the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 15 Sep 2025
Targeted Adaptive Chaos Control of Regimes and Eddy Strength in Two Lorenz Models
Abstract. Extreme weather events present growing challenges as climate changes. "Weather Jiu-Jitsu" is a proposal to nudge atmospheric circulation to redirect or defuse these extreme events by leveraging the sensitivity of chaotic atmospheric dynamics to initial conditions. We demonstrate an optimal control strategy to stabilize two low-order models of atmospheric dynamics, the Lorenz 63 (L63) and Lorenz 84 (L84). Estimated local Lyapunov exponents (LLE) are used to decide when to apply control. In L63, regime transitions are treated as model analogs of persistent circulation states of concern, while in L84, large eddy amplitudes serve as conceptual surrogates for synoptic-scale moisture transport events such as atmospheric rivers. The timing and amplitude of nudges is solved over a forecast horizon to minimize the total energy applied, while ensuring that the trajectory remains within predefined bounds to avert undesirable consequences. We explicitly incorporate multiplicative noise, randomly selecting a trajectory from an ensemble forecast to apply control, thus reflecting the mismatch between model and reality that would arise in operational applications.
- Preprint
(1296 KB) - Metadata XML
-
Supplement
(681 KB) - BibTeX
- EndNote
Status: open (until 13 Nov 2025)
-
CC1: 'Comment on egusphere-2025-3997', Justin Finkel, 19 Sep 2025
reply
-
AC1: 'Reply on CC1', Moyan Liu, 20 Sep 2025
reply
Thank you for your feedback! They are really valuable! Here are some answers for your feedback:- Overall CommentsI agree that the extremes we defined now are natural intrinsic, and the more practical ways for application maybe more physically intrinsic. I am currently working on defining hidden states as a trigger for control, i think this may be useful in the sense of more generalized intrinsic. But adding a sentence to emphasize will definitely be helpful.Yes, varying the variables will be a great help, and i was wanting to do it but there is word limits. But I will consider to put a section for test in the supplementary.- Minor Comments:A. the reason why I use this initial state is just for comparison with previous work as cited. They deliberately use this initial state but no explanation. Since we have the same goal (confine l63 to one wing), I decide just to make it same.C. the idea using surrogate is to decrease the time for calculating RK4, and it can be regarded as a forecast model in the real world, which is our future goal of control the real weather model. And yes, I will make more explicit description.D. You are rightE. Yes. They are from surrogate modelF. We added ensembles is to mimic between model and real world, which is also part of reason why we have re-optimization step.G. We want to see the relative energy usage comparing with intrinsic energy, and also for easier comparison with L63 and L84H. good idea thanksI. Yes its fixedJ. that's interesting, I would say 1) we probably cannot afford the continuously perturbation 2) the more perturbation will generate more uncertainty.K. YesL. ExactlyCitation: https://doi.org/
10.5194/egusphere-2025-3997-AC1
-
AC1: 'Reply on CC1', Moyan Liu, 20 Sep 2025
reply
-
RC1: 'Comment on egusphere-2025-3997', Henry Schoeller, 26 Sep 2025
reply
General Comments
The provided manuscript presents an original addition to the well-established field of control theory with meteorological application in mind. It gives convincing evidence that the implemented approach achieves the worthwile and reasonable goals mentioned in the introduction. There are several aspects that I suggest could be improved upon to make the documentation of the procedure more understandable and reproducible. Additionally, the presented results spark some questions for further investigation that surely warrant forthcoming studies, but could also be adressed already here. In particular, you have touched upon the possiblity of the algorithm not achieving the set goal depending on initial conditions for the Lorenz 63 system, but have omitted a systematic analysis of this failures dependence on the algorithms parameters. E.g. could it be prevented if the perturbation energy was allowed higher? Also, I would appreciate the code be made public to be able to reproduce results and better understand the implementation. Please also make sure your mathematical notation is consistent and thorough (see below).
Specific Comments
- 54ff: You mention control theory has been studied extensively with the L63 system, but is there also literature on control theory applied to the L84 system?
- 90ff: Since you later use the energy of the system, I suggest you introduce the concept already upon introduction of the Lorenz system. As an ad-hoc measure, I think using the norm of the state vector as a measure of the energy of the system makes sense, but introducing and explaining why might make your manuscript more understandable.
