Rate-induced tipping in natural and human systems

Ritchie, Paul; Alkhayuon, Hassan; Cox, Peter; Wieczorek, Sebastian

doi:https://doi.org/10.5194/egusphere-2022-1176

Preprints

https://doi.org/10.5194/egusphere-2022-1176

Preprints

11 Nov 2022

| 11 Nov 2022

Rate-induced tipping in natural and human systems

Paul Ritchie, Hassan Alkhayuon, Peter Cox, and Sebastian Wieczorek

Abstract. Over the last two decades, tipping points have become a hot topic due to the devastating consequences that they may have on natural and human systems. Tipping points are typically associated with a system bifurcation when external forcing crosses a critical level, causing an abrupt transition to an alternative, and often less desirable, state. The main message of this review is that the rate of change in forcing is arguably of even greater relevance in the human-dominated anthropocene, but is rarely examined as a potential sole mechanism for tipping points. Thus, we address the related phenomenon of rate-induced tipping: an instability that occurs when external forcing varies across some critical rate, usually without crossing any bifurcations. First, we explain when to expect rate-induced tipping. Then, we use three illustrating examples of differing complexity to highlight universal and generic properties of rate-induced tipping in a range of natural and human systems.

Received: 28 Oct 2022 – Discussion started: 11 Nov 2022

Paul Ritchie et al.

Status: final response (author comments only)

CC1: 'Comment on egusphere-2022-1176', Richard Rosen, 12 Nov 2022

I think this paper covers an extremely important topic. But as a physicist who is not familiar with the many references provided, I hope that the paper could be expanded to include more common sense wording and clearer explanations for figures 1, 2, and 3. Those figures might have to be modified to make them more understandable to the scientific reader who is not familiar with the extensive literature. I think this would make this very good and important paper more accessible to the broader readership of this journal. In addition, the text for each of the three examples provided should be expanded and made simpler without needing to understand the set of equations provided for each.

Citation: https://doi.org/10.5194/egusphere-2022-1176-CC1
RC1:
'Comment on egusphere-2022-1176', Niklas Boers, 26 Nov 2022
The authors submitted a great, concise review of the phenomenon of rate-induced tipping and showcase it nicely using conceptual models from ecology, climate, and powergrids. I very much enjoyed reading this. The authors accomplished to explain rate-induced tipping in a very accessible manner while being technically fully accurate and still giving sufficient details to allow for an easy reproduction of the examples. The authors might want to take into account some of the very minor comments below; but the manuscript could also be accepted as is I think.

Some minor comments / questions / suggestions (and most are really very minor):

l6: "an instability that occurs when external forcing varies across some critical rate" - to me it seems this could be misunderstood, maybe "varies faster than some critical rate"

l11: I wouldn't necessarily say that the "changes" are referred to as "tipping points", the latter are rather the points (in forcing or time) at which such changes occur?

l19: I think that the seminal paper by Stocker Schmittner (https://www.nature.com/articles/42224) should be cited here as well - to my knowledge this is the first paper describing rate-induced tipping effects, at least in the climate context

l52: replace "vanishing" by "sufficiently small"? Also in the next line, "instantaneously" to "fast enough"?

l76: istability -> instability

Fig.3: is the axis of (c) logarithmic?

l109: could this sentence be simplified?

l125: the system has too much intertia?

l164: cite some more papers, including some of the older ones, on AMOC collapse here as well?

Fig.4: show additional panel similiar to the one in Fig.5c here as well?

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Niklas Boers
Citation: https://doi.org/10.5194/egusphere-2022-1176-RC1
- AC1: 'Reply on RC1', Paul Ritchie, 10 Feb 2023
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2022/egusphere-2022-1176/egusphere-2022-1176-AC1-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2022-1176-AC1
RC2:
'Comment on egusphere-2022-1176', Anonymous Referee #2, 27 Dec 2022
The article “Rate-induced tipping in natural and human systems” explains the phenomenon that occurs when an accelerating parameter change drives a system’s state across an unstable equilibrium into another dynamical regime. The authors demonstrate this using two idealized (artificial) systems, and three simple models supposed to represent ecological, climatological and technological systems, each based on a few coupled ordinary differential equations.

I think that the content is relevant and within the scope of the journal. The article also does a relatively good job at explaining rate-induced tipping. On the other hand, I also have one very major point of concern:

To put it bluntly: Do we really need another article that demonstrates what rate-induced tipping is? What makes this one different? Scanning the authors reference list, I find several previous articles with a similar aim and scope, for example Alkhayuon and Ashwin 2018, Ashwin et al. 2012, Ashwin and Newman 2021, Scheffer et al. 2008, Wieczorek et al. 2011, Wieczorek et al. 2021.

In its current state, the article has a flavor of an “understandable science” communication or teaching essay for academics. I sympathize a lot with such educational articles, but I am also a bit skeptical if a scientific journal is the best place for such a piece, if there is not also some new content. I hence believe that the aim of the article should go beyond a mere demonstration of the phenomenon.

I also believe that there are elements in the article already that can fulfil this requirement, but that could be worked out more explicitly. The authors should therefore reconsider what is the main objective and the type of this article, and make this clearer.

