This paper by Klose et al examines how tipping cascades can emerge through interactions between the Greenland Ice Sheet and the Atlantic Meridional Overturning Circulation. They use a simple, but physically motivated, model of these systems to show that tipping cascades can emerge due to bifurcations and also due to rate induced tipping. I thought the analysis of the results was good, but the paper could be improved with regard to its presentation and framing.

I organise the review as follows: I give some broad comments about the paper, followed by some specific points.

Broad Comments:

What is the relation of this work to that of Sinet 2023, which also examines a simple model of the interaction of the AMOC with ice sheets?

The paper distinguishes between overshoot and rate induced cascades. I think it would be better to call overshoot cascades "bifurcation cascades" instead, as this reflects their tipping mechanism. The term overshoot is connected to notions of reducing the forcing back to a safe level.

How do the rates and magnitudes of the changes in the control parameter relate to the rates and magnitudes of changes in observations and future projections under different climate scenarios?

I often think of hosing as being related to the melting of the Greenland ice sheet, yet here it is included as an additional process. What process is this capturing and what are plausible values of H?

For the first part of the paper the parameter d_{oa}, which controls the thermal coupling of the AMOC and GIS is set to zero. Would it not be clearer to maintain it at the non-zero level throughout the paper, particularly as the interaction between the tipping elements is a focus of the paper.

The model is described as process based, but I don't think that's quite right. For example the box model of the AMOC does not represent real processes in the system. I think this modelling approach is fine, but it is probably fairer to describe the model as 'physically motivated' for example.

Specific comments:

Is line 51 consistent with line 39?

Lines 59,91 and 405 are too strong. The IPCC has low confidence in historical reconstructions of the AMOC, and the findings of Boers and Rypdal are consistent with the approach of a critical threshold but other explanations are possible.

Line 190: What is meant by analogously here?

Line 199: What is A_i?

Line 200: Why should fluxes be balanced? If water was added in the North Atlantic box I wouldn't a priori expect it to be removed elsewhere.

Line 232 "As exemplarily indicated" should read "As indicated"

Line 265 "For exemplary" should read something like "For different"

Lines 296 to 305 Does figure 4 refer to figure 3?

Line 355 "As exemplarily shown" should read "As shown"

Figure 4 a,b,d,e could have a zoomed in x-axis, possibly each with a different scale as most of the interesting dynamics happen early in the simulation.

Line 386: "Pending" should read something like "yet"

Rate-induced tipping cascades arising from interactions between the Greenland Ice Sheet and the Atlantic Meridional Overturning Circulation

This manuscript investigates the different manners  in which cascade tipping between the Greenland ice sheet and the Atlantic meridional overturning circulation can take place, depending on the rate of forcing resulting from Greenland melting and the existence of a negative feedback of the reduced AMOC on Greenland temperature. The manuscript is in general well written and the results are interesting, although limited by the simplicity of the model used. I think it deserves to be published once a few issues are solved. Most notably, the writing is a bit confusing in some points and the Results section is a bit entangled, so I would recommend restructuring it somewhat.

Specific issues:

Line 12: replace “breaching” by “surpassing”

Line 20: references should be included to refer to the melt-elevation feedback and the positive salinity advection feedback (e.g. Robinson et al. 2012 and Rahmstorf 1996, both of which are already included in the Reference list)

Lines 30, 38: “The Greenland Ice Sheet and Atlantic Meridional Overturning Circulation are strongly linked via freshwater fluxes into the North Atlantic originating from a melting GIS on the one hand, and via a relative cooling around Greenland with a slowdown of the AMOC on the other hand (Kriegler et al., 2009; Sinet et al., 2023).” This statement refers to studies consisting of expert elicitation and a conceptual model. I would compel the authors to try to find more physically-based support in the literature (actually these are provided in lines 36-37).

Line 35: the question as to whether the AMOC is already decreasing is controversial, I would recommend having a more balanced discussion. This does not preclude that projections all indicate an AMOC reduction in the future, so the question is relevant.

Line 91: same as above, the question of whether the AMOC is approaching a tipping point is controversial. 

Lines 101, 132: suppress “compare”

Lines 133: Besides Robinson et al. you could refer also to the recent study by Bochown et al:

Bochow, N., Poltronieri, A., Robinson, A. l. Overshooting the critical threshold for the Greenland ice sheet. Nature 622, 528–536 (2023). https://doi.org/10.1038/s41586-023-06503-9

Line 143: insert commas before and after “based on the shallow-ice approximation (Hutter 1983)”

Lines 148-149: consider rewriting this as “the ice flux F and the mass balance at the surface as (first and second term on the right hand side of Eq (1), respectively”.

