Ice clouds as nonlinear oscillators

Hannah Bergner; Peter Spichtinger

arxiv: 2507.03475 · v1 · submitted 2025-07-04 · ⚛️ physics.ao-ph

Ice clouds as nonlinear oscillators

Hannah Bergner , Peter Spichtinger This is my paper

Pith reviewed 2026-05-19 06:29 UTC · model grok-4.3

classification ⚛️ physics.ao-ph

keywords ice cloudsnonlinear oscillatorHopf bifurcationlimit cycledynamical systemsscaling behaviorradiative effects

0 comments

The pith

A simple ice cloud model constitutes a nonlinear oscillator with two Hopf bifurcations.

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

The paper develops a simple but physically consistent model for pure ice clouds and subjects it to analysis from dynamical systems theory. The model is shown to constitute a nonlinear oscillator that undergoes two Hopf bifurcations within the range of atmospheric parameters. Scaling behaviors are identified for the bifurcations and the associated limit cycle, which reduces the effective parameter space substantially. The model reproduces features of real measurements, supporting its use as a tool for investigating the radiative effects of ice clouds.

Core claim

We develop a simple but physically consistent ice cloud model, and analyze it using methods from the theory of dynamical systems. We find that the model constitutes a nonlinear oscillator with two Hopf bifurcations in the relevant parameter regime. In addition to the characterization of the equilibrium states and the occurring limit cycle, we find scaling behaviors of the bifurcations and the limit cycle, reducing the parameter space crucially. Finally, the model shows very good agreement with real measurements, indicating that the main physics is captured and such simple models are helpful tools for investigating ice clouds.

What carries the argument

The simple ice cloud model analyzed as a dynamical system whose equilibria and limit cycle arise from two Hopf bifurcations.

Load-bearing premise

The simple model is physically consistent and captures the main physics of ice clouds, as evidenced by its very good agreement with real measurements.

What would settle it

Long-term observations of ice cloud properties that show no oscillatory behavior or scaling relations in the predicted parameter regimes would falsify the central claim.

Figures

Figures reproduced from arXiv: 2507.03475 by Hannah Bergner, Peter Spichtinger.

**Figure 1.** Figure 1: Equilibrium states for the ice cloud model at p = 300 hPa. Colors indicate different temperatures, i.e. red: 230 K, grey: 210 K, blue: 190 K. Black lines indicate approximations as given in Equations (83)-(85). Left: Number concentration (in 1/kg), middle: mass concentration (in kg/kg), right: saturation ratio. with an absolute error of approximation smaller than 4 · 10−4 . Hence, the proof of uniqueness u… view at source ↗

**Figure 2.** Figure 2: Example of real (left) and imaginary (right) parts of numerically determined eigenvalues at the equilibrium state of the system for a fixed set of parameters p = 300 hPa, T = 230 K, F = 1 We calculate the bifurcation points numerically. For a fixed temperature T we scan through the w-interval, calculate the eigenvalues of the respective Jacobi matrix J, and determine the position where the sign of the rea… view at source ↗

**Figure 3.** Figure 3: Bifurcation diagram in the parameter space T − w for p = 300 hPa and F = 1. The red and blue dots indicate the numerically calculated transition in the real parts of the complex-conjugate eigenvalues λ1,2, i.e. the two Hopf bifurcations. The black lines indicate fits to the bifurcations using a quadratic polynomial in T. Above the red points and below the blue points, the unique equilibrium state is stable… view at source ↗

**Figure 4.** Figure 4: Examples of solutions for different regimes. Left: damped oscillator (w = 0.01 m s−1 ), middle: undamped oscillator (w = 0.1 m s−1 ), right: damped oscillator (w = 1 m s−1 ) first and undisturbed nucleation event followed by a (more or less) strong reduction of the supersaturation, and afterwards subsequent nucleation events, which are less vigorous since pre-existing ice slows down the explosion term. Thi… view at source ↗

