There is little agreement on either surface- or deep-layer slopes across the five GCMs over most parts of the Atlantic basin (Figure 1). This may reflect limited skill in simulating strong western boundary currents (e.g., the Atlantic Meridional Overturning Circulation (AMOC)) in the GCMs, which have a relatively coarse (~1˚) spatial resolution in ocean component (Small et al., 2014) and so poorly capture non-linear mesoscale processes in the ocean current (Hallberg, 2013). Near the eastern coast of North America, DSL is closely related to AMOC (Goddard et al., 2015), which is expected to weaken in a warming climate (Caesar et al., 2018). Low skill in capturing AMOC behavior can affect the DSL projections in the Atlantic basin as well as its coasts (van Westen et al., 2020). As the coefficients of pattern scaling depend on the simulations by the GCMs, they also do not explicitly resolve the non-linear mesoscale process of the ocean current. Therefore, we should interpret the DSL changes predicted by the two-layer emulator with cautions over the regions where non-linear mesoscale effects of ocean current are strong.
Projecting DSL Using FaIR-2LM and Patterns
We use two steps to generate a probabilistic ensemble of DSL projections. First, we generate an ensemble of surface and deep-ocean temperature pairs using FaIR-2LM. The planetary energy balance at the top of the atmosphere (Zelinka et al., 2020) is:
\[N = \mathcal{F} + \lambda T   (6)\]
where N is the radiative imbalance at the top of the atmosphere. The equilibrium climate sensitivity (ECS) is given by T when N = 0, and mathcal{F} = F_{2 times CO_2}. Therefore, lambda is related to F_{2 times CO_2} and ECS by
lambda = - F_{2 times CO_2} / ECS  (7)
The uncertainty of F_{2 times CO_2} is small relative to the spread of lambda, while ECS largely determine the uncertainty of lambda. Therefore, we adopt the best estimation in the Intergovernmental Panel on Climate Change Fifth Assessment Report (AR5) for F_{2 times CO_2} = 3.71 W m^{-2} (Collins et al., 2013). We produce initial distributions of ECS, gamma, and gamma_{epsilon} based on the literature constraints (Figure S4) outlined below:
ECS: Based on multiple lines of evidence, the uncertainties of ECS estimated by AR5 are likely in the range 1.5˚C-4.5˚C with high confidence, extremely unlikely less than 1˚C and very unlikely greater than 6˚C (Collins et al., 2013). In the AR5 terminology, likely denotes a probability of at least 66%, very unlikely a probability of less than 10%, and extremely unlikely a probability of less than 5% (Mastrandrea et al., 2010). Therefore, we construct a log-normal distribution for ECS with parameters optimized to match a 5th percentile of 1˚C, a 17th percentile of 1.5˚C, an 83rd percentile of 4.5˚C, and a 90th percentile of 6˚C.
gamma: We treat gamma as normally distribution, with mean 0.67 W m^{-2} K^{-1} and standard deviation 0.15 W m^{-2} K^{-1} derived from the 16 GCMs in the CMIP5 archive (Geoffroy et al., 2013).
gamma_{epsilon}: As the efficacy factor of heat uptake is related to deep-ocean heat uptake (Held et al., 2010), we use gamma_{epsilon} instead of epsilon to maintain the covariance between gamma and epsilon. We calculate the mean of 0.86 W m^{-2} K^{-1} and standard deviation 0.29 W m^{-2} K^{-1} of gamma_{epsilon} based on the products of gamma and epsilon from 16 GCMs in CMIP5 archive (Geoffroy, et al., 2013a). The distribution of gamma_{epsilon} is constructed as a normal distribution with the multi-model mean and the multi-model standard deviation.
TCR: Under a zero-layer approximation which considers the 1%/yr increase in CO_2 until doubling scenario occurring on a timescale long enough that the upper ocean is in approximate equilibrium and short enough that the deep-ocean temperature has not yet responded substantially, Transient Climate Response (TCR) can be obtained by (Jimenez-de-la-Cuesta & Mauritsen, 2019):
{TCR} = - frac{F_{2 times CO_2}}{lambda - gamma_{epsilon}}  (8)
Uncertainty in the TCR can be constrained adequately by varying only lambda and gamma_{epsilon} under the zero-layer approximation. Therefore, although there are significant uncertainties of C and C_0 among GCMs, we use fixed values because the uncertainties of these parameters are not necessary to represent uncertainty in the TCR. In this study, we set C = 8.2 W m^{-2} K^{-1} and C_0 = 109 W m^{-2} K^{-1} based on the multi-model mean of GCMs from Coupled Model Intercomparison Project Phase 5 (CMIP5) archive (Geoffroy, et al., 2013a).
We then generate a 100,000-member ensemble of ECS, gamma and gamma_{epsilon} based on these distributions via Monte Carlo sampling. As gamma_{epsilon}} should be larger than 0, we discard parameter sets in which gamma_{epsilon}} < 0 or gamma_{epsilon}} > 2 times 0.86 to keep the mean of gamma_{epsilon}} in parameter sets to be 0.86 W m^{-2} K^{-1}. Therefore, 99734 parameter sets are kept. An ensemble of lambda is then computed by the best estimation of F_{2 times CO_2} and the ensemble of ECS based on Equation 6 (Figure S4). The median (central 66% range) of lambda is −1.39 (−2.4 to −0.8) W m^{-2} K^{-1}. As the likely range of ECS estimated by AR5 is equivalent to the central 90% range of ECS estimated by CMIP5 GCMs, the uncertainty range of lambda estimated by FaIR-2LM is larger than that estimated by ensemble of GCMs (Geoffroy et al., 2013). The spread of TCR is estimated as a diagnostic by substituting the ensemble of lambda, gamma_{epsilon}, and best estimation of F_{2 times CO_2} into Equation 8. The uncertainty of TCR is in a central 66% range of 1.1–2.3˚C, with a 95th percentile of 2.9˚C. This is consistent with but slightly narrower than the TCR estimated by AR5, which is likely between 1˚C and 2.5˚C, and is extremely unlikely greater than 3˚C.
We apply Latin hypercube sampling (LHS, Stein, 1987) to the parameter sets of lambda, gamma, gamma_{epsilon} by sampling 1,000 sets from the 99,734 parameter sets. For each parameter, LHS divides the probability density function of the 99,734 samples into 1,000 portions that have equal area. A sample is taken from each portion randomly so that the 1,000 sample sets cover the multidimensional distribution of the three parameters. Finally, we applied 1,000 parameter sets together with the fixed parameters (F_{2 times CO_2}, C, C0) to the FAIR-2LM and generate a 1,000-member probabilistic ensemble of temperature pair time-series.
We compare the spread in GSAT projected by FaIR-2LM with the likely ranges estimated by AR5 for four different periods (Collins et al., 2013) (Table 3 and Figure S5). The mean of the probabilistic ensemble is slightly lower than the mean estimate of GSAT from AR5 in all four periods of RCP2.6 and RCP4.5, and in the 21st century for RCP8.5. Compared with AR5 likely ranges, the central 66% probability range of GSAT from FaIR-2LM is generally consistent: narrower in all four periods of RCP2.6, narrower in the first two periods but wider in the last two periods in RCP4.5, and wider in the first two periods but narrower in the last two periods in RCP8.5.
We project GMTSLR based on Equation 3 using the probabilistic ensemble of surface and deep-ocean temperature projections from FaIR-2LM. The C, C_0 and expansion efficiency of heat sigma (0.113 times 10^{-24 m}) used here are adopted from the multi-model ensemble mean of CMIP5 archive (Geoffroy, et al., 2013a; Kuhlbrodt & Gregory, 2012).


We apply Latin hypercube sampling (LHS, Stein, 1987) to the parameter sets of lambda, gamma, gamma_{epsilon} by sampling 1,000 sets from the 99,734 parameter sets. For each parameter, LHS divides the probability density function of the 99,734 samples into 1,000 portions that have equal area. A sample is taken from each portion randomly so that the 1,000 sample sets cover the multidimensional distribution of the three parameters. Finally, we applied 1,000 parameter sets together with the fixed parameters (F_{2 times CO_2}, C, C0) to the FAIR-2LM and generate a 1,000-member probabilistic ensemble of temperature pair time-series.
