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\begin{document}

\title{Removing numerical pathologies in a turbulence parameterization through
convergence testing}



\author[1]{Shixuan Zhang}%
\author[2]{Christopher James Vogl}%
\author[3]{Vincent E Larson}%
\author[2]{Quan M. Bui}%
\author[4]{Hui Wan}%
\author[4]{Philip J. Rasch}%
\author[5]{Carol S. Woodward}%
\affil[1]{Pacific Northwest National Laboratory}%
\affil[2]{Lawrence Livermore National Laboratory}%
\affil[3]{University of Wisconsin-Milwaukee}%
\affil[4]{Pacific Northwest National Laboratory (DOE)}%
\affil[5]{Center for Applied Scientific Computing}%


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  \date{May 18, 2024}


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\begin{abstract}
Discretized numerical models of the atmosphere are usually intended to
faithfully represent an underlying set of continuous equations, but this
necessary condition is violated sometimes by subtle pathologies that
have crept into the discretized equations. Such pathologies can
introduce undesirable artifacts, such as sawtooth noise, into the model
solutions. The presence of these pathologies can be detected by
numerical convergence testing. This study employs convergence testing to
verify the discretization of the Cloud Layers Unified By Binormals
(CLUBB) model of clouds and turbulence. That convergence testing
identifies two aspects of CLUBB's equation set that contribute to
undesirable noise in the solutions. First, numerical limiters (i.e.
clipping) used by CLUBB introduce discontinuities or slope
discontinuities in model fields. Second, this noise can be amplified by
an advective term in CLUBB's background diffusion. Smoothing the
limiters and removing the advective component of the background
diffusion reduces the noise and restores the expected first-order
convergence in CLUBB's solutions. These model reformulations improve the
results at coarser, near-operational grid spacing and time step in
cumulus cloud and dry turbulence tests. In addition, convergence testing
is proved to be a valuable tool for detecting pathologies, including
unintended discontinuities and grid dependence, in the model equation
set.%
\end{abstract}%



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