**Figure 5.** Figure 5: Left: Periods of limit cycles in the bifurcation diagram. Right: Example of section through the parameter space at fixed temperature T = 210 K, representing the oscillation period as calculated from the imaginary part of the complex eigenvalues (red line), and the period of the respective limit cycle in the unstable regime (black line). a timestep, adaptively but constant determined by the empirical relati… view at source ↗

**Figure 6.** Figure 6: Scaling of bifurcations for different values of the sedimentation parameter F. Black dots show the numerically derived values, whereas colored lines represent the scaling of the fit curves as derived for the parameter F = 1. Reddish colors show the change in the upper bifurcation, whereas bluish colors show the change in the lower bifurcations; in both cases, sedimentation parameters of F = 1, 0.1, 0.01 ar… view at source ↗

**Figure 7.** Figure 7: Scalings of limit cycle’s periods. Left: periods of limit cycles for different temperatures; middle: numerical scaling of the data together with the reference curve for T = 210 K; right: scaled data using the scaling functions from Equations (111) and (112). 30 [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗

**Figure 8.** Figure 8: Fit of exponents of scaling functions. Grey dots: empirical scaling as derived from the numerical calculations. Dark red/blue curves: exponents of the empirical fits using scalings of the functional form f(x) ∼ x a , exact values are given in Equation (109). Light red/blue curves: exponents of the empirical fits using scalings of the functional form f(x) ∼ x a−b(x−1), exact values are given in Equations (1… view at source ↗

**Figure 9.** Figure 9: Comparison of the model results with measurements. Colors indicate the frequency of occurrence of ice crystal number concentrations in measurements (Krämer et al., 2016; Krämer et al., 2020). Bluish colors show the values of the model, i.e. equilibrium states and range of the limit cycle calculations. Dark blue lines indicate conditions for F = 1, light blue indicate conditions for F = 0.1, respectively. B… view at source ↗

**Figure 10.** Figure 10: Examples of solutions for w = 0.01 m s−1 . Left: number concentration, middle: mass concentration, right: saturation ratio, red line; black lines indicate the equilibrium values. 0 60 120 180 240 300 360 420 480 540 600 660 720 time (min) 0 10 20 30 40 50 60 70 number concentration (1/L) 0 60 120 180 240 300 360 420 480 540 600 660 720 time (min) 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 mass concentra… view at source ↗

**Figure 11.** Figure 11: Examples of solutions for w = 0.10 m s−1 . Left: number concentration, middle: mass concentration, right: saturation ratio, red line; black lines indicate the equilibrium values. 0 30 60 90 120 150 180 210 240 time (min) 0 500 1000 1500 2000 number concentration (1/L) 0 30 60 90 120 150 180 210 240 time (min) 0.00000 0.00002 0.00004 0.00006 0.00008 0.00010 0.00012 0.00014 mass concentration (kg/kg) 0 30 6… view at source ↗

**Figure 12.** Figure 12: Examples of solutions for w = 1.00 m s−1 . Left: number concentration, middle: mass concentration, right: saturation ratio, red line; black lines indicate the equilibrium values. 44 [PITH_FULL_IMAGE:figures/full_fig_p044_12.png] view at source ↗

read the original abstract

Clouds are important features of the atmosphere, determining the energy budget by interacting with incoming solar radiation and outgoing thermal radiation, respectively. For pure ice clouds, the net effect of radiative effect is still unknown. In this study we develop a simple but physically consistent ice cloud model, and analyze it using methods from the theory of dynamical systems. We find that the model constitutes a nonlinear oscillator with two Hopf bifurcations in the relevant parameter regime. In addition to the characterization of the equilibrium states and the occurring limit cycle, we find scaling behaviors of the bifurcations and the limit cycle, reducing the parameter space crucially. Finally, the model shows very good agreement with real measurements, indicating that the main physics is captured and such simple models are helpful tools for investigating ice clouds.

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.

