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Publication Date
1 May 2023

Applying a Mathematical Diagnostic Tool to Detect Numerical Pathologies in Atmospheric Physics Parameterizations

A success story of applying convergence testing to detect and address issues of numerical discretization in nonlinear representations of turbulence and clouds.
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Numerical models can help researchers develop an understanding of atmospheric physics. However, results computed by numerical models may be misleading if their mathematical algorithms cause large inaccuracies. Computational scientists employ a diagnostic tool called convergence testing to help verify if the behavior of the numerical results is consistent with the characteristics of the underlying equations that describe the physics. In this study, researchers applied this tool to a sophisticated numerical model of turbulence and clouds. Such atmospheric physics parameterizations are not usually subjected to convergence testing of numerical discretization, partly because of the difficulty in identifying the cause of non-convergent results. However, guided by the convergence tests, the researchers identified and addressed issues in the numerical algorithms used in this physics parameterization, improving the overall fidelity of the results.


When convergence testing suggests proper behavior, researchers can have more confidence that the numerical model, if it is run at adequate resolution, faithfully represents the physics described by the model equations. This is a necessary foundation for further work to increase numerical accuracy.


Atmospheric physics parameterizations are simplified descriptions of atmospheric processes that cost too much computing power to simulate in detail. Typically, parametrization development focuses on the formulation of underlying mathematical equations rather than the numerical algorithms that solve those equations. Resolution convergence testing assesses the behavior of the numerical results by varying the resolution across a wide range. 

In this study, researchers applied resolution convergence testing to a sophisticated turbulence and cloud parameterization and found non-convergent behavior in several test cases. Through a close collaboration between atmospheric model developers and applied mathematicians, the team identified and reformulated problematic components of the numerical algorithms used in the parameterization. After reformulation, the model produced the expected convergence behavior in four test cases covering a diverse range of weather regimes. This enhances confidence in the trustworthiness of the numerical results. It also provides a necessary foundation for future improvements to the numerical accuracy. Both the method of testing and the numerical issues detected are expected to be relevant to other atmospheric models.

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