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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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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RC1: 'Comment on egusphere-2026-2648', Anonymous Referee #1, 08 Aug 2026
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The author analyses a simple model of moist convection using both analytic and numerical techniques. They first propose and anzats for a stationary solution and prove its existence and uniqueness. They then present a numerical solution for this steady state solution and provide a numerical simulation of the time-depending problem which itself presents a damped oscillatory behaviour to approach the steady-state solution at large time. The crux of the work resides in the introduction of a delay equation for the water loading that incorporates rain-depletion process with a continuum lag. This leads to an oscillatory mechanism due to episodic water loading/rain depletion process that acts on the buoyancy of the raising parcel leading to acceleration and deceleration episodes. The author then attempted an analogy to damped oscillations occurring in a Volterra integro-differential equation.
Unfortunately, the paper is poorly structured, too long and provides little to no observational support to their solution. For instance while they explain the oscillatory behaviour as being triggered by the episodic water loading it is unclear whether such behaviour is observed in nature. It is unclear while the oscillation amplitude decreases (damped oscillations). Perhaps, the biggest weakness of the work is that it does not connect to the convection parameterization problem. There is little to no mention of the convection parameterization literature and the author made zero effort to provide pathways on how their work could help improve climate models. It is true that their model is very simplistic as it doesn't take into account things like stability of atmospheric layers (neutral buoyancy, free condensation level, level of free convection), energetics (CAPE and CIN), and more importantly mixing. Typical mass flux schemes to which the model studied is closest to utilize the notion of entraining plumes to account for continuous environmental-mixing that the raising air experiences leading to pre-mature loss of buoyancy and a sustained increase in updraft mass flux; this could been reflected in the author's solution in terms of a convection surface area S(z) that increases with z. But in the author’s solution the increase in S(z) is purely a by-product of mass conservation with the area increase compensating for the parcel’s expansion with height.
Many things are unclear in this work.
- The intro section should discuss not only the history of this simple model but how it connects to practical parameterization schemes used in climate model. The lack of entrainment is clearly an issue but perhaps there is something that can be gained from the community in terms of steady plume models. Perhaps the most compressive paper to start with is Arakawa and Schubert (1974) but there is also Zhang and McFarlne (1990), Tidelka (1974), etc. Here are some references: The Plant and Craig review can be useful,
Arakawa, A., & Schubert, W. H. (1974). Interaction of a cumulus cloud ensemble with the large-scale environment. Part I. Journal of the Atmospheric Sciences, 31, 674–701.
Emanuel, K. A. (1991). A scheme for representing cumulus convection in large-scale models. Journal of the Atmospheric Sciences, 48, 2313–2335.
Plant, R. S., & Craig, G. C. (2010). A review of the theoretical basis for bulk mass flux convective parameterization. Atmospheric Chemistry and Physics, 10, 3529–3544.
- The paper is too long and poorly organized. The analysis part (existence and uniqueness of stationary solution) could be written as a single section, with the main result highlighted and presented first as a theorem or one proposition of two. Some of the proofs and lemma could be also be included but the proofs should be brief. Technical details (such as Section 6 which was all about proving a lemma) could be included in an appendix or a series of appendices.
- The section on numerical solution should be written better. There is no need for a lengthy description of the code used to solve the steady-state system. A few lines will suffice. However, a description of the scheme used to solve the unsteady system is needed. Also a subsection or a clean up paragraph describing the simulation is welcome.
- The section on the Volerra model is not necessary. The analogy and the properties of this equation could be summarized in a paragraph or so. Readers interested can look it up in the literature.
Specific comments:
- Abstract (line 16): Not sure if this is the case without comparison to observations or comprehensive model simulations such as LES, etc..
- Lines 22-23: Simply say that these atmospheric phenomena are driven by deep-moist convection or accurately explain what moist convection pertain to. Saying that thunderstorms are primarily caused by upward air flow and water vapor condensation may be interpreted as if those two processes are separable while in reality they are not; one cannot be sustained without the other!
- Lines 29-31: A statement like this would make the reader expect that the present work is going to suggest an alternative that leads to realistic results. However, the author hasn't provided any physical evidence to back up such claim. The damped oscillations and Volterra dynamics are still theoretical views based on mathematical models.
- Lines 34-35: Is there any physical evidence to justify that a constant vertical velocity profile is more realistic.
- Line 93+1: This statement is arbitrary and doesn't provide any useful information. Plus v is not the vertical motion but the velocity of the vertical motion. Doesn't the model invoked simply expresses conservation of water substance for the raising parcel of air under the assumptions that (1) the parcel remains saturated and (2) lateral entrainement and time variations are neglected.
- Line 93: Why “turbulent condensation”? Where is the turbulence in the model?
- Eqn 8: If I understand correctly, Eqns 4-7 are for the air flow inside the chimney, with S(z) being the updraft area while (8)-(9) describe the fluid motion in the surrounding environment. However, by neglecting horizontal motions and lateral mixing (entrainment and detrainment) I don't see how the two systems communicate. On the other hand Eqns 8 has a condensation term on the RHS (with a tuning parameter vartheta in front). This seems to indicate that (8)-(9) is the model for the large-scale average and not the "external environment". If so I'd expect vartheta to depend on S(z) but this doesn't seem the case. Also, using the same variable name for environment and plume variables doesn't help!
