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# Committed sea-level rise for the next century from Greenland ice sheet dynamics during the past decade

Edited by Hans-Joachim Schellnhuber, Potsdam Institute for Climate Impact Research, Potsdam, Germany, and approved April 19, 2011 (received for review November 22, 2010)

## Abstract

We use a three-dimensional, higher-order ice flow model and a realistic initial condition to simulate dynamic perturbations to the Greenland ice sheet during the last decade and to assess their contribution to sea level by 2100. Starting from our initial condition, we apply a time series of observationally constrained dynamic perturbations at the marine termini of Greenland’s three largest outlet glaciers, Jakobshavn Isbræ, Helheim Glacier, and Kangerdlugssuaq Glacier. The initial and long-term diffusive thinning within each glacier catchment is then integrated spatially and temporally to calculate a minimum sea-level contribution of approximately 1 ± 0.4 mm from these three glaciers by 2100. Based on scaling arguments, we extend our modeling to all of Greenland and estimate a minimum dynamic sea-level contribution of approximately 6 ± 2 mm by 2100. This estimate of committed sea-level rise is a minimum because it ignores mass loss due to future changes in ice sheet dynamics or surface mass balance. Importantly, > 75% of this value is from the long-term, diffusive response of the ice sheet, suggesting that the majority of sea-level rise from Greenland dynamics during the past decade is yet to come. Assuming similar and recurring forcing in future decades and a self-similar ice dynamical response, we estimate an upper bound of 45 mm of sea-level rise from Greenland dynamics by 2100. These estimates are constrained by recent observations of dynamic mass loss in Greenland and by realistic model behavior that accounts for both the long-term cumulative mass loss and its decay following episodic boundary forcing.

In its Fourth Assessment Report (AR4), the Intergovernmental Panel on Climate Change (IPCC) estimated a rise in global sea levels of 0.18–0.59 m by 2100, but the report explicitly avoided accounting for ice dynamical effects because of a perceived limit in understanding at that time (1). This failing was, in part, due to oversimplified numerical models of ice flow and the glaciological and climate modeling communities have responded with efforts toward developing “next generation” numerical ice sheet models with improved predictive skill (2, 3). However, the list of necessary model improvements is nontrivial (3) and more reliable sea-level projections from next generation models are still likely to be several years away.

Since AR4, there have been several efforts to make a reasonable assessment of the effects of ice dynamics on future sea-level rise (SLR). These efforts largely fall into two categories, those that extrapolate current trends in dynamic mass loss into the future (e.g., 4, 5) and those based on semiempirical models (e.g., 6, 7). For the latter, although there is no explicit accounting for ice dynamics, the historically constrained response of ice sheets to temperature forcing is assumed to include both surface mass balance (SMB) and ice dynamical effects. Both methods lead to estimates for total SLR (SMB plus dynamics) by 2100 that are significantly larger than the IPCC AR4 estimates,* which account primarily for changes in SMB. However, both of these methods have their inherent limitations. Extrapolation of current trends in dynamic mass loss is problematic because periods of significant dynamic mass loss may be episodic in nature, making it unclear if current trends are representative of the future. Similarly, semiempirical models rely on historical relationships between globally averaged temperatures and ice sheet volumes and it is unclear if these relationships will hold in the future, particularly if geographically local phenomena are important components or drivers of dynamic mass loss.

Indeed, the latter appears to be the case for the majority of dynamic mass loss from the Greenland Ice Sheet (GIS) during the past decade.^{†} Whole ice sheet and basin scale studies indicate that the GIS was loosing mass at an accelerating rate during the late 1990s and early to mid-2000s (8, 9). Between 2000 and 2008, this imbalance was split approximately equally between losses as a result of surface melting and dynamic thinning (8, 10). The majority of the dynamic thinning is thought to be due to outlet glacier acceleration during the late 1990s and early to mid-2000s (11–20). Although the cause for this thinning has been debated (17, 18, 21), there is strong evidence that changes in the balance of forces at the glaciers’ marine termini are responsible (13, 14, 17–20, 22) and that these changes were indirectly triggered by relatively warm ocean waters (15, 20, 22). In most cases speed-up events have been short-lived (13, 16, 20) and their long-term impact on ice sheet mass balance and SLR remains unclear.

