A Cellular Automaton Model of Tropical Oceanic Rain Clusters with Criticality

Cheung, Kevin K. W.; Teo, Chee-Kiat; Koh, Tieh-Yong

doi:10.5194/egusphere-2026-2469

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https://doi.org/10.5194/egusphere-2026-2469

Preprints

22 May 2026

| 22 May 2026

A Cellular Automaton Model of Tropical Oceanic Rain Clusters with Criticality

Kevin K. W. Cheung, Chee-Kiat Teo, and Tieh-Yong Koh

Abstract. The distributions of the cluster area, A, and total rain rate, R, for tropical oceanic rain clusters from a cellular automaton (CA) are analysed for their scaling exponent ζ_A, ζ_R, and β where f(s)~s-^ζ_^S;S∈{A,R}; f(s) the probability distribution of S. The CA only includes a few simple rules representing a small set of dynamics thought to be important for convective organization. These rules represent large-scale destabilization of the atmosphere under the moisture static energy framework with a slow driving timescale, as well as convective cells interaction through propagating gravity waves with a fast relaxation timescale. The CA exhibits percolation-like criticality, and the ζ_A is estimated to be near the 2-dimensional percolation value of 187/91. This agrees well with the ζ_A estimates over the Indian Ocean warm pool and the tropical Atlantic reported in previous modelling study. Although other critical exponents of the rain cluster distributions from the CA, namely the η_S (scaling exponent of characteristic scale with driving force) and D_S (cluster fractal dimension), S∈(A, R}, depend on the adjustable parameter of the CA, the ζ_A is robust to the adjustable parameter. Although the CA cannot account for the observed ζ_A ~ 5/3 reported elsewhere based on observations, further tuning of it such as through the convective cells interaction strength or manner may make it approach the state of self-organized criticality.

Received: 29 Apr 2026 – Discussion started: 22 May 2026

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Journal article(s) based on this preprint

04 Sep 2026

A cellular automaton model of tropical oceanic rain clusters with criticality

Kevin K. W. Cheung, Chee-Kiat Teo, and Tieh-Yong Koh

Atmos. Chem. Phys., 26, 12543–12564, https://doi.org/10.5194/acp-26-12543-2026,https://doi.org/10.5194/acp-26-12543-2026, 2026

Short summary

Kevin K. W. Cheung, Chee-Kiat Teo, and Tieh-Yong Koh

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2026-2469', Anonymous Referee #1, 01 Jun 2026

The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2026/egusphere-2026-2469/egusphere-2026-2469-RC1-supplement.pdf

Citation: https://doi.org/10.5194/egusphere-2026-2469-RC1
- AC1: 'Reply on RC1', kevin cheung, 04 Jun 2026
  
  We are glad to have received your positive and encouraging comments to our manuscript. We found your suggestions, including addition of historical context in the literature review, clearer distinction between power-law scaling behavior and SOC, and better physical interpretation of results, useful for us to further improve the comprehensibility of our work by readers.
  When comments from the other reviewer(s) are available, we will provide detailed, point-by-point responses together. Thanks.
  
  Citation: https://doi.org/10.5194/egusphere-2026-2469-AC1
RC2:
'Comment on egusphere-2026-2469', Claudia Stephan, 13 Jun 2026

The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2026/egusphere-2026-2469/egusphere-2026-2469-RC2-supplement.pdf

Citation: https://doi.org/10.5194/egusphere-2026-2469-RC2
- AC2: 'Reply on RC2', kevin cheung, 15 Jun 2026
  
  Dear Claudia,
  Thanks for the positive comments on our manuscript. Your recommendations for us to clarify and improve on the following points: (1) the simulated scaling exponents versus the observed and theoretical values, (2) more background information on percolation/SOC especially past studies we didn't review and the causality issue, (3) representation of gravity waves in the model, (4) model size and initialization, and (5) rain representation in the CA model have been noted. By these, we think our work can be better understood by readers.
  When the system is ready, we will provide a revised version of manuscript and detailed responses to all your points.
  
