In their manuscript, the authors revisit existing methods to decompose the aerosol forcing into different terms and compare them against each other. In particular they flag some errors in the implementation of the APRP method by Smith et al. (ACP, doi: 10.5194/acp-20-9591-2020, 2020). As Chris Smith is a co-author of this manuscript, I assume he agrees with this! Overall this is a technical yet interesting manuscript that also documents the aerosol forcings of CMIP5 and CMIP6 models. I recommend publication after my comments below are accounted for.

Comments (major and minor interspersed)

Title: I did not find the title to be very informative. May I suggest instead “On methods to estimate aerosol effective forcings in CMIP models” or something that conveys the idea that several methods are looked at and compared in this manuscript?  

Abstract and possibly elsewhere, “forcing from aerosol scattering”: By scattering solar radiation, aerosols generally tend to increase the mean photon path and therefore gaseous absorption. I would argue that “forcing due to aerosol scattering” or “forcing induced by aerosol scattering” is a more appropriate term than “forcing from aerosol scattering”. Likewise for cloud scattering and absorption.

Equation 1: there is an additional term in this equation as ΔR also depends on the climate response over land (SST are fixed in these experiments but Land Surface Temperatures or LST are not). This is well explained in e.g. Sherwood et al (BAMS, 2015) as, I’m sure, the authors are aware. I would suggest that this additional term is included in Eq. 1. How it can be corrected or reasons why it can be neglected should be discussed.

Line 78: is “temperature” here meant to be “atmospheric temperature” or both “atmospheric” and “surface” temperature? See my comment above on Equation 1.

Line 78 and elsewhere: IPCC AR6 is too vague, please cite explicitly the relevant Chapter or Chapters.

Equation 5: Aerosols may also change surface albedo by changing the distribution between direct and diffuse radiation. Indeed albedo is not an intrinsic property of the surface but depends on the properties of the incoming solar radiation. Several models now include different albedoes for direct and diffuse radiation, in particular over the ocean. By increasing the fraction of diffuse radiation at the surface in clear sky and by changing the amount of cloudiness, aerosols may thus modify the surface albedo. This additional term is probably small, and maybe negligible, but worth mentioning.

Line 115: should it be IPCC ari and aci rather than direct and indirect effects?

APRP method, section 2.2.2: I would strongly encourage the authors to include a bit more description of the APRP method so that this manuscript can be read as a stand alone piece without the need to refer back to Taylor et al (2007) to make sense of it.

Equation 15: although there is no change in notation, it may be worth repeating here that ΔR is the difference in R between two simulations. 

Line 160: see my comments above. Does it mean that the ΔR term due to the change in LST is negligible? Can you provide evidence of this? 

Lines 165-169: this paragraph needs more context to be understood. A more detailed section 2.2.2 would help the reader to appreciate fully this paragraph.

Figure 1: It may be worth saying in the caption that each circle represents a different CMIP model.

Section 3.3: It may be worth saying that the forcing year is different for CMIP5 and CMIP6 (2014 for the latter, I’m not sure for the former). However this should not have much of an impact as the aerosol forcing is thought to be pretty flat over that period.

A Table summarizing all the symbols / notations used in the manuscript would be useful.

Summary:

In their manuscript, the authors reveal biases in previously published aerosol effective radiative forcing estimates in CMIP6 models (Smith et al., 2020; doi: 10.1002/2014JD021710) which arise from two coding errors in the application of the approximate partial radiative perturbation (APRP) technique proposed by Taylor (2007; doi: 10.1175/JCLI4143.1). The authors show the impact of each of these errors on aerosol ERF components, correct them and compare the results with the ones derived using the classic double radiation call method (Ghan, 2013; doi: 10.5194/acp-13-9971-2013). Further, they describe, compare and relate the various aerosol radiative forcing terms from the two approaches and list the corrected estimates of aerosol ERF and its components for CMIP6 and CMIP5 models.

General Comments:

Since this manuscript revisits existing methods to compute aerosol ERF, I find it particularly useful that they disassemble the aerosol ERF defined in IPCC AR6 and show how the various components relate to each other. Further they provide a detailed comparison of the various forcing terms and components from the two methods mentioned above, explaining differences and similarities in definitions and nomenclatures which is a great help to understand the involved processes and to be able to make meaningful comparisons.

Overall, the manuscript is well structured and logically. They start with describing the two coding errors in APRP application and demonstrating their impact. I am interested on how these coding errors emerged, and if you did something to the code to adjust this? Did you e.g. make the code more user friendly so that this does not happen again when users get your code from Zenodo? Otherwise, in this section, my main concerns were just a few explanations and figures to be adjusted.

