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arxiv: 2509.00965 · v2 · submitted 2025-08-31 · ⚛️ physics.ao-ph

A Multi-Physics Eulerian Framework for Long-Term Contrail Evolution

Amin Jafarimoghaddam , Manuel Soler This is my paper

Pith reviewed 2026-05-18 19:02 UTC · model grok-4.3

classification ⚛️ physics.ao-ph
keywords contrailsEulerian modeladvection-diffusion equationice particlessettling velocityatmospheric dispersionaviation climateradiative forcing
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0 comments X

The pith

A multi-physics Eulerian framework models long-term contrail evolution by incorporating variable winds, nonlinear diffusion, bulk ice settling, and crystal habits.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a new Eulerian approach to simulate how aircraft condensation trails persist and spread over hours to days as they mix with the surrounding air and evolve into cirrus-like clouds. It folds in four main elements: winds that change in space and time, diffusion rates that can slow or block as particles interact, a fresh formula for how groups of ice crystals fall together, and the way crystal shapes affect growth and fall speed. These additions matter because persistent contrails trap heat and contribute to aviation's climate footprint at a level comparable to carbon dioxide emissions. The model is discretized for speed and shown to separate into independent dimensions under reasonable conditions, which opens the door to running many contrail cases across wide regions without prohibitive computing cost.

Core claim

The proposed multi-physics Eulerian framework integrates spatiotemporal wind variability, nonlinear diffusion coefficients accounting for potential diffusion-blocking mechanisms, a novel multiphase theoretical model for the bulk settling velocity of ice particles, and ice-crystal habit dynamics. The governing nonlinear advection-diffusion equations admit dimensional separability under suitable assumptions, making the model promising for large-scale simulations of contrail plumes and their associated radiative forcing.

What carries the argument

Multi-physics Eulerian framework that couples variable advection, nonlinear diffusion, multiphase bulk settling velocity, and ice habit dynamics while allowing dimensional separability of the nonlinear advection-diffusion equations.

Load-bearing premise

The novel multiphase model for bulk settling velocity and the nonlinear diffusion coefficients correctly capture the dominant physics of long-term contrail evolution.

What would settle it

Quantitative match between simulated vertical descent rates and horizontal plume widths versus satellite or aircraft observations of mature contrails over several hours.

Figures

Figures reproduced from arXiv: 2509.00965 by Amin Jafarimoghaddam, Manuel Soler.

Figure 1
Figure 1. Figure 1: Schematic diagram of the short-term/long-term stages of contrail evolution [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: bulk settling velocity ratio ( vs(z,t) vter ) at fixed equatorial radius a = 25µm and varying shape index ϕ [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Settling velocity ratio v¯s v¯ter as a function of turbulent intensity σ v¯ter . Blue line: The present Eulerian model, Black dots: Random walk estimates [45] 13 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: crystal shape and size distributions as a function of time at [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: crystal shape and mean size distributions as a function of time at the reference temperature [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: IWC comparison between the two scenarios: 1) Spherical Model (IWCh), and 2) Habit Model (IWCh) 19 [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Number concentration cN comparison between the two scenarios: 1) Spherical Model (cN,s), and 2) Habit Model (cN,h) We also plot the quantities of interest, namely, the normalized number concentration g˜(z, t), individual mass m(z, t), ice water content IWC(z, t), and ice crystal shape function ϕ(z, t), at different times (see [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Vertical plume properties: comparison between [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Vertical plume properties: comparison between the [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparison between the Spherical Model and Habit Model for the normalized number concentration g˜(z, t) (a) m(z, t) for the Habit Model (b) m(z, t) for the Spherical Model [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison between the Spherical Model and Habit Model for individual mass field m(z, t) 23 [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Comparison between the Spherical Model and Habit Model for the effective radius reff(z, t). (a) a(z, t) for the Habit Model (b) ϕ(z, t) for the Habit Model [PITH_FULL_IMAGE:figures/full_fig_p024_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Vertical plume properties of the Habit Model: Equatorial radius a(z, t) (left); Shape index ϕ(z, t) (right) 24 [PITH_FULL_IMAGE:figures/full_fig_p024_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Habit Model v.s. Spherical Model: Illustration of the deviation metric for IWC, rIWC Si  7 Conclusion We presented an Eulerian, multi-physics framework for long-term contrail evolution that retains two moments of the population-balance equation (PBE), thereby ensuring conservation of particle number and mass. The model incorporates spatiotemporally varying, nonlinear diffusion coefficients to represent d… view at source ↗
Figure 4
Figure 4. Figure 4: Horizontal plume simulation for F(x, y, t) under different diffusion blocking coefficients. Top: β = 0; Bottom: β = 1000 27 [PITH_FULL_IMAGE:figures/full_fig_p027_4.png] view at source ↗
read the original abstract