- 124: "... we consider them to be particular seasonal condition"; not sure what is meant here
- 151: Not sure what is meant with "spatially extended systems"
- 158: The sources you cite above upon introducing the local Lyapunov exponents do not give the formula you mention. In particular, the maximum real part of the eigenvalues of the jacobian characterize the instantaneous growth rate in the direction of the respective eigenvector rather than the maximum perturbation growth. As far as I understand, the local Lyapunov exponents are the finite-time Lyapunov exponents for one time step. As such, they are the singular values of the instantaneous finite time propagator of the linear tangent equation and as $\Delta t \rightarrow 0$ they are approximated by the eigenvalues of the symmetric part of the Jacobian $(J+J^T)/2)$ rather than the eigenvalues of $J$ itself. See, for example, Ott, E. (2002) Chaos in Dynamical Systems. 2nd edn. Cambridge: Cambridge University Press.
- 164: I am not sure, I fully understand the following descriptions. Do you generate a new 50 member ensemble every time step or do you select a new ensemble member every time step?
- 171: So if I understand correctly, your assumption is that model and observation uncertainty scale with system energy? Maybe add a sentence about this.
- Figure 1: While the figure is helpful, I would appreciate a little more verbosity in the caption. This could improve the quality and make the content even more helpful and more easily understandable. Also, the figure makes it seem like different algorithms are applied between the first and latter optimization attempts (Re-Optimization), but this is not really the case is it? Isnt the re-optimization really a part of the control algorithm? And would it not be more accurate if the "Apply Control" was on the arrow between "Sampled Trajectory" and "Outside the bound"? I also feel like the whole "Optimization" box does not fit the flow chart since there is not really a "flow" involved but rather the two "Constraints" are properties of the algorithm that are considered simultaneously. In addition, I am not sure if I understand how an element of the flow chart is chosen to have a rectangular or a diamond shape and how its colour is decided.
- equation 9: is E_control = u_t^2 ? why introduce a new variable? And shouldnt the variables in eqn. 9 also get a t index?
- equation 10 & 11: so $\delta X_{t} = (dx, dy, dz)$? Also, at this point the reader assumes lambda signifies the LLE, but later you mention it is a penalty weight. Please introduce new variables immediately upon use and avoid naming conflicts.
- 203: what exactly is meant by "chance constrained or probabilistic constraint set"?
- Figure 2: Consider merging subfigures (c) and (d) by creating a second y axis on the right, but I concede that this is a matter of taste.
- 238: "best balance" seems subjective. Please elaborate in what sense this is the best balance; i.e. according to what metric?
- 247: Why not simply use the 90th percentile of the LLE?
- 249: In the formulation of the penalty term in equation 14, you provide the penalty as it applies to the Lorenz 63 system only. The description here suggests the penalty for the Lorenz 84 system is related to the eddy magnitude (i.e. circular as opposed to square like in the previous sections). Please explicitly provide the penalty terms for both systems.
- In contrast to the Lorenz 63 system, you do not mention the algorithms dependence on initial conditions. Does it not exist or have you not studied it?
- 274ff: The control approaches you mention here were not mentioned in the introduction and the ones you mention in the introduction are not mentioned here. Please give a survey of relevant literature and methods in the introduction and only summarize and compare them with regard to the results of your study in the final section.
- 345: The source Huang et al. is incomplete
- Supplement: the surrogate model is verified appearently only for one test case for each model and not in a systematic way (e.g. splitting a dataset into training and verification, applying common metrics, etc.). I understand that the surrogate model is not the focus of the study and the successful control justifies its use, a more rigorous verification might still strengthen your manuscript.
Citation: https://doi.org/10.5194/egusphere-2025-3997-RC1 -
AC2: 'Reply on RC1', Moyan Liu, 01 Oct 2025
reply
Thank you so much for your valuable feedback, and here are answers for your feedback:
For the general comments, “could it be prevented if the perturbation energy was allowed higher?”, the answer is both yes and no. The system can be controlled with sufficiently high energy, but this is not our objective. Our focus is on using small perturbations to prevent extreme outcomes. In practice, some LLEs are so large that effective control within the phase space is extremely difficult and would require high energy.
For Specific Comments:
- 54ff: You mention control theory has been studied extensively with the L63 system, but is there also literature on control theory applied to the L84 system?
- As far as we know, there is no existing literature on control applied to the L84 system.- 90ff: Since you later use the energy of the system, I suggest you introduce the concept already upon introduction of the Lorenz system. As an ad-hoc measure, I think using the norm of the state vector as a measure of the energy of the system makes sense, but introducing and explaining why might make your manuscript more understandable.