Here, we make a first effort toward assessing the impact of GIS dynamics on future SLR using an appropriate thermomechanical, three-dimensional, higher-order ice flow model and a realistic initial condition of the GIS. Because the higher-order momentum formulation accounts for horizontal stress gradients, we are able to simulate the flow of outlet glaciers in a realistic way, and following perturbations at the ice sheet margin, account for the transfer of those perturbations into the ice sheet interior. We apply the model to perturbation experiments that mimic the dynamic changes observed on Greenland’s three largest outlet glaciers, Jakobshavn Isbræ (JI), Helheim Glacier (HG), and Kangerdlugssuaq Glacier (KG) (Fig. 1) during the last decade. Following the perturbations, we run the model forward in time to estimate the cumulative dynamic thinning in these basins from 2000–2100.

An ice sheet continues responding to a dynamic perturbation long after that perturbation ceases and our model is well-suited to capture this behavior. By applying only well-constrained dynamic perturbations from the past decade in Greenland, we obtain minimum estimates for future mass loss due to dynamics, and thus minimum estimates for Greenland’s dynamic contribution to future SLR. Following the concept of committed climate change (23), we refer to this minimum estimate as the committed sea-level rise (i.e., the long-term mass loss from an ice sheet in response to past perturbations). We show how recent observations and the modeling conducted here can be combined to estimate the committed future SLR from all dynamic thinning in Greenland during the past decade. Finally, we use our model results and some reasonable assumptions about future dynamic forcing and response in Greenland to make an upper-bound estimate for Greenland’s dynamic contribution to SLR during the next century.

## Results

### Flow Modeling.

Our three-dimensional ice sheet model solves the first-order momentum balance equations and the advective–diffusive temperature equation for ice sheets (24, 25). We use our model, 5-km resolution ice sheet geometry (26) and balance-velocity fields (27), and a tuning procedure to derive a realistic, steady-state initial condition for the GIS. At the fronts of JI, HG, and KG, we apply a stress boundary condition appropriate for ice in contact with seawater adjusted so that steady-state model fluxes agree with flux estimates from the mid-1990s (8), prior to the initiation of large perturbations at the termini of these glaciers (Table 1). The balance velocities and steady-state, depth-averaged model velocities are shown in Fig. 1. Root-mean-square differences for the entire GIS are 38 m y^{-1} and 3–8 m y^{-1} for the JI, HG, and KG basins (Fig. S1). The flow model, boundary conditions, tuning procedure, and applied perturbations are discussed in further detail in *Methods*, *SI Text*, and Fig. S2.

We perturb the initial, steady-state ice sheet by altering the stress boundary condition over time at the fronts of JI, HG, and KG, so that modeled and observed flux changes are in agreement. Physically, the perturbations can be interpreted as a reduction in resistive stress near the glacier termini resulting from, e.g., the loss of a floating ice tongue (12, 15, 19, 22, 28), or from an increase in the longitudinal strain-rate near the glacier front following a retreat into deeper water (13, 14). The resulting instantaneous acceleration of ice at the glacier front propagates inland as a diffusive wave of thinning. When integrated in time and space, this thinning represents the committed future SLR from GIS outlet glacier dynamics during the past decade. For HG and KG, we continue to alter the applied perturbation for 2–3 y after the initial perturbation to match observed discharge changes (13). The modeled and observed discharge anomalies^{‡} for HG and KG are shown in Fig. 2*A*. For JI, the applied perturbation is simpler for two reasons. First, a published record of its discharge since it began accelerating in the mid to late 1990s (12) does not exist. Second, and more importantly, JI has undergone near continuous acceleration and retreat (29) since that time for reasons that are poorly understood. Because the resolution, physics, and simplified perturbations applied in the current model cannot recreate this complex behavior we apply a one-time, step-function to the stress boundary condition for JI, which approximates the initial doubling in discharge (see Table 1 and Fig. 2*A*) observed on JI from the mid to late 1990s (8, 12, 29). Although the model cannot fully mimic the observed behavior of JI with this simplified perturbation, the simulations of JI do provide a useful comparison between the model behavior and the observations, as discussed further below.

