Improving Time Step Convergence in an Atmosphere Model With Simplified Physics: The Impacts of Closure Assumption and Process Coupling

Abstract

Convergence testing is a common practice in the development of dynamical cores of atmospheric models but is not as often exercised for the parameterization of subgrid physics. An earlier study revealed that the stratiform cloud parameterizations in several predecessors of the Energy Exascale Earth System Model (E3SM) showed strong time step sensitivity and slower-than-expected convergence when the model's time step was systematically refined. In this work, a simplified atmosphere model is configured that consists of the spectral-element dynamical core of the E3SM atmosphere model coupled with a large-scale condensation parameterization based on commonly used assumptions. This simplified model also resembles E3SM and its predecessors in the numerical implementation of process coupling and shows poor time step convergence in short ensemble tests. We present a formal error analysis to reveal the expected time step convergence rate and the conditions for obtaining such convergence. Numerical experiments are conducted to investigate the root causes of convergence problems. We show that revisions in the process coupling and closure assumption help to improve convergence in short simulations using the simplified model; the same revisions applied to a full atmosphere model lead to significant changes in the simulated long-term climate. This work demonstrates that causes of convergence issues in atmospheric simulations can be understood by combining analyses from physical and mathematical perspectives. Addressing convergence issues can help to obtain a discrete model that is more consistent with the intended representation of the physical phenomena. Plain Language Summary Computer codes that simulate the time evolution of a physical system produce errors in the results due to the finite step sizes used to advance the calculations in time. These errors are expected to decrease as the time steps are shortened, at a rate determined by the characteristics of the equations and numerical methods. An earlier study revealed that the error reduction in several predecessors of the Energy Exascale Earth System Model (E3SM) was at a rate slower than expected. This study creates a simplified configuration of those models and investigates the causes of the unexpected behavior. We show that slow error reduction can be understood and improved by combining analyses from physical and mathematical perspectives. The required code modifications can lead to significant changes in the simulated long-term behavior of a full-fledged climate model. Furthermore, ensuring a proper rate of error reduction can help to obtain a computer code that is more consistent with the intended representation of the corresponding physical system.

Improving Time Step Convergence in an Atmosphere Model With Simplified Physics: Using Mathematical Rigor to Avoid Nonphysical Behavior in a Parameterization

“…Despite the fact that the rate of time step convergence has not been used as a standard metric in the evaluation of physics parameterizations, especially in the context of global simulations, achieving the expected convergence can help to build confidence that the simulations correspond to a correct implementation of the numerical methods. In addition, as is shown here and in Wan et al (2020), poor convergence can be an indicator of nonphysical behavior, and addressing convergence issues can lead to changes in the long-term climate of an AGCM that are sizable and nonnegligible to atmospheric model developers. Finally, proper convergence rates allow global atmospheric models to achieve the expected accuracy gain from time step and grid refinement opportunities provided by upcoming increases in computational power.…”

“…As in the companion paper by Wan et al (2020), this work demonstrates the key methodology using a simplified global AGCM that considers only the fluid dynamics (dynamical core) and its interaction with the large-scale condensation of water vapor (or reversely, the evaporation of cloud liquid). Other physical processes typically included in an AGCM, such as radiation and cloud microphysics, are not present.…”

“…Wan et al (2013) showed that the underlying numerical issues that triggered clipping limiters for the sulfuric acid gas concentration in a climate model led to significant impacts on the simulated aerosol life cycle. In the companion paper to this work, Wan et al (2020) showed that a large-scale condensation parameterization derived from assumptions on grid cell average quantities following Rasch and Kristjánsson (1998) and M. Zhang et al (2003) leaves room for inconsistency between the simulated cloud fraction and liquid condensate amount. That inconsistency can result in large, nonphysical values of in-cloud liquid concentration.…”

“…Note also that because f (t + τ) < f (t) implies that 1 − f (t + τ)/f (t) < 1 in (27) , this average b q l t; p q l decreases as p q l increases, which is consistent with higher-order polynomial profiles concentrating more liquid concentration outside of the annihilation region (into the persistent cloudy region). To compare (26) to the grid cell average condensation rate studied in Wan et al (2020), consider using piecewise-defined constant polynomials for liquid concentration and concentration tendency p Al ¼ p q l ¼ 0 . In the limit as τ → 0, both grid cell average condensation rates are identical.…”

“…The C term here has a clear physical meaning, namely the moistening and cooling that overcome saturation deficit leads to condensation. In contrast, the C terms in Rasch and Kristjánsson (1998), M. Zhang et al (2003), and Wan et al (2020), assume newly formed clouds have the same in-cloud liquid concentration as the existing clouds, which is clearly a simplification with room for improvement.…”