- Yes, we will introduce and explain this concept earlier in the manuscript.- 124: "... we consider them to be particular seasonal condition"; not sure what is meant here
- We use fixed parameters for the forcing parameters (F&G) that are not seasonally varying- 151: Not sure what is meant with "spatially extended systems"
- We will change to ‘spatio-temporal dynamics’ instead to make it more clear in the paper.- 158: The sources you cite above upon introducing the local Lyapunov exponents do not give the formula you mention. In particular, the maximum real part of the eigenvalues of the jacobian characterize the instantaneous growth rate in the direction of the respective eigenvector rather than the maximum perturbation growth. As far as I understand, the local Lyapunov exponents are the finite-time Lyapunov exponents for one time step. As such, they are the singular values of the instantaneous finite time propagator of the linear tangent equation and as $\Delta t \rightarrow 0$ they are approximated by the eigenvalues of the symmetric part of the Jacobian $(J+J^T)/2)$ rather than the eigenvalues of $J$ itself. See, for example, Ott, E. (2002) Chaos in Dynamical Systems. 2nd edn. Cambridge: Cambridge University Press.
- Thank you for pointing this out. You are correct about the LLE definition. We have updated the formulation accordingly, and we confirmed that the results are not substantively changed.- 164: I am not sure, I fully understand the following descriptions. Do you generate a new 50 member ensemble every time step or do you select a new ensemble member every time step?
- Each time we need to control, 50 new members will be generated and 1 member will be randomly selected. The idea is to mimic the process that would be followed in practice. The emergent trajectory will be one from a possible ensemble and on observing which one it is, we need to take the next decision.- 171: So if I understand correctly, your assumption is that model and observation uncertainty scale with system energy? Maybe add a sentence about this.
- Yes, we assume that both model and observation uncertainty scale with system energy, and we will add a clarifying sentence.- Figure 1: While the figure is helpful, I would appreciate a little more verbosity in the caption. This could improve the quality and make the content even more helpful and more easily understandable. Also, the figure makes it seem like different algorithms are applied between the first and latter optimization attempts (Re-Optimization), but this is not really the case is it? Isnt the re-optimization really a part of the control algorithm? And would it not be more accurate if the "Apply Control" was on the arrow between "Sampled Trajectory" and "Outside the bound"? I also feel like the whole "Optimization" box does not fit the flow chart since there is not really a "flow" involved but rather the two "Constraints" are properties of the algorithm that are considered simultaneously. In addition, I am not sure if I understand how an element of the flow chart is chosen to have a rectangular or a diamond shape and how its colour is decided.
- Optimization and re-optimization are using the same algorithm. We will make a description in the caption to make the points clear. For the diamond shape, they are decisions or question mark, and the boxes are actions made.- equation 9: is E_control = u_t^2 ? why introduce a new variable? And shouldn't the variables in eqn. 9 also get a t index?
- I wanted to make the control and total energy constant in format easier to understand. And yes t index will be added- equation 10 & 11: so $\delta X_{t} = (dx, dy, dz)$? Also, at this point the reader assumes lambda signifies the LLE, but later you mention it is a penalty weight. Please introduce new variables immediately upon use and avoid naming conflicts.
- Yes, sure we will rename it for better understanding.- 203: what exactly is meant by "chance constrained or probabilistic constraint set"?
- This is the idea why we chose re-optimization because it is computationally simpler and more realistic, instead of using a probabilistic method.- Figure 2: Consider merging subfigures (c) and (d) by creating a second y axis on the right, but I concede that this is a matter of taste.
- We will adjust that.- 238: "best balance" seems subjective. Please elaborate in what sense this is the best balance; i.e. according to what metric?
- Supporting Information S4 shows the trade off between total control energy and percentage of time steps with control applied. We choose the one with less total energy and less time for control applied among others. But it is subjective.- 247: Why not simply use the 90th percentile of the LLE?
- This is chosen to represent the high eddy amplitude. We do not want a very high percentile where only few points need to be controlled, nor too many points to control.- 249: In the formulation of the penalty term in equation 14, you provide the penalty as it applies to the Lorenz 63 system only. The description here suggests the penalty for the Lorenz 84 system is related to the eddy magnitude (i.e. circular as opposed to square like in the previous sections). Please explicitly provide the penalty terms for both systems.
- yes will add l84 penalty term for better understanding- In contrast to the Lorenz 63 system, you do not mention the algorithms dependence on initial conditions. Does it not exist or have you not studied it?
