Sea-level change does not occur uniformly. Dynamic ocean processes (Levermann et al 2005), changes to temperature and salinity (i.e. steric processes), and changes in the Earth’s rotation and gravitational field associated with water-mass redistribution (e.g. land-ice melt; Mitrovica et al 2011), as well as glacial isostatic adjustment (GIA; Farrell and Clark 1976) and other drivers of vertical land motion cause local relative sea levels to differ from the global mean. We model local relative sea level using the K14 framework, but make modifications to accommodate the stratification of atmosphere–ocean general circulation models (AOGCMs) and RCPs into groups that meet GMST stabilization targets (see section 2.2). AOGCM output from the Coupled Model Intercomparison Project (CMIP) Phase 5 archive (Taylor et al 2012) forced with the RCPs (to 2100) and their extensions (to 2300) are used directly for global mean thermal expansion (TE) and local ocean dynamics, and as a driver of a surface mass balance (SMB) model of glaciers and ice caps (GICs; Marzeion et al 2012). Antarctic ice sheet (AIS) and the Greenland ice sheet (GIS) contributions are estimated using a combination of the Intergovernmental Panel on Climate Change’s (IPCC) Fifth Assessment Report (AR5) projections of ice sheet dynamics and SMB (table 13.5 in Church et al 2013) and expert elicitation of total ice sheet mass loss from Bamber and Aspinall (2013). As in AR5, ice sheet SMB contributions are represented as being dependent on the forcing scenario, while ice sheet dynamics are not. A spatiotemporal Gaussian process regression model is used with tide gauge data to estimate the long-term contribution from non-climatic factors such as tectonics, GIA, delta processes (e.g. sediment compaction), and human-induced subsidence. Changes in the rate of human-induced subsidence are not considered. Global mean land water storage effects are modeled using relationships between population and groundwater removal and impoundment (Kopp et al 2014). To generate probability distributions of global and local mean sea level for each GMST scenario at tide gauges (table S-1), we use 10 000 Latin hypercube samples of probability distributions of individual sea level component contributions.

The RCP-driven experiments in the CMIP5 archive are not designed to inform the assessment of climate impacts from incremental temperature changes. As such, we construct alternative ensembles for 1.5 °C, 2.0 °C, and 2.5 °C scenarios using CMIP5 output filtered according to each AOGCM’s 2100 GMST. Specifically, we create ensembles for 1.5 °C, 2.0 °C, and 2.5 °C scenarios with AOGCMs that have a 21st century GMST increase (19 year running average) of 1.5 °C, 2.0 °C, and 2.5 °C (±0.25 °C). For consistency with the K14 framework, which models 19 year running averages of SLR relative to 2000, GMST is anomalized to 1991–2009 and then shifted upward by 0.72 °C to account for warming since 1875–1900 (Hansen et al 2010, GISSTEMP Team 2017). Selection of the AOGCMs for each scenario ensemble is made irrespective of the AOGCM’s RCP forcing. For model outputs that end in 2100, we extrapolate the 19 year running average GMST to 2100 based on the 2070–2090 trend. While we chose 2100 as the determining year for which AOGCMs are selected for each ensemble, it should be noted that Article 2 of the UNFCCC (UNFCCC 1992) does not require that GMST stabilization be achieved within a particular timeframe. The Paris Agreement likewise does not specify a timeframe for GMST stabilization, though its goal of bringing net greenhouse gas (GHG) emissions to zero in the second half of the 21st century implies a similar timeframe for stabilization. We make the assumption that AOGCM outputs that end at 2100 either stay within the range of the target ±0.25 °C or fall below by any amount (i.e. undershoot). For AOGCMs that have GMST output available after 2100, only those that undershoot the target are retained. However, we make an exception to this rule for the 2.5 °C scenario ensemble in order to include AOGCMs for generating post-2100 projections. For RCP4.5 and RCP6, GMST stabilization should not occur before 2150, when GHG concentrations stabilize (Meinshausen et al 2011b) and so SLR projections after 2100 may not be representative of conditions under true GMST stabilization. The GMST trajectories and GMSL contributions from TE and glacial ice from selected CMIP5 models that are binned into 1.5 °C, 2.0 °C, and 2.5 °C GMST categories are shown in figures 1 and S-2, respectively. Table S-2 lists the AOGCMs employed in each GMST scenario ensemble and the sea-level components used. Given the paucity of CMIP5 output after 2100, the range of TE and GIC contributions to SLR in the 22nd century is likely underestimated relative to the 21st century. Total ice sheet contributions from AR5 are calculated for each GMST scenario by randomly sampling AIS and GIS ice sheet distribution for each RCP (table 13.5 in Church et al 2013) in proportion to the representation of each RCP in the groups of CMIP5 models selected for each GMST scenario⁸.

