the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 20 Jul 2026
The fractional Rothermel model: Mittag-Leffler kinetics of wildfire front propagation from a stochastic hierarchical parcel cascade, with application to heading fire collapse
Abstract. In the Rothermel model the heading wind factor is an unbounded power law of midflame wind speed, unphysical at high wind and patched with an ad hoc cap, and its projected-wind extension collapses the heading front above a low fuel-dependent threshold. We recast the rate law as the fractional Rothermel model, a fractional ignition kinetics in which, with memory in the ignition waiting time, unburned-fuel survival solves a Riemann-Liouville relaxation equation with Mittag-Leffler solution, giving a wind factor that recovers the power law at low wind and saturates at high wind. We fix the fractional order rather than fitting it, grounding it in a stochastic parcel cascade whose random branching number yields distributed-order dynamics and whose effective order factorizes into a cascade exponent and a subordinator index. We prove a critical-plateau theorem governing collapse: it occurs only above a fuel-dependent threshold, is confined to a bounded wind window, vanishes for coarse low-exponent fuels, and becomes classical persistent collapse only in the unbounded-plateau limit, and we prove these structural results for every fractional order in the relevant range. Grounding the plateau in documented reaction intensities, short grass is the standard fuel whose window onsets and is cured within operational winds, and there the predicted spread rate matches in shape the field-based grassland relationship of Cheney and colleagues, which the classical power law overshoots. A fractional front framework adds ignition memory and long-range spotting.
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Status: final response (author comments only)
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RC1: 'Comment on egusphere-2026-3823', Anonymous Referee #1, 02 Sep 2026
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AC1: 'Reply on RC1', Farrukh Chishtie, 14 Sep 2026
I thank the referee for a candid report and for the willingness to withhold judgment pending my reply. The report identifies a real failure on my part: the manuscript does contain a physical derivation of the fractional order, but that derivation is not presented in a way a geophysical reader can follow, and several of its key terms are used without definition. I take full responsibility for that. Below I address each concern and describe the revisions I would make, so the referee can judge whether they would resolve the objections.
1. Why and how a fractional time derivative enters the Rothermel model
I agree that a fractional derivative added by fiat would not be justifiable. It is not added by fiat here, and I will restructure Section 2 so that this is unmistakable. The argument, which I will present in full with every step named, is as follows.
Rothermel's rate law is an algebraic, memoryless relation between spread rate and wind. The physical process it summarizes is a sequence of discrete ignitions of fuel elements ahead of the front (Frandsen 1971): heat is transferred intermittently, and the time for a given fuel element to ignite is a random variable. I model the sequence of ignitions as a renewal process, meaning simply that the front advances through a series of ignition events separated by random waiting times. Nothing about time itself is renewed; the term refers to the counting process restarting at each ignition, and I will say so explicitly.
If the ignition waiting times are exponentially distributed (no memory), the fraction of unburned fuel decays exponentially, the counting process is Poisson, and one recovers an integer-order relaxation equation whose steady-state wind response is exactly Rothermel's power law. This is the classical case, and I will display it as such.
If instead the waiting-time distribution has a heavy tail, which is the empirical signature of intermittent heat transfer in a heterogeneous fuel bed, the standard continuous-time random-walk limit (Metzler and Klafter 2000; Mainardi 2010) replaces the exponential decay by a Mittag-Leffler decay and the integer-order equation by a fractional one of order α equal to the tail exponent. The Mittag-Leffler function interpolates between exponential decay at short times and power-law decay at long times; it is the unique solution of the fractional relaxation equation, and it is the origin of the wind-factor saturation. The derivative appears as a consequence of the ignition statistics, not as an assumption.
The remaining question is the value of α, and I did not fit it. I obtain it from the branching structure of co-igniting fuel parcels (the "participating particles" the referee rightly flagged as undefined; these are fuel elements, not ash) using a cascade framework developed in my earlier atmospheric and hydrological work. I will state the resulting value, the range it spans, and the fact that all structural conclusions of the paper hold for any order in (0,1) and do not depend on its precise value. I will also add a table mapping each physical quantity (fuel-bed heterogeneity, ignition waiting-time tail, parcel branching number, reaction intensity, wind exponent) to the model parameter it controls, so that the referee's request that fuel and ignitions be explicitly introduced into the model is met by construction rather than by assertion.
2. Fuel and ignition treated only qualitatively
The referee is right that Section 3.1 reads as phenomenology. That section was intended as motivation for the derivation, but with the derivation unclear the motivation floats. In revision I would shorten the qualitative discussion, place it before the derivation as a physical setup, and tie every physical statement in it to a specific quantity in the equations that follow. Fuel enters the model through the wind exponent B, the reaction intensity, and the saturation plateau; ignitions enter through the waiting-time distribution and the cascade branching number. The revised text will make each of these links explicit at the point of introduction.