### Comparison with Observations.

To demonstrate that the perturbed model response is reasonable we compare modeled and observed profiles of the thinning rate in the JI, HG, and KG drainage basins. Our goal is not for the model to accurately match the observed profiles at any particular point in time, as such a comparison is beyond the model resolution and the simplifying assumptions made here. Rather, we aim to demonstrate that the perturbed model response—the magnitude and spatial pattern of thinning—is reasonable when compared to the observed response. Fig. 3 shows observed thinning rates calculated from differencing satellite laser altimetry- and satellite imagery-derived digital elevation models (30). From observations (13, 15, 20), we assume the trigger for acceleration and thinning of HG and KG occurred between 2003 and 2005, and thus we compare our modeled thinning rates to observations from 2004 to 2005. For both HG and KG, the modeled magnitude and spatial pattern of thinning are similar to the observations except that the observations show larger thinning rates farther inland. We attribute this difference primarily to the fact that deep bedrock troughs, which focus and channel the thinning inland (13, 30), are absent from the model bedrock topography. Thus, the pattern of modeled thinning is more radial than observed. Importantly, Fig. 3 shows that the model does not overestimate the amount of thinning relative to observations. As such, estimates of dynamic thinning will be biased toward zero relative to the observations.

For reasons discussed above, a comparison between the model and observations for JI is more difficult. In particular, we expect to significantly underestimate the observed thinning because of our simplified, one-time perturbation. Further, the initial perturbation applied to JI is based on discrete observations from 1996 to 2000 (8), but the earliest profile of inland thinning for JI is from 2003 to 2005, 6–8 y after the model perturbation is applied. Over this time period, the modeled perturbation has decayed significantly relative to reality and the match between modeled and observed thinning is poor; the maximum observed thinning on JI from 2003 to 2005 is approximately an order-of-magnitude larger than that predicted by the model. However, if we scale the modeled thinning rates by the observed, maximum thinning rate, we find that the observed and (scaled) modeled thinning profiles are remarkably similar (Fig. 3). We interpret this similarity as indicating that our model captures the essential long-term, spatial response of thinning on JI, while underpredicting its magnitude because of the simplified perturbation.

### Committed Sea-Level Rise from JI, HG, and KG.

Following the initial perturbations to JI, HG, and KG, we step the model forward in time for 100 y. At each timestep we spatially integrate modeled thinning rates within each basin, which we assume contribute directly to the rate of SLR. The discharge anomaly (the cause for the thinning) and the corresponding rate of SLR for the first 10 y of the simulation are shown in Fig. 2. From the modeled rate of SLR curves we calculate cumulative SLR curves (Fig. 4*A*). By 2100 the combined, cumulative SLR from JI, HG, and KG is 1.1 ± 0.35 mm (the uncertainty estimate is discussed further below). We stress that this estimate is a conservative, minimum estimate for two reasons. First, we have not fully accounted for the continued acceleration and thinning of JI, in which case the flux anomalies for JI in Fig. 2*A* are too small. Second, we have only considered dynamic thinning resulting from perturbations on these glaciers during the past decade. No accounting is made for perturbations on other outlet glaciers or for future perturbations that might result in additional thinning. Thus, these simulations provide estimates for the sea-level contributions from JI, HG, and KG over the next century in response to forcing during the past decade.

### Committed Sea-Level Rise from all of Greenland.

From the rate of SLR and cumulative SLR curves in Fig. 2*B* and Fig. 4*A* we make two observations. First, if we attribute all of the SLR for the three years^{§} following the perturbation to the perturbation itself then ≥75% of the total SLR over 100 years results from long-term, diffusive thinning of the ice sheet. Second, we note that the exact timing of the perturbation applied to JI vs. HG and KG, makes little difference to the long-term, cumulative SLR (Fig. 4*A*); for multiple perturbations within a 10 year time period the cumulative SLR after 20 years is similar to if those perturbations occurred simultaneously.