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We generate estimates for GMSL for 2000–2200 using the SESL model from Kopp et al (2016a) and Bittermann et al (2017) driven with both GMST trajectories from CMIP5 models (figure 1) and GMST trajectories from the reduced-complexity climate model MAGICC6 (Meinshausen et al 2011a, as employed in Rasmussen et al 2016) for 2100 GMST targets of 1.5 °C, 2.0 °C, and 2.5 °C (±0.25 °C) (figure S-3). The MAGICC6 GMST trajectories are selected from all RCP-grouped projections using the same criteria as in section 2.2. The SESL model is calibrated to the common era temperature reconstruction from Mann et al (2009) and the sea level reconstruction of Kopp et al (2016a). The historical statistical relationship between temperature and the rate of sea-level change is assumed to be constant; not included are nonlinear physical processes or critical threshold events that could substantially contribute to SLR, such as ice sheet collapse (Kopp et al 2016b, Levermann et al 2013). Threshold behavior is partially incorporated in the K14 framework through expert assessments of future ice sheet melt contributions (Bamber and Aspinall 2013), which may be one reason why the K14 framework produces higher estimates in the upper tail of the SLR probability distribution.

The heights of historical ESL events that result from tropical and extra-tropical cyclones, extreme astronomical tides, and other processes are recorded in sub-daily tide gauge observations. Extreme value theory can be used with these tide gauge measurements to estimate the historical return levels of ESL events, including events that occur less often, on average, than the length of the observational record. For example, one could use extreme value theory to estimate the height of the present-day 500 year (or 0.2% average annual probability) ESL event from a record that is <500 years in length. Assuming no non-linear relationships between SLR and ESL events and no change in the frequency and intensity of processes that cause ESLs (e.g. tropical and extra-tropical cyclones), the estimated return levels of historical ESL events can be combined with local SLR projections to estimate the return levels of future ESL events.

2.4.1. Estimation of historical return levels of extreme sea levels

Here, we use extreme value theory with daily maximum sea levels at tide gauges archived by the University of Hawaii Sea Level Center (see supplementary data; Caldwell et al 2015) to estimate historical return levels of ESL events. Specifically, we follow Tebaldi et al (2012) and Buchanan et al(2016, 2017) and employ a generalized Pareto distribution (GPD) and a peaks-over-threshold approach (Coles 2001b, 2001a). The GPD describes the probability of a given ESL height conditional on an exceedance of the GPD threshold. We use the 99th percentile of daily maximum sea levels as the GPD threshold, which is generally both above the highest seasonal tide and balances the bias-variance trade-off in the GPD parameter estimation (Tebaldi et al 2012). The number of annual exceedances of the GPD threshold is assumed to be Poisson distributed with mean λ. Tide gauge observations are detrended and referenced to mean higher high water (MHHW)⁹ and the GPD parameters are estimated using the method of maximum likelihood (see supplementary data). Uncertainty in the GPD parameters is calculated from their estimated covariance matrix and is sampled using Latin hypercube sampling of 1000 normally distributed GPD parameter pairs. For a given tide gauge, the annual expected number of exceedances of ESL height z is given by N(z):

