We solve the sea-level equation to investigate the pattern of the gravitationally self-consistent sea-level variations (fingerprints) corresponding to modeled scenarios of future terrestrial ice melt. These were obtained from separate ice dynamics and surface mass balance models for the Greenland and Antarctic ice sheets and by a regionalized mass balance model for glaciers and ice caps. For our mid-range scenario, the ice melt component of total sea-level change attains its largest amplitude in the equatorial oceans, where we predict a cumulative sea-level rise of ~ 25 cm and rates of change close to 3 mm/yr from ice melt alone by 2100. According to our modeling, in low-elevation densely populated coastal zones, the gravitationally consistent sea-level variations due to continental ice loss will range between 50 and 150% of the global mean. This includes the effects of glacial-isostatic adjustment, which mostly contributes across the lateral forebulge regions in North America. While the mid range ocean-averaged elastic-gravitational sea-level variations compare with those associated with thermal expansion and ocean circulation, their combination shows a complex regional pattern, where the former component dominates in the Equatorial Pacific Ocean and the latter in the Arctic Ocean.
 The non-uniform effect of terrestrial ice melt (TIM) on relative sea-level (RSL) was recognized over a century ago [Woodward, 1888]. The modern theory [Farrell and Clark, 1976] has been further developed more recently to include, for example, changes in Earth rotation and shoreline migration [Milne and Mitrovica, 1998] and is generally termed the sea-level equation (SLE). This equation has been used to investigate the impact of idealized melt scenarios for Greenland and Antarctica and of observed volume changes for glaciers and ice caps [Mitrovica et al., 2001] and to examine the RSL pattern resulting from observed recent TIM [Bamber and Riva, 2010]. This latter study found that maxima occurred at low latitudes, in the Western Pacific in particular, and had a marked zonal gradient driven, primarily, by the dominant sources in both polar regions. To date, however, the SLE has not been used to examine the RSL pattern resulting from prognostic model predictions of future land ice melting nor to examine the relative importance of TIM and steric effects regionally.
 Here, we combine predictions from numerical models for the evolution of the Greenland (GrIS) and Antarctic (AIS) ice sheets with a regionalized model for glaciers and ice caps to investigate the gravitationally consistent signature of future TIM based on the SRES (Special Report on Emissions Scenarios) A1B scenario and the fully coupled atmosphere-ocean GCM (Global Climate Model) ECHAM5/MPI-OM (hereafter referred to as ECHAM5) [Meehl et al., 2007]. Steric and ocean dynamic processes are also non-uniformly distributed, and we examine the relative importance of these with respect to TIM, using the same GCM and greenhouse gas forcing.
2 Data Processing and Methods
 For the Greenland and Antarctic ice sheets, volume changes are caused by both ice dynamics and surface mass balance (SMB). SMB is driven by accumulation and surface melting in Greenland and just the former in Antarctica. These fields were obtained from two regional climate models (RCMs): MAR for Greenland [Fettweis, 2007] and RACMO for Antarctica [Lenaerts et al., 2012]. Both were forced by ECHAM5 under scenario A1B, projecting a mid-range global temperature change [Solomon et al., 2007]. In a model inter-comparison [Reichler and Kim, 2008] ECHAM5 performed well in reproducing a suite of atmospheric and oceanic variables. Annual SMB anomalies were calculated with respect to the baseline period 1989–2008 and re-gridded to a spatial resolution of 1°. The period 1992–2000 was appended to the scenarios using re-analysis data (ERA-Interim for Greenland, ERA–40 for Antarctica) and downscaled using the same RCMs. Ice dynamics, represented as grounding line flux anomalies with respect to 1992, were taken from ice sheet model simulations using atmospheric and oceanic forcing from the same RCMs. SMB and ice dynamics sources are shown in Figure 1 in Auxiliary Material.