We now propose a simple conceptual model for scaling the cumulative SLR estimates from the individual outlet glaciers modeled here to the entire GIS.^{¶} The scaling model relies on the following three assumptions: (*i*) the majority of the dynamic thinning in Greenland over the past decade occurred as a result of dynamic changes on outlet glaciers and as a result of perturbations like those applied in the modeling reported on here (8, 12–20, 22, 30); (*ii*) the time span over which those perturbations were initiated was short relative to some longer time period of interest with respect to future SLR (8, 12–20, 22, 30) (e.g., 10 vs. 100 y); and (*iii*) the modeling conducted here provides a reasonable estimate for the mean, long-term dynamic response of GIS outlet glaciers to perturbations initiated at their marine margins (e.g., Fig. 3).

Based on the modeling conducted here, the rate of dynamic SLR from individual outlet glaciers (e.g., Fig. 2*B*) can be fit to a function of the form *R*(*t*) = *rf*(*t*), where *r* is the initial rate of SLR from the perturbation and *f*(*t*) is a time decaying function describing the normalized rate of SLR. The cumulative SLR as a function of time is then given by [1]where *F*(*t*) gives the normalized, cumulative SLR (in units of years); multiplying *F*(*t*) by an initial rate of SLR *r* gives the cumulative SLR as a function of time. To extend Eq. **1** to multiple outlet glaciers, recall that the long-term, cumulative SLR is insensitive to small offsets in the timing of the initial outlet glacier perturbations (e.g., Fig. 4*A*) and assume that dynamic thinning was triggered simultaneously on *n* of Greenland’s outlet glaciers. At some later time, the total, cumulative SLR from those *n* perturbations is estimated by [2]If is a representative mean for the individual *F*(*t*) in Eq. **2**, such that , then [3]Assumptions (*i*) and (*ii*) from this conceptual model are inherent in the summation term on the right-hand side of Eq. **3** and in the simplification . Assumption (*iii*) from our conceptual model is that can be estimated from the modeling discussed above. We take to be the mean of the normalized, cumulative SLR curves modeled here noting that, over a 100 y time period, *F*(*t*) for the individual outlet glaciers vary from the mean by no more than ± 35%. We take this variation with respect to the mean as an estimate for the uncertainty introduced by assuming that (see additional discussion in *SI Text* and Fig. S3).

Although we do not have a direct way to estimate the first term on the right-hand side of Eq. **3** we can estimate its mean value over the time period of interest here from observations. Recent estimates (10) give a mean SLR of 0.46 mm y^{-1} from the GIS from 2000 to 2008, split equally between surface mass balance and ice dynamics (8, 10). We constrain the first term on the right-hand side of Eq. **3** by requiring that the mean rate of SLR predicted from the model for that same time period is equal to 0.23 mm y^{-1}. Conceptually, our modeled, cumulative mass discharge for Greenland over the last decade is scaled so that it agrees (on average) with that from (10) (see figure 2A in ref. 10). In future decades this scaled discharge decays according to our model. The resulting cumulative SLR estimate is shown in Fig. 4*B*. By 2100, the lower-bound estimate for SLR from GIS dynamics during the past decade is 5.8 ± 2.1 mm.

## Discussion and Conclusions

From perturbation experiments with a higher-order ice flow model we estimate a minimum, committed dynamic sea-level contribution from JI, HG, and KG of 1.1 ± 0.35 mm by 2100. We stress the minimum nature of this estimate, which accounts only for ice sheet thinning on these outlets as a result of known and well quantified past perturbations and their resulting long-term, diffusive response. Based on support from numerous observational and modeling studies, we scale these results to the entire GIS for a minimum SLR of 5.8 ± 2.1 mm (from dynamics) by 2100. When added to an estimate for the cumulative SLR (∼40 mm) from changes in SMB following a midrange future emissions scenario,^{∥} the total cumulative SLR increases to approximately 46 mm, with 13% resulting from dynamic perturbations during the past decade.

Of the total estimated dynamic SLR by 2100, ≥75% is due to the long-term, diffusive response of the ice sheet. Thus, the majority of the SLR from perturbations to Greenland’s outlet glaciers during the past decade is yet to come. Put another way, the hundred-year mean rate of SLR from dynamic perturbations amounts to approximately 5% of the initial rate; an initial perturbation giving 1 mm y^{-1} of SLR will contribute a hundred-year mean rate of 0.05 mm y^{-1}. Given some initial rate of SLR associated with future dynamic perturbations, this order-of-magnitude estimate provides a means by which the additional, diffusive contribution to SLR may be estimated.

