the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 22 May 2026
Stationary Solutions and Oscillatory Dynamics in a Mathematical Model of a Saturated Moist Atmosphere
Abstract. This paper investigates the mathematical properties of a simplified atmospheric system describing vertical air flows with water condensation. We provide a rigorous proof for the existence and uniqueness of the stationary solution under specific technical conditions. Numerical simulations reveal damped oscillations in the air flow intensity and liquid water content, interpreted as a self-regulating physical cycle driven by latent heat and droplet accumulation. By establishing a qualitative analogy with a Volterra integro-differential equation, we demonstrate that these oscillations are governed by memory effects. The convergence of the evolutionary variables toward the stationary values provides a robust validation of the model's consistency.
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Status: open (until 31 Jul 2026)
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CC1: 'Comment on egusphere-2026-2648', Fayssal Benkhaldoun, 22 Jun 2026
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AC1: 'Reply on CC1', Dalila Bourega, 23 Jun 2026
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We sincerely thank Pr. Fayssal Benkhaldoun for his highly constructive and positive assessment of this work, and in particular for raising this important technical point.
Regarding the numerical scheme in Section 7: the comment correctly points out the need for validation of the explicit Euler integration, particularly concerning the integral defining $\alpha$. To ensure the reliability of the computed profiles, we carried out a systematic grid refinement study using three spatial step sizes: $\Delta z = 10$ m, $5$ m, and $1$ m.
At $\Delta z = 10$ m, the nonlinear solver produced convergence warnings during the computation of the stationary state, indicating numerical instability at this coarser resolution. Reducing the step to $\Delta z = 5$ m fully resolved these instabilities.
The table below summarizes the key outputs for the converged runs ($\Delta z = 5$ m and $\Delta z = 1$ m):
Humid ($\Delta z=5$ m)
94.091261
0.015331
32.875402
Humid ($\Delta z=1$ m)
94.101857
0.015331
32.877689
Dry ($\Delta z=5$ m)
85.523085
0.056212
120.049870
Dry ($\Delta z=1$ m)
85.533782
0.056212
120.062326
Variations ($\Delta z=5$ m $\rightarrow$ $\Delta z=1$ m):
- Humid: $\Delta q_0 = 0.011\%$, $\Delta \Sigma = 0.000\%$, $\Delta \alpha = 0.007\%$
- Dry: $\Delta q_0 = 0.013\%$, $\Delta \Sigma = 0.000\%$, $\Delta \alpha = 0.010\%$
The intersection point $\Sigma$ is stable up to six decimal places between $\Delta z = 5$ m and $\Delta z = 1$ m. Furthermore, the parameter $\alpha$ shows a relative variation of less than $0.02\%$. This confirms that the spatial resolution of $\Delta z = 5$ m is largely sufficient to capture the dynamics accurately and without numerical artifacts.
We will add a brief paragraph and this convergence summary to Section 7 in the revised manuscript to provide this useful technical information for readers wishing to reproduce the calculations.
To further support reproducibility, the complete Python scripts used for the numerical simulations - including the stationary solver, the explicit Euler integration scheme, and the grid convergence study - are provided as a supplementary archive (.zip) attached to this comment.
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AC1: 'Reply on CC1', Dalila Bourega, 23 Jun 2026
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CC2: 'Comment on egusphere-2026-2648', Abbas Belfar, 23 Jun 2026
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Were the mass and speed of the water droplets taken into account in this model.
Citation: https://doi.org/10.5194/egusphere-2026-2648-CC2 -
AC2: 'Reply on CC2', Dalila Bourega, 23 Jun 2026
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We sincerely thank you for this insightful question regarding the microphysical assumptions of the model.
To answer your question directly: the mass of the droplets is accounted for on a macroscopic level, while their individual falling speeds are modeled implicitly through a statistical approach.
1. Mass of the droplets: The model does not track individual droplets, but rather the total mass density of condensed liquid or solid water in the air, denoted by the variable Σ(t). This bulk mass plays a crucial mechanical role: it acts as a gravitational drag in the momentum equation, represented by the term -g[Σ + ρ]. This accumulated weight is the primary physical mechanism that slows down the upward air flow and drives the oscillatory cycle.
2. Speed of the droplets: The individual kinematic falling speeds (terminal velocities) of the droplets are not explicitly calculated. Instead, the model captures the effect of droplet fallout using a probabilistic macroscopic approach via the memory kernel φ(τ). The speed at which droplets leave the system is governed by the parameter b, which represents the mean droplet residence time in the air column. The function φ(τ) = exp( -π τ² / 4b² ) essentially dictates the probability that a droplet remains suspended after a time τ from its formation.
By treating the mass globally and the falling speed statistically, the model avoids the heavy microphysical equations (such as droplet collision or friction) that would make the system mathematically intractable. This macroscopic simplification is precisely what allows us to rigorously prove the existence of the stationary solution and to draw the meaningful analogy with the Volterra integro-differential equation.
We hope this clarifies the physical framework of our model, and we thank you again for your interest in our work!
Citation: https://doi.org/10.5194/egusphere-2026-2648-AC2
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AC2: 'Reply on CC2', Dalila Bourega, 23 Jun 2026
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This work offers a remarkable contribution to the mathematical modeling of convective atmospheric processes. The author rigorously establishes the existence and uniqueness of the stationary solution for a system of equations derived from a model of moist air ascent with condensation, under clearly stated technical conditions. The proof, which relies on subtle comparison and monotonicity arguments, is both elegant and robust. Moreover, the analogy with the Volterra integro-differential equation, introduced to interpret the damped oscillations of the key variables (α,Σ), is particularly insightful and opens up promising perspectives on the role of memory effects in thunderstorm dynamics. The consistency between the theoretical results and the numerical simulations—both for the stationary profiles and the temporal dynamics—lends significant credibility and depth to the study. This work constitutes a substantial advance in the mathematical physics of the atmosphere.
Question : In Section 7, the numerical integration is performed using an explicit Euler scheme. Could you briefly specify whether an error control or a convergence study with respect to the spatial step size has been carried out, in order to ensure the reliability of the computed profiles, particularly for the integral involved in the expression of αα ? This would provide useful technical information for readers wishing to reproduce the calculations.