We compare the spread in GSAT projected by FaIR-2LM with the likely ranges estimated by AR5 for four different periods (Collins et al., 2013) (Table 3 and Figure S5). The mean of the probabilistic ensemble is slightly lower than the mean estimate of GSAT from AR5 in all four periods of RCP2.6 and RCP4.5, and in the 21st century for RCP8.5. Compared with AR5 likely ranges, the central 66% probability range of GSAT from FaIR-2LM is generally consistent: narrower in all four periods of RCP2.6, narrower in the first two periods but wider in the last two periods in RCP4.5, and wider in the first two periods but narrower in the last two periods in RCP8.5.
We project GMTSLR based on Equation 3 using the probabilistic ensemble of surface and deep-ocean temperature projections from FaIR-2LM. The C, C_0 and expansion efficiency of heat sigma (0.113 times 10^{-24 m}) used here are adopted from the multi-model ensemble mean of CMIP5 archive (Geoffroy, et al., 2013a; Kuhlbrodt & Gregory, 2012).
A projection of DSL is constructed as follows: 1) a pair of alpha and beta is randomly picked with replacement from the pool of two-layer patterns produced in Section 2.3; 2) a temperature pair from the 1000 members is combined with the pair of alpha and beta in an equation:
{DSL}(t, x, y) = alpha(x, y) T(t) + beta(x, y) T_0(t) + b(x, y)  (9)
Evaluation of Two-Layer Pattern Scaling
To evaluate the prediction skill of the two-layer pattern scaling, we compare the DSL changes simulated from a GCM with the DSL changes emulated by the two-layer pattern scaling (widehat{{DSL}}) based on FAIR-2LM using the key parameters (i.e. parameters in Table 1) in the same GCM. Two metrics are used: (1) absolute values of the residual differences between widehat{{DSL}} changes and DSL changes during a period at each grid point, and (2) global average of the absolute values obtained from the metric 1 (Table S1). These two metrics are applied to both bivariate pattern scaling and univariate pattern scaling, to examine the improvement of bivariate approach comparing with the univariate approach.
In 2271-2290, for instance, the global-averaged climatology of | DSL - widehat{{DSL}} | (score obtained by the second metric) from the two-layer pattern scaling is less than that from the univariate pattern scaling (bottom row Figure 2), with a reduction of 36%, 24%, and 34% in RCP2.6, RCP4.5, and RCP8.5, respectively. The spatial pattern of R = DSL - widehat{{DSL}} is derived from both approaches are various across GCMs (Figures S6-S10). The 5-model ensemble averaged climatology of | R | in both approaches is higher over high latitudes (e.g., Arctic, subpolar Northern Atlantic, Southern Ocean) than over middle to low latitudes, but is generally lower in two-layer pattern scaling than in univariate pattern scaling (first two rows Figure 2). As the pattern-scaling method cannot resolve DSL change due to unforced variability, the relatively large | R | over the high latitudes may be due to the relatively high unforced variability over these regions (Bilbao et al., 2015).
We further compare the time evolution of widehat{{DSL}} predicted by the two-layer pattern scaling approaches with the evolution of DSL in corresponding GCMs through the period 1981-2290. As case studies, we pick two grid cells: one in the western Pacific near the Philippines (14.5˚N, 127˚E), and the other over the North Atlantic near the coast of New York City [NYC] (40˚N, 73˚W) (solid black dots in Figure 2). The grid point near the Philippines is selected because it is in the tropical Pacific, where DSL rise associated with the deep ocean temperature rise is strongest, while the grid point near the NYC is selected represent a coastal area that some projections find experiences significant DSL changes in response to changes in AMOC.
Figure 2: Differences between DSL simulated by GCMs (zos) and DSL (widehat{{zos}}) predicted by univariate pattern scaling (univariate, first row) and two-layer pattern scaling (two-layer, second row) over the period 2271-2290 for the ensemble mean of 5 GCMs in three scenarios: RCP2.6, RCP4.5, RCP8.5 (Units: m). The third row shows the global mean of the | {zos} - (widehat{{zos}} | in 1TS and 2TS, respectively. Black dots on maps denote the two grid cells used for the plot in Figure 3.
At the western Pacific grid cell, in RCP 2.6, the relationship between DSL and GSAT anomaly displays a hook-like shape, indicating continued rise in DSL as GSAT stabilizes and declines in response to negative emissions (Figure 3a). The delayed adjustment of DSL may be due to the continuous warming of deep layer (T_0) when GSAT is stabilized, because the ocean is not yet equilibrated with the elevated forcing. In response to changes in T_0, the deep ocean density is still changing even without changing circulation in the deep ocean (England, 1995), so DSL continues to change. This hook-like shape is captured by the two-layer pattern scaling approach but not by the univariate pattern scaling. Compare to the DSL simulated by a GCM, the RMSE of the predicted DSL is smaller if using the two-layer pattern scaling approach than using the univariate pattern scaling approach. The average RMSE across the five GCMs is reduced 26% if we use the approach from the two-layer pattern scaling instead of the univariate pattern scaling (Table 3). Across the five GCMs, although the relationship between DSL and GSAT is diverse in RCP4.5 and RCP8.5, DSL projected by the two-layer technique is consistently closer than that predicted by the univariate technique to the DSL simulated by the GCMs. The sole exception is for bcc-csm1-1 in RCP8.5, for which the simulated DSL projection is quite linearly associated with GSAT. The average of RMSEs across the 5 GCMs decrease from the univariate pattern scaling approach to the two-layer pattern scaling approach by 35% in RCP4.5 and by 33% in RCP8.5 (Table 3).
Figure 3: widehat{{zos}} predicted by univariate pattern scaling and two-layer pattern scaling at the grid cell (a) over Western Pacific (14.5˚N, 127˚E) and (b) over the North Atlantic (40˚N, 73˚W) for the five models in the three scenarios. The zos simulated by corresponding GCMs is shown by scatters in which colors indicate years. Root mean square errors between the widehat{{zos}} and zos determined over the entire simulating period for each GCM are shown in parentheses of legend (units: m).


Figure 3: widehat{{zos}} predicted by univariate pattern scaling and two-layer pattern scaling at the grid cell (a) over Western Pacific (14.5˚N, 127˚E) and (b) over the North Atlantic (40˚N, 73˚W) for the five models in the three scenarios. The zos simulated by corresponding GCMs is shown by scatters in which colors indicate years. Root mean square errors between the widehat{{zos}} and zos determined over the entire simulating period for each GCM are shown in parentheses of legend (units: m).
At the North Atlantic grid cell, the relationship between DSL and GSAT also displays non-linear features for all the five models, especially in low- and moderate-emission scenarios (Figure 3b). These non-linear features, which may arise from the delayed response of deep branch of AMOC, cannot be captured by univariate pattern scaling but are captured to a large extent by the two-layer pattern scaling (lines in Figure 3). The value of the two-layer approach is highlighted by the clear non-linearity of the DSL response when viewed as a function of GSAT anomaly. Compare to the univariate approach, the RMSE between the DSL simulated by GCMs and the DSL predicted by the two-layer pattern scaling is smaller, with a reduction of 19%, 16%, and 13% for RCP2.6, RCP4.5, and RCP8.5, respectively. The method of two-layer pattern scaling generally has a better performance in emulating the DSL from the corresponding GCM than the univariate pattern scaling, as the two-layer pattern scaling includes one more predictor than univariate pattern scaling, allowing it to capture the delayed adjustment of DSL. The delayed adjustment of DSL due to the deep-ocean warming is important for the DSL projections, as different areas present different features that may reveal the regional variation in deep-ocean circulation (Held et al., 2010).
Projections of DSL
The procedure described in Section 2.4 allows us to produce a 1000-member probabilistic ensemble of DSL projections not only for the three CMIP5 scenarios: RCP2.6, RCP4.5, and RCP8.5, but also for any other scenarios with an emission pathway between these three scenarios. We demonstrate this capability using SSP3-7.0, a CMIP6 scenario that has forcing intermediate between RCP4.5 and RCP8.5 (O’Neill et al., 2016) and is closer than either to no-policy reference scenarios from most integrated assessment models (Riahi et al., 2017). The emission pathway of SSP3-7.0 used to drive the FaIR-2LM is taken from the Reduced Complexity Model Intercomparison Project (Nicholls et al., 2020).
The five projections using parameters calibrated to the five GCMs respectively are within the 66% range of the 1000-member ensemble for both surface and deep-ocean temperature in the three RCPs (Figure 4). By 2300, the median estimates (66% range) of the surface temperature relative the period of 1986-2005 are 0.5˚C (0.2-1.0˚C) for RCP2.6, 2.2˚C (1.2-3.6˚C) for RCP4.5, 7.4˚C (4.5-11.7˚C for RCP8.5, and 5.3˚C (3.2-8.6˚C) for SSP3-7.0.