Then they compare the results with the standard double call method and emphasize the good agreement and usefulness of APRP. This could be shortened since this was shown in previous studies. However, since APRP only works in the SW, they used proxies for the LW components to be compared. Even though this has nothing to do with correcting Smith et. al. (2020), it is nice that they included this to give their study a complete picture and to be able to compute the net aerosol ERF.

Finally they conclude with listing the corrected APRP results for all CMIP6 models and even CMIP5 models. In my opinion this part is the main point of this manuscript. The corrected estimates should be compared to the biased CMIP6 estimates, as well as with CMIP5. The latter has been too brief and should be given more weight.

I can recommend to publish this manuscript after minor revisions.

Specific Comments:

Title: I suggest to add something like “Correction to” to the title to make clear that you update something that is already published

Abstract:

L.7 – 10: You need to be more specific. I suggest to add “CMIP6 models” and “two coding errors” to hint where the bias is coming from.

L. 10: The word “ground-truth” is a bit too strong for my taste – we are still talking of a model here, not reality. I suggest to reword.

I would delete the last sentence and instead add some main conclusion on corrected CMIP6 vs biased CMIP6 and/or CMIP5 estimates.

Text:

L.20: “cease”? Isn’t that too strong? Why should they do that? Maybe change to “decrease”

L. 40: “If a small degradation of absolute accuracy can be tolerated” Like how much?

L. 45: “made different choices that have quantitative impacts on the results” What choices? Please be more elaborate on this point in the “Data and Methods” Section, since this is what has led to the erroneous results.

L. 165 – 172: Can you relate the two features to an equation already given in this manuscript? E.g. is K^α in EQ. 5 & 10 the albedo sensitivity term that has not been scaled correctly in Smith et al. (2020)? Can you name a specific code parameter or module that needed correction?

L. 209: “whereas the ERFalb SW bias vanishes because it is not affected by the albedo sensitivity formulation error” Can you explain this to me? How is it related to Eq. 5 & 10? Can you also state where the grey markers are in Fig. 3B? I assume they are under the blue ones, because correcting the albedo sens. has no effect?

L. 285: You state that you did the same as in Zelinka et al. (2014), so I assume you recomputed all values? So a difference by one in the second decimal place is ok. How come the difference in net ari+aci for MRI-CGCM3 is 0.03 (-1.16 vs -1.13)? See and cite! Zelinka et al. (2014), Table 1

Figures:

Fig. 2, 3 & 4: I know you want to focus on the bias if this or that is wrongly implemented. However I find it a bit confusing. So if you show Smith et al as being the wrong graph, then why don't you show what would happen if you correct this or that? So, instead of writing e.g. “w/erroneous insolation” I suggest to write “w/ corrected albedo sens.” Instead of going backwards to the erroneous results, you can go forward from erroneous to correct result. This means of course, to logically turn around graphs and corresponding text. Please discuss if this could be a better and more logical presentation, it is not a must-do!

Technical Comments:

L. 39: “Fortunately, aerosol direct and indirect effects primarily operate in the SW with much smaller effects in the longwave (LW),” Please give some references for this!

L. 49: change to “APRP-derived”

L. 79: “cloud particle number “ I would reword to “cloud droplet number”, otherwise it is not clear whether you include aerosols or not

EQ. 8: Is this the same as EQ. 6, just for SW?

Fig. 1: Are all the red points overlain? Maybe you could state that somewhere, so that readers don’t get confused and don’t think you might have just used one model. Caption: Delete “true”, Figure text: change to “TOA SW Residuals”

L. 180: Maybe state as in-line equation instead of text

L. 215: change to “clouds scatter more SW radiation than they absorb”.

L. 240: make clear you mean the corrected APRP from this study and Ghan’s double call

L.256: “opposite-signed errors” change to “opposite-signed differences”, errors are between Smith (2020) – APRP and this study APRP, but these I wouldn’t state as errors

Fig. 3(a,b,c), Fig. 4(a,c,d), Fig. 5B: Please make the scaling equal between x and y axis. If there is not enough space on the x axis, reduce ticking but start and end with the same numbers as on y-axis. This way is a bit confusing.

Fig. 5C: Make grey line dashed.

Fig. 6: Insert the same color bar between column 2 and 3 and adjust the rightmost color bar (e.g. ranging from -2 to 2) for the difference plot. Please state that for Fig. 6 only the corrected APRP from this study is used.

Fig. 7: Extend dashed grey 1:1 line all the way through the plot just as in the other plots.

Fig. 8: instead of “Amount” write “Cloud amount”, change title to “Corrected CMIP6 mean aerosol ...”