When compared to other recent estimates of SLR from GIS dynamics by 2100 we find that our estimate is approximately 2–12% of that from semiempirical models (6, 7) and approximately 5–25% of that based on the extrapolation of currently observed trends in dynamics (4, 5) (see *Methods* and *SI Text* for more discussion). This comparison is not straightforward to interpret; our method explicitly accounts for long-term SLR from dynamical changes that have already taken place but makes no attempt to account for future dynamical changes, whereas the methods we are comparing to do exactly the opposite. Nevertheless, our estimate is not an insignificant fraction of these other estimates and, assuming the methodologies behind these other estimates are sound, we conclude that they may underestimate future SLR from dynamics by as much as 25%. This additional SLR is the result of perturbations that have already taken place and are thus already locked into the long-term evolution of the present-day ice sheet.

The modeling conducted here and some reasonable assumptions can be used to make approximate upper-bound estimates for future SLR from GIS dynamics, without accounting for future dynamical changes explicitly. As discussed above, numerous observations indicate that the trigger for the majority of dynamic thinning in Greenland during the last decade was episodic in nature, as the result of incursions of relatively warm ocean waters. By assuming that similar perturbations occur at regular intervals over the next century and that the ice sheet responds in a similar manner, we can repeatedly combine (sum) the cumulative SLR curve from Fig. 4*B* to arrive at additional estimates for SLR by 2100. For example, if perturbations like those during the last decade recur every 50, 20, or 10 y during the next 100 y, we estimate a cumulative SLR from GIS dynamics by 2100 of approximately 10, 25, and 45 mm, respectively (see *Methods* and Figs. S4–S6). This range of periodic forcing is reasonable with respect to natural modes of climate variability (e.g., the Atlantic multidecadal oscillation and the North Atlantic oscillation) that have been implicated in causing the warm ocean water incursions responsible for outlet glacier acceleration during the past decade (e.g., 15, 22). The upper-bound estimate of approximately 45 mm allows for the maximum influence of GIS dynamics on SLR while also accounting for the episodic and decaying nature of the causative perturbations (*Methods*). Although this upper-bound estimate relies on extrapolation of current observations of dynamic imbalance (as in refs. 4 and 5) it does so in a way that takes advantage of realistic, prognostic model behavior. Thus, if we assume that the previous decade of dynamic mass loss in Greenland is representative of the future, then we should expect no more than 45 mm of SLR from dynamics by 2100. For comparison, this estimate is approximately one half of the upper-bound estimate for the contribution from GIS dynamics from ref. 5. Addition of the estimated 40 mm of SLR from changes in SMB by 2100 (31) would result in a total SLR from Greenland of 85 mm by 2100.

Lastly, we emphasize that our modeling implicitly assumes no change in the magnitude or spatial distribution of basal sliding parameters in the future. For example, if basal sliding becomes more widespread on the GIS in the future, then both the lower- and upper-bound SLR estimates presented here are biased low.

## Methods

### Three-Dimensional, Higher-Order Ice Sheet Model.

Our three-dimensional, dynamic ice flow model (25) obeys conservation of linear momentum according to the first-order approximation to the nonlinear Stokes flow equations (24, 32–34). The model participated in the ice sheet model intercomparison project for higher-order models (35) and model output for tests A–E fall within one standard deviation of the mean defined by all participating models of the same type. Conservation of energy is expressed through the standard nonsteady, advective–diffusive heat equation for ice sheets (24, 36) with appropriate adjustments to account for higher-order dynamical terms (e.g., internal- and boundary-friction energy dissipation terms include contributions from horizontal stress- and strain-rate gradients). For geometric evolution of the ice sheet, mass conservation is enforced using incremental remapping (37), a conservative, explicit, high-order upwinding scheme. Timesteps of 10 y were used during thermomechanical evolution in the tuning procedure discussed below (for which the geometry was held constant). For perturbation experiments with both thermal and geometrical evolution, timesteps of 0.1 y were used. In both cases, results were found to be insensitive to a halving of the timestep.

