January 4, 2021

Using Mathematical Rigor to Increase the Physical Realism of Models

Rigorous mathematical methods avoid unnecessary simplifications in numerical cloud models

Clouds in front of a sunset-filled sky

Incorporating newly derived equations into numerical models of atmospheric clouds produces more accurate results, particularly at high temporal resolution.

(Image courtesy of Magda Ehlers | Pexels.com)

The Science

Atmospheric clouds play important roles in weather and climate but are relatively small in size and complex in nature. As a result, numerical models require simplified representations of clouds. This study demonstrates the benefit of using rigorous mathematical methods, like Taylor series approximation and sub-grid reconstruction, to derive more detailed equations featuring more accurate descriptions of the modeled physical phenomena. Such an approach avoids oversimplifications or inconsistencies found in models based primarily on intuitive conceptualizations, both of which can produce modeled behaviors that are physically impossible and therefore contain large numerical errors.

The Impact

Basing numerical models on a more rigorous mathematical approach produces more realistic results, providing a more accurate basis for weather and climate predictions. These models also possess superior mathematical properties, including a higher error reduction rate, as they generate more frequent results, a concept known as increasing temporal resolution. This better positions them to make use of upcoming advances in computational power.


A previous study shows that the numerical representation of clouds in a computer model of the Earth’s atmosphere can suffer from inadvertent oversimplifications that lead to physically invalid behavior and slow error reduction upon increasing temporal resolution. This study first derives equations for the water vapor condensation and cloud liquid evaporation processes associated with cloud growth or decay at small, unresolvable spatial scales. The employed sub-grid reconstruction methodology makes assumptions about the small-scale details of atmospheric temperature, humidity etc., and detailed temperature and humidity information is used to calculate condensation and evaporation rates at unresolvable scales. These rates are then aggregated to scales resolvable by the numerical model. This methodology provides a flexible framework that helps avoid the previously observed oversimplifications and inconsistencies, leading to improved numerical accuracy in short-term simulations and significant differences in the long-term statistics of simulated cloud amounts.

PNNL Contact

Hui Wan, Pacific Northwest National Laboratory, Hui.Wan@pnnl.gov 


This work was supported by the U.S. Department of Energy (DOE), Office of Science, Biological and Environmental Research (BER) program, and the Advanced Scientific Computing Research (ASCR) program via the Scientific Discovery through Advanced Computing (SciDAC) program. Computing resources were provided by the National Energy Research Scientific Computing Center, a DOE Office of Science user facility supported by the DOE Office of Science under Contract DE‐AC02‐05CH11231. Additional computing resources were provided by Livermore Computing at Lawrence Livermore National Laboratory and Research Computing at Pacific Northwest National Laboratory. The work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.  LLNL-MI-817756.

Published: January 4, 2021

Vogl, C. J., et al., “Improving time step convergence in an atmosphere model with simplified physics: Using mathematical rigor to avoid nonphysical behavior in a parameterization.” Journal of Advances in Modeling Earth Systems, 12, e2019MS001974 (2020), [DOI:10.1029/2019MS001974]