Based on the projections of temperature pairs, we also produced projections of GMTSLR for the four scenarios (Figure 4). The spread of GMTSLR ensemble encapsulates the GMTSLR time series from the 5 GCMs (Figure 4). During the period of 2081-2100, the median estimates (66% range) of the GMTSLR relative the period of 1986-2005 are 0.12 (0.07-0.18) m for RCP2.6, 0.16 (0.10-0.24) m for RCP4.5, 0.19 (0.12-0.27) m for SSP3-7.0, and 0.24 (0.15-0.34) m for RCP8.5. This compares to Oppenheimer et al. (2019)’s projected median estimates (66% ranges) of 0.14 (0.10-0.18) m for RCP2.6, 0.19 (0.14-0.23) m for RCP4.5, and 0.27 (0.21-0.33) m for RCP8.5. By 2300, the median estimates (66% range) of GMTSLR relative to the period of 1986-2005 are 0.20 (0.12-0.33) m for RCP2.6, 0.43 (0.25-0.68) m for RCP4.5, 0.85 (0.50-1.33) m for SSP3-7.0, and 1.15 (0.69-1.76) m for RCP8.5. As climate warms, the projections of DSL changes increase along with the increase in GMTSLR. In the five GCMs, although the contributions of DSL changes to the local sterodynamic sea level (DSL+GMTSLR; Gregory et al., 2019) changes are small at some locations (i.e., regions marked by the light and dark gray shadings in Figure S11), in others the ratio of DSL change with respect to the GMTSLR changes are fairly significant. For instance, DSL changes at some regions (e.g., Arctic, North Atlantic, and Southern Ocean) are greater than 50% of the GMTSLR during the period of 2271-2290 (identified by yellow contours in Figure S11).
Figure 4: Ensemble projections of CO_2 concentrations (first row), GSAT (second row), deep-ocean temperature (third row), and GMTSLR (fourth row) changes relative to the baseline period 1986-2005 under the four scenarios. Shadings represents the 66% range, dark blue lines the median of probabilistic ensemble projections. The projection calibrated to the five GCMs in the three RCP scenarios are shown on top of the shadings (orange lines).
Compared with the GSAT and GMTSLR spread in 2300 estimated by M. D. Palmer et al. (2018), the FaIR.2LM projections have a slightly lower median for all the three RCPs. The 66% range of both surface temperature and GMTSLR estimated by FaIR-2LM is comparable to the 90% range of that estimated by M. D. Palmer et al. (2018) because we adopt a distribution of lambda based on the AR5 assessment of equilibrium climate sensitivity (Collins et al., 2012), which is broader than the 90% range estimated by the CMIP5 multi-model ensemble emulated by M. D. Palmer et al. (2018).
Comparing the DSL projections between the period of 2081-2100 and the period of 2271-2290 (Figure 5), the median estimate is lower and the 66% range of uncertainty is narrower at the end of 21st century than that at the end of 23rd century in moderate-to high-emission scenarios (RCP4.5, SSP3-7.0 and RCP8.5). But in RCP2.6, the median estimate and 66% uncertainty range are comparable in magnitude between these two periods. In both periods, the median DSL anomaly projections across the four scenarios share many similar features (Figure 5). Over the Arctic region, a weak increase in DSL is observed over the Chukchi Sea and the Beaufort Sea in RCP2.6. In the higher emission scenarios, the increase in DSL extends to the whole Arctic basin with intensified amplitudes. The changes in DSL over the North Atlantic are dominated by a negative anomaly under RCP2.6, and display positive anomalies over much of the North Atlantic under RCP8.5 and SSP3-7.0. The ensemble spread of the 5th-95th range of DSL projections are relatively large over the Southern Ocean, Arctic and Subpolar Atlantic than other areas. The large uncertainties over these areas, consistent with previous literatures (M. D. Palmer et al., 2020; Perrette et al., 2013; Yin, 2012), may be interpreted by the diverse characteristics simulated by GCMs that do not explicitly resolve non-linear mesoscale processes of the ocean current over these areas (van Westen et al., 2020).


Comparing the DSL projections between the period of 2081-2100 and the period of 2271-2290 (Figure 5), the median estimate is lower and the 66% range of uncertainty is narrower at the end of 21st century than that at the end of 23rd century in moderate-to high-emission scenarios (RCP4.5, SSP3-7.0 and RCP8.5). But in RCP2.6, the median estimate and 66% uncertainty range are comparable in magnitude between these two periods. In both periods, the median DSL anomaly projections across the four scenarios share many similar features (Figure 5). Over the Arctic region, a weak increase in DSL is observed over the Chukchi Sea and the Beaufort Sea in RCP2.6. In the higher emission scenarios, the increase in DSL extends to the whole Arctic basin with intensified amplitudes. The changes in DSL over the North Atlantic are dominated by a negative anomaly under RCP2.6, and display positive anomalies over much of the North Atlantic under RCP8.5 and SSP3-7.0. The ensemble spread of the 5th-95th range of DSL projections are relatively large over the Southern Ocean, Arctic and Subpolar Atlantic than other areas. The large uncertainties over these areas, consistent with previous literatures (M. D. Palmer et al., 2020; Perrette et al., 2013; Yin, 2012), may be interpreted by the diverse characteristics simulated by GCMs that do not explicitly resolve non-linear mesoscale processes of the ocean current over these areas (van Westen et al., 2020).
Figure 5: Projection of DSL changes at median estimation (first column) and range of 17th-83rd percentile averaged over the period of 2081-2100 in four scenarios (a) relative to the baseline period 1986-2005. (b) is the same with (a) except for the period of 2271-2290. Units are m.
At the illustrative grid point near Philippines over western Pacific (Figure 6a), the 66% range of the probabilistic ensemble encapsulates DSL projections from 2 of the 5 GCMs in the three RCPs, while the 90% range of the probabilistic ensemble contains DSL projections from all the 5 GCMs in the three RCPs, except for HadGEM2-ES in RCP2.6. At the grid point near NYC, the projected DSL changes estimated by the probabilistic ensemble exhibits a fat tail, with a median projection in RCP 8.5 of 0.13 m and a 95th percentile projection of 0.8 m by the end of 23rd century. By contrast, RCP 2.6 exhibits a much narrower range, with a median of 0 m and a 95th percentile of 0.08 m. The 66% range of the projected DSL uncertainties encapsulates 2 of 5 GCM projections. The 90% range of the probabilistic ensemble only encapsulates the DSL projections from three over five GCMs in RCP2.6 and RCP4.5, but encapsulates the DSL projections from all five GCMs in RCP8.5. The emulator fails to capture multidecadal variability in DSL, a limitation which would be expected because the emulator is constructed based on the pattern scaling approach.
Figure 6: Ensemble projections of DSL changes relative to the baseline period 1986-2005 at the grid cells near Philippines over the western Pacific (upper panel) and near NYC over the North Atlantic (lower panel) for the four scenarios: RCP2.6, RCP4.5, SSP3-7.0, and RCP8.5. Shadings represent the projections produced by the two-layer emulator, with light and dark shadings indicating the 90% and 66% range, respectively, and dark blue lines the median of probabilistic ensemble projections. The black lines represent the projections produced by the univariate emulator, with the dashed and solid lines indicating the 90% and 66% range, respectively. The projection of DSL changes smoothed by 20-years running average in the five GCMs are shown on top of the shadings (colored lines). Units: m.
