Approximate glacier models are routinely used to compute the future evolution of mountain glaciers under any given climate-change scenario. A majority of these models are based on statistical scaling relations between glacier volume, area, and/or length. In this paper, long-term predictions from scaling-based models are compared with those from a two-dimensional shallow-ice approximation (SIA) model. We derive expressions for climate sensitivity and response time of glaciers assuming a time-independent volume–area scaling. These expressions are validated using a scaling-model simulation of the response of 703 synthetic glaciers from the central Himalaya to a step change in climate. The same experiment repeated with the SIA model yields about 2 times larger climate sensitivity and response time than those predicted by the scaling model. In addition, the SIA model obtains area response time that is about 1.5 times larger than the corresponding volume response time, whereas scaling models implicitly assume the two response times to be equal to each other. These results indicate the possibility of a low bias in the scaling model estimates of the long-term loss of glacier area and volume. The SIA model outputs are used to obtain parameterisations, climate sensitivity, and response time of glaciers as functions of ablation rate near the terminus, mass-balance gradient, and mean thickness. Using a linear-response model based on these parameterisations, we find that the linear-response model outperforms the scaling model in reproducing the glacier response simulated by the SIA model. This linear-response model may be useful for predicting the evolution of mountain glaciers on a global scale.

In the coming decades, shrinking mountain glaciers will contribute significantly to global eustatic sea-level rise

Instantaneous (annual) glacier surface mass balance can be calculated readily using climate model outputs. In contrast, any prediction of the long-term evolution of a glacier requires simulating the slow (decadal) changes in glacier area and geometry. Ideally, this is to be done by solving the dynamical ice-flow equations

While some of the approximate parameterisations of glacier dynamics are empirical prescriptions for adjusting the hypsometry of the transient glaciers

Theoretically, the scaling exponent

It is the above statistical interpretation of the scaling relation, where a best-fit time-invariant

The performance of scaling models in simulating the transient glacier response has previously been tested against various dynamical ice-flow models (e.g. SIA, higher-order approximations, or Stokes' model) in one to three dimensions using both idealised

to obtain analytical predictions for climate sensitivity and response time of glaciers in a scaling model;

to compare the climate sensitivity and response time of a large number of synthetic glaciers with realistic geometries, as obtained from a scaling model and a 2-D SIA model;

to investigate the possibility of long-term biases in scaling model estimates of changes in glacier area and volume with respect to corresponding SIA results;

to find convenient parameterisations of glacier-response properties obtained from the SIA simulations and develop an accurate linear-response model.

The paper is organised as follows. First, we theoretically derive the glacier-response properties within a time-invariant scaling assumption (Sects. 2.1 and 3.1). Then, we compare the performance of a representative scaling model

For a theoretical analysis of the glacier-response properties implied by a scaling model, we consider a set of hypothetical glaciers that respond to a warming climate such that the volume–area scaling relation (Eq.

We simulated the response of an ensemble of synthetic clean glaciers with realistic geometries to a hypothetical step change in ELA using three different methods (scaling, SIA, and linear-response models). For this exercise, we considered all the 814 glaciers larger than 2 km

The ice-flow dynamics was implemented within a two-dimensional SIA

The value of Glen's flow-law exponent was assumed to be 3

The model was integrated using a linearised implicit finite-difference scheme

The SIA simulation was run starting with an empty bedrock, with the initial

Out of the total 810 simulated glaciers from the Ganga basin, on 98 glaciers the fractional change in glacier area at

Finally, we were left with an ensemble of 703 synthetic Himalayan glaciers (Fig. S1), with area in the range of 2.2–156.0 km

The response of the above set of 703 steady-state glaciers to a 50 m instantaneous rise in ELA was also computed with a scaling model

For each of the 703 glaciers, the time series of volume and area as obtained using the SIA and scaling models were separately fitted to linear-response forms (e.g. Eq.

Note that applying a step change in ELA of a steady-state glacier to obtain the step-response function is a standard prescription for obtaining glacier-response properties

The best-fit linear-response properties obtained from the scaling model results for the 703 glaciers were used to verify the corresponding theoretical expressions obtained from scaling theory (Eqs.

The best-fit empirical parameterisations for climate sensitivity and response time obtained by fitting the SIA results as described above (given in Eqs.

To assess the uncertainty of the linear-response model output, the uncertainty of each of the fit parameters was set equal to the corresponding standard error, and the 95 % uncertainty band for the linear-response model outputs was generated using a Monte Carlo method.

To test the applicability of the above linear-response model that was calibrated using SIA results for the 703 central Himalayan glaciers, the same model was applied to a different set of 204 glaciers from the western Himalaya. The parameterisations developed for the central Himalayan glaciers as discussed above (given in Eqs.

