From the Main section or the paper:
Greenland’s ice budget deficit emerged after the 1980s from increases in surface meltwater run-off1,2 and ice flow discharge from its tidewater sectors3,4. Yet, despite its importance for future sea-level rise (SLR), our capacity to accurately predict Greenland’s response to climate change is hampered by process limitations in ice sheet models and their imprecise coupling to land, atmosphere and ocean boundaries5,6,7. Given these constraints, we pursue a complementary approach to obtain Greenland’s thus far lower-bound-committed SLR contribution.
Our approach does not directly solve the transient ice flow equations but rather calculates the committed areal and volumetric changes incurred by the up-to-present ice geometrical disequilibrium with climate8,9. The method determines the ice extent and thickness perturbations required to bring the current ice sheet into equilibrium with surface mass balance (SMB). Changes in flow dynamics are implicitly accounted for by a glaciological power law scaling function that relates imposed areal changes in ice extent to an implied ice volume10. To account for marine-terminating sectors and tidewater outlets where the ablation area is truncated by iceberg calving, we introduce an effective ablation area treatment. For application to Greenland—including its peripheral ice masses—the essential empirical requirements are met with new multi-year inventories of: (1) tidewater glacier discharge4; (2) SMB (that is, snowfall accumulation minus run-off) from observational reanalysis and regional climate data11; and (3) the accumulation area ratio (AAR): the glacierized area with net annual mass gain divided by its total area, readily retrieved from optical satellite imagery12,13.
For grounded ice masses with an ablation area, the maximum snow line elevation at the end of each melt season marks the transition between the lower-elevation dark bare ice and the bright upper-elevation snow accumulation areas. This equilibrium line and its corollary, the AAR, conveniently integrate the competing effects of surface mass loss from meltwater run-off and mass gain from snow accumulation. Minimum AAR each year demarcates hydrological years on a sector basis (Extended Data Fig. 1). By regressing annual AAR and mass balance, we obtain the statistical property of AAR in the condition of mass balance equilibrium (AAR0) that is necessary for the current ice surface morphology to be in dynamic equilibrium with climate (Fig. 1). The ratio of the observed AAR to AAR0 yields the fractional imbalance () that quantifies the area perturbation required for the ice mass to equilibrate its shape to an imposed climate shift away from that associated with AAR0 (ref. 8). This disequilibrium approach exploits how climatically driven SMB perturbations are at least an order of magnitude faster than the associated dynamic adjustment of the ice mass14. The resulting derivation for the adjustment in ice volume (V) and committed eustatic SLR follows glaciological scaling theory relating the glacierized area change to ice volume perturbation using a power law function10 (Fig. 1) with exponent () (Methods).
While under the most up-to-date ice thickness and subglacial topography mapping15, Greenland’s current ice sheet configuration implies an area–volume scaling exponent of =1.24 that closely abides the theoretically derived value of 1.25 (ref. 10), we apply a linear exponent of 1 to avoid the mathematically intractable regional case in which some ice flux between adjacent flow sectors is inevitable. The choice of a linear exponent represents an absolute minimum committed loss, encompassing flow interaction between adjacent ice sheet sectors, since it accounts for how scaling techniques are best applied to ensembles of many ice masses10,16, which we accomplish by summing the volumes from 473 subregions of the ice sheet. While it is possible to scale the entire ice sheet with an exponent of 1.25, yielding volume changes roughly 20% higher than our results, this is an ill-posed inversion with potentially large random errors that grow exponentially with the size of the ice mass, a problem shared by both numerical models and scaling theory17. Our aggregate sums from many regions exploit the law of large numbers to dramatically reduce these errors10. As such, while an exponent of 1 underestimates the volume change for each region, such an approach guarantees a mathematically sound lower bound of the ice volume loss along with the resulting SLR while minimizing methodological errors.
While the method has previously only been applied to assess mountain glacier and ice cap disequilibria8,9,18, the theoretical basis and derivation of ice area–volume scaling analysis can be applied independently of size and area8,9,18. Hence, we apply the method to Greenland’s entire glacierized area through summing over sectors to obtain lower-bound estimates of its committed mass loss and SLR resulting from its imbalance with recent climate. We calculate ice disequilibrium for three reference climate scenarios applied in perpetuity: ‘recent’ (2000–2019 average) climate to determine Greenland’s already committed ice loss and then take the respective high- and low-melt years of 2012 and 2018 to assess potential future area and volume changes under extreme end-member climate states, with the proviso that no long-term reversal in climate warming trends are anticipated this century.