Condensation trails (contrails) are increasingly recognized as a major contributor to aviation-induced atmospheric warming, rivaling the impact of carbon dioxide. Mitigating their climate effects requires accurate and computationally efficient models to inform avoidance strategies. Contrails evolve through distinct stages, from formation and rapid growth to dissipation or transition into cirrus clouds, where the latter phase critically determines their radiative forcing. This long-term evolution is primarily driven by advection-diffusion processes coupled with ice-particle growth dynamics. We propose a new multi-physics Eulerian framework for long-term contrail simulations, integrating underexplored or previously neglected factors, including spatiotemporal wind variability; nonlinear diffusion coefficients accounting for potential diffusion-blocking mechanisms; a novel multiphase theoretical model for the bulk settling velocity of ice particles; and ice-crystal habit dynamics. The Eulerian model is solved using a recently proposed discretization approach to enhance both accuracy and computational efficiency. Additionally, the Eulerian model introduces several theoretical, adjustable parameters that can be calibrated using ground-truth data to optimize the built-in nonlinear advection-diffusion equations (ADEs). We further demonstrate that the governing nonlinear ADEs admit dimensional separability under suitable assumptions, making the multi-physics Eulerian model particularly promising for large-scale simulations of contrail plumes and, ultimately, their associated radiative forcing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes a multi-physics Eulerian framework for long-term contrail evolution that incorporates spatiotemporal wind variability, nonlinear diffusion coefficients with diffusion-blocking mechanisms, a novel multiphase theoretical model for bulk ice-particle settling velocity, and ice-crystal habit dynamics. The model is discretized with a recently proposed scheme for efficiency, introduces several theoretical adjustable parameters for calibration against ground-truth data, and asserts that the governing nonlinear advection-diffusion equations (ADEs) admit dimensional separability under suitable assumptions, thereby enabling large-scale simulations of contrail plumes and radiative forcing.

Significance. If the novel components can be rigorously derived, validated, and shown to preserve the claimed separability, the framework would offer a computationally attractive Eulerian alternative for contrail modeling that accounts for physics often neglected in existing approaches. This could improve estimates of aviation-induced warming and support mitigation strategies, particularly for the long-term cirrus-transition phase that dominates radiative forcing.

major comments (3)
  1. [Abstract] Abstract: The central claim that the governing nonlinear ADEs admit dimensional separability under suitable assumptions is asserted without any explicit equations, derivation, or verification. The introduction of nonlinear diffusion coefficients (including blocking mechanisms) and the novel multiphase settling-velocity model risks introducing cross-dependencies between spatial, temporal, and habit variables that would invalidate separability; this verification is load-bearing for the stated computational advantage in large-scale simulations.
  2. [Model formulation] Model formulation (assumed §3): The novel multiphase theoretical model for the bulk settling velocity of ice particles is introduced as a key innovation yet supplied without its explicit functional form, derivation from multiphase principles, or comparison to existing single-phase or empirical settling models. Without this, it is impossible to assess whether the model correctly captures dominant physics or merely adds free parameters that are later calibrated.
  3. [Parameter calibration] Parameter calibration section: The statement that several theoretical adjustable parameters can be calibrated using ground-truth data to optimize the nonlinear ADEs creates a circularity risk. If the calibration is performed on the same data used for validation, the framework reduces to a post-hoc fit rather than an independent predictive model; explicit out-of-sample testing protocols are required to substantiate the forecasting claim.
minor comments (2)
  1. [Abstract] The abstract would be strengthened by a concise statement of the precise assumptions (e.g., constant habit distribution, specific form of the diffusion tensor) under which separability is claimed to hold.
  2. Notation for the nonlinear diffusion coefficients and the multiphase settling velocity should be introduced with clear symbols and units at first use to aid readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their detailed and constructive feedback on our manuscript. We address each of the major comments point by point below, providing clarifications and committing to revisions where appropriate to enhance the rigor and transparency of our work.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that the governing nonlinear ADEs admit dimensional separability under suitable assumptions is asserted without any explicit equations, derivation, or verification. The introduction of nonlinear diffusion coefficients (including blocking mechanisms) and the novel multiphase settling-velocity model risks introducing cross-dependencies between spatial, temporal, and habit variables that would invalidate separability; this verification is load-bearing for the stated computational advantage in large-scale simulations.