where the shape parameter (ξ) governs the curvature and upward statistical limit of the ESL event return curve, the scale parameter (σ) characterizes the variability in the exceedances caused by the combination of tides and storm surges, and the location parameter (μ) is the threshold water-level above which return levels are estimated with the GPD, here the 99th percentile of daily maximum sea levels. Meteorological and hydrodynamic differences between sites give rise to differences in the shape parameter (ξ). ESL frequency distributions with ξ > 0 are ‘heavy tailed’, due to a higher frequency of events with extreme high water (e.g. tropical and extra-tropical cyclones). Distributions with ξ < 0 are ‘thin tailed’ and have a statistical upper bound on extreme high water levels. Events that occur between λ and 182.6/year (i.e. exceeding MHHW half of the days per year) are modeled with a Gumbel distribution, as they are outside of the support of the GPD. Note that ESL events at tide gauges are not referred to as floods as the occurrence of an actual flood depends on the level of coastal flood protection, terrain, infrastructure, and other local factors.

2.4.2. Extreme sea level event frequency amplification factors

The frequency amplification factor (AF) quantifies the increase in the expected frequency of historical ESL events (e.g. the 100 year ESL event) due to SLR (Buchanan et al 2017, Hunter 2012, Church et al 2013). Due to variation in the local storm climate and hydrodynamics, the height of ESL event return levels are unique to each location (SI figure S-4). The calculation of the expected AF includes both the uncertainty in the estimates of the return periods of historical ESL events and uncertainty in SLR projections. Following Buchanan et al (2017), we define the expected ESL event frequency amplification factor AF(z) for ESL events with height z as the ratio of the expected number of ESL events after including uncertain SLR to the historical expected number of ESL events:

where N(z–δ) is the annual expected number of exceedances of ESL height z after including SLR (δ), E [] is the expectation operator applied to the full probability distribution of SLR projections, and N(z) is the historical annual expected number of exceedances of ESL height z.

Table 1. GMSL projections. All values are cm above 2000 CE baseline. AIS = Antarctic ice sheet, GIS = Greenland ice sheet; TE = thermal expansion; GIC = glaciers and ice caps; LWS = land water storage. AIS and GIS ice sheet distributions for each RCP from the Intergovernmental Panel on Climate Change’s Fifth Assessment Report (Church et al 2013) are randomly sampled in proportion to the RCP representation in the CMIP5 model filtering (table S-2). K16: SESL model from Kopp et al (2016a) driven with GMST trajectories from MAGICC (see SI figure S-3) and CMIP5 GMST trajectories (see figure 1); J18: Jackson et al(2018), S16: Schleussner et al (2016a).

	1.5 °C			2.0 °C			2.5 °C
cm	50	17–83	5–95	50	17–83	5–95	50	17–83	5–95
2100—Components
AIS	6	−4–17	−8–35	6	−5–17	−8–34	6	−5–16	−8–34
GIS	7	4–12	3–19	8	4–14	2–22	8	4–15	2–22
TE	19	14–23	10–27	25	15–34	7–42	26	20–31	16–35
GIC	11	8–13	6–15	11	7–16	2–21	13	11–15	9–17
LWS	5	3–7	2–8	5	3–7	2–8	5	3–7	2–8
Total	48	35–64	28–82	56	39–76	28–96	58	45–75	37–93
Projections by year
2050	24	20–28	18–32	25	20–32	15–37	26	22–30	19–34
2070	34	27–41	24–50	38	28–48	21–58	38	32–47	27–55
2100	48	35–64	28–82	56	39–76	28–96	58	45–75	37–93
2150	69	42–107	28–151	88	50–133	25–181	86	54–126	35–171
2200	93	43–161	20–241	120	57–197	20–281	118	62–189	31–268
Other projections for 2100
K16a	38	33–43	30–47	45	39–52	35–58	54	47–62	42–68
K16b	41	36–48	32–53	48	41–56	36–62	51	45–59	41–65
J18	44	30–58	20–67	50	35–64	24–74	–	–	–
S16	41	29–53	–	50	36–65	–	–	–	–
Other projections for 2200
K16a	58	45–72	36–81	79	65–93	56–104	100	85–115	75–127
K16b	59	45–75	37–88	81	68–95	59–105	101	87–116	78–128