To compare with the DSL projections derived from two-layer pattern scaling, we also produced the DSL projections based on univariate pattern scaling following the same procedure (Figures S12 and S14). The median and 17th-83rd range of DSL projections derived from univariate pattern scaling are similar in patterns (Figure S12). However, differences of both median and spreads of the DSL projections between univariate and two-layer pattern scaling vary across regions, especially over high latitudes in high-emission scenarios (Figure S13). Specifically, compared to the univariate emulator, the median of DSL projected by the two-layer emulator is lower over the Pacific and Indian Ocean, and higher over Arctic, Atlantic and Southern Ocean in the period of 2081-2100, but the opposite in the period of 2271-2290. In addition, the 17th-83rd range of the DSL projected by the two-layer emulator is wider than that by the univariate emulator over the Arctic. The area-weighted increases in DSL spread over the Arctic are 0.02 m in RCP2.6, 0.03 m in RCP4.5, and 0.04 m in RCP8.5 during 2081-2100, and are 0.006 m in RCP2.6, 0.008 m in RCP4.5, 0.009 m in RCP8.5 during 2271-2290. For the grid cell near Philippines, despite the shift in median, the distributions of DSL projections derived from univariate pattern scaling exhibit a different shape from that derived from two-layer pattern scaling (Figure 6). In 2290, the 90% range of DSL projection from the univariate emulator is close to that from the two-layer emulator, slightly narrower by 0.03 m for RCP2.6, 0.02 m for RCP4.5, and 0.01 m for SSP3-7.0 and RCP8.5. The two-layer pattern scaling leads to not only a shift of the distribution but also a different shape of the distribution of the DSL projections, compared to that derived from univariate pattern scaling. There are more DSL projections simulated by GCMs encapsulated within the 90% range of the probabilistic ensemble of DSL projections derived from two-layer pattern scaling than that by the univariate pattern scaling (Figure 6). For the grid cell near the NYC, the shape of DSL distributions derived from univariate pattern scaling is similar to that derived from two-layer pattern scaling, except the spread of the DSL projections derived from univariate pattern scaling is slightly narrower than that derived from the two-layer pattern scaling, with the 90% range of DSL projection in 2290 narrower by 0.03 m for RCP2.6, 0.01 m for RCP4.5, and 0.03 m for SSP3-7.0 and RCP8.5 (Figure 6). The greatest difference is in RCP 2.6, where the difference in the 90% range is by far the largest compared to the overall range.
Table 2: The Averaged RMSE Between the DSL Simulated by GCMs and the DSL Predicted by Univariate/Two-Layer Pattern Scaling Across Five Models. Note: The averaged RMSEs and the reduction of RMSE from univariate pattern scaling approach to two-layer pattern scaling are calculated for the three RCP scenarios, respectively.
Discussion and Conclusions
We have developed a probabilistic ensemble of DSL projections through 2300 using a novel two-layer emulator. Replacing the climate module in the FaIR simple climate model with a two-layer energy-balance model, we developed FaIR-2LM, which produces projections of global average temperature in the well-mixed upper layer (T) for rapid responses to radiative forcing, and in the deep ocean layer (T_0) for delayed responses. Calibrated by the parameters for each GCMs, the GSAT (Figure S2) and GMTSLR (Figure S3) emulated by FaIR-2LM generally follow that from the corresponding GCM, with RMSE <0.43 K for GSAT and <0.05 m for GMTSLR. A two-layer pattern scaling based on surface and deep-ocean temperature is used to project DSL. During the period 2271-2290, for instance, the DSL predicted by the two-layer pattern scaling are closer to the DSL simulated by the corresponding GCM than that predicted by the univariate pattern scaling (Figure 2). At two selected grid cells (near the coast of Philippines and NYC), the time evolution of DSL projections predicted by the two-layer emulator more accurately reflects GCM behavior and captures non-linearities and non-stationarity in the relationship between DSL and global-mean warming, comparing with that predicted by the univariate technique (Figure 3).


Table 2: The Averaged RMSE Between the DSL Simulated by GCMs and the DSL Predicted by Univariate/Two-Layer Pattern Scaling Across Five Models. Note: The averaged RMSEs and the reduction of RMSE from univariate pattern scaling approach to two-layer pattern scaling are calculated for the three RCP scenarios, respectively.
Discussion and Conclusions
We have developed a probabilistic ensemble of DSL projections through 2300 using a novel two-layer emulator. Replacing the climate module in the FaIR simple climate model with a two-layer energy-balance model, we developed FaIR-2LM, which produces projections of global average temperature in the well-mixed upper layer (T) for rapid responses to radiative forcing, and in the deep ocean layer (T_0) for delayed responses. Calibrated by the parameters for each GCMs, the GSAT (Figure S2) and GMTSLR (Figure S3) emulated by FaIR-2LM generally follow that from the corresponding GCM, with RMSE <0.43 K for GSAT and <0.05 m for GMTSLR. A two-layer pattern scaling based on surface and deep-ocean temperature is used to project DSL. During the period 2271-2290, for instance, the DSL predicted by the two-layer pattern scaling are closer to the DSL simulated by the corresponding GCM than that predicted by the univariate pattern scaling (Figure 2). At two selected grid cells (near the coast of Philippines and NYC), the time evolution of DSL projections predicted by the two-layer emulator more accurately reflects GCM behavior and captures non-linearities and non-stationarity in the relationship between DSL and global-mean warming, comparing with that predicted by the univariate technique (Figure 3).
Table 3: Comparison of the Distributions of GSAT Anomaly (Relative to 1986-2005) Projected by FaIR-2LM With the Distributions of Global-Mean Surface Temperature Assessed by AR5 (Collins et al., 2013) in RCP 2.6, RCP4.5, SSP3-7.0 and RCP 8.5 (the 2nd and 3rd Columns, units: ˚C). Note: Means are given without parentheses; likely range (for AR5) and 17th-83rd percentile range (for FaIR-2LM) are given in parentheses. The fourth column is similar with previous columns except for the GMTSLR projected by FAIR-2LM (Units: m).
By perturbating the key parameters, FaIR-2LM allows emulation of projected global-mean surface and deep-ocean temperature pairs and GMTSLR for emissions scenarios (e.g., SSP3-7.0; Figures 4 and 5) beyond those run by the GCMs to which it is calibrated. Compared with the likely ranges assessed by AR5 in the RCP 2.6, 4.5 and 8.5, the FaIR-2LM performs well in emulating the GSAT spread (Table 2 and Figure S5). By 2300, the ensembles of GSAT and GMTSLR estimated by FaIR-2LM have a slightly lower median and a slightly wider 90% range than the estimations by M. D. Palmer et al. (2018), likely because we use the uncertainty of ECS from AR5, which has a larger range than that estimated by CMIP5 multi-model ensemble.
We produce probabilistic ensembles of DSL projections for four different emissions scenarios. Characteristics of median DSL projections during 2271-2290 include increases in DSL along most of the coast around the Pacific and Indian Oceans and a decrease in DSL over the Southern Ocean in all four scenarios, as well as increased DSL over the Arctic and along the North Atlantic Current in moderate to high emissions scenarios (Figure 5). The 66% range (17th-83rd percentile) of uncertainties are small over the middle and low latitudes, and are relatively large over the Southern Ocean, Arctic and North Atlantic, where the simulations of GCMs are diverse due to the challenges of capturing the complex physical processes, such as deep water formation in the subpolar Atlantic, the Antarctic circumpolar current, and ice-albedo feedback in polar regions (Flato et al., 2013; Landerer et al., 2014; Wang et al., 2014). The ensemble of DSL projections also allows us to examine the trajectories of the DSL projections and their uncertainties at specific locations (Figure 6). At selected locations in the North Atlantic and Western Pacific, the 90% range of DSL spread generally encapsulates the time series of DSL changes relative to the baseline period from the 5 GCMs.
The two-layer emulator provides a useful tool to explore the uncertainty of DSL projections over multiple centuries with computational resources that are much less than a GCM requires. It can be calibrated to match assessments of key values like the equilibrium climate sensitivity, and allows the flexibility of simulating forcing conditions intermediate between the RCPs as the patterns are common for different scenarios. However, we should note that the errors between the DSL predicted by two-layer emulator and DSL simulated by the corresponding GCMs are small in middle and low latitudes but relatively large in high latitudes (e.g., the Southern Ocean, Arctic, and subpolar Atlantic). In addition, the two-layer emulator cannot explicitly resolve the non-linear mesoscale effects of the ocean current due to the coarse resolutions of the CMIP5 GCMs that the two-layer pattern scaling relies on. Comparing with the predicted DSL derived from univariate approach, the improvement of using the two-layer approach on predicting DSL has a similar magnitude with the uncertainty of DSL projections over the middle and low latitudes in RCP2.6 and RCP4.5 scenarios during the period of 2271-2290. But the improvement is limited comparing to the uncertainty of DSL projections in RCP8.5. As the non-linear responses of DSL are more obvious in the RCP2.6 and RCP4.5 than in RCP8.5, the notable improvement of using the two-layer approach over the middle- and low-latitudes in the RCP2.6 and RCP4.5 highlight the advantage on improving the DSL projections in these two scenarios.