- Eqn 10: why the time derivative term in kept in rho but neglected in T?
- Out of curiosity, what is the motivation behind the form of the "memory kernel" in Eqn 15?
- Eqn 18: why is the subscript "cond" is suddenly added to H_{tr}? What changed?
- Line 129+3: Eqn (18) is also a conservation law. In fact none of these equations are conservation laws; they are conservation laws with source terms!
- Eqn 127+3: “(derived in Section 2, but …)”. No need to say this.
- The elementary calculations leading from Eqn 18 to the steady solution in (19) can be deleted. The reader of the journal can workout these details on the own if needed. Otherwise they are just distractive.
- Line 131: use math mode for w(z).
- Line 135-136: Another and perhaps a better way to say this is that by assuming constant vertically velocity, the vertical variations in updraft mass flux is all carried by the increase in convection area fraction. Also S(z) is not a volume but a surface area. However, environmental mixing and entrainment in particular is neglected.
- Eqn 24: how q0 and z1 are being determined. In realistic mass flux schemes the plume is integrated till it reaches its level of neutral buoyancy.
- Why Section 4 is called preliminaries for the stationary solution?
- Eqn 25: isn't this the same as Eqn 2? Why the need for writing it again here?
- Line 15: There is no parameter \vartheta on the right hand side of Equation 20. This sentence is poorly written. Later, in the numerical examples, the authors uses the value \vartheta=2/3, somehow assuming a partially moist environment. This approach is however unrealistic in many respects. The dynamics between the moist of dry regimes do not just average out. The transition between moist and dry regimes is non smooth and best represented by a Heaviside function.
- Line 162: “lemmata” which lemmata? There isn’t a lemma or several lemmas called “monotonicity lemma”. Precision in writing is key. Also use lemmas instead of lemmata.
- Line 207+2: This paragraph is not needed or could be written differently. The introduction of the auxiliary variable is done only for the purpose of deriving the solution in (40). It is just an elementary technique for the integration of (38), which is a separable equation.
- Line 211: Why by virtue of (36)? Isn't (36) what's being used to bound the two exponentials? Don't you need the quantity T_0- g/(c_v +R1)z to be non-negative also to make the statement?
- Isn’t it better to call Lemma 1 a proposition or theorem instead of having intertwined lemmas that makes the paper hard to read?
- Section 7: The description of the numerical method for the stationary solution which is presented more like the description of the numerical code in a typical code-documentation literature while useful for people who would have to run the code it is completely uncalled for in a scientific article. The whole idea could be summarized in a short paragraph. The author may choose to provide a documental for the code that could be made available together with the code itself on a public data-sharing website.
- Line 374: What does this mean? Wasn't the author able to solve these equation using simple ode integration combined with basic root finding techniques?
- Line 400: The author needs to clarify and describe the numerical method(s) used to the solve the evolution equation and which equations they have exactly solved.
- Line 403 and elsewhere: what are the units of b? Based on its definition b? Based on how it is defined I am expecting second! Is this the case? How does it represent the mean residence time of a droplet? Can the author confirm whether a 7.5 minute residence time is realistic or not? This should be easy to verify from cloud physics literature. Also what happens when is varied (smaller or larger than 7.5 min)? Does the oscillatory behaviour break down?
- Figure 4d: This rather a peculiar plume behaviour. The author needs to explain/justify what's the meaning of these damped oscillations. An explanation (without physical justification) was provided earlier for the oscillations as being due to episodic changes in water loading but it is not why the amplitude of these oscillations would decay as the plume matures!
- Figure 4: why is there a w(z) and w(t)? Isn't w a function of both t and z?
- The section on the Volterra model is unnecessary. Oddly, here the author discuss in the details the Volterra equation and describes the numerical methods used to solve it. I assume that the same methods were used for the convection problem, however this is not clear. The analogy with the Volterra equation could be made in one sentence. It could maybe used as a motivation beforehand, i..e, to justify the choice of numerical or afterwards to explain the oscillation but not as a separate section. Beside the analogy is poorly drawn. Readers will wonder what the variable y(t) represents and also some care should be taken because the storm system has multiple variables and it is a PDE while the Volterra problem is a scalar ODE.
- Line 510: The code availability "upon reasonable request" policy is rather odd. Shouldn't the code and be made all publicly available for all EGU publications.
Citation: https://doi.org/10.5194/egusphere-2026-2648-RC1 -
AC3: 'Reply on RC1', Dalila Bourega, 10 Aug 2026
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Dear Editor and Anonymous Referee #1,
We sincerely thank the referee for the thorough, critical, and highly constructive review.
Please find attached our detailed point-by-point response (PDF),
as well as the Supplementary Material archive containing the detailed mathematical proofs and the Python scripts for reproducibility.
Best regards,
Dalila Remaoun Bourega
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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.