 For Antarctica, from an ensemble of 81 model runs, two simulations were selected: a “mid–range” (MR) scenario contributing ~ 7 cm of mean sea-level rise (SLR) by 2100 and a “high-end” (HE) scenario contributing ~ 30 cm. The accelerations in ice loss required to obtain these contributions are approximately 2.5 and 19.1 Gt yr−2 for the MR and HE scenarios, respectively. To put this into perspective, the estimated observed acceleration over 1992–2010 is 14.5 Gt yr−2 [Rignotet al., 2011b]. Only volume changes of ice above flotation (i.e., contributing to SLR) are taken into account here. Antarctica is divided into 15 major drainage basins, and in each basin, the volume change is evenly distributed over all 1° grid cells with an average velocity [Rignot et al., 2011a] exceeding 50 m/yr, corresponding to an approximate coastal zone that is affected by ice dynamics. For Greenland, flow-line model simulations were carried out for three outlet glaciers—Jakobshavn isbræ, Petermann, and Helheim glaciers—and upscaled to obtain total volume changes due to calving for three sectors of the GrIS including all outlet glaciers using a linear relation between flux anomaly and glacier width [Alley et al., 2008]. For the MR scenario, the model was calibrated against present-day observations. The HE scenario resulted from a lowering of the bedrock by its two-sigma error estimate.
 Within each sector, volume changes are distributed evenly over all grid cells with average velocity [Moon et al., 2012] exceeding 100 m/yr. We assume the ice sheets were close to balance until about 1992 [Nerem and Wahr, 2011], when we prescribe the calving flux anomalies to be zero, and interpolate linearly between 1992 and the initial scenario values in 2001. For Greenland, volume changes in the MR and HE scenarios contribute ~ 4 and ~ 6 cm SLR by 2100, respectively, which is small compared to the present day trends in mass imbalance. This is due, primarily, to the relatively moderate warming response in the GCM resulting in a concomitant modest reduction in SMB.
 Volume changes for glaciers and ice caps (GIC) (Figure 1c in Auxiliary Material) are derived from a regionalized glacier mass balance model that uses temperature and precipitation anomalies for 19 glacierized sectors globally. The same GCM forcing was used as for the ice sheets and steric response. The sensitivities of the regional glacier responses were calibrated using automatic weather station data for 80 benchmark glaciers [Giesen and Oerlemans, 2012]. Whereas the MR scenario was produced by a model run using calibrated parameters, for the HE scenario the model was run with perturbed parameters reflecting increased longwave radiation and turbulent fluxes, decreased albedo, increased initial volume, and climate forcing from a different but nearby RCM grid cell that would produce 75% higher future sea level rise.
 In the above, it should be noted that the two scenarios relate only to differences in their ice dynamic response. The same atmospheric warming scenario (A1B) is used for the SMB response in both the MR and HE cases. Details of the numerical method employed to solve the SLE and of the Glacial Isostatic adjustment (GIA) model adopted are given in section 2.1 of the Auxiliary Material.
 Figure 1 shows the TIM component of RSL expected for the year 2100, relative to 1992, for the MR (a) and HE (b) scenarios. The maps clearly indicate that a SLR is expected almost everywhere, except in the near field of areas of large TIM: predominantly Greenland and West Antarctica. The geometry of the RSL variation is a consequence of the elastic regional uplift caused by ice un-loading and the decreased gravitational force between the depleting ice and the surrounding ocean. With increasing distance, the amplitude of vertical deformation decreases and SLR dominates the global pattern, reaching values in excess of the eustatic amplitude generally at latitudes below about 30° (the eustatic SLR represents the spatially uniform response for a rigid, non-self-gravitating Earth and is obtained by ocean-averaging the fingerprints). The RSL patterns in Figure 1 qualitatively agree with Mitrovica et al. . The global pattern is affected by the irregular shape of the shorelines, but it is, however, fairly insensitive to the localized distribution of ice loss except in the near field of the sources [Bamber and Riva, 2010; Spada et al., 2012].