Loss of land ice (e.g., Greenland Ice Sheet and Antarctic Ice Sheet) is an important contributor not only to GMSL but also to RSL. Glacio-isostatic adjustment (GIA) caused by changes in ice sheet mass loading induces local vertical land motion and associated changes in local sea level. Mass loss of an ice sheet also reduces the gravitational attraction that pulls sea water toward it, causing water to migrate away with a distinct spatial pattern, or “fingerprint”, of sea level change to the global ocean (Mitrovica et al., 2009). Freshwater flux from a melting ice sheet may drastically alter the salinity profile of the near-by ocean, bringing about complex feedbacks involving near-surface ocean stratification, sea ice formation, and corresponding changes in surface temperature, winds, and ocean currents (Bronselaer et al., 2018; Sadai et al., 2020) - each process could have its impact in RSL. Among these factors, only freshwater flux from the ice sheet can significantly affect both global climate and DSL (e.g., Golledge et al., 2019). Despite the importance of polar ice sheets, their contributions to DSL are not included in the current generation coupled climate models. Our study, relying on outputs from climate models participating the CMIP5 project, thus cannot take into account the effects of evolving ice sheets on DSL. In more comprehensive analyses, the effect of land-ice loss should be considered.
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Church et al. (2013)
Title:
Evaluating the ability of process based models to project sea-level change
Key Points:
Evaluation of CMIP5 and other process-based models using observations
Process-based models, when properly calibrated, can reproduce a significant portion of the observed sea-level rise.
Ocean thermal expansion and glacier melting are the two dominant contributors to sea-level rise
The models’ ability to reproduce observed sea-level rise has improved significantly, increasing confidence in their projections for the 21st century.
Keywords:
sea level, climate change, projections
Corresponding author:
John A. Church
Citation:
Church, J. A., Monselesan, D., Gregory, J. M., & Marzeion, B. (2013). Evaluating the ability of process based models to project sea-level change. Environmental Research Letters, 8(1), 014051. doi:10.1088/1748-9326/8/1/014051
URL:
https://iopscience.iop.org/article/10.1088/1748-9326/8/1/014051
Abstract
We evaluate the ability of process based models to reproduce observed global mean sea-level change. When the models are forced by changes in natural and anthropogenic radiative forcing of the climate system and anthropogenic changes in land-water storage, the average of the modelled sea-level change for the periods 1900–2010, 1961–2010 and 1990–2010 is about 80%, 85% and 90% of the observed rise. The modelled rate of rise is over 1 mm yr−1 prior to 1950, decreases to less than 0.5 mm yr−1 in the 1960s, and increases to 3 mm yr−1 by 2000. When observed regional climate changes are used to drive a glacier model and an allowance is included for an ongoing adjustment of the ice sheets, the modelled sea-level rise is about 2 mm yr−1 prior to 1950, similar to the observations. The model results encompass the observed rise and the model average is within 20% of the observations, about 10% when the observed ice sheet contributions since 1993 are added, increasing confidence in future projections for the 21st century. The increased rate of rise since 1990 is not part of a natural cycle but a direct response to increased radiative forcing (both anthropogenic and natural), which will continue to grow with ongoing greenhouse gas emissions.
Introduction
A complete understanding of 20th century sea-level rise has been lacking, with the observed rise over recent decades being larger than projections in the Intergovernmental Panel on Climate Change (IPCC) Third (TAR, Church et al 2001) and Fourth Assessment Reports (AR4, Hegerl et al 2007). As a result, sea-level projections for the 21st century and beyond have been controversial. The omission of ‘rapid’ dynamic ice sheet contributions from the AR4 projections, because of the lack of a published basis for estimating them, compounded this problem. Rahmstorf et al (2007) argued that the observed sea-level rise since 1990 was at or above the upper limit of the TAR projections, and Rahmstorf et al (2012a) argued that both the TAR and AR4 projections were biased low. However, using an improved understanding of the impact of volcanic eruptions on sea level, Church et al (2011a) described the comparison slightly differently, finding that while the observed rise was in the upper quartile of the AR4 projections there was no inconsistency between observations and projections from 1990. Semi-empirical models (SEMs; Rahmstorf 2007, Horton et al 2008, Vermeer and Rahmstorf 2009, Grinsted et al 2010, Jevrejeva et al 2010, 2011, Rahmstorf et al 2012b) have been proposed as an alternative way to estimate future sea-level rise.
Using observations of 20th century temperatures, Bittermann et al (2013) [B13] compared SEM forecasts of 20th century sea-level rise to the observations. They concluded that, when both tide-gauge reconstructions of global mean sea-level and paleo sea-level data were used to estimate the parameters of their SEMs, their forecasts for 1900–2000 (range of 13–30 cm) were in agreement with the observations (14–26 cm). For the period 1961–2003, they also found that when the SEM was trained using the Church and White (2011) [CW11] estimates of GMSL and paleo data up to 1960, the predicted rate of 2.0 mm yr−1 (range of 1.9–2.3 mm yr−1) agreed with the observations within the uncertainties. However, using the Jevrejeva et al (2008) estimate of GMSL, with or without the paleo data, resulted in sea-level projections that were biased high by up to 70%. B13 challenged the sea-level process model community to test their projections for the same period.
Here we evaluate the ability of process based models (the basis of 21st century projections) to simulate 20th century global averaged sea-level rise. These process based models are dependent on our physical understanding of the climate system built up over many years. Modelling sea level requires simulations of oceanic and atmospheric global and regional temperatures. In contrast to the SEMs, the process based models are forced by greenhouse gas concentrations and simulate temperature changes, rather than using observed temperature or radiative forcing changes as input, and they are not trained with observed sea levels and are therefore not sensitive to uncertainties in them. This study builds upon recent progress in understanding the 20th century sea-level budget (Church et al 2011b, Moore et al 2011, Gregory et al 2013a) but focuses clearly on model results and their ability to project sea-level change. We compare individual contributions to sea level calculated with the World Climate Research Programme Coupled Model Intercomparison Project Phase 5 (CMIP5) model results with observations (section 2) and compare the sum of these terms to the observed sea-level rise since 1900, 1961 and 1990, as in B13. We discuss the possibility that some fraction of 20th century sea-level change is possibly due to internally generated variability that is unlikely to be simulated in phase and amplitude by the models (section 3). We also discuss the implications of the results for the observed increase in the rate of rise and for future projection of 21st century sea-level rise (section 4).
Sea-level response to historical radiative forcing and anthropogenic intervention in the water cycle
Ocean thermal expansion
Ocean thermal expansion (figure 1(a)) is available for 25 atmosphere–ocean general circulation models (AOGCMs) participating in the CMIP5 experiment. The simulations used here have been forced with the best estimates of historical radiative forcings up to 2005 and then radiative forcing from the RCP4.5 scenario (Moss et al 2010, Taylor et al 2012) until 2010. However, the preindustrial spin-up and the control simulation for these models assumed zero volcanic forcing and thus the sudden imposition of the negative volcanic forcing in the historical simulations from 1850 results in a negative bias in the estimated ocean thermal expansion (Gregory 2010). To overcome this bias, we have added 0.1 mm yr−1 (±0.05 mm yr−1) ocean thermal expansion to the model results (Gregory 2010, Gregory et al 2013b). We compare the model results to the observational estimates based on the analysis of Domingues et al (2008), updated to 2012 for the upper 700 m, the Levitus et al (2012) analysis from 700 to 2000 m, and a linear trend from 1992 to 2011 for the ocean below 2000 m (Purkey and Johnson 2010). From 1970, when the amount of observational data increases significantly, the models and the observations are not significantly different and the observations are near the centre of the model simulated range.
Figure 1: Comparisons of modelled and observed (a) ocean thermal expansion (observations in blue), (b) glacier contributions, (c) changes in terrestrial storage (the sum of aquifer depletion and reservoir storage) and (d) and the rate of change (10 year centred average) for the terms in (a) to (c). Individual model simulations are shown by grey lines with the model average shown in black (thermal expansion) and purple (glaciers). The estimated glacier contributions estimated by Cogley (2009, green), Leclercq et al (2011, red) and using the model of Marzeion et al (2012, dark blue) forced by observed climate are also shown in (b). All curves in (a) and (b) are normalized over the period 1980–1999 and the colours in (d) are matched to earlier panels.


Figure 1: Comparisons of modelled and observed (a) ocean thermal expansion (observations in blue), (b) glacier contributions, (c) changes in terrestrial storage (the sum of aquifer depletion and reservoir storage) and (d) and the rate of change (10 year centred average) for the terms in (a) to (c). Individual model simulations are shown by grey lines with the model average shown in black (thermal expansion) and purple (glaciers). The estimated glacier contributions estimated by Cogley (2009, green), Leclercq et al (2011, red) and using the model of Marzeion et al (2012, dark blue) forced by observed climate are also shown in (b). All curves in (a) and (b) are normalized over the period 1980–1999 and the colours in (d) are matched to earlier panels.