- The initial condition is intrinsic, so I make two distinct explore figure 2&3 to show how the initial condition affects the control. In reality, the idea is that the control is unfeasible if the initial state is too chaotic.- 274ff: The control approaches you mention here were not mentioned in the introduction and the ones you mention in the introduction are not mentioned here. Please give a survey of relevant literature and methods in the introduction and only summarize and compare them with regard to the results of your study in the final section.
- Thank you for the comments, we will update it- 345: The source Huang et al. is incomplete
- This is our perspective paper that currently under review, we will fix this and here is the link if you are interested to read: https://arxiv.org/abs/2508.09376Citation: https://doi.org/10.5194/egusphere-2025-3997-AC2
-
AC2: 'Reply on RC1', Moyan Liu, 01 Oct 2025
reply
-
RC2: 'Comment on egusphere-2025-3997', Anonymous Referee #2, 08 Oct 2025
reply
This manuscript presents controllability results for two different Lorenz models. The material is interesting, and the presentation, including figures is pretty decent. My central issue is with the applicability and generalizability of the results. There are many Lorenz models, all generally being some sort of Galerkin truncation. Both models the authors choose are three ODE truncations. This means these are toy models, and regardless of what words (e.g. eddies) are attached to them, the models are really simple ODE models that yield hard chaos in certain regimes.
While some Lorenz models are often quoted in the popular press, claiming that actual climate change mitigation strategies should be based on them is either fanciful or frightening (depending on one’s persuasion). For a technical journal like NPG, this means some perspective on the models is necessary (for example, the monograph on Mantle Convection by Schubert, Turcotte and Olson provides extensive discussion of how different truncation levels of the original Lorenz 63 model yield very different behaviour). There is likely a similar discussion somewhere for L84, though I cannot think of an appropriate reference.
This issue pops up again in constructing the controller. Since what is being evolved are coefficients of the modal functions one is implicitly assuming that the controller can be constructed to act on the modal function. While a relatively trivial assumption mathematically, this may be impossible from a physical point of view, and even when possible may be prohibitively expensive (from either an energy or a monetary point of view).
The actual work done appears to me to be sound as a fun bit of applied mathematics. I do find the data availability statement at odds with current notions of open science. There are new computational methods here, and the software, in my opinion, should be available to the community. Stating that no new data was used is getting away on a technicality (this is a theoretical exercise), and if one considers the main statement then the authors are basically following the practice of the 1990s when my career began. I too write software, and understand the fine line between sharing and avoiding getting scooped. But every once in a while, I think we have to take the jump and assume collegiality.
Finally I am not sue I get the whole “jiu-jitsu” terminology. Is there some analogy between 3 ODE systems and martial arts? Between the climate system and martial arts? Why not kung fu instead of jiu-jitsu? I like clever acronyms and fun ways to label our science, but I am afraid this one didn’t land for me.
In summary, I think this manuscript should be published, but it needs work. In particular work on bridging the gap between trivial 3 ODE systems and larger climate models and concepts of larger climate models is essential.
Citation: https://doi.org/10.5194/egusphere-2025-3997-RC2 -
AC3: 'Reply on RC2', Moyan Liu, 08 Oct 2025
reply
Thank you very much for your thoughtful and constructive review.
We agree that both Lorenz-63 and Lorenz-84 are toy models. Our purpose in applying control to these systems is to test the feasibility of our control concept within an idealized chaotic framework before extending it to higher-dimensional climate models.
Regarding the energy concern, this is precisely why we formulate the control as an optimization problem, designed to minimize control energy. This embodies the main idea behind our term “Weather Jiu-Jitsu.” Quoted from google: "The core concept of jiu-jitsu, especially Brazilian Jiu-Jitsu (BJJ), is using leverage and technique to control and submit a larger, stronger opponent, rather than relying on brute force'. Therefore, our approach seeks to use small, well-timed perturbations to achieve large and beneficial changes in the system trajectory. For additional context, our related perspective paper (currently under review) elaborates on this philosophy, a preprint is available at 'https://arxiv.org/abs/2508.09376'.
We also appreciate your comments on open science. We are finalizing the code and will release it on our GitHub repository.
Finally, your suggestion to bridge the gap between toy models and realistic climate systems is very valuable. We will add a paragraph discussing the limitations of idealized toy models and the relevance of higher-dimensional climate systems, and we note that our ongoing work applies the same control framework to more complex and data-driven climate models.
