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

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.

💡 Research Summary

This paper tackles a long‑standing paradox in phase‑change thermodynamics: once a system reaches a melting (or freezing) temperature, the observable temperature is “clipped” at that threshold, while any additional energy input is diverted into latent heat. Consequently, the temperature record alone cannot reveal the underlying energetic fluctuations that drive melt. To resolve this information loss, the authors introduce the concept of a latent temperature θ(t), a counterfactual trajectory the system would follow if phase change were energetically suppressed.

The latent temperature is not an arbitrary construct; it is derived from a variational principle that minimizes the deviation from a linear relaxation process while respecting the phase‑change constraint. The latent heat flux Q_m(t) is decomposed into three exact bookkeeping terms: (i) a storage term ρC_s · θ̇, (ii) a dissipation term (θ − θ_f)/τ, and (iii) a residual redistribution term R(t). Here τ is a characteristic surface‑dissipation timescale, and R(t) accounts for any energy redistribution not captured by the linear relaxation approximation. Substituting this decomposition into the surface energy balance (ρC_s · θ̇_obs = Q_net − Q_m) and enforcing the melt‑season constraints (θ(t_s)=θ(t_e)=θ_f, ∫R dt = 0) yields the key integral identity

\