Bell-Shaped Curve Captures Cloud System Variability

Lamb, P. J., University of Oklahoma

Cloud Distributions/Characterizations

Cloud Properties

Journal of Geophysical Research, 110, D18205, doi:10.1029/2005JD006158.


Figure 1. Reflectivity standard deviation PDFs, resampled as a function of timescale and contoured by equal values of probability, show an increase in variability with scale. The PDF modes lie mostly along the mean value line (black line), and the PDF distribution are largely symmetrical.


Figure 2. Blue diamonds denote relative error in grid volume mean precipitation flux, computed using a Gaussian PDF approximation. Black crosses represent error from assuming zero variability, i.e., the subgrid variability bias. A perfect representation of subgrid variability would have zero error.


Figure 1. Reflectivity standard deviation PDFs, resampled as a function of timescale and contoured by equal values of probability, show an increase in variability with scale. The PDF modes lie mostly along the mean value line (black line), and the PDF distribution are largely symmetrical.

Figure 2. Blue diamonds denote relative error in grid volume mean precipitation flux, computed using a Gaussian PDF approximation. Black crosses represent error from assuming zero variability, i.e., the subgrid variability bias. A perfect representation of subgrid variability would have zero error.

Standard climate and cloud models typically assume clouds and radiation are constant across each grid cell, or vary in a highly unrealistic way. We know, of course, that clouds and radiation are highly variable when grid cells are larger than about 1 km. Neglecting this variability inside model grid cells can systematically bias radiative and cloud microphysical quantities. Advanced models try to approximate this subgrid variability using analytic probability distribution functions (PDFs). The most famous PDF is the familiar bell-shaped curve, derived by the famous mathematician Gauss. Gauss' curve is the basis for much of statistics because the sum of independent random variables like dice rolls can be rigorously proved to converge to that curve. Thus, when faced with the necessity of guessing a PDF, a good first guess is always a Gaussian curve.

To evaluate how good a Gaussian PDF is for characterizing subgrid variability of clouds, we analyzed 2 years of cold season low-altitude cloud systems observed by the Millimeter-Wave Cloud Radar (MMCR) located at the Central Facility of the ARM Climate Research Facility's Southern Great Plains site. The radar data were classified into two low-cloud categories and stratified by scale. Scale is important because certain physical phenomena, such as internal gravity waves and shallow convection, have characteristic scales. In figure 1, the symmetric nature of the contoured PDFs at all scales suggests that the Gaussian curve well captures cloud variability.

Subgrid precipitation flux was calculated to evaluate the ability of the Gaussian PDF to approximate the observational PDFs. When the distribution mean and standard deviation parameters were well constrained (either diagnosed or predicted), a truncated Gaussian approximation accounted for a large degree of the subgrid variability and removed much of the bias in grid-mean quantities. Variability parameters are likely dependent on grid cell dimensions.

These results provide a degree of confidence in cloud schemes that rely on simple analytic PDFs for subgrid variability.