Modelled thermal expansion (figures 1(a), (d)) falls slightly following the volcanic eruption of Santa Maria in 1902. The rate of expansion is then relatively constant up until the eruption of Mt Agung in 1963 when there is a significant fall in sea level. There are similar falls in sea level following the eruptions of El Chichon in 1982 and Mt Pinatubo in 1991. The increase in sulfur dioxide emissions by more than a factor of two from 1950 to 1975 (Smith et al 2011) results in an increasingly negative aerosol forcing that partially offsets the increasing greenhouse gas concentrations leading to a slower rate of warming (and thermal expansion) after 1960 (Church et al 2011b). Over the 20th century, there is a clear increase in the rate of rise with the fastest rate occurring from 1993. This latter increase is a result of increasing greenhouse gas concentrations, recovery from impacts of the Mt Pinatubo eruption (Gregory et al 2006, Gleckler et al 2006, Church et al 2005, Domingues et al 2008, Gregory et al 2013a) and falling sulfur dioxide emissions from 1975 to 2000 (Smith et al 2011).
Glacier contributions
Marzeion et al (2012) use an (offline) glacier model forced by regional surface temperatures and precipitation from AOGCM CMIP5 climate simulations to estimate glacier contributions for the 20th century (figures 1(b), (d), grey and purple lines). Their model is calibrated using individual glacier mass balance observations also used by Cogley (2009), such that the Marzeion et al (2012) results are neither strictly independent from the Cogley (2009) results, nor from observations. However, the dependency is very weak, since only about 0.1% of all the world’s glaciers are included in these observations (substantially less in terms of surface area and ice volume, and with a mean time series length of only 15 years). Two different validation methods in Marzeion et al (2012) show that the model is able to reconstruct observed glacier changes independent of the observations, such that the combined contribution of all the glaciers to sea level is in fact only weakly dependent on the observations of individual glacier contributions. From 1950, the model results (15 available models) used by Marzeion et al are not significantly different from the observed estimated changes in glacier mass of Cogley (2009) and the estimates based on glacier length from Leclercq et al (2011). The Marzeion et al modelled rate of rise is almost constant during the first half of the century up until 1960 but larger than the Leclercq et al estimate (that also relies on Cogley), then smaller until the 1990s (figure 1(d)), after which it increases. Loss of glacier area at low altitudes combined with the stabilization of temperatures for the 1950–1975 period could have contributed to this slowing of the rate of glacier contribution.
Greenland and Antarctic contributions
The contributions of the Greenland and Antarctic Ice Sheets for the 20th century are poorly determined. Observational estimates for 1993–2011 (Shepherd et al 2012) indicate a net contribution of about 11 mm, about two thirds of this from Greenland. Models of surface mass balance (using AOGCM results) for Greenland agree with the increased surface mass loss over the last two decades but indicate little impact on sea level over previous decades, and with divergent results for the first half of the 20th century. (See Gregory et al (2013a) for a full discussion.) Recent model results for four major Greenland outlet glaciers (Helheim, Jakobshavn, Petermann and Kangerdlugssuaq) forced by changes in ocean temperatures (Nick et al 2009, and personal communication) indicate a contribution of order 0.6 mm yr−1 for 2000–10, consistent with the observational estimates. However, we are unaware of any completed model simulations of the Greenland Ice Sheet contribution for the 20th century using the new generation of ice sheet models.
For Antarctica, Levermann et al (2012) have recently completed an ice sheet model simulation for the 20th century using ocean temperatures on the shelf near Antarctica and atmosphere–ice exchange from the CMIP5 AOGCMs, with an allowance for a delay of the warming to penetrate underneath the ice shelves. The 20th century contribution is only 4 mm, mostly since 1990, similar to recent observational estimates (Shepherd et al 2012).
These results indicate significant progress in modelling ice sheet response to climate and ocean forcing. However, as they are as yet incomplete, we have not included these new model results here. Instead, we include estimates of these terms in section 3 and discuss the implications in section 4.
Land-water storage
Internally generated climate variability influences the amount of water stored as soil moisture and in lakes, rivers and reservoirs. On short timescales, the rate of change in the storage can be several millimetres leading to rapid rates of sea-level change (Boening et al 2012). However, over decadal timescales the net contribution is small (Ngo-Duc et al 2005) and hence for the comparison we ignore this contribution.
There are also direct human related interventions in the hydrological cycle that impact the amount of water stored on land. This occurs principally through the building of reservoirs (Chao et al 2008, Lettenmaier and Milly 2009) and the depletion of groundwater (Konikow 2011, Wada et al 2012). For reservoir storage, we use the estimates of Chao et al with no allowance for seepage (as in Gregory et al 2013a). We assume that the reservoirs are on average 85% full (with a range of 70–100%). For groundwater depletion, we average the observational estimates of Konikow (2011) and model results of Wada et al (2012). Over the first half of the 20th century, both of these terms are small (figure 1(c)). After 1950, the significant increase in the rate of dam building leads to negative contribution to sea-level change. From the 1980s, a slowing in the rate of dam building and an increase in the rate of groundwater depletion leads to a small positive contribution to sea-level rise (figures 1(c) and (d)).
Observed and modelled sea-level change 1900–2012
We compare the sum of the ocean thermal expansion, glacier and estimated land-water contributions (available for 13 models) with observational estimates of global mean sea level of CW11 and Ray and Douglas (2011) [RD]; figure 2. Both estimates are similar over the 20th century (RD has a slightly larger trend), with a broad maximum in the rate of rise from 1930 to 1950, a minimum about 1960 and then a rising trend to the end of the records. Both series have a minimum in the rate of rise in the 1920s and a maximum in the 1970s, but it is unclear if these two features are robust or an indication of the inadequacy of the available sea-level data.


There are also direct human related interventions in the hydrological cycle that impact the amount of water stored on land. This occurs principally through the building of reservoirs (Chao et al 2008, Lettenmaier and Milly 2009) and the depletion of groundwater (Konikow 2011, Wada et al 2012). For reservoir storage, we use the estimates of Chao et al with no allowance for seepage (as in Gregory et al 2013a). We assume that the reservoirs are on average 85% full (with a range of 70–100%). For groundwater depletion, we average the observational estimates of Konikow (2011) and model results of Wada et al (2012). Over the first half of the 20th century, both of these terms are small (figure 1(c)). After 1950, the significant increase in the rate of dam building leads to negative contribution to sea-level change. From the 1980s, a slowing in the rate of dam building and an increase in the rate of groundwater depletion leads to a small positive contribution to sea-level rise (figures 1(c) and (d)).
Observed and modelled sea-level change 1900–2012
We compare the sum of the ocean thermal expansion, glacier and estimated land-water contributions (available for 13 models) with observational estimates of global mean sea level of CW11 and Ray and Douglas (2011) [RD]; figure 2. Both estimates are similar over the 20th century (RD has a slightly larger trend), with a broad maximum in the rate of rise from 1930 to 1950, a minimum about 1960 and then a rising trend to the end of the records. Both series have a minimum in the rate of rise in the 1920s and a maximum in the 1970s, but it is unclear if these two features are robust or an indication of the inadequacy of the available sea-level data.
The sum of the modelled ocean thermal expansion, glacier, and terrestrial storage contributions from 1900 to 2010 (figure 2(a)) ranges from 110 mm to almost 200 mm with a model average of 153 mm. The spread of models encompasses the GMSL estimate of CW11 but is slightly less than RD. The average of the model results explains about 80% of the observed rise. The average modelled rate of sea-level rise (figure 2(b)) is more than 1 mm yr−1 prior to 1950, as a result of early 20th century warming and thermal expansion and increased glacier melting, but is somewhat less than the observed rate over 1930–50. The average modelled rate of rise decreases to less than 0.5 mm yr−1 in the 1960s before increasing again to reach a maximum of 3 mm yr−1 in 2000, about double the 20th century average and substantially greater than the modelled rate of rise in the first half of the 20th century. The slower rate of rise from 1950 to 1980 is likely a result of the impact of volcanic eruptions, the increase in tropospheric aerosol loading (emissions peak in the 1970s) on the modelled ocean thermal expansion and glacier melting contributions, a loss of glacier area following early 20th century melting and an increase in the rate of reservoir storage.