Citation: https://doi.org/10.5194/egusphere-2025-3997-AC3
-
AC3: 'Reply on RC2', Moyan Liu, 08 Oct 2025
reply
Viewed
| HTML | XML | Total | Supplement | BibTeX | EndNote | |
|---|---|---|---|---|---|---|
| 795 | 127 | 17 | 939 | 14 | 17 | 18 |
- HTML: 795
- PDF: 127
- XML: 17
- Total: 939
- Supplement: 14
- BibTeX: 17
- EndNote: 18
Viewed (geographical distribution)
| Country | # | Views | % |
|---|
| Total: | 0 |
| HTML: | 0 |
| PDF: | 0 |
| XML: | 0 |
- 1
I. Overall comment: I think these dynamical systems and control experiments are a great place to start with this agenda. But I want to flag two particular choices made here, which make total sense but carry some danger of misleading us in more complex systems. (And I don't think this is news for you --- just highlighting my own leading concerns.)
A. The choice of the target variable to control (the wing to avoid in L63, or the eddy amplitude |Y|+|Z| in L84) are in these cases very natural intrinsic coordinates, and they happily correlate with LLEs. In spatially extended systems where the target of interest may be local in space and a more complex compound function of different physical fields (like mortality risk as a function of wet bulb temperature and electric grid failure), the relationship between "physically intrinsic" extremes and "stakeholder-defined" extremes might break down. I think the key is to generalize the "trigger" variable beyond LLEs to something that is less intrinsic, and more tailored to the event. I don't know what it is yet.
B. Similarly, the choice of actuation variable is not up to us in the real world; it is pretty arbitrary.
C. As a result, in future studies, I feel that a premium should be placed on varying both the target variable and the actuation space very rigorously to verify algorithmic robustness.
II. Minor comments
A. L104-105: is the high precision of initial conditions warranted in light of downstream errors induced by stochastic noise and surrogate models? My initial reaction (which might be flawed) is that those errors would dominate initial condition errors. I wonder how much precision is needed to ensure the relative stability at early stages, which you cite as the rationale.
B. L124 add "a" before "particular"
C. L135: I didn't quite understand the role of the surrogate model here. Is it to subject your method to the additional challenge of dealing with model error? That would be reasonable (especially in follow-up work with more complicated systems), but since the present paper focuses on the control algorithm and doesn't quantify the role of model error, the use of the surrogate just seems like a distraction. Alternatively, it would make sense if you used a *finite-time* surrogate, but the supplement shows it to be a *tendency* surrogate, so you still need to do the numerical integration. As the "truth" models are quadratic, it seems to me the surrogate isn't helping with efficiency either. Also, in the second equation of the supplement defining W\*, shouldn't the objective involve $W\Phi$, as in the first equation, instead of $\Phi W$? Perhaps capital $X$ refers to a whole dataset of lower case $x$s arrange in rows? Overall this section could benefit from more explicit description of the datasets used.
D. L150: add "finite" before "spatial", and clarify whether "spatial" here refers to geographical space or model state space. Relatedly, I think that FSLEs are relevant in all systems, but especially ones with multiple timescales, and spatially extended systems are only one example where multiple timescales occur. Is my understanding correct?
E. L154-156: I assume that $f$ and $J(x)$ are both from the surrogate model?
F. L163-165: What is the role of the ensemble, if only one member is selected? More generally I'm not sure about the role of noise in thie study. Just like the surrogate model, it appears to be an extra layer of difficulty for the algorithm, but one that is not quantified in any way. Given the relatively small time horizons and small number of interventions applied, I question whether the extra randomness has been explored enough to even merit inclusion at this stage. (Of course noise will be critical for later experiments for representing model uncertainty and extrinsic factors.)
G. L178: why use the ratio of control to total energy in the objective, when the absolute control energy is really what would drive operational costs?
H. L192: I recommend including the control $u_t$ as an extra argument to $f()$, just for clarity's sake.
I. L198: is the noise realization fixed over subsequent iterations of SLSQP?
J. L210-212: just a research speculation: are there cases where applying repeated controls actually sets up the system for a later, bigger catastrophe, analogously to how putting out small fires lets vegetation build up and set the stage for a catastrophic blaze? Perhaps "excitable" is the right category in which to put such systems, and this deserves some dedicated research.
K. L224-225: the high cost of late intervention also indicates that LLEs are not the end-all of triggers; we should prefer *leading* indicators (and I know this was acknowledged before, maybe should be referenced again)
L. L242: "To determine an appropriate state space for specifying control": Does this mean "to determine a useful leading indicator to trigger control"?