Corresponding author: K. Richter, Geophysical Institute, University of Bergen, Allégaten 70, 5007 Bergen, Norway. (Kristin.Richter@gfi.uib.no)
 We investigate the effect of self-attraction and loading (SAL) induced by the projected accumulation of sea water on shallow continental shelf areas. Using output from a climate model, we compute 21st century changes in regional steric sea surface height and find that steric changes are largest over the deep ocean and relatively small on the shallow continental shelves. The resulting redistribution of sea water towards the shelf areas leads to mass accumulation on the shelves and therefore to increased gravitational attraction as well as increased loading on the sea floor. We find that, depending on the scenario and region, SAL effects may result in an additional sea level rise of 1–3 cm on the world's continental shelf areas by the end of the 21st century. These estimates are at most 15% of the combined changes in sea surface height induced by redistribution of water masses and steric expansion.
 Global sea level is rising [Church et al., 2011] and is expected to continue to rise on a multi-centennial to millennial time scale [Li et al., 2012; Meehl et al., 2012; Yin, 2012]. Due to a variety of complex and interconnected processes in the ocean, on land and in the atmosphere, the observed increase in global mean sea level shows large short-term variations and regional changes in sea surface height vary significantly [Cazenave et al., 2008]. The rate of change in global mean sea level and regional differences superimposed on the long-term change are expected to continue in the future [Yin et al., 2010; Slangen et al., 2012; Yin, 2012]. To accurately assess regional rates of changes in sea level, it is important to quantify all processes that contribute to regional sea level variability.
 Warming of the ocean and the resulting thermal expansion lifts the sea surface and the water column's center of mass. The warming that takes place below the depth of an ocean shelf, typically a few hundred meters, will induce a horizontal pressure gradient with elevated pressure over the deep ocean compared to the shallow shelves. To balance this pressure gradient, water masses will flow from areas of larger water depths (ocean interiors) onto shallow continental shelf areas [Landerer et al., 2007; Yin et al., 2010]. The resulting change in the water mass loading will be manifested as changes in ocean bottom pressure.
 Up to now, most of the ocean warming is found in the uppermost 500–1000 m of the water column. However, Purkey and Johnson  reported an abyssal warming equivalent to 0.053 ± 0.017 mm/yr global sea level change in the 1990s and 2000s. In a recent study, Levitus et al.  show that substantial amounts of heat are found in the intermediate and deep ocean, well below the depth of coastal shelf areas. Even if, through aggressive mitigation measures, global average temperature could be stabilized, the deep ocean will continue to warm thus contributing to rising sea levels far into the future [e.g., Li et al., 2012; Meehl et al., 2012].
 Any change in the mass distribution within the ocean and/or solid earth will modify the Earth's gravitational field and will result in a further readjustment of the mass distribution. In addition, accumulation of mass on shallow shelf areas will increase the loading and lead to a deformation of the solid earth surface. The combination of these processes is referred to as self-attraction and loading (SAL) effect [Gordeev et al., 1977]. Focus so far has been on the effect of melting land ice, i.e., the redistribution of mass from land to the oceans, and the subsequent adjustment of the sea surface to the perturbed gravity field [Tamisiea et al., 2003; Riva et al., 2010]. Recently, Tamisiea et al.  and Vinogradova et al.  extended the SAL effect resulting from changes in hydrology, atmospheric loading and ocean dynamics, while Kuhlmann et al.  explicitly incorporated SAL effects in a baroclinic ocean model.
 In this study, the shelf mass loading effect due to redistribution of ocean mass between the deep and shallow regions of the ocean is examined in 21st century projections of a state-of-the-art atmosphere-ocean general circulation model. The magnitude of SAL effects induced by the redistribution of ocean mass is then estimated. Since most of the shelf regions are found at high northern latitudes, the net effect is expected to be largest there. In a recent study, Gregory et al.  estimated the SAL effect to vary from − 4 to + 14% of the global mean sea level rise due to thermal expansion. They do not, however, go into details as to how SAL effects are modeled.
 The following section describes the data and methods employed in this study. The results are presented in Section 3 and discussed in Section 4.