 The dominant localized sources of loss in both scenarios are from West Antarctica (mainly the Amundsen Sea sector) and the GrIS (Figure 1 in Auxiliary Material). Although the integrated GIC response is similar in magnitude to the AIS and larger than the GrIS, it is spread over a large part of the Earth's surface and has, therefore, a smaller localized effect on RSL. This explains the large sea-level fall (Figure 1) in the region surrounding Greenland and the Svalbard archipelago, and off the Antarctic Peninsula which, according to our computations, for the year 2100 will be subject to elastic uplift rates of ~ 10 mm/yr (MR scenario) and of ~ 25 mm/yr (HE) in response to ice un-loading. The sea-level fingerprints of Figure 1 are characterized by a distinct zonal pattern with a strong equatorial symmetry, which reflects the dipole pattern of the major concentrations of TIM in Figure 1 in Auxiliary Material. For both scenarios, the largest increases are expected in the equatorial oceans where SLR exceeds the eustatic value shown by the green contour. In these regions, the maxima are around 25% greater than eustatic. This RSL pattern is broadly similar to the present-day fingerprint due to TIM [Bamber and Riva, 2010]. Maps of the rates of sea-level variation expected for the year 2100 are shown in Figure 2 in Auxiliary Material.
 Inspection of Figure 1 indicates that the cumulative RSL along European coastlines does not exceed the eustatic value (this is also observed for the trends in Figure 2 in Auxiliary Material). This results specifically from mass loss from the GrIS and other Arctic GICs. The rate is close to eustatic in North America and largely above in Southeast Asia and Africa. A more quantitative analysis of future sea-level variations at coastal sites is shown in Figure 2. This shows the location of the Revised Local Reference (RLR) Permanent Service for Mean Sea Level (PSMSL) tide gauges (TG) and, in particular, those 23 sites obeying specific criteria useful for the assessment of secular SLR [Douglas, 1997]. Figure 2d shows 35 towns and cities in low-elevation (<10 m) densely populated coastal zones (LECZ), generally considered to be the most vulnerable to risks resulting from climate change [see, e.g., McGranahan et al. 2007]. The TG and LECZ form complementary sets, with a reasonably full geographical coverage globally.
 In Figures 2b and 2e, the sea–level fingerprints of Figure 1 are projected on the TG and LECZ, respectively. The spread of predictions associated with the elastic deformations and the gravitational effects are immediately evident. We obtain a sea level rise at all TG stations discussed here. For both scenarios, the maximum SLR is expected at Honolulu (Hawaii). The lowest values are predicted for Newlin (English Channel) and Quequen (South America) for the MR and HE scenarios, respectively. In particular, we note that Honolulu is located in the broad area in the Pacific Ocean where the sea-level fingerprint is expected to attain its largest (super-eustatic) amplitude (see Figure 1). According to Figure 2 in Auxiliary Material, here the rates of sea–level change will exceed 8 mm/yr during the second half of this century in the HE scenario. The SLR curves for the LECZ locations, shown in Figure 2e, are broadly similar to those pertaining to the TG. European cities (blue) will be less affected by future TIM, as a consequence of mass loss in the Arctic.
 For the MR scenario, in Figures 2c and 2f, we combine the TIM component of future sea-level variations with the GIA component (see Auxiliary Materials section) for year 2100. Since the future TIM is not considered in GIA modeling, the ocean-averaged GIA fingerprint vanishes. Nevertheless, its local amplitude can be significant (up to several mm/yr), especially in the surroundings of ice-covered regions at the Last Glacial Maximum. Figure 2c shows that the GIA component is particularly important (it amounts to 25% of the total SLR) for TG located along the North American West coast (e.g., San Francisco) and those in South East North America (e.g., Pensacola). These regions are (and will be, during the next century) still subject to the collapse of the isostatic peripheral bulge adjacent to the former Laurentian ice sheets, which acts to increase the sea-level variations associated with TIM. Results for cities and towns in the LECZ (Figure 2f) show a modest contribution of GIA for Asia, Africa, and Europe, while it is apparent that for US cities, from Miami to New York, GIA exacerbates the current TIM sea-level variations. For European cities, the expected total SLR is smaller than the eustatic values from TIM. Interestingly, the largest cumulative SLR is predicted for Venice, irrespective of the GIA component (note, however, that these estimates do not include steric and dynamic contributions).
