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, nonlinear numerical diffusion employed for improving numerical stability can introduce unintended small-scale features into the solution of the model equations. Smoothing the limiters and using linear diffusion (low-order hyperdiffusion) reduces the noise and restores the expected first-order convergence in CLUBB’s solutions. These model reformulations enhance our confidence in the trustworthiness of solutions from CLUBB by eliminating the unphysical oscillations in high-resolution simulations. The improvements in the results at coarser, near-operational grid spacing and timestep are also seen in cumulus cloud and dry turbulence tests. In addition, convergence testing is proven to be a valuable tool for detecting pathologies, including unintended discontinuities and grid dependence, in the model equation set.

We investigate whether or not the default ocean-atmosphere turbulent flux algorithm in the Energy Exascale Earth System Model version 2 (E3SMv2) converges to unique surface fluxes. We demonstrate that under certain conditions (i) discontinuities in the underlying equations result in the lack of a solution for the algorithm to converge to, and (ii) more than one set of surface fluxes may satisfy the aforementioned equations, some of which may have non-physical interpretation. These issues underpinning the theoretical foundations of the parameterization have significant impacts on the accuracy and convergence of turbulent fluxes in E3SM.   We address issues of non-existence and non-uniqueness of surface fluxes in E3SM’s default algorithm by (a) regularizing discontinuous exchange coefficients to enforce continuity and allow the algorithm to converge to a solution of the underlying equations, and (b) utilizing an adaptive procedure for selecting limiting values of the Monin-Obukhov length to ensure the underlying equations have a unique solution. The proposed revisions result in significant changes to model latent and sensible heat fluxes which are most notable in boreal winter in the Northern Hemisphere.

It is still challenging to reproduce marine boundary layer (MBL) clouds well in large-scale models despite their importance to the Earth’s radiation budget and hydrological cycle. This study evaluates MBL and clouds in the Energy Exascale Earth System Model (E3SM) version 2. The E3SM simulation results are compared with remote sensing and reanalysis data during the Cloud System Evolution in the Trades (CSET) field campaign to better understand stratocumulus to cumulus cloud transition (SCT) over the northeast Pacific. E3SM results are extracted along the CSET Lagrangian trajectories. The comparison shows that the E3SM simulation applying horizontal wind nudging performs well in reproducing thermodynamic variables of the MBL and evolution trends of cloud variables along the trajectories. However, substantial overestimations of aerosol and cloud drop number ($N_d$) are observed, which is explained as an issue with version 2 of the model. Cloud fraction (CF) does decrease from the Californian coast to Hawaii in the E3SM simulation, but most CF values indicate an overcast or almost clear sky, which differ with satellite and reanalysis data. The effect of $N_d$ overestimation on CF evolution is assessed via prescribed $N_d$ simulations. Those simulations with $N_d$ modifications show negligible CF changes. A comparison of estimated inversion strength (EIS) also shows that the simulated EIS values are similar to those of reanalysis data. Our study suggests that cloud macrophysics and boundary layer processes are more important in improving the simulation rather than improving the model’s dynamics or cloud microphysics to capture SCT better in the model.

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.

Stochastic parameterizations are used in numerical weather prediction and climate modeling to help capture the uncertainty in the simulations and improve their statistical properties. Convergence issues can arise when time integration methods originally developed for deterministic differential equations are applied naively to stochastic problems. (Hodyss et al 2013, 2014) demonstrated that a correction term to various deterministic numerical schemes, known in stochastic analysis as the Itô correction, can help improve solution accuracy and ensure convergence to the physically relevant solution without substantial computational overhead. The usual formulation of the Itô correction is valid only when the stochasticity is represented by white noise. In this study, a generalized formulation of the Itô correction is derived for noises of any color. It is applied to a test problem described by an advection-diffusion equation forced with a spectrum of fast processes. We present numerical results for cases with both constant and spatially varying advection velocities to show that, for the same time step sizes, the introduction of the generalized Itô correction helps to substantially reduce time integration error and significantly improve the convergence rate of the numerical solutions when the forcing term in the governing equation is rough (fast varying); alternatively, for the same target accuracy, the generalized Itô correction allows for the use of significantly longer time steps and hence helps to reduce the computational cost of the numerical simulation.