3. Collapse is not removed
I would like to clarify this point, since I believe the manuscript did not state its own result plainly enough. The paper does not claim, and does not attempt, to eliminate heading-fire collapse. Collapse here means that, in the projected-wind extension of Rothermel's model, the computed heading spread rate falls to zero above a fuel-dependent wind speed while flanking and backing rates remain finite, so the downwind-facing front is predicted to stall while the flanks continue. In the classical model this occurs for every wind speed above the threshold. What I prove is that under the Mittag-Leffler wind factor collapse occurs only if the saturation plateau exceeds a critical value; when it does, it is confined to a bounded window of wind speeds and is cured at higher wind; and for coarse fuels the critical value diverges, so collapse never occurs at all. The persistent collapse of the classical model is recovered only in the limit of an unbounded plateau. This is a qualitative change in the model's behaviour, and I will state it in a single plain sentence in the abstract and again immediately after the theorem.
On the proof itself: it is elementary, a single-crossing argument for a monotone function, and requires no mathematics beyond calculus. I would move the technical steps to an appendix and leave in the main text only the statement, its physical meaning, and the phase diagram placing the seven standard fuel models relative to the critical curve.
4. Wildfire as a phenomenon
This criticism is well taken and I accept it fully. The revised introduction would open with the physical process: a fire front advancing through a fuel bed by radiative and convective preheating, the dependence of the spread rate on wind and fuel, the empirical observation that spread rate does not grow without limit at high wind (the reason Rothermel imposed a wind limit in the first place), and the operational stakes of predicting heading-fire behaviour. Only after that would the model be introduced. I will also revise every occurrence of "wildfire model" where "wildfire" alone is meant.
5. Terminology and writing
I will add a glossary table defining every term the referee lists, and paraphrase each at first use. As a preview of the definitions that would appear:
Fractional: a derivative of non-integer order, which encodes memory of past states; the integer order is the memoryless special case.
Unbounded power law: Rothermel's wind factor grows as U^B without limit as wind increases, so the predicted spread rate has no ceiling, which is unphysical and is why an empirical cap is imposed.
Projected-wind extension: the modification in which the wind entering the rate law is the component normal to the local front, used to compute spread in directions other than heading.
Fuel-dependent threshold: the wind speed above which the projected-wind model predicts collapse; its value depends on the fuel model.
Fractional ignition kinetics: the rate equation for unburned fuel when ignition waiting times carry memory.
Riemann-Liouville relaxation equation: the fractional generalization of exponential decay.
Mittag-Leffler function and solution: the function solving that equation, exponential at short times and power-law at long times.
Mittag-Leffler wind factor: the resulting wind response, which follows Rothermel's power law at low wind and saturates at high wind.
Collapse: as defined in point 3.
Fractional moment theory: a framework relating the fractional order to the scaling of statistical moments; I will cite it and drop the phrase from the text.
Continuous-time renewal process: as defined in point 1.
Hierarchical parcel cascade: the picture of ignition proceeding through nested groups of fuel parcels at successive scales.
Convex duality: a term from the proof that will be removed from the main text.
Normalized spread rate: the spread rate divided by its no-wind value.Every instance of "ground" and "grounded" will be replaced by "derive," "justify," or "anchor" as appropriate, and the manuscript will be edited throughout for readability by a geophysical rather than mathematical audience. "I show" at L113 will be replaced with a forward reference to the specific section and equation.
6. Suitability for NPG
I respectfully submit that the paper belongs in this journal. NPG's scope includes stochastic and scale-free descriptions of geophysical processes, anomalous kinetics, and the nonlinear dynamics of fronts and interfaces, and the paper's argument is precisely that wildfire spread is such a process: intermittent, heterogeneous, and carrying memory. I agree that the current draft reads as applied mathematics, and I believe the revisions above would bring the presentation into line with the physical content. I would welcome the referee's view on whether a revised manuscript along these lines would address the concerns.
Citation: https://doi.org/10.5194/egusphere-2026-3823-AC1
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AC1: 'Reply on RC1', Farrukh Chishtie, 14 Sep 2026
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RC2: 'Comment on egusphere-2026-3823', Yicun Zhen, 04 Sep 2026
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2026/egusphere-2026-3823/egusphere-2026-3823-RC2-supplement.pdf
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AC2: 'Reply on RC2', Farrukh Chishtie, 14 Sep 2026
I thank the referee for a careful and constructive reading, and for the accurate summary of the manuscript's mechanism. The three major comments identify genuine gaps in the rigour of the presentation, and I address each below together with the revision I would make. On the referee's framing question, the manuscript is intended as a theoretical foundation whose claims are structural, and I accept the suggestion to state those claims more precisely wherever the present argument does not support them in full generality.
