Thermodynamic Analysis of Snowball Earth Hysteresis Experiment: Efficiency, Entropy Production, and Irreversibility

We present an extensive thermodynamic analysis of a hysteresis experiment performed on a simplified yet Earth-like climate model. We slowly vary the solar constant by 20% around the present value and detect a substantial bistability: for a large range of values the realization of snowball (SB) or of warm (W) climate conditions depend on the history of the system. Using recent results on the global climate thermodynamics, we show that that the two regimes feature radically different properties. The efficiency of the climate machine increases with decreasing solar constant in W climate conditions, whereas the opposite takes place in the SB regime. Instead, entropy production is increasing with the solar constant in both branches of climate conditions, and its value is about 4 times as large in the W branch than in the corresponding C state. Finally, the degree of irreversibility of the system is much higher in the W conditions, with an explosive growth in the upper range of the considered values of solar constants. Whereas in the SB regime a dominating role is played by changes in the meridional albedo contrast, in the W climate regime changes in the intensity of latent heat fluxes are crucial for determining the observed properties. This substantiates the importance of addressing correctly the variations of the hydrological cycle in a changing climate. An interpretation of the transitions at the boundary of the bistable region based upon macro-scale thermodynamic properties is also proposed. Our results support the adoption of a new generation of diagnostic tools based on the 2nd law of thermodynamics for auditing climate model and outline a set of parameterizations to be used in conceptual and intermediate complexity models or for the reconstruction of the past climate conditions.

💡 Research Summary

The paper presents a comprehensive thermodynamic investigation of a hysteresis experiment performed with a simplified Earth‑like climate model. By slowly varying the solar constant (S) by ±20 % around its present value (S₀), the authors uncover a broad bistable region in which the model can reside either in a Snowball Earth (SB) state or in a Warm (W) state, depending on its prior history. The study proceeds in four logical stages.

First, the experimental design is described. The model includes atmospheric, oceanic, cryospheric, and radiative components and is run in a quasi‑static manner: S is increased stepwise from 0.8 S₀ to 1.2 S₀, then decreased back, allowing the system to equilibrate at each step. Within roughly 0.9 S₀ – 1.1 S₀ a hysteresis loop appears; for the same S the climate can be either SB or W, illustrating true multistability.

Second, the authors introduce three macroscopic thermodynamic diagnostics derived from the second law of thermodynamics: (i) the climate‑machine efficiency η = W_out/Q_in, where W_out is the mechanically available work extracted from the heat engine formed by the atmosphere‑ocean system and Q_in is the absorbed solar radiation; (ii) the total material entropy production σ, which quantifies irreversible processes such as radiative diffusion, turbulent heat transport, and especially latent‑heat cycles; and (iii) the irreversibility parameter α = σ/η, measuring the proportion of dissipated energy relative to the useful work. These quantities are computed directly from model output (radiative fluxes, latent heat fluxes, temperature fields) following the methodology of Lucarini et al. (2011) and subsequent extensions.

Third, the paper contrasts the two climate branches. In the Warm regime, decreasing S leads to a larger meridional temperature gradient, which raises η because the atmospheric heat engine operates over a larger temperature difference. Conversely, in the Snowball regime the dominant response is an increase in planetary albedo; the reflected solar radiation reduces the absorbed energy, the temperature gradient shrinks, and η declines. Thus η varies oppositely with S in the two branches, reflecting distinct physical controls.

Entropy production σ grows monotonically with S in both branches, but its magnitude in the Warm state is roughly four times that in the corresponding Snowball state. The authors attribute this disparity to the vastly stronger hydrological cycle in the Warm climate: higher water‑vapor content, more vigorous evaporation‑condensation cycles, and enhanced cloud radiative effects generate far larger irreversible heat fluxes. Near the upper end of the S range, σ exhibits an “explosive” increase, indicating a threshold‑like amplification of latent‑heat processes.

Irreversibility α follows a similar pattern. In the Warm branch α reaches values above 0.6, signifying that a substantial fraction of the absorbed solar energy is dissipated irreversibly, whereas in the Snowball branch α remains near 0.2, indicating that most of the energy loss is reversible (mainly albedo‑driven reflection). This contrast underscores that the Warm climate is dominated by latent‑heat mediated irreversibility, while the Snowball climate is governed by radiative albedo feedbacks.

The fourth part focuses on the transitions at the edges of the bistable region. The Snowball‑to‑Warm transition is marked by a sudden surge in latent‑heat fluxes, a rapid recovery of η, and a steep rise in σ, reflecting the activation of a water‑vapor feedback that converts the system into a high‑efficiency, high‑entropy‑production regime. The reverse Warm‑to‑Snowball transition is driven by a sharp increase in meridional albedo contrast, which collapses the absorbed solar budget, causing simultaneous drops in η and σ. The asymmetry of these transitions is captured quantitatively by the thermodynamic diagnostics, offering a macro‑scale interpretation of climate tipping points.

Finally, the authors argue that these second‑law based diagnostics constitute powerful tools for model evaluation and inter‑comparison. They propose a set of parameterizations for intermediate‑complexity models that explicitly link latent‑heat fluxes and albedo changes to η, σ, and α, thereby improving the representation of hysteresis and tipping‑point behavior. The study also suggests that paleo‑climate reconstructions and future climate projections should incorporate efficiency, entropy production, and irreversibility as diagnostic metrics to assess the physical plausibility of simulated climate states.

In summary, the paper demonstrates that the Warm and Snowball climate regimes are thermodynamically distinct: the Warm state exhibits higher efficiency, markedly larger entropy production, and greater irreversibility, all controlled primarily by the hydrological cycle, whereas the Snowball state is dominated by albedo feedbacks. By framing climate dynamics in terms of the second law, the authors provide a novel, physically grounded perspective on bistability, tipping points, and the design of more reliable climate models.