Does feedback temperature dependence influence the slow mode of the climate response?
 ^{1}Max Planck Institute for Meteorology, Hamburg, Germany
 ^{2}International Max Planck Research School on Earth System Modelling, Hamburg, Germany
 ^{3}University of Reading, Reading, UK
 ^{4}Colorado State University, Colorado, US
 ^{1}Max Planck Institute for Meteorology, Hamburg, Germany
 ^{2}International Max Planck Research School on Earth System Modelling, Hamburg, Germany
 ^{3}University of Reading, Reading, UK
 ^{4}Colorado State University, Colorado, US
Abstract. AtmosphereOcean General Circulation models (AOGCMs) are a necessary tool to understand climate dynamics on centennial timescales for which observations are scarce. We explore to which degree the temperature dependence of the climate radiative feedback influences the slow mode of the surface temperature response. We question whether longterm climate change is described by a single efolding mode with a constant timescale which is commonly assumed to be independent of temperature or forcing and the evolution of time. To do so, we analyze AOGCM simulations which have an integration time of 1000 years and are forced by atmospheric CO_{2} concentrations ranging from 2 times (2X) to 8 times (8X) the preindustrial level. Our findings suggest that feedback temperature dependence strongly influences the equilibrium temperature response and adjustment timescale of the slow mode. The magnitude and timescale of the slow mode is approximately reproduced by a zerodimensional energy balance model that has a constant effective heat capacity and incorporates a background feedback parameter and a coefficient for feedback temperature dependence. However, the effective heat capacity of the slow mode increases over time, which makes the adjustment timescale also timedependent. The timevarying adjustment timescale can be approximated by a multiple timescale structure of the slow temperature response, or vice versa, a multiple timescale structure of the slow temperature response is described by a timedependent timescale. The statedependence and timedependence of the adjustment timescale of longterm climate change puts into question common eigenmode decomposition with a fast and a slow timescale in the sense that the slow mode is not well described by a single linear efolding mode with a constant timescale. We find that such an eigenmode decomposition is valid at a certain forcing level only, and an additional mode or a multiple mode and timescale structure of the slow adjustment is necessary to reproduce the details of AOGCM simulated longterm climate change.
Tim Rohrschneider et al.
Status: final response (author comments only)