Below, we derive some relevant consequences of the time-invariant scaling assumption, including expressions for the climate sensitivity and response time of glacier area and volume. These results are expected to be generally valid for all scaling models that are based on Eq. (

Equation (

To compute the area response time, let us consider a constant perturbation, i.e., a step change in ELA applied to a steady glacier for time

The instantaneous change in volume (

After a step change in ELA, as the ablation zone shrinks, the initial net negative balance of a glacier gradually decays to zero over a period determined by the corresponding response time. A longer area response time in SIA implies that this reduction in the ablation zone is slower here than that in a scaling model. A corresponding feedback of a larger ablation zone on the net mass balance should then lead to a higher long-term volume loss in an SIA model than that in a scaling model. This indicates the possibility of a low bias in scaling model estimates of the climate sensitivity of volume, or equivalently, that in the long-term changes in glacier volume due to any rise in ELA.

An expression for the climate sensitivity of glacier area (

The corresponding expression for

Strictly speaking, the climate sensitivity of area and volume with respect to a change in ELA should be defined as

Following Eq. (

The slow and systematic decline in

The dependence of the glacier-specific scale factor on the mean slope is known

The theoretical predictions for glacier area and volume response time (Eq.

Scaling model simulations of the 703 synthetic Himalayan glacier show that

For SIA simulations, the data showed that

Results from the SIA simulations of the 703 synthetic Himalayan glacier show that

Apart from the overall underestimation of area and volume response times by the scaling model, another serious limitation of scaling models that emerges from the above analysis is that here the area and volume response times are equal to each other (Eq.

For the 703 glaciers simulated by the scaling model, the fitted asymptotic fractional changes in area and volume, or equivalently the corresponding (fractional) climate sensitivities, were proportional to each other (Fig.

In contrast, the SIA simulations obtained

Figure

The above relations suggest that the climate sensitivity of volume in the SIA simulation was about 2.6 times larger than that in the scaling model. Similarly, the climate sensitivity of glacier area obtained from the SIA model was also about 1.9 times larger than that obtained from the scaling model. This trend of a relatively large (by about a factor of 2) underestimation of climate sensitivity of glacier volume and area by the scaling model is consistent with the effects of a relatively faster shrinkage of the ablation zone in the early stages of the response as discussed in Sect. 3.1.3 and 3.2.2.

Starting with an initial volume (area) of 847 km

The evolution of the total

The low bias in the long-term changes of glacier area and volume computed with the scaling model is consistent with the underestimation of corresponding climate sensitivities by this model (Sect. 3.2.3). On shorter timescales of multiple decades, an underestimation of response times by about a factor of 2 (Sect. 3.2.2) partly compensates for a corresponding underestimation of the climate sensitivities (Sect. 3.2.3), and the deviations between the SIA and the scaling model are not that prominent (Fig.

Note that, depending on the details of the scaling and SIA models compared, or the set of glaciers simulated, the actual magnitude of the biases in scaling-model-derived climate sensitivity, response time, and long-term glacier change could be different from these here. However, based on the theoretical arguments and numerical evidence presented, similar qualitative trends are expected if the above exercise were to be repeated with a more detailed model and/or for a more realistic set of glaciers.

The above results indicate the possibility of a negative bias in scaling model estimates of future changes in mountain glaciers and the corresponding contribution to sea-level rise. As an example, let us consider a recent comparison

The above results show that the linear-response model outperformed the scaling model, producing a closer match with the SIA results for the 703 synthetic glaciers from the Gangetic Himalaya. However, this linear-response model was calibrated using the SIA results for the same set of glaciers. Therefore, this match is not enough to establish the effectiveness of the linear-response model. To confirm the improved performance of the linear-response model compared to that of the scaling model, we applied both the models to simulate a different set of 164 glaciers in the western Himalaya (Fig. S1). The best-fit linear-response properties obtained from SIA simulation of the 703 central Himalayan glaciers were first fitted to obtain four equations (Eqs.

Can the biases in the scaling model described above be artefacts arising out of some peculiarities of the geometry of the specific set of glaciers being simulated and not relevant in general for scaling model computations of global-scale mass loss of mountain glaciers? To rule out this possibility, we simulated the response of a set of highly idealised synthetic glaciers using both a flow-line model

The above flow-line-model experiment provides an additional piece of evidence that the scaling-model biases discussed in this paper are, in general, expected to be present in scaling model simulations of any set of glaciers. We re-emphasise that even though the biases are expected to be qualitatively similar to those presented here, the magnitudes of the biases are likely to depend on the detailed characteristics (related to geometry, flow, and mass-balance processes) of the glaciers studied and the models used.