### Initial Conditions.

We tune our ice flow model to match a target velocity field using the following procedure: (*i*) With no-slip and zero-flux basal and lateral boundary conditions, respectively, we step the model forward in time while holding the ice sheet geometry steady until the temperature, velocity, and effective viscosity fields come to equilibrium. A map of mean annual surface temperatures (38) is applied as the surface boundary condition for the heat balance model. At the base of the ice, we specify a constant and steady geothermal flux of 55 mW m^{-2}. (*ii*) Using the equilibrated velocity fields from the first step, we calculate the depth-averaged speed and the basal traction magnitude ‖**t**‖ at each *x*, *y* point within the horizontal extent of the ice sheet. The depth-averaged model speed is subtracted from the depth-averaged speed of the target balance velocity field (27) to determine the sliding speed ‖**u**_{b}‖ needed to bring the target and modeled speeds into agreement. We then use the sliding law ‖**t**‖ = *β*‖**u**_{b}‖ to deduce the distribution of the traction parameter, *β*(*x*,*y*) = ‖**u**_{b}‖/‖**t**‖, required to enforce sliding at the desired speed ‖**u**_{b}‖. (*iii*) Using the traction parameter field determined in the second step, allowing for basal sliding, and applying a boundary condition for floating ice (discussed below) at the fronts of JI, HG, and KG, the model is again run to equilibrium. (*iv*) Steps (*ii*) and (*iii*) are iterated on with underrelaxation of changes to β to bring model velocities into closer agreement with their desired target values. The final map of the traction parameter is held steady at its initial value for the duration of the time-dependent perturbation experiments. Additional discussion of the tuning procedure can be found in *SI Text*.

### Outlet Glacier Boundary Conditions.

We first adjust the lateral boundary conditions in the model so that the initial, steady-state flux from JI, HG, and KG is similar to observed values from the mid-1990s (8), which we take as representing steady state in the modeling conducted here. At the marine termini of these outlets we apply the standard boundary condition for ice in contact with ocean water, where **T** is the stress tensor, **n** = **n**(*x*,*y*) is the outward pointing normal vector to the marine terminating ice front, *g* is the acceleration due to gravity, *H* is the ice thickness, *ρ* and *ρ*_{w} are the densities of ice and ocean water, respectively, and *C* is a tuning constant (e.g., for *C* > 1 ice flux through the boundary increases). The right-hand side of the equation is the depth-average of the depth-integrated normal stress at the ice shelf front, applied to each vertical level in the ice column if that column is in contact with the ocean. For *C* = 1, the boundary condition has been verified** by comparing the modeled velocity field to those from the well-established European Ice Sheet Modeling Initiative Ross Ice Shelf benchmark (39). The value of *C*, used here for model tuning as in previous outlet glacier modeling studies (e.g., 14), is composed of two multiplicative factors, *C* = *C*_{0}*C*_{1}, where *C*_{0} is a steady part and *C*_{1} = *C*_{1}(*t*) is a time-dependent part. For each outlet glacier under consideration, *C*_{0} is initially adjusted to achieve the desired steady-state flux in step (*iii*) of the tuning procedure discussed above (i.e., to give the values shown in Table 1), while *C*_{1} is held constant at a value of 1. From this initial steady state, a perturbation is introduced by holding *C*_{0} at its initial value while changing *C*_{1} over time. At any timestep, the applied value of *C*_{1} is that required to enforce agreement between modeled and observed outlet glacier flux anomalies (e.g., Fig. 2*A*). See *SI Text* for additional discussion.

### SMB Contribution to SLR by 2100.

The estimate for Greenland’s contribution to SLR by 2100 as a result of changes in SMB is taken from ref. 31, in which an SMB model for the GIS was calibrated to precipitation and temperature anomaly data for the period 1970–1999. The calibrated SMB model was then applied to temperature and precipitation anomaly data from 23 atmospheric and oceanic general circulation models participating in the IPCC AR4 (1) under the midrange, A1B future emissions scenario. The median SMB prediction was used for the value of the cumulative SLR by 2100 quoted here (for the 25 and 75% quartiles, the predicted cumulative SLR by 2100 is approximately 32 and 45 mm).