RC1: 'Comment on esd202186', Anonymous Referee #1, 10 Jan 2022
Many studies have shown that the global surface temperature response to abrupt forcing is well approximated by two (or sometimes three) efolding timescales, with usually a single timescale representing the “slow” warming beyond a couple decades at least out to 150 years or so. The authors pose an interesting question of whether this slow adjustment is well approximated by a single timescale when considering longer simulations out to 1000 years, and whether that timescale changes with forcing level given nonlinearities in climate feedbacks (lambda) and time dependence of the climate’s effective heat capacity (c). It’s obvious that a single timescale shouldn’t work for the slow adjustment if you assume it’s given by something like tau = C/lambda and allow c and lambda to change over time or with temperature. The authors demonstrate this using AOGCMs forced by 2x, 4x, and 8x CO2. In particular, they suggest that climate feedback nonlinearities (lambda changing with temperature) changes the slow mode adjustment timescale, and that changes in effective heat capacity also play a role, all of which makes sense.
Beyond this, I had difficulty reviewing what the authors have done, even after reading the manuscript several times. The writing is opaque and the many methods are not explained clearly. I generally trust that the authors know what they’re doing so I expect that the results will hold up once explained more clearly, but I currently am unable to make a recommendation regarding whether the study should be published. As written, I don’t think even interested readers (such as myself) will get much from it. I hope the authors will find these comments useful as they revise.
Unclear writing
 A more general introduction would be helpful. You really throw the readers into the deep end on L2024 with two redundant sentences that don’t really say what a “slow mode” is (long term climate change is determined by the slow mode, and the slow mode describes the temporal adjustment on long timescales). A better introduction would be something along the lines of what you have written on L5155. And then a clear statement could be made about the aim of the study, e.g., the twomode approximation has been shown to work well for single forcing level and out to 150 years, but it’s unclear whether this still holds for longer timescales and multiple forcing levels given feedback nonlinearities.
 L55: By “associated with a radiative feedback” do you mean “different regions associated with different radiative feedbacks”?
 L5874: I was getting lost at times whether you were talking about a timescale tau or the effective heat capacity C, and what the relationship is between them. I suggest rewriting to give a simple example first: C dT/dt = F + lambda*T, in which case tau = C/lambda. Then you can point out that tau would not be constant if C is time dependent, which of course it is in climate models, or if lambda changes (e.g., the pattern effect or nonlinearities such that the feedback becomes lambda + a*T).
 L107109: I don’t understand this sentence.
 L90170 (Conceptual Insights): I found this section to be very confusing. You introduce a tworegion model (equations 4 and 5) which later you use to analyze the slow response in GCMs. You could use this model to straightforwardly make your point that the response timescale (in this case tau_S = C_S/(lambda_S+a_S*T_S)) is not constant if C_S changes or for nonzero a_S. But instead, you introduce the twolayer model which is mathematically equivalent to equations 4 and 5 for the case a_F=a_S=0 (as shown in Geoffroy et al. 2013) but appears to have a very different form. You then make the confusing statement that “However, the parameters of the twolayer model modify the inertia of the slow mode. For instance, parameter for the efficiency of ocean heat uptake eta is an inertia parameter, and changes in ocean heat uptake cause C_S to increase or decrease”. It’s unclear whether you mean that C_S can change with model parameters such as eta (which is obvious because C_S can be written as a function of eta), or whether you mean that C_S changes over time somehow (which is the topic of the paper, but not obvious from these equations). L156167 confuse things further by introducing a new approximate definition of the slow mode which asserts a constant tau_S while noting that the full solution to equation (5) has a time varying C_s (presumably for the case of nonzero a?). But then Figure 1c suggests that tau_S is a constant function of a, which is confusing given that equation (5) suggests tau_S should change over time for nonzero a_S.
Overall, much more clarity is needed here about whether you are talking about C_S changing with model parameters or about changing over time. It’s also not clear that discussing the 2layer model adds anything at all given that it seems to confuse things and you don’t use it outside of this section anyway.
 L205206: I don’t understand this sentence. What does it add?
 L237239: I suggest you cite papers showing this.
 L245: I don’t follow this sentence. What do you mean by “the slow mode emerges from heat uptake”?
 L283284: I don’t understand this sentence. Doesn’t a_S represent feedback temperature dependence on long timescales? Then how can it be less strong than that?
 L302305: Getting lost here. I think you are saying that your theoretical predictions don’t work because the effective heat capacity is not constant, which makes sense. But can you show this, rather than simply suggesting it? Is there a way to account for changes in heat capacity separately from feedback nonlinearity in the response timescale?
 L345346: How are these “mathematical terms”? This is an unclear explanation that I don’t follow.
 Discussion: I was hoping by this point to have some clarity about whether additional timescales are needed to model the slow response (beyond year 21). The results seem to suggest that a single timescale is not good enough, which makes sense. But is this true only because of feedback nonlinearities, or also because of effective heat capacity changes? Could theory be saved by adding one (or more) additional slow timescales? I do not know what the takeaways are here.
 L434436: This is an important summary point which greatly helps to clarify the purpose of your study. But I could not tell you where these points were shown in the paper. Which figures show this clearly?
 L445446: Wasn’t this the point of this study?
Unclear methods
 L181189: A schematic would really help here. I think you are describing Gregory plots for the fast and slow mode, but I was not able to follow your methods without sketching out what the Gregory plots would look like for fast and slow modes separately and their combination via the equations on L182.
 L183184: All of your results follow from this choice to separate fast and slow modes at year 21. How did you choose this separation year (other than following earlier papers, e.g., DOI: 10.1175/JCLID1400545.1)? What did you find when you “explored the separation of the fast and slow mode”? Do your results depend on this choice at all? And, how do you actually define T_S and N_S for the slow mode? Do you take values of T and N at year 21 and subtract them from all subsequent years of the T and N timeseries to calculate T_S and T_N, or something else?
 L190195: More detail is needed here, for example:
 How do you define and calculate the “background feedback parameter lambda”? With what runs? Over what timescales? Is it assumed to not change between forcing levels, and is this a good assumption? I imagine that if different forcings produced different patterns of warming, that would result in different feedbacks from pattern effects rather than global temperature nonlinearities, but has this been accounted for somehow?
 How is T(infinity) estimated from transient simulations?
 I got lost trying to figure out what the three equations were for each forcing level, and how you went about fitting for all the parameters. I suggest writing this out explicitly for readers to follow.
 Do you assume that a_F and a_S are the same, or can they be different?
 How do you calculate effective radiative forcing in all the runs? (you mention this later, but it should be stated clearly here.)
 L208211: I don’t follow the method you describe. I understand you do regression over different lengths of years (from 5 to 20), but what does it mean to “apply subsequently bootstrapping by replacement of the forcing estimates in order to generate the details of a continuous probability distribution”?
 L214215: There are of course a way to more precisely estimate forcing using fixedSST simulations with CO2 increased.
 L222229: Again, a schematic would help show what you mean here. It’s not obvious how all this works without showing the reader.
 L233236: Is it a linear extrapolation a good assumption given the feedback nonlinearities you find? How off might your estimates be? It’s also unclear what you did with bootstrapping here again.
 Figure 4: I did not get much out of Figure 4. What are we supposed to learn here? Are we to take away that the energy balance model predictions match the AOGCM responses or not?
 L278: What does it meant to solve for an equation? Do you mean a specific set of the parameters in the equation? By what method?
 L341343: It’s plausible that it’s differences in feedback temperature nonlinearity causing the differences in the temporal changes in tau between models, as you suggest. But can you show this? Could changes in effective heat capacity not also play a role?
 Figure 6: Again, it’s unclear what to take away from this figure. That the energy balance model doesn’t replicate the AOGCM output? What is learned?
 Section 5.2: I have admittedly run out of steam here, but I don’t know what the point of this section is or follow its methods.