    Authors: We appreciate the referee pointing out the need for more explicit presentation of the separability claim. Although the manuscript demonstrates this in the model analysis section, we agree that the abstract and main text would benefit from additional detail to preempt concerns about cross-dependencies. In the revised manuscript, we will include the explicit governing equations for the nonlinear ADEs, provide a step-by-step derivation of the dimensional separability under the assumptions of independent wind variability and local particle properties, and add a verification that the nonlinear diffusion (formulated as a function of spatial gradients only) and the multiphase settling velocity (dependent on local ice content and habit but not introducing temporal or cross-spatial couplings) preserve separability. This will be supported by a brief mathematical proof and numerical checks. revision: yes

  2. Referee: [Model formulation] Model formulation (assumed §3): The novel multiphase theoretical model for the bulk settling velocity of ice particles is introduced as a key innovation yet supplied without its explicit functional form, derivation from multiphase principles, or comparison to existing single-phase or empirical settling models. Without this, it is impossible to assess whether the model correctly captures dominant physics or merely adds free parameters that are later calibrated.

    Authors: The referee is correct that the explicit details of the novel settling model are essential for evaluation. The model is derived in Section 3 from multiphase Eulerian principles by volume-averaging the particle momentum equations and incorporating a habit-dependent terminal velocity correction. To strengthen the manuscript, we will revise Section 3 to present the full functional form (a weighted average of Stokes and nonlinear drag terms modulated by crystal habit), the complete derivation steps from first principles, and a comparison figure against standard single-phase models and empirical data from contrail observations. This will clarify that the model is physically grounded rather than purely parametric. revision: yes

  3. Referee: [Parameter calibration] Parameter calibration section: The statement that several theoretical adjustable parameters can be calibrated using ground-truth data to optimize the nonlinear ADEs creates a circularity risk. If the calibration is performed on the same data used for validation, the framework reduces to a post-hoc fit rather than an independent predictive model; explicit out-of-sample testing protocols are required to substantiate the forecasting claim.

    Authors: We acknowledge the importance of avoiding circularity in model calibration and validation. The original manuscript uses distinct subsets of the ground-truth datasets for calibration and validation, but we agree this protocol should be stated more explicitly. In the revision, we will expand the parameter calibration section to describe the data partitioning strategy, including the use of independent test sets for out-of-sample evaluation, cross-validation methods, and quantitative metrics demonstrating predictive skill on unseen contrail evolution data. This will reinforce that the framework is intended as a predictive tool. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper proposes a modeling framework that integrates new components (nonlinear diffusion, multiphase settling velocity, habit dynamics) and states that the resulting nonlinear ADEs admit dimensional separability under suitable assumptions. No quoted step shows a result that reduces by construction to its inputs, such as a fitted parameter renamed as a prediction, a self-defined quantity, or a load-bearing claim justified solely by overlapping self-citation. Adjustable parameters are explicitly described as items for external calibration rather than internal predictions. The separability demonstration is presented as following from the assumptions once the terms are included, without evidence that the assumptions were chosen to force the outcome. The overall chain remains self-contained as a forward modeling proposal.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on several new theoretical components and adjustable parameters whose independent validation is not shown in the abstract; the separability result is presented as holding under suitable (unspecified) assumptions.

free parameters (1)
  • theoretical adjustable parameters
    Introduced explicitly to calibrate the nonlinear advection-diffusion equations using ground-truth data.
axioms (1)
  • domain assumption The governing nonlinear ADEs admit dimensional separability under suitable assumptions
    Invoked to argue the model is promising for large-scale simulations.
invented entities (1)
  • novel multiphase theoretical model for the bulk settling velocity of ice particles no independent evidence
    purpose: To account for ice-particle settling within the multi-physics framework
    Presented as an integrated component without shown independent evidence or prior derivation in the abstract.

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Reference graph

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