Desk Editor's Note private letter to a colleague

This paper takes a stripped-down ice cloud model and shows it behaves as a nonlinear oscillator with two Hopf bifurcations plus scaling laws that shrink the parameter space, plus decent match to measurements.

read the letter

The main point for you is that the authors build a low-dimensional ODE model for pure ice clouds and treat it as a dynamical system. They identify two Hopf bifurcations in the relevant regime, characterize the limit cycle, and derive scaling relations that collapse much of the parameter space. They also report that the model reproduces observed cloud properties reasonably well. That combination of bifurcation analysis and scaling is the concrete new piece here, rather than just another cloud parameterization tweak. The scaling step looks practically useful because it lets you explore the behavior without scanning a high-dimensional space. If the derivations hold up in the full equations, it gives a compact way to think about oscillatory radiative effects in ice clouds. The agreement with measurements is presented as evidence that the main physics is in there, which is a reasonable claim for this level of simplicity. On the downside, the model is deliberately minimal, so it probably leaves out things like particle size distributions or mixing with other cloud types. The abstract does not spell out the exact error metrics or which datasets were used for validation, so a referee would want to see those details to judge how much the fit depends on parameter choices. Still, nothing in the description suggests the bifurcation results are forced by construction. This is aimed at atmospheric physicists who already work with reduced cloud models or who like dynamical-systems approaches to radiative transfer. A reader focused on subgrid ice cloud processes or on finding simpler ways to capture variability would get something out of it. The work is coherent enough on its own terms to warrant a serious referee rather than a desk reject, even if revisions will be needed on the validation side.

Referee Report

2 major / 1 minor

Summary. The manuscript develops a simple but physically consistent model for pure ice clouds and analyzes it with dynamical systems methods. It claims the model behaves as a nonlinear oscillator exhibiting two Hopf bifurcations in the relevant parameter regime, characterizes the equilibrium states and limit cycle, identifies scaling behaviors of the bifurcations and limit cycle that reduce the parameter space, and reports very good agreement with real measurements indicating that the main physics is captured.

Significance. If the central claims hold with proper validation, the work could provide a useful reduced-order dynamical framework for ice clouds, whose net radiative effects remain uncertain. The Hopf bifurcations, limit-cycle characterization, and scaling laws that collapse parameters would represent a novel application of nonlinear dynamics to cloud microphysics, potentially aiding simpler yet consistent modeling of atmospheric processes.

major comments (2)

[Abstract and results section] Abstract and results section: the claim of 'very good agreement with real measurements' is presented without any equations, error bars, specific datasets, data exclusion rules, quantitative fit metrics (e.g., RMSE or R²), or validation figures. This directly supports the assertion that the model is physically consistent and captures the dominant physics, yet the supporting evidence is not shown.
[Model and dynamical analysis sections] Model and dynamical analysis sections: the free parameters for cloud processes are introduced without discussion of their constraints, sensitivity tests, or whether the two Hopf bifurcations and scaling behaviors persist across reasonable ranges rather than being tuned to match observations.

minor comments (1)

[Throughout] Notation for state variables and parameters could be summarized in a table for clarity, especially when discussing the reduced parameter space after scaling.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive review and for highlighting areas where the presentation of evidence and parameter discussion can be strengthened. We address each major comment below and will revise the manuscript accordingly to improve clarity and rigor.

read point-by-point responses

Referee: [Abstract and results section] Abstract and results section: the claim of 'very good agreement with real measurements' is presented without any equations, error bars, specific datasets, data exclusion rules, quantitative fit metrics (e.g., RMSE or R²), or validation figures. This directly supports the assertion that the model is physically consistent and captures the dominant physics, yet the supporting evidence is not shown.

Authors: We agree that the supporting evidence for the agreement with measurements should be presented with greater detail and quantitative rigor. In the revised manuscript we will expand the results section to specify the observational datasets employed, the data exclusion rules applied, quantitative metrics including RMSE and R² values, error bars on both model and data, and dedicated validation figures that directly compare model output to measurements. These additions will make the claim of physical consistency more transparent and verifiable. revision: yes
Referee: [Model and dynamical analysis sections] Model and dynamical analysis sections: the free parameters for cloud processes are introduced without discussion of their constraints, sensitivity tests, or whether the two Hopf bifurcations and scaling behaviors persist across reasonable ranges rather than being tuned to match observations.

