the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 28 Apr 2026
Using a two-stage Rosenbrock solver to improve surface ozone prediction and increase computational efficiency in the DOE's Exascale Earth System Model version 1 (E3SMv1)
Abstract. The integration of stiff atmospheric chemical systems is a major computational cost in Earth System Models and poses challenges for numerical stability and scalability as chemical complexity and temporal resolution increase. Current implementations commonly rely on fully implicit Newton–Raphson-based solvers, which are robust but computationally expensive. In this study, we implement a semi-implicit two-stage Rosenbrock method (ROS2) in the Energy Exascale Earth System Model version 1 (E3SMv1) and evaluate its performance as an alternative chemistry time-integration scheme. The ROS2 solver is applied to two interactive chemical mechanisms of differing stiffness and size: chemUCI (60 prognostic species) and trop_strat_mozart_mam4 (155 prognostic species). Numerical experiments include short simulations to quantify computational cost and year-long integrations to assess numerical stability, solution consistency, and long-term behavior. Solver performance is evaluated under identical model configurations and timesteps. At a 180-second timestep, ROS2 reduces chemistry integration cost by approximately 33 % relative to the default implicit solver. Differences in global mean surface ozone concentrations are small (≤1.03 ppb), and no systematic drift or degradation in numerical stability is observed over one-year simulations. These results indicate that low-order Rosenbrock methods provide a computationally efficient and numerically stable alternative for stiff atmospheric chemistry integration in E3SMv1. The implementation offers a flexible framework for future model configurations with increased chemical complexity and resolution.
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Status: final response (author comments only)
- RC1: 'Comment on egusphere-2026-1451', Anonymous Referee #1, 14 Jun 2026
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RC2: 'Comment on egusphere-2026-1451', Anonymous Referee #2, 14 Aug 2026
Goal: evaluate the ROS2 solver in E3SMv1 with two different chemistry packages.
ROS2 was previously shown to work well in the CESM.1. Introduction: References describing the status of atmospheric chemistry in
modern Earth system models seems dated. It mentions CMIP3 models (from 2007)
and all the papers regarding chemistry treatment in global models only go
up to 2013. Has the situation changed in the CMIP6 models?2. Line 166: "In this paper (Chien and Fu, 2026b), ..."
It is odd to self-site the actual paper I'm reading. Seems like an AI thing.3. Line 198 "All simulations are conducted on the Cray XC-based Cori supercomputer..."
This computer was decommissioned in 2023 and the architecture did not prove to be successful,
so it's not representative of leadership-class architectures in 2026. No objections to
using this old data to draw computational speed conclusions, but the authors should
drop the last sentence in this paragraph and the previous claims about exascale modeling.4. Figure 2: Confusing. x-axis label "number of halven". Part of my
confusion was my own ignorance - I was not familiar with the word
"halven". However, for this types of plots I think it is more
conventional to choose an x-axis so the slope is k. I think with the
choice here, the slope is -1/k? If the authors prefer their current
plot, at least mention in the text how the slope is related to k.5. Figure 6: "In both cases, the amplitude, phase,and variability of the
diurnal cycle are nearly identical across solvers, indicating that
ROS2 preserves key temporal characteristics while improving numerical
efficiency."How do the authors conclude this from Figure 6? ROS2_180 seems
significantly different than IMP_1800 and IMP_180. How do we know it
is more accurate? Maybe if the timestep was reduced, ROS2 would
quickly converge to the IMP_180 result - which would mean that it is
actually less efficient than IMP_180.6. For figure 7 - same question. For plot (a) (c) and (d), we see that
ROS2_180 matches the IMP_180 result, which we can assume is more
accurate than IMP_1800, and thus ROS2_180 is more efficient (since it
is faster than IMP_180. But in panel (b), IMP_1800 and IMP_180 agree
- suggesting the IMP method has converged, and leaves open the
possibility that ROS2_180 needs a smaller timestep to converge as
well, making it less efficient.
I actually would expect ROS2_180 to be more accurate than IMP_180, and
thus I think the authors conclusions are probably correct, but I dont
see how the evidence presented justifies the conclusion. If the authors can
better justify these claims, the results would be a nice contribution.7. The conclusions section is not concise and is somewhat rambling, seems
like AI.Citation: https://doi.org/10.5194/egusphere-2026-1451-RC2 -
RC3: 'Comment on egusphere-2026-1451', Anonymous Referee #3, 20 Aug 2026
This manuscript reports the first implementation of ROS2, a numerically accurate and computationally efficient integration scheme widely used in regional-scale atmospheric chemistry models, into the E3SMv1 Earth system model. The motivation is to prepare for future high-resolution global Earth system simulations in which atmospheric chemistry is fully coupled with dynamical and radiative processes. Although the same scheme has previously been implemented in another global Earth system model (CESM), reporting its implementation in a different ESM could still be considered within the scope of GMD. However, the manuscript suffers from numerous serious deficiencies in both the presentation and interpretation of the results, as detailed below. In its current form, I do not believe that the manuscript is suitable for publication in GMD. Rejection, or alternatively resubmission after major revision, would be appropriate.
