A knowledge gap related to debris-covered glacier dynamics affects our understanding of the past and future of Himalayan glaciers in a changing climate . The supraglacial debris cover present over the ablation zone of a glacier induces qualitative changes in its dynamic response due to a suppressed melt rate under a thick debris layer , whereas a thin debris cover is expected to accelerate melt due to its low albedo. While responding to a warming climate, debris-covered glaciers exhibit a larger climate sensitivity, a longer response time , a decoupling of volume and length changes and the formation of a slow-flowing stagnant downwasting tongue . Despite several efforts to model and understand the dynamics of debris-covered glaciers with various degrees of sophistication , challenges still remain. This task is made more difficult by our limited understanding of the time evolution of the debris extent , the variability of debris thickness, and common occurrences of highly dynamic supraglacial ponds and ice cliffs that cause intense localised melting .

A curious fact that has emerged from large-scale remote-sensing measurements of glaciers in the Himalayas and Karakoram during the first decade of the 21st century is a similar magnitude of thinning of glacial ice irrespective of the presence of supraglacial debris cover and this may seem counter-intuitive. A thick debris cover, due to its insulating properties, significantly inhibits the melting of the underlying ice – so much so that in the debris-covered part of the glacier, the specific melt rate does not increase with decreasing elevation. Rather, it saturates to some lower bound or even decreases downglacier . On the other hand, on a debris-free glacier the melt rate typically increases monotonically as elevation decreases. Why then should both glacier types experience similar rates of thinning as climate warms up?

Heuristic arguments were offered by various authors to reconcile with this apparent paradox. suggested that the insulating effect of the debris cover might be compensated for at the scale of the whole ablation zone due to an enhanced melting from the thermokarst processes, namely supraglacial ponds and ice cliffs that are often present on the debris-covered glaciers. These features, due to an associated discontinuous debris cover, experience large localised melting. Given that these features typically contribute ∼ 10–20 % of the total melt (Sakai et al., 2000; Reid and Brock, 2014), it is unlikely that they can lower the glacier-wide mean melt rate on debris-covered glaciers sufficiently so that it matches that on the debris-free glaciers. Field measurements by seem to confirm this. It was also conjectured that a reduction in ice flux from upstream areas to the stagnant tongue may be behind the larger-than-expected thinning of debris-covered glacial ice . too pointed out the possible role of reduced flux into the low-slope slow-moving stagnant tongues of large debris-covered glaciers. However, a quantification of this flux effect is missing as of yet.

On the other hand, showed that a reduced melt rate on a debris-covered glacier does not affect the volume response of the glacier qualitatively, in stark contrast with its drastic effect on the length response of the glacier. However, their model results (Fig. 3d of ) show a relatively larger thinning rate on the debris-free glaciers in response to a rapid warming. Also, it was reported that in the Pamir–Karakoram–Himalayas, depending on the region chosen, geodetic measurement yielded decadal thinning rates of debris-covered ice that were either larger or smaller than, or similar to that of debris-free ice . The present scenario is summed up neatly by , “This question of area-averaged melting rates over debris-covered or clean glacier ablation areas remains unanswered”.

In this contribution, we analyse the rate of thinning on debris-covered and debris-free glaciers in a warming climate using a one-dimensional flowline model of idealised glaciers . We conduct simple numerical experiments to investigate the role of the magnitude of warming rate, the ice dynamics (i.e. the changes in the flux-gradient profiles or equivalently that in emergence/submergence velocities), and that of the surface mass-balance forcing in controlling the thinning rates on these two glacier types.

An easy-to-analyse piece of this problem is the behaviour of a steady-state debris-covered or debris-free glacier immediately after an instantaneous rise of temperature (or equivalently that of the equilibrium line altitude, ELA). In a steady state, the ice-thickness profile remains constant due to a stable balance between the surface ablation (accumulation) rate and the emergence (submergence) velocities. Dictated by mass conservation of incompressible ice, the emergence or submergence rate equals the negative gradient of the flux, F(x). After an instantaneous change in ELA, the surface mass-balance values change, but the viscous ice flow takes a characteristic longer time to relax. Therefore, the local thinning rate is initially just the difference in specific mass balance, B(x), before and after the change in temperature. However this is valid only over a timescale that is short compared to the flow relaxation time.

Let us consider two idealised model glaciers. Glacier A does not have any debris cover and has a linear mass-balance profile. Glacier B has a supraglacial debris cover on its lower ablation zone where the ablation rate saturates to a value of -2 m yr-1 (Fig. 1b). This idealised mass-balance profile for the debris-covered glacier is motivated by data from a Himalayan glacier . Similar simplified mass-balance profiles have been used to analyse the response of the debris-covered Himalayan glaciers . In a real glacier, the possible variability of debris thickness and ephemeral thermokarst features (ponds and ice cliffs) cause significant spatial variation of the melt rate in the debris-covered parts of the glacier. However, a relatively fast advection of these surface features would imply that a long-term mean melt rate at a specific location is a well-defined quantity. This justifies the simplified mass-balance profile employed here. Further, the observed thinning rate values in the Himalayas are obtained for a large set of glaciers so that the possible effects of specific details of mass-balance profile of individual glaciers would be averaged out.