 Here, the focus has been on the gravitationally consistent fingerprint of future terrestrial ice loss. For the melt scenarios used, the patterns of SLR are fixed and will only change significantly if the relative contributions of the AIS and Arctic ice masses change significantly. The pattern is a consequence of localized elastic uplift and changes to the geoid caused by mass redistribution. Thermal expansion and ocean circulation also have a non-uniform impact on the pattern of SLR [Yin et al., 2010]. For convenience, we will refer to these as the ocean response. It is, therefore, interesting to consider the relative importance of oceanic and TIM effects on the future pattern of SLR and to investigate where these effects may be compounded or possibly compensating. Slangen et al.  combined GCM model ensemble oceanic and TIM signatures using data from the IPCC AR4 simulations but with a crude estimate of future TIM, resulting in the ocean response being the main source of SLR. As is the case here, their TIM fluxes were not coupled to the AOGCM simulations.
 In this study, we use the ocean response signal from the ECHAM5 A1B simulation [Meehl et al., 2007] for consistency with the TIM forcing, but it should be noted that this was not done in a coupled experiment, which is beyond the scope of this study. Thus, the ocean response is consistent with the greenhouse gas forcing used but not with the TIM fluxes produced by the offline ice sheet and glacier models. For the HE scenario, in particular, the ocean response would likely be altered, due to the impact of the freshwater fluxes on the pattern of steric changes and ocean circulation, if carried out in a fully coupled experiment. For our MR scenario, the eustatic TIM contribution is 24 cm in terms of RSL, which is of similar magnitude to the ocean response for ECHAM5 A1B of 27 cm [Meehl et al., 2007]. For the HE scenario, the TIM contribution is 61 cm, which is approximately double the ocean signal. Thus, in the MR case, there will be areas where the ocean response is larger than TIM and vice versa but not for the HE scenario. The TIM and ocean fingerprints are considered in Figures 3 and 4, showing the total (TIM + ocean) contribution and the fraction of the total SLR due to TIM for both scenarios, respectively. Here, the TIM contribution is expressed in terms of RSL and does not include the GIA component of sea-level change since its importance is limited to formerly glaciated areas and is not an important fraction of the total SLR, even in the MR scenario (see Figure 2). In Figure 4, a fraction greater than 0.5 indicates that TIM is larger than the ocean response and vice versa.
 From the results in Figures 3 and 4, it is apparent that the Southern Ocean is dominated by the TIM signal, even for the MR scenario, because the ocean response is significantly below the global mean but this is also an area where TIM is less than eustatic (Figure 1) and thus a region that experiences considerably less than the mean total SLR response of 48 cm (i.e., TIM plus ocean response; see Figure 3). Across a large swathe of the Southern Ocean, the total SLR is close to zero and it is negative in the vicinity of the Antarctic Peninsula and West Antarctica. Conversely, the Arctic Ocean is a region where the ocean response is greater than the global mean while TIM is less (Figure 1). For the MR scenario, this results in a total SLR that is close to the global mean, except for the Chukchi Sea where it reaches almost a factor two more at about 80 cm (Figure 3). TIM dominates SLR across the Equatorial Pacific Ocean and into a large part of the Indian Ocean, which are all areas where the ocean response is close to, or less than, the global mean. The other region where TIM dominates for the MR scenario is in the vicinity of the Kuroshio Current. For the HE scenario, TIM is dominant everywhere except for a region around the Antarctic Peninsula, West Antarctica, and surprisingly, the Arctic Ocean again. For reasons discussed above, this conclusion is, however, tentative and should be confirmed using a coupled AOGCM forced with the TIM fluxes used here.
 Although the ocean response presented here is for just one model and one SRES scenario, the pattern appears to be relatively robust across the ensemble of GCMS used in the IPCC AR4 [Meehl et al., 2007]. It is also the case that as long as the mass loss is dominated by the WAIS and Arctic TIM, then the areas experiencing the largest SLR from ice melt will remain roughly the same. Thus, we conclude that TIM will likely be of critical importance to regional SLR in the Equatorial Pacific Ocean and, in particular, around Western Australia, Oceania, and the small Atolls and islands in this region. We also conclude that SLR in the Arctic Ocean will be greater than the global mean but in this case dominated by ocean processes with relatively little impact from TIM.
 Work funded by the European Commission's 7th Framework Program through grant number 226375 (ice2sea contribution number 104). We thank David Vaughan, Erik Ivins, Rianne Giessen, and an anonymous reviewer for helpful suggestions. All figures have been drawn using GMT [Wessel and Smith, 1998].