^aSESL model driven with MAGICC6 GMST trajectories shown in SI figure S-3. ^bSESL model driven with CMIP5 GMST trajectories shown in figure 1.

2.4.3. Assessment of population exposure

The GMSL projections for each GMST target from the K14 and SESL method are shown in figure 1and are tabulated along with the component contributions in table 1. For the K14 method, differences in median GMSL between 1.5 °C, 2.0 °C, and 2.5 °C GMST stabilization targets do not appear until after 2050, when the 1.5 °C scenario begins to separate from the 2.0 °C and 2.5 °C trajectories (table 1). The median GMST trajectories diverge earlier, around 2030 (figure S-3). This is consistent with the early to mid-century divergence in the radiative forcing pathways and this study’s allocation of RCPs in the 1.5 °C (primarily RCP2.6), 2.0 °C (primarily RCP4.5), and 2.5 °C (primarily RCP4.5 and RCP6) scenarios (SI, table S-2). Median projections for 2100 GMSL under a 1.5 °C scenario are 48 cm, with a very likely range (90% probability) of 28–82 cm. An additional 8–10 cm of median GMSL rise is found for the 2.0 °C and 2.5 °C GMST scenarios, 56 cm (very likely 28–96 cm) and 58 cm (very likely 37–93 cm), respectively. Prior to mid-century, TE and GIC contributions account for more than half of GMSL projection uncertainty, but by 2100, ice sheet contributions dominate (SI figure S-5). Other studies found similar GMSL results. Using the same framework, Kopp et al (2014) estimated median 2100 GMSL projections under RCP2.6 and RCP4.5 of 50 cm (very likely 29–82 cm) and 59 cm (very likely 36–93 cm), respectively. Jackson et al (2018) also employs the CMIP5 ensemble to estimate probabilistic local SLR projections for GMST stabilizations, but do not consider non-linear ice dynamics (e.g. Bamber and Aspinall 2013). Their median projections for 1.5 °C (44 cm; very likely 20–67 cm) and 2.0 °C (50 cm; very likely 24–74 cm) GMST stabilizations are within 4–6 cm of this study. Using a method that scales SLR component contributions as a function of GMST and ocean heat uptake (Perrette et al 2013), Schleussner et al(2016a) estimated a median 2100 GMSL for 1.5 °C and 2.0 °C scenarios, that is 6–7 cm lower than this study’s K14 framework projections. (table 1).

Despite being warmer by a half-degree, the 2.5 °C GMSL projections largely overlap the 2.0 °C scenario (figure 1). Variation in the transient climate response and ocean heat uptake efficiency across CMIP5 models leads to weak correlation between TE and GMST (r² = 0.10; figure S-6; Kuhlbrodt and Gregory 2012, Raper et al 2002). As such, cooler models may produce more TE than warmer models, and vice versa. Ice sheet contributions are also similar between 2.0 °C and 2.5 °C scenarios (table 1). To test the sensitivity of model-RCP filtering to the choice of GMST stabilization, we additionally calculate GMSL under a 1.75 °C and 2.25 °C GMST scenario. The median 2100 GMSL under the 1.75 °C scenario is 3 cm greater than the 1.5 °C scenario, and the 2.25 °C scenario is 1 cm less than the 2.0 °C scenario (table S-3), suggesting that GMST scenarios that are primarily represented by only one RCP (i.e. the 1.5 °C scenario) may be less sensitive to model filtering.