  Citation: https://doi.org/10.5194/egusphere-2026-2469-AC2
RC3:
'Comment on egusphere-2026-2469', Anonymous Referee #3, 14 Jun 2026

Understanding the distributions of the cluster area and total rain rate for tropical oceanic rain clusters is a challenging problem. This study developed a cellular automation (CA) model for simulating the evolution of cloud clusters by only imposing a few simple rules for convective organization. Among them, the key rules are the large-scale destabilization of the atmosphere and effects of propagating gravity waves. The results reproduces some observed scaling exponent as reported in other studies. Considering its scientific merits, I believe this study matches the broad scope of the journal. That said, I have several major and minor comments regarding its presentation and scientific discussion. Overall, I would suggest a major revision before it is considered for publication.
Major Comment:
1. Section 2: CA for tropical rain clusters and Table 1. This part is very important but poorly presented. I would suggest the authors to rewrite the whole section and clearly present the CA model. The details in Appendix B is also very important, I would suggest to move Appendix B back to the main context.
2. Two time scales. I can understand that the two time scales are introduced in the model to mimic radiative time scale and convective time scale. But it is unclear to me how it works in the model perspective. Please clarify how the variable X changes in terms of two time scales (can tau be arbitrary large when t is fixed? ) and why you choose to do so. It seems to me that using a single time scale should be a more straightforward way.
3. Cellular automation model. It seems to me that the so-called cellular automaton model is just grid models that have been extensively studied before in this field. The authors seem to ignore such previous literatures such as Craig and Mack (2013) and Biagioli and Tompkins (2023). Please go search the literature and add more discussions about the grid models for convective organization.
4. Effects of propagating gravity waves. This model only consider destabilization effects for the nearest-neighboring grids and stabilization effects for the next-nearest-neighboring grids. Why is that? Gravity waves generally can propagate further than that. In fact, a recent study by Wilson and Hearter (2026) (https://meetingorganizer.copernicus.org/EGU26/EGU26-21637.html, presented in EGU this year but probably not published yet) discussed this from a conceptual model perspective. I would suggest the authors to discuss the similarity and differences between this model and other models that incorporates gravity waves.
5. Figure 1. It is better to show some data analysis based on your simulation, beside the schematic diagram. Otherwise, readers have no idea how the actual model simulations matches the schematic diagram in Figure 1. Same comment to Figure 9.
6. It is a bit surprising to me that the cases without interacting lattice still exhibit the power law. Why is that? I would suggest to add the scaling exponent in the panels of Figure 2. Same comments to the remaining figures with the distribution curves such as Figure 3-5.
7. The discussions in Section 3.2 and 3.3 are way too technical. I would suggest the authors to rewrite these discussions, avoid too many numbers, and try to focus more on physical interpretation.
8. Figure 6-10. I had a hard time to follow all the details. I guess at least one thing you need to focus on is the effects of propagating gravity waves when comparing the interacting cases with non-interacting cases.
Minor Comment:
1. Table 1, Equation 3: The symbol "I_j" first occurs in this equation without definition before or after it. You need to first introduce what I_j is. (May be the avalanche intensity ?)
2. Line 104: "coordinates"
3. Line 108: "lattice indices that defines ..."
4. Equation (7): The symbol "Z" first occurs in this equation without explanation before or after it. It is better to first introduce the lattice length Z.
5. Figure 1: There are too many words in this figure and the fontsize is too small. I would suggest first simplifying the descriptions into several words in the figure and then explaining them in detail in the figure caption.
6. Line 145: "Ds is the a ...", "the" is needless.
7. Line 159: "be mapped to"
8. Line 332: The description of the question is too hard to understand and a question mark is missing. How about "Is feedback from convection (i.e., generation of waves and vertical motions by condensation) relevant for the relationship between ... ?"
9. Line 348: "a limited set of ..."
10. Line 355: "exhibits criticality which is similar to ..."
11. Line 378: "A and R", a space is missing.

Citation: https://doi.org/10.5194/egusphere-2026-2469-RC3
- AC3: 'Reply on RC3', kevin cheung, 15 Jun 2026
  
  Thanks for providing detailed comments and suggestions to our manuscript. You comments include some model details including how we treat the two time scales and how it compares with some past studies. The link to the recent study of how convection organizes through gravity waves is very useful for us to show similarity and differences from our model. Moreover, we will try to show more model simulation details other than the scaling exponents, and also improving the flow of discussion for better understanding from readers.
  As in the replies to the other two reviewers, after the open discussion period we will provide point-by-point responses to your comments as well as the revised manuscript.
  