Authors: We will add explicit discussion of the physical constraints and literature-based ranges for each free parameter in the model section. The revised dynamical analysis will include sensitivity tests and additional bifurcation diagrams or parameter sweeps demonstrating that the two Hopf bifurcations and the identified scaling laws remain robust across physically plausible ranges, independent of any specific tuning to observations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; dynamical analysis and scaling emerge from model equations independently of data fit

full rationale

The paper constructs a low-dimensional ODE model for ice clouds from physical balances, then applies standard dynamical-systems methods (equilibrium analysis, linearization, Hopf bifurcation detection, limit-cycle characterization) to reveal nonlinear-oscillator behavior and scaling relations that collapse the parameter space. These properties are direct mathematical consequences of the derived equations rather than being fitted or redefined from observations. The subsequent comparison to measurements functions as external validation of physical consistency, not as the source of the claimed bifurcations or scalings. No self-citation chain, ansatz smuggling, or self-definitional loop is present in the derivation; the central claims remain independent of the validation step.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on a small set of differential equations whose parameters are adjusted to physical regimes and observations; no new particles or forces are introduced.

free parameters (1)

model parameters for cloud processes
Parameters in the differential equations are chosen or fitted to produce physically consistent behavior and match measurements.

axioms (1)

domain assumption Ice clouds can be represented by a simple but physically consistent set of differential equations suitable for dynamical systems analysis.
This premise enables the identification of equilibrium states, Hopf bifurcations, and limit cycles.

pith-pipeline@v0.9.0 · 5645 in / 1185 out tokens · 41649 ms · 2026-05-19T06:29:11.443094+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

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A Comprehensive Habit Diagram for Atmospheric Ice Crystals: Confirmation from the Laboratory, AIRS II, and Other Field Studies

Bailey, M. P. and Hallett, J. (2009). “A Comprehensive Habit Diagram for Atmospheric Ice Crystals: Confirmation from the Laboratory, AIRS II, and Other Field Studies”. In: Journal of the Atmospheric Sciences66.9, pp. 2888–2899.doi: 10.1175/2009JAS2883.1. Baumgartner, M. and Spichtinger, P. (Nov. 2017). “Local Interactions by Diffusion between Mixed-Phase ...

work page doi:10.1175/2009jas2883.1 2009
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A model of coupled oscillators applied to the aerosol– cloud–precipitation system

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work page doi:10.1021/acs.chemrev 2013
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Survival of Ice Particles Falling From Cirrus Clouds in Subsaturated Air

New York: Springer, pp. XVI, 462.doi: 10.1007/978-1-4612-1140-2. Hall, W. and Pruppacher, H. (1976). “Survival of Ice Particles Falling From Cirrus Clouds in Subsaturated Air”. In:Journal of the Atmospheric Sciences33.10, pp. 1995–2006.doi: 10.1175/1520-0469(1976)033<1995:TSOIPF>2.0.CO;2. Hanke, M. and Porz, N. (2020). “Unique Solvability of a System of O...

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Kajikawa, M. and Heymsfield, A. (Oct. 1989). “Aggregation of Ice Crystals in Cirrus”. In: Journal of the Atmospheric Sciences46.20, pp. 3108–3121.doi: 10 . 1175 / 1520 - 0469(1989)046<3108:AOICIC>2.0.CO;2. Kärcher, B. and Lohmann, U. (2002). “A parameterization of cirrus cloud formation: Homoge- neous freezing of supercooled aerosols”. In:Journal of Geoph...