Agreement between central estimates from process-based and semi-empirical projections implies consistency with the observed statistical relationship between GMST and the rate of SLR used to calibrate the SESL model. Across scenarios, median 2100 GMSL projections from the SESL model driven with CMIP5 GMST trajectories are 7–8 cm lower than those from the K14 framework (figure 1 and table 1), but more disagreement exists between the processed-based and SESL projections when driven with the MAGICC GMST trajectories shown in SI, figure S-3 (median projection differences of 4–11 cm; table 1). These differences are smaller in magnitude relative to the differences in the median RCP2.6 and RCP4.5 projections from Kopp et al (2014) and the SESL projections from Kopp et al (2016a) (8–12 cm). After 2100, the differences between projections from the K14 framework and the SESL model become larger. Across scenarios, median 2200 GMSL projections from the K14 framework are higher by 34 cm (1.5 °C), 39 cm (2.0 °C) and 17 cm (2.5 °C) than those from the SESL model driven with CMIP5 GMST trajectories (figure 1 and table 1). These differences are largely attributed to the treatment of ice sheets in each approach. The K14 framework accounts for non-linearities in crossing threshold ice sheet behavior by drawing from AR5 and Bamber and Aspinall (2013), but the SESL model does not because these events are absent from the calibration period.

Under the median projected GMSL for a 2.0 °C GMST stabilization, lands currently home to about 60 million people are projected to be permanently submerged by 2150, including lands currently home to over half a million inhabitants of United Nations defined Small Island Developing States (SIDS). Aggregation of all SIDS can mask important risks. For instance, local SLR projections for 2150 under a 2.0 °C GMST stabilization place lands currently home to almost a quarter of the current population of the Marshall Islands at risk of being permanently submerged. In comparison to these totals, under the median projection for the 1.5 °C stabilization scenario, lands currently home to about 5 million people, including 60 000 in SIDS, avoid inundation (table 2), but little difference is found for the Marshall Islands.

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We assess the effects of different GMST stabilizations on the frequency of ESL events by highlighting three cities: (1) New York, New York, USA, (2) Kushimoto, Wakayama, Japan, and (3) Cuxhaven, Lower Saxony, Germany (figure 2). Estimates of the historical 10-, 100-, and 500-year ESL events (expected frequency of 0.1/year, 0.01/year, and 0.002/year, respectively) and the future ESL frequency AF for all sites are provided in SI tables S-4 to S-6. Under 2.0 °C GMST stabilization, the 2100 median local SLR for New York City is 69 cm (likely 44–98 cm). In figure 2, median local SLR under the 2.0 °C scenario (SL₅₀ ) shifts the expected historic ESL event return curve to the right (i.e. N(z), the heavy gray curve, becomes N+SL₅₀ 2.0 °C, the dashed green curve) and increases the expected annual number of historical 10 year ESL events from 0.1/year to ~10/year. However, when both the uncertainty in the GPD fit and the SLR projections are considered in the calculation of the projected future ESL event return curve (i.e. N_e 2.0 °C; the heavy green curve), the expected frequency of the current 10 year ESL event increases from 0.1/year to 36/year (i.e. 3/month, on average). GHG mitigation that stabilizes GMST at 1.5 °C reduces projected median local SLR at New York City to 55 cm (likely 35–78 cm), and reduces the expected number of current 10 year ESL events by half (15/year). By 2150, the reduction in projected 10 year ESL events from the 2.0 °C to the 1.5 °C scenario is still ~50% (99/year reduced to 59/year; table S-4).