Major comment 1: status of Lemma 1(iv)
The referee is correct. Statement (iv) of Lemma 1 is supported numerically on α ∈ [0.3, 1] and not by proof, and it enters the proofs of Theorem 1(a), (b) and (d). The claim in the abstract and at lines 205 and 445 that the results hold for every α ∈ (0, 1] is therefore overstated. I will do two things. First, I will attempt an analytical proof of (iv); the statement concerns monotonicity of a ratio of Mittag-Leffler functions through the unit level, and the asymptotic expansions of the Mittag-Leffler function give the behaviour at both ends of the α range, so a proof may be within reach. Second, whether or not that succeeds, I will follow the referee's suggestion: (iv) will be removed from Lemma 1 and stated as a conjecture, Theorem 1 will carry an explicit hypothesis that the conjecture holds, and every "for every α" claim will be replaced by "for every α in the range on which the conjecture is verified." I note that the value of α delivered by the cascade argument, approximately 0.7, and the full range spanned by the cascade lie inside [0.3, 1], so the physical conclusions of the paper are unaffected by this restriction; I will say so at the point where the conjecture is stated, and I will extend the numerical verification over a finer grid down to smaller α to report exactly where it holds.
Major comment 2: applicability of the hierarchical cascade
The referee's intuition is right that fuel parcels in a spreading fire are not moving particles, and the manuscript as written does not explain what is being transferred from the N-body result. What is transferred is not the dynamics but the combinatorics. Theorem 3 of Chishtie (2026a) is derived for a Hamiltonian system, but the quantity that enters the present paper is the scaling of the branching number across hierarchical levels, which depends only on the hierarchy being a nested partition of interacting units with a fluctuating integer number of active members at the dominant level. For a fire front, the interacting units are fuel parcels, the interaction is heat exchange by radiation and convection rather than momentum exchange, and the active members at a given level are the parcels that co-ignite within one ignition waiting time. The cascade exponent then follows from the branching structure alone.
I acknowledge that this mapping is an analogy at the level of structure and not a derivation from first principles for combustion, and I will present it as such. Specifically, I will restate the connection as an explicit hypothesis (the manuscript already uses a hypothesis environment for exactly this purpose), give the correspondence between the N-body quantities and their wildfire counterparts in a table, and make clear that the cascade is used only to supply a value of α and a plausible range. The structural results of Theorem 1 do not depend on the cascade: they hold for any α in the verified range, and a reader who rejects the cascade hypothesis loses only the specific value of the order, not the theorem. I believe this is preferable to removing the hierarchy, since it is the only part of the paper that says where α comes from, but I will make its provisional status unmistakable.
Major comment 3: the origin of φ∞
This is the most important of the three comments, and I want to answer it directly rather than defend the current text, because the referee has identified a real ambiguity. If φ∞ were a free parameter, the referee's equivalence with a wind cap would be correct in substance.
In the submitted manuscript φ∞ is not chosen; it is computed per fuel model from the documented reaction intensity and fuel parameters of Andrews, Cruz and Rothermel (2013), which is why different fuels fall on different sides of the critical curve in the phase diagram. But the text does not make clear that the plateau has a physical origin, and I will remedy this by deriving it. In Rothermel's own formulation the spread rate is proportional to the reaction intensity times the propagating flux ratio times (1 + φ). The propagating flux ratio is the fraction of the heat release that reaches unburned fuel and cannot exceed unity. That constraint bounds the wind factor from above by a fuel-dependent quantity, which is the plateau: the spread rate cannot exceed the rate at which the fire's own heat release can bring fuel ahead of it to ignition. The classical wind limit is an empirical proxy for this same energy bound. I will add this derivation, compare the resulting bound with the per-fuel values used in the paper, and report the agreement or disagreement. [Note to self before posting: run this comparison and quote the numbers; if the bound does not reproduce the plateaus, say so and state φ∞ as physically constrained rather than derived.]
Beyond the origin of the value, two things distinguish the Mittag-Leffler factor from a cap even at equal plateau. The cap is a non-analytic truncation at a wind speed that must be chosen; the Mittag-Leffler factor is smooth, and its saturation scale is set by the fractional relaxation time of the ignition process, not by a separate parameter. And the functional form of the approach to the plateau, which is what enters the front-normal projection and hence the curvature behaviour the referee notes in the overview, differs between the two. I will add a direct comparison of the capped classical model and the Mittag-Leffler model on the collapse phase diagram so that the reader can see which conclusions depend only on the plateau and which depend on the form of the saturation.