  Citation: https://doi.org/10.5194/egusphere-2026-2469-AC3

Peer review completion

AR – Author's response | RR – Referee report | ED – Editor decision | EF – Editorial file upload

AR by kevin cheung on behalf of the Authors (30 Jul 2026) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (01 Aug 2026) by Shaocheng Xie

RR by Anonymous Referee #3 (10 Aug 2026)

RR by Claudia Stephan (12 Aug 2026)

RR by Anonymous Referee #1 (12 Aug 2026)

ED: Publish subject to minor revisions (review by editor) (15 Aug 2026) by Shaocheng Xie

AR by kevin cheung on behalf of the Authors (18 Aug 2026) Author's response Author's tracked changes Manuscript

ED: Publish subject to technical corrections (22 Aug 2026) by Shaocheng Xie

AR by kevin cheung on behalf of the Authors (25 Aug 2026) Author's response Manuscript

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2026-2469', Anonymous Referee #1, 01 Jun 2026

The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2026/egusphere-2026-2469/egusphere-2026-2469-RC1-supplement.pdf

Citation: https://doi.org/10.5194/egusphere-2026-2469-RC1
- AC1: 'Reply on RC1', kevin cheung, 04 Jun 2026
  
  We are glad to have received your positive and encouraging comments to our manuscript. We found your suggestions, including addition of historical context in the literature review, clearer distinction between power-law scaling behavior and SOC, and better physical interpretation of results, useful for us to further improve the comprehensibility of our work by readers.
  When comments from the other reviewer(s) are available, we will provide detailed, point-by-point responses together. Thanks.
  
  Citation: https://doi.org/10.5194/egusphere-2026-2469-AC1
RC2:
'Comment on egusphere-2026-2469', Claudia Stephan, 13 Jun 2026

The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2026/egusphere-2026-2469/egusphere-2026-2469-RC2-supplement.pdf

Citation: https://doi.org/10.5194/egusphere-2026-2469-RC2
- AC2: 'Reply on RC2', kevin cheung, 15 Jun 2026
  
  Dear Claudia,
  Thanks for the positive comments on our manuscript. Your recommendations for us to clarify and improve on the following points: (1) the simulated scaling exponents versus the observed and theoretical values, (2) more background information on percolation/SOC especially past studies we didn't review and the causality issue, (3) representation of gravity waves in the model, (4) model size and initialization, and (5) rain representation in the CA model have been noted. By these, we think our work can be better understood by readers.
  When the system is ready, we will provide a revised version of manuscript and detailed responses to all your points.
  
  Citation: https://doi.org/10.5194/egusphere-2026-2469-AC2
RC3:
'Comment on egusphere-2026-2469', Anonymous Referee #3, 14 Jun 2026

Understanding the distributions of the cluster area and total rain rate for tropical oceanic rain clusters is a challenging problem. This study developed a cellular automation (CA) model for simulating the evolution of cloud clusters by only imposing a few simple rules for convective organization. Among them, the key rules are the large-scale destabilization of the atmosphere and effects of propagating gravity waves. The results reproduces some observed scaling exponent as reported in other studies. Considering its scientific merits, I believe this study matches the broad scope of the journal. That said, I have several major and minor comments regarding its presentation and scientific discussion. Overall, I would suggest a major revision before it is considered for publication.
Major Comment:
1. Section 2: CA for tropical rain clusters and Table 1. This part is very important but poorly presented. I would suggest the authors to rewrite the whole section and clearly present the CA model. The details in Appendix B is also very important, I would suggest to move Appendix B back to the main context.
2. Two time scales. I can understand that the two time scales are introduced in the model to mimic radiative time scale and convective time scale. But it is unclear to me how it works in the model perspective. Please clarify how the variable X changes in terms of two time scales (can tau be arbitrary large when t is fixed? ) and why you choose to do so. It seems to me that using a single time scale should be a more straightforward way.
3. Cellular automation model. It seems to me that the so-called cellular automaton model is just grid models that have been extensively studied before in this field. The authors seem to ignore such previous literatures such as Craig and Mack (2013) and Biagioli and Tompkins (2023). Please go search the literature and add more discussions about the grid models for convective organization.
4. Effects of propagating gravity waves. This model only consider destabilization effects for the nearest-neighboring grids and stabilization effects for the next-nearest-neighboring grids. Why is that? Gravity waves generally can propagate further than that. In fact, a recent study by Wilson and Hearter (2026) (https://meetingorganizer.copernicus.org/EGU26/EGU26-21637.html, presented in EGU this year but probably not published yet) discussed this from a conceptual model perspective. I would suggest the authors to discuss the similarity and differences between this model and other models that incorporates gravity waves.
5. Figure 1. It is better to show some data analysis based on your simulation, beside the schematic diagram. Otherwise, readers have no idea how the actual model simulations matches the schematic diagram in Figure 1. Same comment to Figure 9.
6. It is a bit surprising to me that the cases without interacting lattice still exhibit the power law. Why is that? I would suggest to add the scaling exponent in the panels of Figure 2. Same comments to the remaining figures with the distribution curves such as Figure 3-5.
7. The discussions in Section 3.2 and 3.3 are way too technical. I would suggest the authors to rewrite these discussions, avoid too many numbers, and try to focus more on physical interpretation.
8. Figure 6-10. I had a hard time to follow all the details. I guess at least one thing you need to focus on is the effects of propagating gravity waves when comparing the interacting cases with non-interacting cases.
Minor Comment:
1. Table 1, Equation 3: The symbol "I_j" first occurs in this equation without definition before or after it. You need to first introduce what I_j is. (May be the avalanche intensity ?)
2. Line 104: "coordinates"
3. Line 108: "lattice indices that defines ..."
4. Equation (7): The symbol "Z" first occurs in this equation without explanation before or after it. It is better to first introduce the lattice length Z.
5. Figure 1: There are too many words in this figure and the fontsize is too small. I would suggest first simplifying the descriptions into several words in the figure and then explaining them in detail in the figure caption.
6. Line 145: "Ds is the a ...", "the" is needless.
7. Line 159: "be mapped to"
8. Line 332: The description of the question is too hard to understand and a question mark is missing. How about "Is feedback from convection (i.e., generation of waves and vertical motions by condensation) relevant for the relationship between ... ?"
9. Line 348: "a limited set of ..."
10. Line 355: "exhibits criticality which is similar to ..."
11. Line 378: "A and R", a space is missing.