Note that the expected number of flood events and the appearance of kinks in the N_e curves in figure 2 are sensitive to the way that the high-end tail of the mean sea level distributions are constructed. For example, for sample sizes explored in this study (<99.9th percentile), this truncation plays an important role in setting the location of the kinks in N_e and for sufficiently heavy tailed distributions, N_e may not converge within this range. Discontinuities in the N_e curves can also arise at the transition between modeling flood events with two different distributions. Specifically, expected ESL frequencies between the historical frequency of the GPD threshold exceedance (i.e. λ) and 182.6 events per year are modeled with a Gumbel distribution and expected ESL frequencies greater than λare modeled with a GPD (section 2.4.1). Note that because the uncertainty in local SLR varies by location, the distance between the N+SL₅₀ and N_e curves also differs by location.

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Sea-level rise will amplify the frequency of all ESL events, but depending on the shape of the GPD, the frequency of some ESL events may amplify more than others (Buchanan et al 2017). For example, by 2100 under a 2.0 °C and 1.5 °C GMST stabilization, respectively, median local SLR for Kushimoto, Japan is projected to be 79 cm (likely 58–103 cm) and 70 cm (likely 52–92 cm), increasing the respective number of historical 10 year ESL events from 0.1/year, on average, to 146/year (AF of 1462) and 128/year (AF of 1277), on average. However, for the same amount of local SLR, the historical number of expected 500 year ESL events for Kushimoto increases from 0.002/year to 83/year (2.0 °C; AF of 41479) and 57/year (1.5 °C; AF of 28 645). When the shape of the return curve is log-linear (as occurs when the shape parameter (ξ) is zero), ESL events amplify equally across return periods. For example, by 2100, under 1.5 °C, 2.0 °C, and 2.5 °C GMST stabilization, respectively, Cuxhaven, Germany is projected to have median local SLR of 43 cm (likely 26–65 cm), 53 cm (likely 29–82 cm) and 51 cm (likely 34–71 cm). The historical 500 year ESL event is projected to become as or more frequent than the historical 100 year ESL event for all scenarios: 0.01/year (1.5 °C; AF of 5.6), 0.03/year (2.0 °C; AF of 13.5), and 0.01/year (2.5 °C; AF of 6.5). Because the shape factor of the Cuxhaven GPD is close to zero, the historical 10 year ESL event also is projected to amplify similarly to the 500 year ESL event: 0.6/year (1.5 °C; AF of 5.6), 1.0/year (2.0 °C; AF of 13.5), and 0.7/year (2.5 °C; AF of 6.5). For some sites, including Cuxhaven, the AF for the 2.0 °C scenario may be greater than the AF for the 2.5 °C scenario. This can be partly attributed to higher SLR projections in the upper tail of the 2.0 °C probability distribution influencing the AF calculation.

We assess regional differences in 100 year ESL event frequency amplification between 2.0 °C and 1.5 °C GMST stabilization by binning ratios of 2.0 °C/1.5 °C expected AFs for 2100 and 2150 (figure 3). Bins on the right side of each graph become filled when there are decreases in the frequency of ESL events at regional groups of tide gauges from 1.5 °C over 2.0 °C GMST stabilization, while bins on the left side of each graph become filled when there are either no changes or increases in ESL event frequency at stations from 1.5 °C GMST over 2.0 °C GMST stabilization. In general, decreases in the frequency of ESL events from a 1.5 °C GMST stabilization grow as GMSL trajectories between scenarios separate from one another (table 1). By 2100 and 2150, substantial decreases in the frequency of ESL events from 1.5 °C GMST stabilization are expected in the East and Gulf Coasts of the United States, where ESL event amplification between GMST scenarios is reduced by roughly half. By 2150, smaller contributions from either local ocean dynamics or GICs in the 2.0 °C scenario attenuate SLR in parts of Europe, leading to lower median local SLR than from 1.5 °C GMST stabilization. Less local SLR in the 2.0 °C scenario causes ESL event frequencies to decrease, relative to the 1.5 °C scenario. We find small decreases or no change in ESL event frequency from achieving a 1.5 °C GMST stabilization over a 2.0 °C GMST stabilization at most tide gauges located in SIDS, as local SLR projections in these areas are similar between GMST scenarios (figure 3).