Abstract
Much attention has been paid for decades to the sensitivity of earth system models (including E3SM) to different spatial grid resolutions. However, a corresponding and equally important issue is the sensitivity of the models to time steps, and this issue has not received much attention. Our earlier work has revealed that the stratiform cloud parameterizations in E3SM and several of its predecessors showed strong time-step sensitivity and slower-than-expected convergence when the model's time step was systematically refined. In this work, a simplified model is configured that consists of the spectral-element dynamical core of the E3SM atmosphere model and a large-scale condensation parameterization based on the scheme in the Community Atmosphere Model version 4 (CAM4). Like E3SM, this simplified model shows poor convergence, and the time-stepping errors are larger than those in the dynamical-core-only configuration.
We present a formal error analysis of the numerical method applied to the simplified large-scale condensation model. This analysis considers the closure assumption and numerical implementation of the parameterization and reveals both the expected temporal convergence rate and the sufficient conditions for obtaining such convergence. Consistent with the theoretical analysis, numerical results show that the original choice of splitting between the resolved dynamics and the parameterized physics leads to an unphysical model of the closure assumption and to consequent singularities in the model and discontinuities in the numerical solution. A revised splitting removes such inconsistency and helps to restore first-order convergence. Also presented is a new model of the closure assumption that is both physically consistent and numerically robust. We will further show that numerical schemes with good or poor convergence can result in significantly different features in the long-term climate.
Although this investigation focuses on a highly simplified model, the issues of singularities and discontinuities are expected to have impact on operational configurations of weather and climate models as well. Preliminary examples from E3SMv1 will be shown.