In Fig. 1a, b we show mass-balance profiles for the idealised model glaciers before and after an instantaneous rise of ELA, ΔE=50 m. It is assumed here that the mass-balance shape remains the same and changes only by a shift of ELA. In practice the debris layer may thicken and the debris-covered area may grow in a warming climate, affecting the shape of the melt rate profile. However, it is known that above a debris thickness of ∼ 10 cm, the decrease in melt rate with a thickening debris layer is small . Therefore such changes can safely be neglected as a first approximation. The possible changes in supraglacial ponds/ice cliffs are neglected at this level of approximation due to a relatively smaller contribution of these features to the total melt, as discussed before. This assumption of an invariant shape allows for the possible increase in debris extent with warming as the upper boundary of the region with saturated melt rate moves up with the ELA. Overall these simplifications allow us to focus on the role of ice flow dynamics in controlling the downwasting of glaciers in a warming climate.

As is clear from Fig. 1a, glacier A responds initially with a uniform glacier-wide thinning rate, 〈dhdt〉A=βΔE, right after the ELA change. Here β is the mass-balance gradient. For glacier B, a uniform thinning operates only on the debris-free upper part of the glacier and the lower part has not thinned at all (Fig. 1b). Thus, glacier B has a lower mean thinning rate to start with that is given by 〈dhdt〉B=(1-fd)βΔE, where fd is the debris-covered fraction. Remarkably these expressions do not involve the lengths of the glaciers. Also, the initial lack of thinning on the debris-covered glacier is independent of the actual value of the melt rate under the thick debris layer (assumed to be 2 m yr-1 here) and depends only on the general shape of the melt curve (Fig. 1b).

A more general mass-balance profile for a debris-covered glacier than the one considered above would involve a smaller or inverted mass-balance gradient in the debris-covered parts . Even then, the mean initial thinning rate on such a glacier would be less than that of a corresponding debris-free one. This delayed thinning of the debris-covered terminus is consistent with the formation of a slow-flowing stagnant tongue with very little retreat as observed on debris-covered glaciers in the Himalaya–Karakoram . This raises confidence in the minimal description of such glaciers that is being used here. In case of an inverted mass balance, a transient thickening of the lower ablation zone is observed, though this is likely to be an artefact of the assumed fixed shape of the mass-balance curve.

Thus, a debris-covered glacier starts with a lower value of mean thinning rate compared to a debris-free one (as 〈dhdt〉A>〈dhdt〉B). The ice fluxes then respond to the mass-balance change and subsequent evolution of the flux-gradient profile, or equivalently that of the emergence velocity profile alters the distribution and magnitude of the thinning rate. Though the detailed spatial and temporal pattern of such changes are difficult to predict, at some later stage the thinning rate on glacier B is likely to become larger than that on glacier A. This is because (1) the debris-covered glacier B has a larger climate sensitivity compared to glacier A and thus loses more mass for the same change in the ELA; (2) on glacier B, the lower ablation zone responds to the perturbation with a delay. There must be an intermediate crossover period as well, where the thinning rates on both the glaciers have a similar magnitude within measurement errors.

To verify the above claims on the nature of the evolution of thinning rate on glaciers A and B, we perform a set of numerical experiments with one-dimensional flowline models of glaciers A and B. The model glaciers have a bedrock slope of 0.1 and mass-balance gradient β=0.007 yr-1. See for further details of the flowline model used. Note that these glaciers are identical above the debris-covered region (Fig. 1a, b). The initial steady states are prepared by running the models with a fixed value of ELA for 500 (900) years for glacier A (B). The steady-state lengths of the simulated glaciers are in the range 6–14 km. Subsequently, the following ELA perturbations are switched on at t=0.

An instantaneous rise by 50 m.

In all three experiments the net warming is similar, but the rates and durations of the ELA perturbations are different (1 – an instantaneous warming; 2 – a rate of 10 m decade-1 for 50 years; 3 – a rate of 2 m decade-1 for 150 years). In experiment 3, we restrict the total ELA rise to 30 m so as to limit the duration of the experiment to 150 years to facilitate comparison with the other two experiments.

We provide very general arguments that debris-covered glaciers, while responding to a warming climate, can have smaller, larger or similar thinning rates compared to corresponding debris-free glaciers. The thinning of glaciers is controlled by a competition between a changing mass balance and the emergence velocity profile. A debris-covered glacier starts with a smaller glacier-averaged thinning rate, but overtakes that of a debris-free glacier at later stages of evolution. The initial difference in the corresponding thinning rates depend on the balance gradient and the debris-covered fraction. The changes in local melt rates control the thinning of glacial ice immediately after an instantaneous warming, whereas a stronger variation of the corresponding emergence velocity profile dictates the evolution of the thinning of ice at subsequent stages. Our arguments are validated against results from flowline model simulations of idealised glaciers.