Line 150: rewrite as “from x = -L to x = L” or “between x = - L and x = L”

Line 152: insert commas before and after “Eq(1)-(2)”

Lines 167, 170: I would write these inequalities in the inverse sense (e.g. a0 > a0gc > 0 for the first one)

Line 173: replace “to include” by “including”

Equation 11: explain Fi0 and Ai

Line 203: introducing freshwater as salinity fluxes was a practice done in old OGCMs using the rigid lid approximation.

Line 212: suppress new paragraph here

Lines 209-232: This is not really part of the Results section, but should be part of section 3.

Lines 232-241: Here there is a discussion with references to Figure 2a which corresponds to the case without the negative feedback; therefore I think it should appear later on, within 4.2. Also, how do you know the bifurcation is a Hopf bifurcation? If this is based on the analysis by Alkhayoun et al. (2019) it should be made more clear, otherwise the reader thinks they have missed something.

Equation 13: I understand that the role of H here is simply to bring the system closer to its critical threshold in order to study the sensitivity of the response to the starting point, right? If so I would try to say it more clearly.

Line 226: I would suppress “It eventually acts as a virtual salinity flux, while assuming a constant ocean volume (compare Section 3.2)”, no need to state again.

Lines 240-241: I would rewrite “it is eventually attributed to a time–dependent decline of the GIS and in fact turns into a state variable in transient experiments” as “it is actually a state variable in transient experiments that represents the freshwater forcing into the North Atlantic due to the time–dependent decline of the GIS”

Lines 242-248: I would move this paragraph to the beginning of the next section, because it contains important information concerning the experimental setup followed.

Line 249: I think you need to include “without the negative feedback” in the title

Line 251: “complementing previous freshwater experiments” - which??

Line 258: I would suppress “ The AMOC hosing H is kept constant”, there is no need to say this again.

Line 264: FGIS = 0 Sv is reached one the GIS has completely melted, right? If so I would say so.

Line 265: “For exemplary melting patterns…”: this wording is a bit confusing. When one reads “melting patterns” it  reminds of spatial patterns which have nothing to do with this study. “Exemplary” also sounds strange. Why not just say that there are two possible modes of cascade tipping and give one example for each?

Line 272: I think this part would be better understood with a bit more information on the parameters H, rao and a0max, which currently are only shown in the caption of Figure 2. The same goes for the discussion in lines 278-285.

Lines 298, 304: I think this should be Figure 3 rather than Figure 4.

Lines 304-305: I would move this sentence to the end of the discussion rather than anticipating the existence of rate-induced AMOC collapse already.

Line 316: same issue with “melting patterns” as above.

Line 332-334: I would delete “The negative feedback…”; this is clear at this point.

Lines 342-346: where is this to be inferred? Also, would it be possible to give a specific example, with values that can be read from figure 4?

Lines 352-354: “ Hence, for a strong surface mass balance decrease the tipping outcomes in terms of the final GIS and AMOC states when neglecting the negative feedback via the temperature are qualitatively resembled (Fig. 4(c))” : The meaning of this sentence is unclear to me.

General:

Acronyms for the Greenland ice sheet (GIS) and the Atlantic meridional overturning circulation (AMOC) are introduced but not systematically used subsequently, this should be corrected (e.g. lines 30, 41, 48, 86, 94, 97, etc) 

This might be a matter of taste, but I feel the text abuses a bit of using parentheses to provide explanations; I would recommend replacing those by inserting the sentence between commas. 

e.g. in line 5 I would replace   “(with a destabilizing effect on the AMOC due to ice loss and subsequent freshwater flux into the North Atlantic as well as a 5 stabilizing effect of a net–cooling around Greenland with an AMOC weakening) 

by  “, with a destabilizing effect on the AMOC due to ice loss and subsequent freshwater flux into the North Atlantic as well as a 5 stabilizing effect of a net–cooling around Greenland with an AMOC weakening, “

The same applies to lines 8, 19, 152, etc.

Figures

Figure 1b: please make this panel make  larger

Figure 2: 

1. Green lines indicating the thresholds are barely seeable 
2. and d) should have the same scale for the x axis

Note: A pdf version of this review is uploaded as a supplement.