As described above, we have used results from the 2-D SIA model simulations of the response of 703 synthetic Himalayan glaciers to a 50 m step change in ELA, to obtain the following best-fit parameterisations of the glacier-response properties (i.e.,

With the estimated glacier-specific response properties obtained from Eqs. (

Note that the above formulation does not require the initial state to be steady. As long as the glacier is close to a steady state, a linear-response theory will be a good approximation

Since the above parameterisation of linear-response properties (Eqs.

Due to the noise present in the fits (Fig.

Because of the idealised descriptions of ice flow and the mass-balance profile (as discussed in Sect. 2.2), and the absence of model calibration to match the available observed data of surface velocity, ice thickness, recent mass balance, etc., the glaciers simulated here are not faithful copies of the Himalayan ones. For a set of more realistic glaciers, the magnitudes of the corresponding biases in scaling-model-derived climate sensitivity and response time could be different from those obtained here. However, based on the theoretical arguments and numerical evidence presented, similar qualitative trends are expected if the above exercise were to be repeated for a more realistic model that includes higher-order mechanics, a more realistic mass-balance model, and so on. Similarly, the parameterisations for the linear-response properties given here are obtained from 2-D simulations of 703 synthetic Himalayan glaciers with some idealisations (Sect. 2.2) and without any tuning of model parameters. The fit parameters in Eqs. (14)–(17) may be different for a different set of glaciers. The parameterisations may also change if a more detailed and calibrated model of the same glaciers is used. However, the protocol used here to obtain the parameterisation for linear-response properties can be directly applied without any change for any set of glaciers and for any ice-flow and/or mass-balance model. While applying the linear-response model to any other region, it may be useful to obtain the response properties of a few tens of representative glaciers using flow-model simulations and check if any recalibration of the parameterisation as given in Eqs. (14)–(17) is necessary.

We performed a theoretical analysis of the response of mountain glaciers within a time-independent scaling assumption. In addition, the step response of 703 steady-state synthetic Himalayan glaciers with realistic geometries and idealised mass-balance profiles were simulated with three different models: a scaling model, a 2-D SIA model, and a linear-response model. The results obtained are as follows.

Analytical expressions for climate sensitivity and response time of glacier area and volume are derived within a time-independent scaling assumption. These expressions are validated using results from the scaling model simulation of the ensemble of 703 glaciers.

The response of the glaciers simulated with the 2-D SIA model reveals that the initial steady states and the transient states follow the volume–area scaling relation, with the best-fit scale factor reducing slowly with time.

For the ensemble of glaciers studied, the scaling model obtains relatively smaller climate sensitivities of glacier area and volume by a factor of about 1.9 and 2.6, respectively, compared to those obtained from the SIA model. This results in a low bias in the long-term changes predicted by the scaling model.

For the ensemble of glaciers studied, the scaling model underestimates volume (area) response time by a factor of

For the scaling model,

The relatively larger ratio of the two response times in the SIA simulations, along with an initial slow change in the area, leads to curved

A linear-response model based on the parameterisations of SIA-derived response properties helps reduce the biases in the long-term glacier changes predicted by the scaling model for the idealised central Himalayan glaciers. The improved performance of this model is validated on an independent set of 164 glaciers in the western Himalaya.

Based on the theoretical arguments and numerical evidence presented here, it is possible that qualitatively similar biases may generally be present in the long-term glacier changes computed with scaling models. However, the actual magnitudes of such biases in scaling models may be different from those obtained here for a set of synthetic Himalayan glaciers with idealised mass-balance profiles. Possible biases in scaling models may, in turn, lead to a low bias in the corresponding estimates of the long-term sea-level rise contribution from shrinking mountain glaciers. On a multidecadal scale, a faster response due to shorter response times in the scaling model can compensate for the effects of smaller climate sensitivities to some extent. However, the low biases in scaling-model-derived changes in glacier area and volume are likely to become apparent over longer timescales of multiple centuries. The linear-response model presented above could potentially be useful in predicting the long-term global glacier change and sea-level rise due to its accuracy and numerical efficiency.

The glacier model codes are available in the repository:

The supplement related to this article is available online at:

AB designed the study, did the theoretical analysis, and wrote the paper. AJ and DP wrote the codes. AJ, DP, and AB ran the simulations. All three authors contributed to the analysis of the simulated data and discussions.

The authors declare that they have no conflict of interest.

The authors acknowledge valuable inputs from reviewer Eviatar Bach, the anonymous reviewer, and editor Valentina Radić. Deepak Suryavanshi has contributed to the initial development of the SIA code.

This research has been supported by the Ministry of Earth Sciences, Government of India (grant nos. MoES/PAMC/H&C/80/2016-PC-II and MoES/PAMC/H&C/79/2016-PC-II).

This paper was edited by Valentina Radic and reviewed by Eviatar Bach and one anonymous referee.