I can imagine three options to achieve this:

Extend the content toward a comprehensive review article. The authors already call it a “review”, but in my view it is too superficial and selective to deserve that name. The title and abstract promise an extremely broad scope, but what we get is an explanation of the phenomenon with selective examples from very simple models. There is hardly any discussion of R-tipping in other systems and models. From a review I would expect a more comprehensive coverage of phenomena and the state of research.

Focus on arguments why R-tipping has been underrated. The authors say that rate-induced tipping is “arguably of even greater relevance” than tipping at a critical level. But I wonder why that should be the case in practice. If the authors want to focus on such a particular statement, like in a “perspective” article, then they should present arguments that support their statement.

The study could also be a normal research article. However, there have been many studies about rate-induced tipping already, as can be seen from the reference list. So the question arises what is new. The phenomenon shown in Fig 3 and 4, that merely increasing the rate of forcing can cause a complex sequence of regimes, looks interesting enough to me, and could possibly be the main result in that case, but it should be checked if/how it has been captured by previous studies. The authors could demonstrate this behavior in some more detail, adjust the abstract and conclusions accordingly, also including examples and a discussion of generality and relevance of the phenomenon. What properties do dynamical systems need to have in order to see such a complex sequence? And how likely would it be to see something like that in complex models and the real world?

Option 1 or 3 make the most sense to me, but I don’t wish to push the authors (or journal) into one particular direction, as long as the revised article is convincing (i.e. unique, focused, and with well-defined and sufficiently comprehensive content).

I have two related points of major importance:

1. Choice of methods

The examples the authors show are very general and simple, and rather repetitions of conceptual models instead of independent / emergent phenomena from process-based models or observations. I would like to read some more statements about their relevance: Has rate-induced tipping been observed in ecosystems, climate or power grids, in a way that gives credibility to the applied models?

The plant-herbivore model is particularly vague. What kinds of species and ecosystems should I think of? Horses in a steppe? Sea urchins in marine kelp forests? Slugs on the salad in my garden? What is the observed behaviour that the model is supposed to represent? I guess there is information in the cited literature, but at least a few lines would help here.

The climate example looks more convincing to me because it is based on physical processes and has variables with a specific meaning. However, a bridge to more complex models is missing, where R-tipping has long been studied as well.

And how well does the power grid model simulate behavior in actual (inter-)national power grids? Is there evidence for R-tipping in these grids?

In general, the linkages between the conceptual models and the real world should be discussed and substantiated more.

2. Title

The title suggests an extremely broad scope – “natural and human” is virtually everything. It could be OK for a very comprehensive review, but it mislead me a bit in case of the current draft. I suggest to make it more precise to better match the content. “Natural systems” here refers to a brief example from climate research and one from ecology. “Human systems” again is quite vague; I first expected something like societal networks here. A better title for the current article might be “rate-induced tipping in climate, ecological and technological systems”? Of course, the new title should reflect what choice the authors make regarding the aim and scope as discussed above.

The selection of content the authors want to focus on could be better justified. What are the criteria? Why exactly grazing, ocean circulation, and power grids? The Authors should both limit the scope and extend the content substantially in order to have comprehensive content within the scope.

List of more minor points:

Abstract: “hot topic” is arguably somewhat informal.

Can it actually be distinguished properly what is tipping at a critical level versus a critical rate? The control parameter in a model could represent a flux (like freshwater input per year into the North Atlantic). I suppose that the unit alone cannot be essential for the difference, which should rather be the mathematical structure of the problem. It becomes clear later in the paper that this is indeed the case, but the notion of “rate” in the beginning can be a bit confusing.

I like Fig. 1 in principle. One could add an arrow to indicate movement of the potential landscape to the left. What is a bit unintuitive: It seems that a critical rate alone is still not enough, but the movement of the potential has to be large enough as well (if it moves infinitely fast but the ball stays close to the minimum, nothing happens). In the text, it reads like a critical rate alone is sufficient. Probably this is also the difference to a “B-tipping” where the control parameter represents a rate of change in physical units?

Something that I find confusing about Fig. 1 is: I have to assume that the ball has no mass (in the sense that I don’t need energy to move the potential and/or lift the ball to the hill)? But it does have inertia (otherwise I could not shift the potential left or right)? And: If it has inertia, it would oscillate around the minimum, unless there is large friction. But if there is large friction, how can I pull away the potential? I guess it is hard to find a physical model that is a better analogy, but at least the essentials and limitations of the analogy should be mentioned.

Line 29: what is a “forced system”? One with boundary conditions, or one where boundary conditions change over time, or even where they accelerate? It seems to me that the latter is needed for rate-induced tipping, but acceleration is not mentioned anywhere. In general, “forcing” is used a lot in the article but not well-defined in the beginning (though I got the idea later on that forcing is the control parameter’s value while “forced system” implies it’s changing over time?).

Fig. 2: It could make this figure more understandable by showing how the forcing (and state) change in time.