Review of Klose, Donges, Feudel, and Winkelmann: “Rate–induced tipping cascades arising from interactions between the Greenland Ice Sheet and the Atlantic Meridional Overturning Circulation." (Earth System Dynamics, Paper: esd-2023-20)

The manuscript of Klose and others presents the results of a conceptional model system where a global ocean interacts with the Greenland Ice Sheet (GrIS). They allow tipping cascades of the Atlantic Meridional Overturning Circulation (AMOC) stability by freshwater released into the North Atlantic. Besides a hosing flux, the meltwater contribution of disintegrating GrIS disturbs the stability of the AMOC and, via a potential feedback loop, controls Greenland's meltwater contribution. The authors access overshot and rate-induced tipping cascades in this highly idealized system.

The first conceptional model renders the global ocean by hydrographic properties in five boxes: Two dedicated Atlantic surface ocean regions (North Atlantic: N, Tropical Atlantic: T) beside surface boxes for the Southern Ocean (S) and the Indo-Pacific Ocean (IP). Those surface boxes communicate with a global bottom ocean box (B). In this model, the density difference between the Northern Atlantic and Southern Ocean drives the AMOC, which controls the North Atlantic temperature and determines the temporal salt flux between different boxes (Eq. 6 – 10).

The other conceptional model represents the Greenland Ice Sheet by a flowline model solving the shallow ice approximation. It has a (half) width of 1000 km, which equals approximately its actual latitudinal extent. The ice loss and gain are exclusively described by the surface mass balance (SMB), where the Lapse rate effect constitutes the melt elevation feedback.

The coupling between these two models is unidirectional or bidirectional. The unidirectional coupling considers only the meltwater flux of a shrinking GrIS into the Atlantic Ocean. This meltwater flux plus an additional freshwater hosing flux decreases the salinity in the Atlantic. In the bidirectional setup, the North Atlantic temperature feeds back on Greenland's surface mass balance.

The authors detect overshoot and rate-induced cascading tipping in their model system with a focus on the GrIS and the AMOC. These cascades are analyzed in the uni- and bidirectional setups where the bidirectional coupling considers the feedback loop between the North Atlantic temperature and the ice loss. This thermal coupling stabilizes the AMOC and the ice sheet because an enhanced meltwater flux reduces, via the density difference, the AMOC strength, lowering the North Atlantic temperatures, which, ultimately, damps additional melting.

This study is highly relevant since it addresses outstanding questions about the AMOC stability while considering the interaction between the AMOC and the Greenland Ice Sheet. At the same time, Greenland's meltwater release grows in a warming world. The authors highlight that it is not sufficient to consider a "fixed" threshold (overshoot tipping) beyond which the AMOC breaks down or GrIS disintegrates. They also underscore that changing rates could drive components in the coupled system beyond stable conditions (rate-induced tipping). In addition, a stabilizing feedback, such as reduced North Atlantic warming due to a weakened AMOC, may not be strong enough to offset the disintegration of the GrIS. Here, the reduced model complexity allows scanning the phase space for numerous tipping cascades. Although, these are not necessarily representative of the natural world.

In general, it was a pleasure to read the well-structured manuscript. The figures are of high quality, necessary, and informative.

I recommend the publication of the manuscript after minor corrections.

General comments

Although the manuscript is well organized and generally well written, section 4.2, including its subsections, could be better written. The language of section 4.2 is less clean than the remaining section. Therefore, I suggest revising this whole section.

Furthermore, the authors presented additional lengthy information in brackets, which may disturb the reader. I suggest the authors integrate this information into the general text or drop it if applicable.

In Figures three and four, small schematic icon-like figures of the remaining ice-thickness across Greenland seem to indicate remaining ice. If this is the case, please state more clearly where this pattern comes from because I do not see how the applied flowline ice sheet model can provide this pattern.

Occupationally, terms/variables are introduced, which are defined later in the text. In some cases, this needs to be clarified. For example, the text refers (Page 9, Line 232) to Figure 2, where the variable appears in the caption, while it is later introduced on Page 9, Line 246.

Specific comments

Main document

Page 3, Line 79: Add comma in “... for a limited forcing, given that the …”

Page 4, Line 112: Please introduce the not SI-unit Sverdrup, for example, by a footnote or additional text.

Page 2, Line 36 – 37; Page 4, Line 107 – 198; Page 4, Line 109 – 111, Page 4, Line 118, Page 5, Line 1124 – 125: Missing citation (Madsen et al. 2022)⁠.

Page 5, Line 132: Do you mean the citation “(compare e.g. Lohmann and Ditlevsen, 2021)?

