On the attribution of industrial-era glacier mass loss to anthropogenic climate change

Abstract. Around the world, small ice caps and glaciers have been losing mass and retreating since the start of the industrial era. Estimates are that this has contributed approximately 30 % of the observed sea-level rise over the same period. It is important to understand the relative importance of natural and anthropogenic components of this mass loss. One recent study concluded that the best estimate of the magnitude of the anthropogenic mass loss over the industrial era was only 25 % of the total, implying a predominantly natural cause. Here we show that the anthropogenic fraction of the total mass loss of a given glacier depends only on the magnitudes and rates of the natural and anthropogenic components of climate change and on the glacier's response time. We consider climate change over the past millennium using synthetic scenarios, palaeoclimate reconstructions, numerical climate simulations, and instrumental observations. We use these climate histories to drive a glacier model that can represent a wide range of glacier response times, and we evaluate the magnitude of the anthropogenic mass loss relative to the observed mass loss. The slow cooling over the preceding millennium followed by the rapid anthropogenic warming of the industrial era means that, over the full range of response times for small ice caps and glaciers, the central estimate of the magnitude of the anthropogenic mass loss is essentially 100 % of the observed mass loss. The anthropogenic magnitude may exceed 100 % in the event that, without anthropogenic climate forcing, glaciers would otherwise have been gaining mass. Our results bring assessments of the attribution of glacier mass loss into alignment with assessments of others aspects of climate change, such as global-mean temperature. Furthermore, these results reinforce the scientific and public understanding of centennial-scale glacier retreat as an unambiguous consequence of human activity.


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On the attribution of industrial-era glacier mass loss to anthropogenic climate change: Supplementary material. S1. Alternative synthetic little ice ages.     Here we evaluate the impact of retaining the quadratic term in the mass loss: % $ & + ′ ( )+. Errors in cumulative mass loss from simply integrating the linear term will vary by glacier, depending on how much of its preindustrial area it has lost 20 (i.e., ′ ( ) cf. & ). They will be greatest for glaciers that have nearly vanished (or have already); however, this is not the norm in the global collection of length records (e.g., Leclercq et al., 2014). Many of the largest glaciers have retreated a small amount compared to their total area, in which case integrating ′ is a good approximation.
We demonstrate this with two limiting cases. In Fig. S5, we show a small glacier with = 33 yr subject to a gradual millennial 25 cooling, and then rapid anthropogenic warming (in the FULL) case, which causes the glacier to lose nearly all its area by 2020 (panel (b)). Panel (c) shows the yearly volume change, which is % & for the linear approximation (dashed) and ′ ( & + % ) including the evolving area. We assume a constant width . In this case, the absolute volume changes become small as the glacier area shrinks. The cumulative loss is shown in panel (d), and is overestimated by the linear approximation. In reality, the total volume is weighted towards the response to early forcing when the glacier was bigger. However, this has no effect on 30 the anthropogenic fraction of cumulative mass loss (panel (e)), because there is no natural component of mass loss in this scenario.
Accounting for the changing glacier area affects cumulative loss if a large preindustrial disequilibrium is assumed. Figure S6 shows the same analysis, but for a case similar to Figs. 4 and 5 (main text), where the long cooling anomaly naturally ends in 35 1850. In this case, !"#$ for cumulative volume loss rises more slowly when the changing glacier area is taken into account (panel e). Thus, the linear approximation would overestimate !"#$ . However, S7 shows the other limiting case, for a large glacier where ′ is small compared to & . Here, there is very little error associated with the linear approximation and essentially no effect on !"#$ for cumulative loss.

