We thank the reviewer for their time and effort in reviewing our work, posing some important questions, and catching a number of grammatical mistakes that we missed (and will address in a revision). The comment about line numbers is well taken: we did not realize that EGU would point to our existing pre-print on arXiv (which forbids line numbers in submissions) instead of hosting a line-numbered version on their own servers, and we will therefore consider a different preprint server that does allow line numbers for future EGU submissions.

With regard to the reviewer's skepticism about the direct application of the semi-discrete framework to a full-discrete model, we agree that LTE caused by splitting is only one contributor to the overall error in a fully discretized system. In situations where the time integration of the individual processes is the dominant source of error, one should not necessarily expect reducing a process splitting LTE to result in an observably more accurate solution. Comprehensive guidance to the choice of the coupling should ideally include consideration of all these factors, and we believe a crucial yet missing piece towards obtaining such a comprehensive guide is an analysis on the errors caused by splitting alone. As such, the manuscript is focused on the splitting error. We note that the dust life cycle problem gives a good example of the role splitting error can play. The earlier studies by Wan et al. (2021, doi: 10.5194/gmd-14-1921-2021) and Santos et al. (2021, doi: 10.1029/2020MS002359) showed empirical evidence of process splitting being a major error source in various clouds regimes in EAMv1. In a broader context, the review paper by Gross et al. (2018, MWR, doi: 10.1175/MWR-D-17-0345.1) points out that process splitting/coupling has been a largely overlooked topic in the development of weather, climate, and Earth system models.

The framework presented in this manuscript can be extended to include time integration errors from individual processes (by replacing exact integrals with discrete sums reflecting quadratures specific to the process integration methods). Such a more comprehensive analysis can be performed when the time integration methods used in individual processes are sufficiently documented, and we are generally interested in performing such analyses for EAM. When performing the more complete analysis, we imagine that the results about splitting error will be useful building blocks, similar to how the distinction between isolation-induced and input-induced errors can provide an intuitive understanding of the LTE caused by splitting. We will include some additional discussion on this point in the revised manuscript.

With regard to the reviewer's comment about reducing the total LTE, that comment highlighted the need to better clarify in the revised manuscript that we are not claiming the revised coupling scheme has a smaller total LTE; rather, we intend to claim the revised coupling scheme has a smaller LTE(B). Indeed, reducing the LTE of a single process does not necessarily reduce the total LTE of a coupling scheme, except when that single process is dominating the total LTE. We have seen empirical evidence for both situations in EAMv1 simulations. In the dust life cycle problem, LTE(A) is negligible as emission does not directly depend on aerosol mixing ratio, whereas the sign and magnitude of LTE(C) can vary in time and space. Because EAM is a global model, we generally do not expect to find a simple and robust relationship between LTE(B) and LTE(C) for all grid cells at all times. Therefore, we believe a more reliable approach to reducing the total LTE is to reduce the LTE associated with each and every individual process. This is what we had in mind when saying in the introduction, ``Reducing splitting errors in the process rates can help avoid compensating errors from different physical processes and, thus, help ensure the model provides good predictions of the prognostic variables for the right reasons." Discussions on this point will be expanded in the revised manuscript. We also note that we have some ongoing follow-up work with the goal of reducing the LTE of all individual terms, which we hope to report on in a separate paper.

With regard to the reviewer's invitation to consider applying the framework to the full process integration, we agree that including the fully-discrete analysis would very nicely complement the semi-discrete analysis. At this time, a fully-discrete analysis is difficult due to the limited documentation on the integration approaches used for the individual processes. For this reason, we have elected to focus our manuscript on the semi-discrete results. That said, all the suggestions from the reviewer are well taken. In our view, the two companion papers have made an initial attempt to address one small (albeit important) part of a much larger and complex problem. We are continuing to work towards more complete and comprehensive solutions to the coupling problem, and we hope publishing the results from our initial attempts will invite more researchers in the weather, climate, and Earth system modeling communities to give more attention to the coupling challenge.

We thank the reviewer for their time and effort in reviewing our work and proving valuable feedback that includes correcting a typo (that we will address in a revision). We are certainly happy to hear the reviewer finds the work a valuable contribution to the climate community.

With regard to the reviewer's request for comment on other factors that go into choosing a coupling technique, we find that multiple factors need to be considered when developing an effective approach to evolving any set of coupled processes together in time. These factors include the time scales of each of the processes, whether the processes together form new time scales, whether the processes depend on each other linearly or nonlinearly, whether one process is highly dependent on another, how accurate the resulting solutions need to be, etc. 

Different coupling approaches may try to more frequently evaluate some processes relative to others and thus address tighter dependencies of one process on another. These strategies will give rise to differing error terms which, when instantiated for a given problem, will show differing values.

With regard to the reviewer's question about whether the framework would adequately calculate errors for cloud macrophysics and microphysics and/or fluid dynamics and sub-grid physics, the mathematical framework presented here provides general expressions for the local truncation errors caused by process splitting. These expressions are generally valid regardless of which physical processes our symbols A and B correspond to. On the other hand, in specific coupling problems like cloud macropyhsics and microphysics or resolved fluid dynamics and parameterized physics, the characteristics of the processes (e.g., the signs, magnitudes, and time scales of A, B, dA/dq, and dB/dq) can differ substantially and also vary significantly in time and space. Therefore, the specific situations may be much more complicated than what we saw in the dust life cycle problem, and the errors of sequential and parallel splitting can be highly dependent on the associated details. Understanding the implications of the error estimates instantiated for a specific situation will continue to require a close collaboration between numerical analysts and climate scientists. Coincidentally, we have done some investigations into coupling problems related to clouds in EAM and plan to report on these in separate papers.