Page 5, Line 151: You may consider adding additional information: “(… continent by the ocean without floating ice shelves in Oerlemans (1981)).”

Page 6, Line 156: “Thereby, the surface mass balance $a_s$ that is the difference between mass gain and mass loss is reduced and …”

Page 6, Equation 3: Is the variable $\tilde{a_0}$a spatial dependent variable? If so, please indicate or mention it. In addition, does this linear equation consider an increased vulnerability by a more than linear increase of the ablation zone for lowering height?

Page 6, Line 165: You may add at the end: "Furthermore, the surface mass balance equals in our setup the total mass balance."

Page 7, Equation 11: I can not find the definition of $A_i$ in the text. Please add it, even if this information is available in the supplement's table. Furthermore, does a hosing of H = 0.2 Sv, as shown in Figure 2b, correspond to an additional freshwater flux of $A_i \cdot H$ ? If so, wouldn’t it correspond to a hosing freshwater flux of 0.014 Sv and 0.1504 Sv into the North Atlantic and Tropical Atlantic box, respectively? Since the manuscript addresses the impact of freshwater on the AMOC stability and the conditions at and around Greenland, it is surprising that the freshwater flux may be ten times larger in the tropics than in the North Atlantic. Anyhow, please clarify this point.

Page 8, Line 217: Does the surface mass balance on the ground correspond to the surface mass balance at the sea-level?

Page 8, Line 218 – 219: You may rephrase “For a declining overturning strength $q$ of the AMOC with $H > H_{ref}$, the temperature $T_N$ in the North Atlantic box declines as well according to Eq. (4)” if applicable, e.g., “For active hosing ($$H > H_{ref}), the AMOC overturning strength $q$ declines as well as the temperature $T_N$  in the North Atlantic box is driven by Eq. (4).”

Page 8, Line 219: You may rephrase “For $d_{oa} = 0$, we recover a …” with “For $d_{oa} = 0$, we obtain a …”

Page 8, Line 220: I find the wording “independent ice sheet” confusing and misleading. Please change it.

Page 9, Line 219 – 223: You may simplify these sentences: "A unidirectional coupling is obtained by $d_{oa} = 0$, where Greenland is not exposed to any changes in the North Atlantic (Eq. 12). In addition, the freshwater flux by the Greenland Ice Sheet resulting from its mass loss (Bamber et al., 2012, 2018; Trusel et al., 2018) is added as $F_{GIS}$ to the combined freshwater into the surface North Atlantic box as:”

Page 9, Line 229: Wouldn’t it be correct to state “($F_{GIS} >0 {\mathrm Sv}$)”?

Page 9, Line 231: You may write "… freshwater flux into the Atlantic Ocean, which increases the …" or?

Page, Line 245 – 247: The sentence “More specifically, the surface …. is reached (Fig. 1(b))” is not clear. Please rephrase.

Page 9, Line 245 – 247: You may change the order in this sentence: “More specifically, with a ramping $r_{ao}$, the ground surface mass balance $a_o$ decreases linearly and, once crossed the deglaction threshold $a_{0_{dgc}}$, the ice sheet stability is not sustainable.” or “More specifically, with a ramping $r_{ao}$, the ground surface mass balance $a_o$ decreases linearly, and the ice sheet becomes unstable, once crossed the deglaction threshold $a_{0_{dgc}}$ is crossed.

Page 10, Line 256: You may rephrase "By decreasing the surface mass balance at the ground level associated … ."

Page 10, Line 256: You may rephrase to “… threshold and eventually disintegrates completely … .”

Page 10, Line 257: You may replace the beginning of the sentence: "In the following, AMOC hosing is kept constant.”

Page 10, Line 260 – 265: You may rewrite it: “The freshwater volume loss resulting from the forced deglaciation of Greenland corresponds to a time-varying GIS freshwater flux $_F_{GIS}$ into the North Atlantic. This time-dependent GIS freshwater flux first increases as the GIS disintegrates. Consequently, the AMOC grows, potentially overshooting its threshold (Ritchie et al., 2021), but eventually returns to $_F_{GIS} = 0 {\mathrm Sv}$ under otherwise constant hosing (Fig. 2(a), the AMOC trajectory approximately follows the black lines)".

Page 10, Line 267: Please consider replacing "observed" with "detected" when describing the results of simulations since model results are not measured and turned into observed properties. Therefore, please rephrase "… to a freshwater flux as detected in previous hosing experiments … ."