Also, for the arguments in the caption to work (e.g.: avoid B-tipping and then cause R-tipping in c), the particular shape of the black curves is important. But these curves are different from typical “saddle-node” bifurcation curves shown in the references. In particular, stable and unstable branches are tilted in the plotted space, and always very close together. So I wonder how generic the “return tipping” is? It looks like a much more special behavior than B or R-tipping in general.

What are the methods used to plot Fig. 2?

Fig. 2b and c: I don’t understand why the state would suddenly drop to 0 instantly after crossing the dotted line. If it has inertia (as is needed for the tipping to occur), it would not care, but continue on a curved continuous line.

Line 80: “then a natural option would be to reverse the external forcing to avoid crossing the critical level.” Why? It would suffice to stop the forcing from changing. For example, I don’t expect that mankind will reverse greenhouse gas forcing with the same rate as the previous increase, which would be even much more difficult than reaching net zero (and probably unnecessary). Maybe for the power grid this matters, but I don’t see the connection between that model and the model used for Fig. 2.

Fig. 3: a nice complement to Fig 2. But could both Figures show the same example? Unintentional return tipping (like in Fig 2c) does not occur here? It would help a lot to also see the stable and unstable equilibria of this system.

Fig 3a: black = blue+purple+orange?

Line 98: “previous research has shown that…” Isn’t that the definition of B-tipping, not a research result?

Fig 3c: How much does this rely on the particular shape of the function forcing versus time? At least it seems to require symmetry in the ramp up and down phases. This is a very strong and, if you think of real-world examples, restrictive assumption.

Line 109-110: I don’t really understand the statement about “multiple critical rates” / slow and fast rates. Wouldn’t any accelerated ramp up require one specific minimum rate of ramp down (given a certain function shape)? But in the system the authors used to generate Fig. 3 (equations would be nice), the ramp up is always assumed to be symmetric to the ramp down? This leads to the “white-green-red-green” regimes when increasing the overall rate. This behavior is indeed interesting; but how generic is it? How does this system differ from the stability diagram in Fig 2?

Line 112 and elsewhere: “fixed maximal change” is confusing. Do the authors mean the amplitude of the Forcing pulse? Or the maximum rate of change? And “Fold level” is the bifurcation point of the static system?

At times, the authors cite rather selectively, e.g. only very recent research, and make the impression that rate-induced tipping phenomena are a rather new field of study. But this is not the case. For instance, as one of the other reviewers points out, rate-induced collapse of the ocean circulation has been a known phenomenon in complex climate models at least since the 1990ies, https://www.nature.com/articles/42224.

As stated above, if the paper is supposed to be a review, the reference list appears rather short.

Line 195-200: unclear to me, could be better explained. There are infinitely many “stable equilibria”, called base states. If I shift the phase by 2pi, don’t I get the same behaviour again, instead of a different solution? Here it says “see Methods”, but I don’t find an answer there. Then, despite the infinite number of stable equilibria there are only two “alternative states”. Each base state has only one specific alternative transient state? Why are the other stable equilibria not also “alternative states”? The blackout is one alternative state to all the base states, correct?

Line 306: grid, not gird

I suspect that most readers will not be familiar with at least two out of the three models because these models describe very different phenomena from different scientific fields. A little more background information about these models and ideally a figure about each would be welcome.

Video supplement: Videos could be a great supplement. However, I was unable to find the github repo referenced in “Ritchie et al., 2022”. Please provide a link that works, and one that works for readers without a github account.
Citation: https://doi.org/10.5194/egusphere-2022-1176-RC2
- AC2: 'Reply on RC2', Paul Ritchie, 10 Feb 2023
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2022/egusphere-2022-1176/egusphere-2022-1176-AC2-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2022-1176-AC2
RC3:
'Comment on egusphere-2022-1176', Anonymous Referee #3, 28 Dec 2022

This manuscript explains the concepts related to the rate-induced tipping and cites the relevant litterature along. As such, it is not really a review, I would rather call it a "pedagogical review", and it is somehow up to the editors to decide if it fit the scope of the present journal. My opinion on the subject is that it does fit.
It is well written and could almost be published as is, but I have some suggestions on the first two figures to help the reader (including me).
For FIgure 1, I would be more descriptive of what is what in the Figure. I had trouble understanding the links between panel a and b at first glance. I have a proposal below:
Anyway, the authors could change it differently, as long as it becomes clearer.
For figure 2, for each panel, I would add besides the external forcing evolution as a function of time for the two different curves, as is done in Figure 4. For it is crucial not to loose the reader at this curcial point.
Note that the resolution of the figures is not sufficient for printing (screen is ok).

Citation: https://doi.org/10.5194/egusphere-2022-1176-RC3
- AC3: 'Reply on RC3', Paul Ritchie, 10 Feb 2023
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2022/egusphere-2022-1176/egusphere-2022-1176-AC3-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2022-1176-AC3

Paul Ritchie et al.

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Short summary

Complex systems can undergo abrupt changes or ‘tipping points’ when external forcing crosses a critical level and are of increasing concern because of their severe impacts. However, tipping points can also occur when the external forcing changes too quickly, without crossing any critical levels, which is very relevant for Earth’s systems and contemporary climate. We give an intuitive explanation of such rate-induced tipping and provide illustrative examples from natural and human systems.


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