Figure 2: The sum of the modelled contributions from ocean thermal expansion, increased glacier melting and changes in land-water storage. The light grey lines are individual models with the black line the model mean. The 20th estimates of global mean sea level are indicated by the blue (CW11) and green (RD) lines with the shading indicating the uncertainty estimates (two standard deviations). The satellite altimeter data since 1993 is shown in red. The adjusted model mean (dashed black line) is the model mean after an allowance for the impact of natural variability on glacier contributions and a potential long-term ice sheet contribution are included. The results are given (a) for the period 1900–2010, (b) the rates of sea-level change for the same period, (c) for 1961–2010, and (d) for 1990–2010. The dotted black line is after inclusion of the Shepherd et al (2012) ice sheet observational estimates but excluding the peripheral glacier contribution (to avoid double counting). The red dot is the average rate from the altimeter record.
For the period since 1961 (figure 2(c)), the modelled rise ranges from about 50 to 110 mm and encompasses the observed rise of close to 90 mm, with the model average rise of 75 mm explaining about 85% of the observed rise. Since 1990 (the start of the projections for the TAR and the AR4; figure 2(d)), the modelled sea-level rise ranges from 30 to 65 mm and encompasses the observed rise of about 55 mm, with the average model rise of 51 mm explaining over 90% of the observed rise. The model average rate over 1993–2010 of 3 mm yr−1 is almost equal to the rate of 3.2 ± 0.4 mm yr−1 observed with satellite altimeters (with both rates being very linear). The increased rate of the modelled rise from 1980 to 2000, and particularly after 1993, is a result of continued increases in greenhouse gas concentrations, the recovery of the climate system from the series of volcanic eruptions (particularly Mt Pinatubo in 1993), decreasing sulfur dioxide emissions from 1975 to 2000 and increasing land-water contributions.
Other effects on sea-level change
There are at least two potential additional contributions to 20th century sea-level change. Firstly, the ice sheets are often assumed to have been in a state of approximate mass balance, hence making zero net contribution to sea level before the major increase in greenhouse gas emissions of the 20th century. However, the long response time of the Antarctic and (to a lesser extent) Greenland ice sheets means that there may be a small ongoing contribution to sea-level change due to climate change in previous centuries or millennia (Huybrechts et al 2011). An ongoing contribution of 0.0–0.2 mm yr−1 was considered in the sea-level budget studies of Gregory et al (2013a) and Church et al (2011b). Here we have added a 0.1 mm yr−1 contribution to the above modelled estimates (10 mm over the 20th century).
Secondly, there may be contributions related to internally generated variability on decadal timescales (Delworth and Knutson 2000). Marzeion et al (2012) have also computed glacier mass changes using observed rather than simulated temperature change (figure 1(b), blue line). An additional contribution of about 20 mm is estimated to occur between 1920 and 1960, with the largest additional contribution in the 1930s (the difference between the blue and purple lines in figure 1(b)). While the Leclercq et al (2011) estimates from measurements of glacier length indicate a smaller overall contribution during the 20th century, they also give a greater rate of mass contribution from 1920 to 1940 than for earlier and later periods. These additional early 20th century contributions are a result of a regional warming over Greenland during this period (Chylek et al 2004).
When these two terms are added to the AOGCM results, the sea-level rise over the 20th century (figure 2(a), dashed line) is 176 mm, which is over 90% of the observed GMSL estimate, and both observed time series lie within the model spread (the individual model results have not been replotted after the addition of these terms). The modelled rate of rise in the first half of the 20th century is now closer to the observed rate and the observed rate lies within the spread of the model rates through nearly all of the century, although the timing of the faster rate of rise occurs slightly earlier in the model results than in the observations. There is little change to the simulations from the additional terms for the periods since 1961 and 1990 and the average model results for these periods remain within 20% of the observed rise (figures 2(c), (d)). The observational estimates (Shepherd et al 2012) indicate a small 20th century ice sheet contribution that would further close the gap between the observed and modelled sea-level rise to about 10% or better, as depicted by the dotted lines in figure 2.
Discussion


Secondly, there may be contributions related to internally generated variability on decadal timescales (Delworth and Knutson 2000). Marzeion et al (2012) have also computed glacier mass changes using observed rather than simulated temperature change (figure 1(b), blue line). An additional contribution of about 20 mm is estimated to occur between 1920 and 1960, with the largest additional contribution in the 1930s (the difference between the blue and purple lines in figure 1(b)). While the Leclercq et al (2011) estimates from measurements of glacier length indicate a smaller overall contribution during the 20th century, they also give a greater rate of mass contribution from 1920 to 1940 than for earlier and later periods. These additional early 20th century contributions are a result of a regional warming over Greenland during this period (Chylek et al 2004).
When these two terms are added to the AOGCM results, the sea-level rise over the 20th century (figure 2(a), dashed line) is 176 mm, which is over 90% of the observed GMSL estimate, and both observed time series lie within the model spread (the individual model results have not been replotted after the addition of these terms). The modelled rate of rise in the first half of the 20th century is now closer to the observed rate and the observed rate lies within the spread of the model rates through nearly all of the century, although the timing of the faster rate of rise occurs slightly earlier in the model results than in the observations. There is little change to the simulations from the additional terms for the periods since 1961 and 1990 and the average model results for these periods remain within 20% of the observed rise (figures 2(c), (d)). The observational estimates (Shepherd et al 2012) indicate a small 20th century ice sheet contribution that would further close the gap between the observed and modelled sea-level rise to about 10% or better, as depicted by the dotted lines in figure 2.
Discussion
Ocean thermal expansion and the increased melting of glaciers are the two dominant contributions to 20th century sea-level rise in the simulations, with a smaller contribution from changes in land-water storage. Each of these components has its own unique temporal dependence. The model results indicate that most of the variation in the thermal expansion and glacier contributions to global mean sea level is a response to radiative forcing of the climate system from changes in concentrations of greenhouse gases, stratospheric volcanic aerosol and tropospheric anthropogenic aerosol. Observations in the latter half of the 20th century provide strong support for and confidence in the model simulations of these components. However, since parameters in the glacier model are estimated from observations in the latter half of the 20th century, the evaluation of the glacier models is not a fully independent test of their skill.
Not all 20th century sea-level rise is necessarily externally forced. There is evidence for an enhanced glacier contribution in the first half of the 20th century (Marzeion et al 2012, Leclercq et al 2011). Since climate models can simulate early 20th century global averaged temperature well (Stott et al 2000), the difference between the two glacier estimates may be partly related to regional climate changes (rather than global averaged temperatures), although natural variability impacts both regional and global averaged temperatures (Delworth and Knutson 2000). The extent to which internally generated climate variability can lead to enhanced sea-level rise deserves further investigation. If the apparent impact during the first half of the 20th century was repeated in the future, it would increase projections for the 21st century by the order of 20 mm. However this additional sea-level rise will enter the calibrations of SEMs that use global averaged temperatures (or radiative forcing) and thus will impact the SEM projections.
The range of the model simulations over the three periods, and particularly since 1961 and 1990, encompasses the observed sea-level rise with the model mean within 20% (about 10% since 1990) of the observed rise. Experience with multi-model ensembles is that they generally outperform individual models (Weigel et al 2008, 2010), but specific results are not available for sea level. The agreement of observations with the model mean represents a significant improvement since the IPCC TAR (Church et al 2001) and AR4 (Hegerl et al 2007) and is a reason for increased confidence in the next generation of global mean sea-level projections. This agreement also means that it should now be possible to attribute 20th century sea-level rise to the various climate forcings. Reasons for the improvement include allowance for the omission of volcanic forcing in the spin-up of the AOGCMs, more complete representation of the radiative forcing driving the AOGCMs, a larger initial glacier mass (Arendt et al 2012) and more complete observations of glacier mass loss (Cogley 2009). Also, the Marzeion et al glacier model is able to reconstruct observed glacier changes independent of the observations and is an important part of the improved representation of 20th century sea-level rise.
Significant challenges remain. It is likely that the model spread does not cover the full range of possibilities because of systematic uncertainties that are common to many models. Hence, the model spread found here (about 40% of the observed rise for the 20th century and more than 50% since 1961) may underestimate the full uncertainty, particularly as it was not possible to include models of the ice sheet components. The observational estimates of ice sheet contributions since 1993 further close the gap between the observed and modelled sea-level rise to 10% or better. This evaluation is also incomplete as the ice sheet contributions to date are only a small fraction for the potential longer-term contributions. Careful comparison of the new generation of ice sheet simulations with observations is required to critically evaluate them.