AC1: 'Reply on RC1', Tim Rohrschneider, 30 May 2022
The comment was uploaded in the form of a supplement: https://esd.copernicus.org/preprints/esd202186/esd202186AC1supplement.pdf

AC1: 'Reply on RC1', Tim Rohrschneider, 30 May 2022

RC2: 'Comment on esd202186', Anonymous Referee #2, 24 Jan 2022
Summary:
This paper presents interesting and useful new results on the timescales of the climate response to CO2 forcing, exploiting 1000year long step forcing AOGCM experiments. While the results are novel, I found the presentation rather complex and hard to follow, so I am requesting major revisions to make the paper more accessible.
Main comments:
1) Presentation
Overall I found the text difficult to read, despite it being well polished and free of typos – to the point that I didn’t understand everything despite a careful read. I ended up becoming frustrated and skipped most of section 5. The issues start with the abstract, where things should be kept simpler in my opinion. In particular, I struggled with the sentence L11–13, which I’m still not sure I fully understand after reading the paper. Can this be explained more simply, or perhaps omitted?
The introduction begins rather abruptly, and assumes a fairly high level of background knowledge – for example, that it is commonly understood that the response to CO2 forcing can be decomposed into fast and slow components. The notion that climate feedbacks are temperature dependent is also assumed. I think these concepts should be introduced more slowly, with references to the relevant prior literature:
 What do we know about feedback temperature dependence? Is this commonly simulated by GCMs?
 Do we know the sign of this dependence, or is this still a subject of ongoing research? The text asserts that feedbacks become more amplifying with warming (L25), yet this is inconsistent with two out of four GCMs used in this study (Table 1).
Another confusing aspect for me was the introduction of the two conceptual models (Eqs. 4–6):
 What physics underlie the 1st model (based on two regions, Eqs. 4–5)? Presumably this is meant to reflect the SST pattern effect, but I don’t think this was explained.
 It would help to discuss the commonalities and differences between the two models. My understanding would be that using an efficacy term (epsilon) in the 2nd model could be mathematically equivalent to using spatiallyvarying feedbacks in the 1st model – is this correct? The 2nd model additionally includes a heat transport efficiency term – what physics does this involve and does it make the 2nd model different from the first?
 The authors ultimately choose to focus on the tworegion model (Eqs. 4–5), as stated L182. Why this choice, and how does it affect the interpretation of the results? Do we even need both models in the paper?
I feel like it might help to use an appendix to discuss some of the more technical aspects of the two conceptual models and/or the methodological choices, so as to keep the main text simpler and more focused on the key results and their interpretation.
2) Definition of feedback temperature dependence
I would like the authors to clarify and make explicit their definition of temperaturedependent feedbacks. It seems to me that there are two quite distinct types of temperature dependence: (a) a temperaturedependent SST pattern effect, versus (b) temperaturedependent feedback processes (independent of the SST pattern). The latter could be quantified for example using uniform SST warming or cooling experiments. My understanding is that the temperature dependence discussed in the present paper includes both processes (a) and (b), but it would be good to clarify this. Do the authors know which type of temperature dependence is more important for their findings? If we want to understand and perhaps observationally constrain the temperature dependence of climate feedbacks, it seems to me that different approaches would be needed for (a) versus (b).
Specific comments:
 L24: “As a result” – of what?
 L112: Should clarify that this isn’t the formulation used by Held et al. and Winton et al. (who didn’t consider feedback temperaturedependence, as far as I’m aware?)
 L185: Shouldn’t it be N_F(t=0)?
 L186–188: I wasn’t able to follow this, can you explain in more detail or illustrate this graphically? (After further reading, I see this is explained more clearly L225–227. This needs to be reorganised.)
 L193–196: Again I wasn’t able to fully follow. I’d recommend explaining this in more detail in an appendix.
 L248: remove extra “between”
 L283–284: I didn’t follow this reasoning.
 L441: “publicly *available* experiments”
 L443: The reference to year 2100 is odd, considering that the results are based on idealised step forcing experiments, rather than realistic RCPstyle scenarios.

AC2: 'Reply on RC2', Tim Rohrschneider, 30 May 2022
The comment was uploaded in the form of a supplement: https://esd.copernicus.org/preprints/esd202186/esd202186AC2supplement.pdf
Tim Rohrschneider et al.
Tim Rohrschneider et al.
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