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We expect that the linear approximation captures the basic evolution !"#$ for cumulative mass loss for the purposes of these analyses. We emphasize again that these issues also only affect conclusions about fractional attribution if a large preindustrial disequilibrium is assumed (Fig. S6). As discussed in the main text, such an assumption is not consistent with estimates of regional-to-global climate history over the last few centuries. Global assessments of total mass loss and contributions to sea level would need to take the evolving glacier geometry into account. However, for the question of attribution of mass loss 45 since 1850, the analyses throughout this study show that the assumed climate history is clearly the more important issue. 7 Figure S6. Same as S5 but for a scenario where the natural cooling trend abruptly ends in 1850. Again, cumulative mass loss is lessened by the vanishing glacier area. In this case, the linear approximation also overestimates the anthropogenic mass loss compared to total mass loss, because later mass loss (predominantly anthropogenic) should in reality be weighted less. Note that the 60 anthropogenic fraction of yearly mass loss begins to dip down again as the glacier vanishes. This is due to the tiny residual imbalance in the counterfactual NAT case, which becomes a larger proportion of NAT/FULL as the glacier area shrinks to zero in the FULL case.
8 Figure S7. Same as Fig. S6 except for a very large glacier with = 100 yr, whose retreat is small compared to its total area. In this case, the linear approximation for cumulative mass loss holds up well (d), and does not affect the anthropogenic mass loss. As described in the main text, is less than 100% because of the assumed natural mass imbalance in 1850, which is slow to diminish 70 due to the 100-year response time.
(q) (p) Figure S9. As for Fig. S8, but zoomed in to the period 1850 to 2020. The CESM underestimates the observed warming, but still shows strongly negative mass balance with anthropogenic forcing included. The percentages inserted into the right-hand columns give the anthropogenic cumulative mass loss relative to the total cumulative mass loss between 1880 and 2020. Note these cumulative percentages will not, in general, be equal to the time average of !"#$ . The = 400 yr case shows a large anthropogenic cumulative mass-loss fraction because the CESM model has cooler temperatures extending into the 1950s, unlike the observations. Hence the cumulative mass loss implied by the CESM output in the FULLcase is small.   On the millennium timescale, precipitation trends in climate models are small (e.g., Fig. 5, S11), and mostly consistent with 85 white noise of internal, interannual variability (Parsons et al., 2017). Precipitation variability also has lower spatial coherence, meaning it will tend to average out at global scales. Environmental proxies provide some indication that more low-frequency variability in precipitation exists than in models (Dee et al., 2017). However, for the most part, precipitation acts as a noise maker for glacier mass balance, and there is no reason to suppose it would lead to widespread mass imbalance on centennial time scales. On scales where climate variability is spatially coherent, mass-balance variability and glacier-length variability 90 will also be coherent. In the global average much of this regional climate variability cancels out; and the response of massbalance and length to the global-scale external climate forcing will be more prominent (Huston et al., in press.) We relate mass-balance forcing to temperature and precipitation (Roe and Baker, 2014): where = & / , and = '() / ; is a melt-factor (which we take to be 0.65 m yr -1 K -1 ), and '() is the length of the 95 equilibrium glacier that experiences some melting during the year. From Eq. (A1), the precipitation anomaly equivalent to a temperature anomaly is: Taking a value of '() / #-#~0 .8, gives %~0 .5 [m yr -1 K -1 ] • ′. In other words, a precipitation anomaly of 0.5 m yr -1 is needed to offset a temperature anomaly of 1 K. Roughly, in terms of interannual variability, a typical standard deviation in melt-season temperature might be ~0.5 K (Roe et al., 2017), and a typical standard deviation in annual precipitation is ~15 to 20% of the 100 annual mean (say 0.5 to 4 m yr -1 , depending on the climatic setting). This means that, on interannual time scales, precipitation variability can be of comparable or greater importance than melt-season temperature for mass-balance variability (e.g., Medwedeff and Roe, 2017). However, whereas observed century-scale changes in temperature are 1 K or greater, precipitation changes are much smaller (a few percent per 1 K, e.g., Stocker et al., 2013). Thus, for the climate of the industrial-era, changes in temperature are generally much more important than changes in precipitation for affecting glacier mass balance.