### Comparison to Other Estimates of SLR by 2100.

We compare our estimate for the committed SLR from ice dynamics by 2100 to other existing estimates in the literature. In most cases, a direct comparison is difficult because (*i*) existing estimates make assumptions about the future influence of ice sheet dynamics, and (*ii*) existing estimates are presented in terms of global SLR, rather than just that from GIS dynamics. In the latter case, we make reasonable estimates for how to partition the total SLR based on values in the published literature. This partitioning suggests that the total SLR contribution from the GIS can be split equally between that due to surface mass balance and that due to ice dynamics and that between 10 and 15% of the global SLR can be assigned to GIS dynamics. For estimates of dynamic SLR from the GIS by the year 2100, we find that our estimate is approximately 2–12% of that estimated using empirical models (6, 7) and approximately 5–25% of that estimated by extrapolation of current observations and trends (4, 5). Our partitioning of published SLR estimates and the comparison between those estimates and ours are discussed in more detail in *SI Text*.

### Upper-Bound Estimates for SLR by 2100.

To make a reasonable upper-bound estimate for future SLR from GIS dynamics by 2100, we make the following simplifying assumptions. First, we assume that the dominant mechanism for dynamic mass loss in the future will be the same as that explored here (i.e., acceleration of marine terminating outlet glaciers following stress-balance perturbations at their termini). Second, we assume that similar perturbations to those during the past decade may occur at regular intervals during the next century. Third, we assume that the ice sheet responds to each of those future perturbations in a self-similar manner to that shown in Fig. 4*B*. For some given recurrence interval, the total SLR by 2100 is given by the cumulative sum of curves like that shown in Fig. 4*B* that have had their origins shifted forward along the time axis (and for which the portion of the shifted SLR curve extending beyond 2100 has been removed). For example, for a single additional perturbation occurring at 2050, the total SLR by 2100 is given by summing the curve shown in Fig. 4*B* with the same curve when its origin has been shifted to a start date of 2050. The individual (colored) and cumulative (black) SLR curves for this example are shown in Fig. S4. Additional examples for recurrence intervals of 20 and 10 y are shown in Figs. S5 and S6. Here, we take the estimate of approximately 45 mm SLR using a 10 y recurrence interval as an upper bound for dynamic SLR from the GIS by 2100. A more frequent recurrence interval cannot be ruled out but is not supported by oceanographic and glaciological observations over the past decade (the only time period for which we have reasonably good observations). Over this time period, observations (15, 17, 18, 20, 22) suggest essentially one period of oceanographic forcing and dynamic ice sheet response in Greenland.

## Acknowledgments

We thank two anonymous reviewers for comments that helped to improve the manuscript. This research was funded by a Los Alamos National Laboratory Director’s Postdoctoral Fellowship (to S.F.P.), a Natural Environment Research Council (London) National Center for earth Observation grant (to A.J.P.), and National Aeronautics and Space Administration Grant NNX08AQ83G (to I.M.H. and B.E.S.).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: sprice{at}lanl.gov.

Author contributions: S.F.P. and A.J.P. designed research; S.F.P. performed research; S.F.P., A.J.P., I.M.H., and B.E.S. contributed new reagents/analytic tools; S.F.P., A.J.P., I.M.H., and B.E.S. analyzed data; and S.F.P., A.J.P., I.M.H., and B.E.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1017313108/-/DCSupplemental.

↵

^{*}By 2100, SLR of 0.5–0.8 m (4, 5) and 0.35–1.6 m (6, 7).↵

^{†}We implicitly assume that the last decade covers the period from 1997 to 2007.↵

^{‡}Plotted discharge anomalies for the model are relative to the steady-state initial condition. For the HG and KG observations, they are relative to the year 2000 (13), and for JI the initial anomaly is relative to the mid to late 1990s (8, 12), at which time we assume that JI was near equilibrium.↵

^{§}From observations (13) and modeling (Fig. 2), we estimate 3 y as an approximate e-folding timescale for decay of the initial perturbations in outlet glacier flux.↵

^{¶}We take this approach rather than attempting to use our model because of the lack of necessary data (e.g., ice thickness and outlet glacier flux time series) available for other outlet glaciers in Greenland.↵

^{**}The chi-squared and maximum velocity for our model are 4,947 and 1,326 m y^{-1}, respectively, which compare well with results from other ice shelf models in the study (39).

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