Chambers et al (2012) argue there is an apparent 60 year cycle in the observed sea-level record. Similar variability is present in the forced simulations of the first half of the 20th century but is enhanced when the additional glacier contributions are included in the sum of terms. For the latter half of the 20th century, the observed minimum in the rate of sea-level rise in the 1960s (a deceleration from 1940 to 1970) and the subsequent increase in rate to about twice the 20th century average at the end of the record is present in the forced sea-level estimates. This increase is principally in response to increasing greenhouse gas concentrations and a combination of changing volcanic forcing and tropospheric aerosol loading, leading to a larger ocean thermal expansion and increased glacier melting. There is an additional contribution of less than 20% from anthropogenic interference in the hydrological cycle (figures 1(c) and (d)). Thus, the observed increased rate of rise since 1990 is not part of a natural cycle but a direct response to increased radiative forcing (both anthropogenic and natural) of the climate system. This radiative forcing will continue to increase with ongoing greenhouse gas emissions. The simulation of the observed 20th century sea-level rise and its variability within the uncertainties is a reason for increased confidence in projections of 21st century sea-level rise in future projections.


Church and White (2011)
Title:
Sea-Level Rise from the Late 19th to the Early 21st Century
Keywords:
Sea level, Climate change, Satellite altimeter, Tide gauge
Corresponding author:
Church
Citation:
Church, J. A. & White, N. J. (2011). Sea-Level Rise from the Late 19th to the Early 21st Century. Surveys in Geophysics, 32, 585–602.
URL:
https://link.springer.com/article/10.1007/s10712-011-9119-1
Abstract
We estimate the rise in global average sea level from satellite altimeter data for 1993Ð2009 and from coastal and island sea-level measurements from 1880 to 2009. For 1993-2009 and after correcting for glacial isostatic adjustment, the estimated rate of rise is 3.2 ± 0.4 mm year$^{-1}$ from the satellite data and 2.8 ± 0.8 mm year$^{-1}$ from the in situ data. The global average sea-level rise from 1880 to 2009 is about 210 mm. The linear trend from 1900 to 2009 is 1.7 ± 0.2 mm year$^{-1}$ and since 1961 is 1.9 ± 0.4 mm year$^{-1}$. There is considerable variability in the rate of rise during the twentieth century but there has been a statistically significant acceleration since 1880 and 1900 of 0.009 ± 0.003 mm year$^{-2}$ and 0.009 ± 0.004 mm year$^{-2}$, respectively. Since the start of the altimeter record in 1993, global average sea level rose at a rate near the upper end of the sea level projections of the Intergovernmental Panel on Climate ChangeÕs Third and Fourth Assessment Reports. However, the reconstruction indicates there was little net change in sea level from 1990 to 1993, most likely as a result of the volcanic eruption of Mount Pinatubo in 1991.
Introduction
Rising sea levels have important direct impacts on coastal and island regions where a substantial percentage of the world’s population lives (Anthoff et al. 2006). Sea levels are rising now and are expected to continue rising for centuries, even if greenhouse gas emissions are curbed and their atmospheric concentrations stabilized. Rising ocean heat content (and hence ocean thermal expansion) is an important element of climate change and sea-level rise. The remaining contributions to sea-level rise come principally from the melting of land ice: glaciers and ice caps (which include the small glaciers and ice caps fringing the major ice sheets) and the major ice sheets of Antarctica and Greenland, with additional contributions from changes in the storage of water on (or in) land. (See Church et al. 2010 for a summary of issues). Correctly estimating historical sea-level rise and representing global ocean heat uptake in climate models are both critical to projecting future climate change and its consequences. The largest uncertainty in projections of sea-level rise up to 2100 is the uncertainty in global mean sea level (GMSL) and thus improving estimates of GMSL rise (as well as regional variations in sea level) remains a high priority.
Since late 1992, high quality satellite altimeters (TOPEX/Poseidon, Jason-1, and OSTM/Jason-2) have provided near global measurements of sea level from which sea-level rise can be estimated. However, this altimeter record is still short (less than 20 years) and there is a need to know how sea level has varied over multi-decadal and longer time scales. Quantifying changes in the rate of sea-level rise and knowing the reasons for such changes are critical to improving our understanding of twentieth century sea-level rise and improving our projections of sea-level change for the twenty first century and beyond.
For the period prior to the altimeter record, estimates of sea-level change are dependent on a sparsely distributed network of coastal and island tide-gauge measurements (Wood.worth and Player 2003). Even today, there are many gaps in the global network of coastal and island sea-level measurements and the network was sparser early in the twentieth century and in the nineteenth century. Many previous studies have used the individual sea-level records (corrected for vertical land motion) to estimate the local rate of sea-level rise (as a linear trend; e.g. Douglas 1991) and some studies attempted to detect an acceleration in the local rate of sea-level rise (Woodworth 1990; Woodworth et al. 2009; Douglas 1992). However, these individual records have considerable interannual and decadal var.iability and thus long records are required to get accurate estimates of the local trends in sea level (Douglas 2001). These authors assumed that these long-term trends are either representative of the global averaged rise or a number of records have been averaged, in some cases regionally and then globally, to estimate the global average rate of rise. However, the modern satellite record has made it clear that sea level is a dynamic quantity and it does not rise uniformly around the globe.
Sea level at any location contains the inßuences of local and regional meteorological effects (including storm surges), modes of climate variability (for example the El Nino-Southern Oscillation) and long-term trends (from both the ocean surface and land move.ments), including the impact of anthropogenic climate change. As the altimeter record has clearly demonstrated, GMSL has much less short term variability (more than an order of magnitude) than sea level at individual locations because while the volume of the oceans is nearly constant the distribution changes with time. While the variability at individual locations can be minimised by low pass filtering, there remains significant energy at yearly to decadal periods that may be either positively or negatively correlated between stations, thus confounding estimates of GMSL rise when few records are available.
To date, there have been two approaches to determining time series of GMSL from coastal and island tide gauges. The first and most straight forward approach averages the sea-level records (corrected for land motion) from individual locations. When there are only a small number of locations with continuous records, this approach is relatively straight forward, although care must be taken to remove data inhomogeneities. When more gauges are used, the records usually have different lengths and starting times. It is then necessary to average the rates of rise over some time step and integrate the results to get the sea-level change. Holgate and Woodworth (2004) used this approach and Jevrejeva et al. (2006) used a virtual station method of averaging neighbouring station sea-level changes in several regions and then averaging to get the global mean sea-level change. No attempt was made to interpolate between the locations of observations and thus to estimate deep ocean sea level. Thus these are essentially estimates of coastal sea-level change. However, note that White et al. (2005) argued that over longer periods the rates of coastal and global rise are similar.
The second approach uses spatial functions which represent the large-scale patterns of variability to interpolate between the widely distributed coastal and island sea-level observations and thus to estimate global sea level (as distinct from coastal sea level). This technique was first developed by Chambers et al. (2002) for interannual sea-level variability and extended by Church et al. (2004) to examine sea-level trends. The Church et al. approach uses the Reduced Space Optimal Interpolation technique (Kaplan et al. 2000) developed for estimating changes in sea-surface temperature and atmospheric pressure. The spatial functions used are the empirical orthogonal functions (EOFs) of sea-level variability estimated from the satellite altimeter data set from TOPEX/Poseidon, Jason-1 and OSTM/Jason-2 satellite data, which now cover the period from January 1993 into 2011. We use data from January 1993 through December 2009 in this paper.
Here, we update previous estimates of GMSL rise for the period of the instrumental record using a longer (17 years) altimeter data set and an expanded in situ sea-level observational data set extending back to 1860. We use the Reduced Space Optimal Interpolation technique to quantify the rate of sea-level rise and the changes in the rate since 1880.
Methods and Data
In Situ Sea-Level Data
We use monthly sea-level data downloaded from the Permanent Service for Mean Sea Level (PSMSL; Woodworth and Player 2003) web site (http://www.psmsl.org) in August 2010. Careful selection and editing criteria, as given by Church et al. (2004) were used. The list of stations used in the reconstruction is available on our web site at http://www.cmar.csiro.au/sealevel/sl_data_cmar.html. Tide gauge records are assigned to the nearest locations (with good satellite altimeter data) on the 1˚-by-1˚ grid of the satellite altimeter based EOFs. Where more than one record is assigned to a single grid point they are averaged. Changes in height from 1 month to the next are stored for use in the reconstruction.