Citation: https://doi.org/10.5194/egusphere-2026-2469-RC3
- AC3: 'Reply on RC3', kevin cheung, 15 Jun 2026
  
  Thanks for providing detailed comments and suggestions to our manuscript. You comments include some model details including how we treat the two time scales and how it compares with some past studies. The link to the recent study of how convection organizes through gravity waves is very useful for us to show similarity and differences from our model. Moreover, we will try to show more model simulation details other than the scaling exponents, and also improving the flow of discussion for better understanding from readers.
  As in the replies to the other two reviewers, after the open discussion period we will provide point-by-point responses to your comments as well as the revised manuscript.
  
  Citation: https://doi.org/10.5194/egusphere-2026-2469-AC3

Peer review completion

AR – Author's response | RR – Referee report | ED – Editor decision | EF – Editorial file upload

AR by kevin cheung on behalf of the Authors (30 Jul 2026) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (01 Aug 2026) by Shaocheng Xie

RR by Anonymous Referee #3 (10 Aug 2026)

RR by Claudia Stephan (12 Aug 2026)

RR by Anonymous Referee #1 (12 Aug 2026)

ED: Publish subject to minor revisions (review by editor) (15 Aug 2026) by Shaocheng Xie

AR by kevin cheung on behalf of the Authors (18 Aug 2026) Author's response Author's tracked changes Manuscript

ED: Publish subject to technical corrections (22 Aug 2026) by Shaocheng Xie

AR by kevin cheung on behalf of the Authors (25 Aug 2026) Author's response Manuscript

Journal article(s) based on this preprint

04 Sep 2026

A cellular automaton model of tropical oceanic rain clusters with criticality

Kevin K. W. Cheung, Chee-Kiat Teo, and Tieh-Yong Koh

Atmos. Chem. Phys., 26, 12543–12564, https://doi.org/10.5194/acp-26-12543-2026,https://doi.org/10.5194/acp-26-12543-2026, 2026

Short summary

Kevin K. W. Cheung, Chee-Kiat Teo, and Tieh-Yong Koh

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This study presents a cellular automaton (CA) model of tropical oceanic rain clusters based on atmospheric stability and gravity-wave interactions. The model reproduces power-law distributions for cluster area and rain rate, exhibiting criticality similar to 2D percolation. The scaling exponent for cluster area is robust to model parameters and matches simulations over the Indian Ocean and Atlantic, though it differs from some observational estimates, suggesting further model tuning is needed.

A Cellular Automaton Model of Tropical Oceanic Rain Clusters with Criticality

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Interactive discussion

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