Numerical demonstration
The referee notes that a foundational paper with gaps or an empirical paper without data are each incomplete. The paper is foundational, and I will state its claims at the precision the above revisions support. I will also add the comparison already available to me, namely that the wind-limit pipeline reproduces the published Table 2 of Andrews, Cruz and Rothermel (2013) across the seven standard fuel models, as a check that the classical limit of the model is implemented correctly, and I will identify comparison against grassfire spread-rate data at high wind as the appropriate next test of the saturating form.
Minor comment
Line 317 will be corrected as the referee suggests.
Citation: https://doi.org/10.5194/egusphere-2026-3823-AC2
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AC2: 'Reply on RC2', Farrukh Chishtie, 14 Sep 2026
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The present manuscript examines the Rothermel model describing the spread of wildfire. The novelty here is to introduce a fractional time derivative to this model. However, as it stands for now, the manuscript does not explain why and how a fractional time derivative is to be added to the Rothermel model. Since such an arbitrary modification to a model is not justifiable, I am much skeptical with the value of this work. Â However, I would like to give a chance to the author to explain, and would like to leave my judgment until I have a response from the author.
As the author's presentation suggests, the model by Rothermel is  developed empirically, mainly based on a dimensional consistency.  Thus, the spread rate, R_H, of fire is merely controlled by a nondimensionalized wind speed, U/U_1. Limitation of such a simplified description of fire spread is even self evident. For this reason, the author spends a substantial space on reviewing various previous efforts for the modifications.
It is suggested that the fuel and the ignitions must be explicitly introduced into the model for a better description of the fire spread. However, the author discusses all those issues only in qualitative and phenomenological manner (see especially Sec. 3.1), without any effort of explicitly considering those key aspects in the model formulation. Instead, the author somehow decides  to add a fractional time derivative, rather in an arbitrary manner.  This is clearly a wrong way of setting up the priority of investigations, making the work less worthwhile to consider.
According to the author, the Rothermel model has various technical difficulties. Notably, the "collapse of fire" (but see below). However, it also appears that this "collapse of fire" is not removed by introducing a fractional time derivative. Thus, the author simply fails to remove a basic defect of the model by a modification. It seems that the author provides a mathematical proof of this "collapse". However, since I am not a mathematician, I cannot judge whether this is something worthwhile to publish.
The manuscript is not well written as a whole. Many technical terms are introduced without any elaborations. Many of the phrases that the author introduces are not understandable either: they must be well paraphrased. Â Most strangely, the author does not introduce a phenomena in concern, i.e., wildfire, in any manner at all: intentionally or not, the word "wildfire" is always associated with the word "model", thus it even gives an impression that the author is more interested with a mathematical model than any geophysical phenomena. Reminds you that this is a journal about nonlinear processes in geophysics.Â
The author is obviously not a native speaker: this fact stands out by frequent use of the words, "ground", "grounded": although what the author intends to say is marginally understandable, in more natural English, different words would be used in the same context.
As a whole, the given materials may be more appropriate to be published in an applied-mathematics journal, also considering the author's writing style.
Partial list of technical terms and phrases that need to be either elaborated or paraphrased:
fractional, title and elsewhere
Mittag-Leffler kinetics, title
unbounded power law, Abstract, L31, L114: what do you mean by "unbounded"? I do not see an explanation anywhere
projected-wind extension, Abstract
fuel-dependent threshold, Abstract
a fractional ignition kinetic, Abstract
Riemann-Liouville relaxation equation, Abstract
Mittag-Leffler solution, Abstract
the downwind facing front collapses, L24: what "collapse" means here?
heading fire collapse, L34: ibid
fractional kinetics, L41
fractional moment theory, L44
participating particles, L46: what kind of particles you are talking about. are these ash particles, for example? and participating to what?
ignitions, L48-49: I think, I know this word, but still need to be paraphrased, because the whole context is not defined at all in presentation
continuous time renewal process, L49: what is "renewed" here? what kind of "physical process" is it? "time renewal" sounds like time itself is renewed, but how you do it: time just goes monotonically without our control
Mittag-Leffler function, L51-52
Mittag-Leffler wind factor, L52
hierarchical parcel cascade, L56
The physical foundation is supported at each step, L85: how is it supported? what are the steps here?
Rothermel heading spread rate, L97: a phrase is just introduced without describing a phenomena or a process in concern
no-wind no-shape spread rate, L98: ibid
fuel properties, L99
normalized spread rate, L102
convex duality, L104: technical jargon
collapse, L105: ibid
I show, L113: show where?
Riemann-Liouville derivative, L129