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Publication Date
1 October 2020

Using Mathematical Rigor to Increase the Physical Realism of Models

Rigorous mathematical methods can help avoid unnecessary simplifications in numerical models of atmospheric clouds.
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Incorporating newly derived equations into numerical models of atmospheric clouds produces more accurate results, particularly at high temporal resolution.

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 subgrid 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.


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 subgrid 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.

Point of Contact
Hui Wan
Pacific Northwest National Laboratory (PNNL)
Funding Program Area(s)