A Thermodynamic Framework for Quantifying Firn Thermal Evolution: Cold Content Depletion, Energy Barrier Reduction and Thermal Commitment of Austfonna Ice Cap, Svalbard (2009-2022)

Svalbard is warming several times faster than the global mean, yet most glaciological studies characterise firn thermal change through temperature trends alone rather than the underlying thermodynamic state. This study develops a linked diagnostic framework, cold content (CCcol), energy barrier (EBrelative), advective Péclet number (Peadv), and thermal relaxation time (τ) and applies it to a 14-year borehole temperature record (2009-2022) from the summit of Austfonna Ice Cap, Svalbard, providing the first continuous multiyear thermodynamic characterisation of this ice cap derived directly from borehole temperature observations rather than a forcing driven model. After correction for multiple comparisons, warming remained significant at 8-10 m using all available years and at 3-10 m using complete years. Near surface trends remained large but statistically uncertain. Between the 2010-2013 reference period and the 2020-2022 late post transition period, mean cold content declined from 9.964 to 4.653 MJ m⁻² (53.3%, -562 kJ m⁻² yr⁻¹, R²=0.57, p=0.007), and the energy barrier fell to 46.7% of its pre transition reference value. Under the reference whole column ice equivalent scaling, Peadv was 8.0, while an alternative scaling based on the assumed mean firn density gave Peadv = 11.3. CCcol, EBrelative are calculated consistently and can be compared between periods. In contrast, Peadv and τ provide site scale context and remain unchanged between periods unless the prescribed values of H, b, or α are updated. Comparisons between sites therefore require consistent parameter definitions and characteristic length scales. Requiring a borehole temperature record, a representative density profile, an accumulation rate estimate, an ice thickness estimate, and a reference thermal diffusivity, the framework is substantially less data intensive than coupled energy balance models and is broadly transferable to other polythermal ice masses beyond the site studied here.