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Water activity as the determinant for homogeneous ice nucleation in aqueous solutions

Koop, T., Luo, B., Tsias, A., and Peter, T. (2000). “Water activity as the determinant for homogeneous ice nucleation in aqueous solutions”. In:Nature 406, pp. 611–614. Koren, I. and Feingold, G. (2011). “Aerosol–cloud–precipitation system as a predator-prey problem”. In:Proceedings of the National Academy of Sciences108.30, pp. 12227–12232. Krämer, M., R...

work page doi:10.5194/acp-2020-40 2000
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The physics of snow crystals

Applied Mathematical Sciences. Springer-Verlag, New York, pp. xx+591. 46 Lamb, D. and Verlinde, J. (2011).Physics and Chemistry of Clouds. Cambridge University Press. doi: 10.1017/CBO9780511976377. Libbrecht, K. G. (Mar. 2005). “The physics of snow crystals”. In:Reports on Progress in Physics 68.4, p. 855.doi: 10.1088/0034-4885/68/4/R03. Murphy, D. and Ko...

work page doi:10.1017/cbo9780511976377 2011
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Pattern formation in clouds via Turing instabilities

Atmospheric and Oceanographic Sciences Library. Dordrecht: Kluwer Academic Publishers. Rosemeier, J. and Spichtinger, P. (2020). “Pattern formation in clouds via Turing instabilities”. In: Mathematics of Climate and Weather Forecasting6.1, pp. 75–96.doi: doi:10.1515/ mcwf-2020-0104. Spichtinger, P. and Cziczo, D. J. (July 2010). “Impact of heterogeneous i...

work page doi:10.1029/2009jd012168 2020
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A trajectory-based classification of ERA-Interim ice clouds in the region of the North Atlantic storm track

Graduate Texts in Mathematics. Springer Science & Business Media, pp. XI, 384.doi: 10.1007/978-1-4612-0601-9. Wernli, H., Boettcher, M., Joos, H., Miltenberger, A. K., and Spichtinger, P. (June 2016). “A trajectory-based classification of ERA-Interim ice clouds in the region of the North Atlantic storm track”. In:Geophysical Research Letters43.12, pp. 665...

work page doi:10.1007/978-1-4612-0601-9 2016
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Effect of crystal size spectrum and crystal shape on stratiform cirrus radiative forcing

1002/2016GL068922. 47 Zhang, Y., Macke, A., and Albers, F. (Aug. 1999). “Effect of crystal size spectrum and crystal shape on stratiform cirrus radiative forcing”. In:Atmospheric Research52.1-2, pp. 59–75.doi: 10.1016/S0169-8095(99)00026-5. 48

work page doi:10.1016/s0169-8095(99)00026-5 1999

[1] [1]

A Comprehensive Habit Diagram for Atmospheric Ice Crystals: Confirmation from the Laboratory, AIRS II, and Other Field Studies

Bailey, M. P. and Hallett, J. (2009). “A Comprehensive Habit Diagram for Atmospheric Ice Crystals: Confirmation from the Laboratory, AIRS II, and Other Field Studies”. In: Journal of the Atmospheric Sciences66.9, pp. 2888–2899.doi: 10.1175/2009JAS2883.1. Baumgartner, M. and Spichtinger, P. (Nov. 2017). “Local Interactions by Diffusion between Mixed-Phase ...

work page doi:10.1175/2009jas2883.1 2009

[2] [2]

A model of coupled oscillators applied to the aerosol– cloud–precipitation system

Feingold, G. and Koren, I. (2013). “A model of coupled oscillators applied to the aerosol– cloud–precipitation system”. In:Nonlinear Processes in Geophysics20.6, pp. 1011–1021. Fu, Q. and Liou, K. (1993). “Parameterization of the Radiative Properties of Cirrus Clouds”. In: J. Atmos. Sci.50.13, pp. 2008–2025.doi: 10 . 1175 / 1520 - 0469(1993 ) 050<2008 : P...

work page doi:10.1021/acs.chemrev 2013

[3] [3]