-The manuscript is excessively verbose. Similar statements and phrases are repeated unnecessarily throughout the text.
-The configurations of the atmospheric model (EAM) and the chemistry schemes (chemUC and MOZART4) are not described clearly enough. As a result, readers cannot properly understand the differences between these two configurations, making it difficult to follow the subsequent discussions.
-Figures 1 and 2 are inadequately documented. For example, the axes are not properly labeled, and the figure captions provide insufficient information. Consequently, the reader cannot accurately interpret the contents of these figures.
-Based on the results shown in Figure 3, does it not follow that IMP_1800 is ultimately the best-performing configuration? The figure does not appear to provide a compelling justification for selecting ROS2_180 instead.
-Figures 4 and 5 are described as showing deviations relative to IMP_1800. If so, it is unclear why the blue line representing IMP_1800 itself does not remain at zero throughout. This inconsistency requires clarification.
-Figures 6 and 7 indicate that the difference between the blue line (IMP_1800) and the red line (IMP_180), i.e., the effect of changing the timestep, is substantially smaller than the difference between the red line and the yellow line (ROS2_180), i.e., the effect of changing the numerical solver. Nevertheless, the conclusion states that "differences in diurnal behavior are shown to arise primarily from timestep selection rather than from the numerical solver itself." This statement appears inconsistent with the results presented in the figures. Is the interpretation of these figures correct?
Citation: https://doi.org/10.5194/egusphere-2026-1451-RC3
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- 1
This manuscript introduces a two-stage Rosenbrock solver (ROS2) into the E3SMv1 model to address the significant computational bottleneck associated with atmospheric chemistry integration. By evaluating the solver across two chemical mechanisms of varying complexity (chemUCI and trop_strat_mozart_mam4), the authors demonstrate that the ROS2 solver can reduce the computational cost by approximately 33% at a 180-second timestep, while maintaining a remarkably low global mean surface ozone bias compared to the default fully implicit solver.
Overall, this paper addresses a critical challenge in Earth System Modeling. The implications of improving the computational efficiency of atmospheric chemistry are broad and highly significant for the development of next-generation, high-resolution climate models. The motivation is strong, the methodology is novel, and the manuscript is generally well-written and clearly structured. I recommend this manuscript for publication after the authors address the following comments.
1. The core contribution of this paper is the enhancement of computational efficiency for atmospheric chemistry modeling within Earth System Models (ESMs), which has broad implications for the modeling community. However, the current Introduction and Conclusion seem to focus too heavily on ozone right from the beginning. I suggest revising the opening paragraphs of both sections to frame the narrative around the broader context, challenges, and historical development of atmospheric chemistry modules in ESMs. Ozone can then be introduced subsequently as a key metric for evaluation, rather than the primary motivation of the paper.
2. The authors note that at a 180-second timestep, the ROS2 solver achieves a ~33% efficiency improvement for both the MOZART and chemUCI mechanisms. Given the significant difference in complexity (number of species and reactions) between these two mechanisms, is this identical 33% reduction a coincidence? Or does it imply that the relative efficiency gain of the ROS2 solver, compared to the default implicit solver, is independent of the chemical network's size? The authors should discuss the theoretical or practical computational reasons behind this consistent scaling.
3. While the global mean surface ozone bias is small, Figure 5 reveals substantial regional discrepancies. The current manuscript lacks a detailed discussion on what drives these specific regional differences (e.g., are they related to specific emission hotspots or distinct meteorological regimes?). Furthermore, focusing almost exclusively on ozone provides a somewhat limited perspective. The spatial differences of other critical atmospheric components, such as NO2, OH and aerosols should also be presented and discussed. I highly recommend strengthening this section to provide a more comprehensive spatial evaluation of the solver’s performance.
4. Although the baseline simulation shows small bias for ozone, the sensitivity of the chemical system to non-linear responses might show larger difference when using the two solvers. To ensure the ROS2 solver behaves consistently with the implicit solver under perturbed conditions, I suggest conducting a simple sensitivity test. For instance, the authors could run a short test where global NOx emissions are reduced by 20%. Comparing the ozone response between the two solvers under this perturbed scenario would robustly demonstrate whether the acceptable bias holds true for non-linear chemical regimes.
Technical Comments: The x and y axes are currently missing Fig.1 labels. I believe they represent the number or index of the chemical species. Please add appropriate axis labels to clarify what is being plotted.