Page 10, Line 269 – 272: You may rephrase “ In particular, the AMOC may transition to its ’off’–state in response to the Greenland Ice Sheet disintegration, which is accompanied by a temporary overshoot of the GIS freshwater flux threshold, resulting in an overshoot cascade (Fig. 2(c)). The increasing GIS freshwater flux puts the AMOC from the ’on’ to the ‘off’-state, while the AMOC does not recover after the decline of the GIS freshwater flux.”

Page 10, Line 272 – 273: The following might be more appropriate: "The surface mass balance decreases substantially… , which results in a complete deglaciation of Greenland."

Page 10, Line 276: I'm unsure, but shouldn't it be: "AMOC weakening without tipping, as commonly detected in hosing … "?

Page 10, Line 278 – 279: Here is an example of avoiding unnecessary brackets: "… within 1000 years driven by a faster and stronger … ."

Page 10, Line 281: What do you think about rephrasing: "...deglaciation of Greenland, the AMOC leaves the stable 'on'-state. Rate-induced transition … ."?

Page 10, Line 284: I suggest replacing the text with "the AMOC to the changing freshwater flux by … ."

Page 10, Line 284: Please add a comma after the preposition: "Here, it is assumed … ."

Page 10, Line 285: Enclose the example by commas: “ disturbances, e.g. in initial box salinities, are always .. .”

Page 10, Line 287: Add a comma for the subordinate clause at the end: “decline as studied, e.g. as scenario-dependent … .”

Page 12, Line 291 – 293: The end of the sentence is unclear; please improve the text.

Page 12, Line 294 – 295: Please extend the text to read: "of the tipping element drivers in our model."

Page 12, Line 309: add missing comma around “thus”: “ lower hosing values and, thus, for the AMOC … .”

Page 12, Line 312: I'm unsure, but I guess a comma is missing: "with a slow ice sheet decline, a high hosing determining the fixed … ."

Page 12, Line 314: The sentence needs to be clarified, or?

Page 12, Line 315 – 316: I would like to suggest: “an overshoot cascade changes by variations of Greenland Ice Sheet’s melting patterns. More … “

Page 13, Line 340 – 342: I suggest: Here, …. of an AMOC weakening, and it may be …. surface mass balance, for a warming … ."

Page 13, Line 344: Do you mean “distinct tipping thresholds” or “different tipping thresholds”?

Page 13, Line 349: Please replace "observed" with "detected."

Page 15, Line 356 – 358: The sentence "With the AMOC tipping …. from the AMOC overturning strength" is unclear to me.

Page 15, Line 378: You may replace "ice sheet melting time" with "ice sheet disintegration time"?

Page 17, Line 389: I guess a comma is missing: “is accelerating (Shepherd et al., 2020), and its … .”

Page 17, Line 398: Since you are apparently using the British syntax predominately, replace "e.g.," with "e.g.".

Page 17, Line 404 – 406: Unclear sentence. Please rephrase.

Figure 1: Please increase the size of the hardly seen green points, which is stated in the text (Page 9, Line 235): "subcritical Hopf bifurcation at $F_{GIS}$ $Hopf (indicated by green points in Fig. 2(a)).”

Figure 1, caption: The introduced variable $r_{a0}$ has to be defined. Please find a way to introduce it and/or refer to the text.

Figure 3, caption: I would like to suggest the following modification to the figure caption:” Shown is the AMOC overturning strength, also taking … “ (drop “now”); “… declining from pink (100 %) to grey (0 %) as indicated by the right colorbar.”; “indicate the AMOC in its ‘on’-state, see bottom colorbar).”

Figure 3: Please define the green arrows in the caption and drop them.

Supplement Material

Table S1: Please add missing units, e.g., “psu” for $S_0$.

Table S1, caption: You may also define the unit Sverdrup in the caption.

Table S2: What are the missing units of the listed salinity contents? Please add.

Table S3: Since the hosing flux $H$ has the unit “Sv” in your figures (e.g., 2b), and the combined freshwater flux according to equation (11) shall result in “Sv” as well, the unit of the parameters $A_i$ are dubious. Please check.

Bibliography

Madsen, M. S., S. Yang, G. Aðalgeirsdóttir, S. H. Svendsen, C. B. Rodehacke, and I M Ringgaard. 2022. “The Role of an Interactive Greenland Ice Sheet in the Coupled Climate-Ice Sheet Model EC-Earth-PISM.” Climate Dynamics, no. 0123456789 (February). https://doi.org/10.1007/s00382-022-06184-6.