Survival of Ice Particles Falling From Cirrus Clouds in Subsaturated Air

New York: Springer, pp. XVI, 462.doi: 10.1007/978-1-4612-1140-2. Hall, W. and Pruppacher, H. (1976). “Survival of Ice Particles Falling From Cirrus Clouds in Subsaturated Air”. In:Journal of the Atmospheric Sciences33.10, pp. 1995–2006.doi: 10.1175/1520-0469(1976)033<1995:TSOIPF>2.0.CO;2. Hanke, M. and Porz, N. (2020). “Unique Solvability of a System of O...

work page doi:10.1007/978-1-4612-1140-2 1976

[4] [4]

Aggregation of Ice Crystals in Cirrus

Kajikawa, M. and Heymsfield, A. (Oct. 1989). “Aggregation of Ice Crystals in Cirrus”. In: Journal of the Atmospheric Sciences46.20, pp. 3108–3121.doi: 10 . 1175 / 1520 - 0469(1989)046<3108:AOICIC>2.0.CO;2. Kärcher, B. and Lohmann, U. (2002). “A parameterization of cirrus cloud formation: Homoge- neous freezing of supercooled aerosols”. In:Journal of Geoph...

work page doi:10.1029/2001jd000470 1989

[5] [5]

Water activity as the determinant for homogeneous ice nucleation in aqueous solutions

Koop, T., Luo, B., Tsias, A., and Peter, T. (2000). “Water activity as the determinant for homogeneous ice nucleation in aqueous solutions”. In:Nature 406, pp. 611–614. Koren, I. and Feingold, G. (2011). “Aerosol–cloud–precipitation system as a predator-prey problem”. In:Proceedings of the National Academy of Sciences108.30, pp. 12227–12232. Krämer, M., R...

work page doi:10.5194/acp-2020-40 2000

[6] [6]

The physics of snow crystals

Applied Mathematical Sciences. Springer-Verlag, New York, pp. xx+591. 46 Lamb, D. and Verlinde, J. (2011).Physics and Chemistry of Clouds. Cambridge University Press. doi: 10.1017/CBO9780511976377. Libbrecht, K. G. (Mar. 2005). “The physics of snow crystals”. In:Reports on Progress in Physics 68.4, p. 855.doi: 10.1088/0034-4885/68/4/R03. Murphy, D. and Ko...

work page doi:10.1017/cbo9780511976377 2011

[7] [7]

Pattern formation in clouds via Turing instabilities

Atmospheric and Oceanographic Sciences Library. Dordrecht: Kluwer Academic Publishers. Rosemeier, J. and Spichtinger, P. (2020). “Pattern formation in clouds via Turing instabilities”. In: Mathematics of Climate and Weather Forecasting6.1, pp. 75–96.doi: doi:10.1515/ mcwf-2020-0104. Spichtinger, P. and Cziczo, D. J. (July 2010). “Impact of heterogeneous i...

work page doi:10.1029/2009jd012168 2020

[8] [8]

A trajectory-based classification of ERA-Interim ice clouds in the region of the North Atlantic storm track

Graduate Texts in Mathematics. Springer Science & Business Media, pp. XI, 384.doi: 10.1007/978-1-4612-0601-9. Wernli, H., Boettcher, M., Joos, H., Miltenberger, A. K., and Spichtinger, P. (June 2016). “A trajectory-based classification of ERA-Interim ice clouds in the region of the North Atlantic storm track”. In:Geophysical Research Letters43.12, pp. 665...

work page doi:10.1007/978-1-4612-0601-9 2016

[9] [9]

Effect of crystal size spectrum and crystal shape on stratiform cirrus radiative forcing

1002/2016GL068922. 47 Zhang, Y., Macke, A., and Albers, F. (Aug. 1999). “Effect of crystal size spectrum and crystal shape on stratiform cirrus radiative forcing”. In:Atmospheric Research52.1-2, pp. 59–75.doi: 10.1016/S0169-8095(99)00026-5. 48

work page doi:10.1016/s0169-8095(99)00026-5 1999