Thermodynamic Origin of Degree-Day Scaling in Phase-Change Systems

TL;DR

The work addresses the degeneracy in thermodynamic state caused by phase transitions that pin temperature at the melting point. It introduces latent temperature $\theta(t)$ as a counterfactual unconstrained trajectory and proves a fundamental duality: the total latent heat absorbed during melting is proportional to the cumulative exceedance of $\theta(t)$ above the melt point, with proportionality set by a dissipation timescale $\tau$ and formalized as the 1D Wasserstein distance $W_1$ between latent and observed trajectories. This yields a first-principles derivation of the Positive Degree Day law, with a physically interpretable PDD coefficient $C_{\rm PDD} = \frac{1}{\tau \rho_i L_f} \approx \frac{C_{\rm sen}+4\sigma \theta_f^3}{\rho_i L_f}$ that agrees with empirical values, and shows how phase change acts as an optimal transport projecting energetic variability onto the melting boundary. The framework generalizes to phase-change systems beyond glaciology, linking surface-energy balance to latent transport and predicting how degree-day factors should covary with boundary-layer coupling and radiative damping, thereby providing a rigorous, tunable basis for threshold-limited thermo-energetic analyses.

Abstract

Phase transitions impose topological constraints on thermodynamic state variables, masking energetic fluctuations at the phase boundary. This constraint is most apparent in melting systems, where temperature remains pinned despite continued energy input. Here we resolve this information loss by introducing a latent temperature-a counterfactual trajectory describing the system's unconstrained thermal evolution. We show that energy conservation alone enforces a rigorous duality between the total latent heat dissipated during phase change and the accumulated exceedance of the latent temperature above the melting point. This duality is mathematically equivalent to the one-dimensional Wasserstein-1 distance between the latent and observed temperature trajectories, with the transport cost set by a characteristic surface dissipation timescale and melting energy. Applied to ice-sheet surface melting, this timescale admits a direct physical interpretation in terms of radiative and turbulent heat loss. The same framework yields a first-principles derivation of the empirical Positive Degree Day law and predicts realistic degree-day factors that emerge from surface energy balance, without ad hoc calibration. More broadly, phase change emerges as an optimal transport process that projects continuous energetic variability onto a constrained thermodynamic boundary.

Figures (2)

• Figure 1: Conceptual framework linking phase change and latent temperature. (a) In phase-change systems, temperature is constrained at the melting point $\theta_f$ (for convenience, $\theta_f=0$ in the diagram), rectifying continuous energy input into latent heat while suppressing observable thermal fluctuations. (b) The latent temperature $\theta(t)$ denotes the counterfactual thermal trajectory the system would follow in the absence of phase change, encoding the energetics sequestered during melting. (c) The cumulative exceedance of the latent temperature above $\theta_f$ is geometrically equivalent to the Wasserstein-1 distance between the latent and observed trajectories and scales linearly with the total melt energy $M$ via a characteristic dissipation timescale $\tau$.
• Figure 2: Variational reconstruction of latent temperature under periodic forcing. The system is driven by a biased sinusoidal energy flux $Q_i$ (gray and pink shading). The observed temperature $\theta_{\mathrm{obs}}$ (solid black) is strictly limited by the melting threshold $\theta_f=0$. To resolve the subtle energetic fluctuations hidden by this clipping, the vertical axis is split: the positive range ($[0, 0.024]$) is magnified relative to the negative range ($[-1.2, 0]$). The reconstructed latent temperature $\theta$ (red) recovers the unconstrained thermal state. The area representing sequestered melting energy (pink) is recovered through the integrated dissipation term (light blue), where the balance between these components defines the optimal relaxation timescale $\tau$. Black dash lines marks the net energy flux in reconstructed system.