversion rate. Although simple, this “single-updraft” method is subject to biases, as the selection of updraft velocity may not always reflect the PDF-averaged value of the cloud property in question.

This study aims at resolving the above problem, and determines the “characteristic” velocity, w*, for calculating CDNC that corresponds to PDF-averaged values of droplet number, autoconversion and cloud droplet effective radius. We first analytically determine the characteristic velocity for a Gaussian vertical velocity distribution and a power law dependence of CDNC on updraft velocity (Twomey , 1959). We find w* that yields the PDF-average droplet number is very close to the average updraft velocity, which is in agreement with cloud droplet number closure studies using in-situ data (Conant et al., 2004; Meskhidze et al., 2005; Peng et al., 2005; Fountouksi et al., 2007). The characteristic velocity for computation of autoconversion rate, however, is much lower, about (50 - 60%) of the average updraft velocity, depending on the autoconversion parameterization used (Khairoutdinov and Kogan, 2000; Liu and Daum, 2004). This means that if the average updraft velocity is used to compute CDNC and autoconversion rate (as is done currently in GCM studies), the latter is systematically underestimated by a factor of 4 to 5. This can partially account for the large ”tuning” factor (~40) required for autoconversion parameterizations to yield realistic autoconversion timescales in GCM simulations.

Repeating the above analysis with a state-of-the-art cloud droplet formation parameterization does not change the above conclusions; in fact, the value of characteristic velocity for computing CDNC and autoconversion rate (consistent with PDF-averaged values) agree with the simplified power-law approach to within 15 % for atmospherically-relevant aerosol.