2 Data and Methods
 Regional changes in ocean bottom pressure (OBP) and steric height are analyzed by using output from version one of the intermediate resolution version of the Norwegian Earth System Model, NorESM1-M [Bentsen et al. 2012; Iversen et al. 2012]. NorESM1-M is based on the Community Climate System Model version 4 [Gent et al. 2011; Vertenstein et al., 2010], but utilizes an ocean module based on the Miami Isopycnic Coordinate Ocean Model (MICOM, Bleck and Smith ; Bleck et al. ). The ocean component of the model conserves mass (does not obey the Boussinesq, or constant density, approximation) and is therefore well suited to capture sea surface height changes associated with changes in the density of sea water, and to analyze changes in OBP. The model has a horizontal resolution of approximately 2 ° for the atmosphere and land components and 1 ° for the ocean and ice components.
 In this study, we use model output from the three representative concentration pathway (RCP) scenarios RCP2.6, RCP4.5 and RCP8.5 [Van Vuuren et al. 2011] for the time period 2006–2100. The RCPs represent, in order, an emission scenario tailored towards the 2 ° target (that the global mean temperature by 2100 should not exceed 2 ° compared to the pre-industrial climate), an emission scenario with rather strong reductions in greenhouse gas emissions, and a business-as-usual scenario. The presented integrations are part of the fifth phase of the Coupled Model Intercomparison Project (CMIP5) [Taylor et al., 2012]. The model output is corrected for climate drift by subtracting the linear trend from a preindustrial control run (the latter with fixed composition of atmospheric greenhouse gases and aerosol particles) from the RCP runs. As shown in Bentsen et al. , this trend is, in general, weak. Correcting for climate drift does not affect the results of this study (not shown).
 To assess the redistribution of water masses within the worlds oceans, we compute the changes in OBP. In this study, its units are mbar, and here, we employ the approximation 1 mbar 蝶 1 cm. Changes in atmospheric loading are not included in the OBP fields used in this study. Steric height is computed by vertical integration of the specific volume anomaly
where the integration is from the bottom to the surface. The reference density ρ0 is evaluated at the temperature 0 °C and the salinity of 35 psu, and ρ is the in situ density of sea water. All spatial fields are remapped to a regular, horizontal grid with 1.0-by-1.0 degree resolution.
 The OBP fields are further interpolated onto a 0.5-by-0.5 degree grid and used as input to a code that computes SAL effects by solving the sea level equation [Farrell and Clark, 1976] through a pseudo-spectral approach [Mitrovica and Peltier, 1991], including the effect of changes in the earth rotation [Milne and Mitrovica, 1998]. In order to allow for the computation of the SAL effects due to ocean dynamics, we have modified the conventional approach to solving the sea level equation by complementing the load function (equation 12 in Mitrovica and Peltier ) with a term representing OBP changes [Tamisiea et al., 2010]. Note that in our approach, the OBP fields are used as a static load, meaning that the resulting SAL effects will determine the equilibrium ocean configuration, but they are not coupled to the general circulation model. The solid earth is modeled as a spherically layered elastic body, with densities and mechanical properties based on the Preliminary Reference Earth Model [Dziewonski and Anderson, 1981]. The solutions are truncated at spherical harmonic degree and order 360.
 The left column of Figure 1 shows the simulated changes in steric height between the first and the last decade of the RCP runs. The global average change is 13 cm for RCP2.6, 18 cm for RCP4.5, and 28 cm for RCP8.5, which are identical to the ensemble mean values from the CMIP5-ensemble of models [Yin, 2012]. All three scenarios exhibit considerable regional variations on top of the global mean steric height change. Common to all scenarios is a strongly reduced sea level rise on the continental shelves compared to the ocean interior. This results in sharp cross-shelf gradients in steric sea surface height. As expected, the gradient increases with increasing emissions (i.e., with increasing warming in the deep ocean). The strongest gradient is found in RCP8.5 along the eastern coast of North America where a local maximum in the steric height anomaly off the coast amplifies the gradient.
 The middle column of Figure 1 displays the corresponding change in OBP for the three RCP runs. All scenarios show an increase in ocean mass on the shallow shelf areas at the expense of the deep ocean regions. For the RCP2.6 scenario, the response is weakest with a maximum increase corresponding to 10 cm on the Arctic shelves. The strongest effect occurs for the RCP8.5 scenario with up to 40 cm equivalent sea level rise along the North American east coast, and 20–30 cm on the shallow water regions in the Arctic, south East Asia, South America and Australia. The increased OBP on the shelves are compensated by a decrease of OBP of around 5 cm equivalent sea surface height in the Atlantic Ocean and somewhat less in the Arctic Ocean, with rather uniform and minor changes in the Pacific Ocean. The drop in OBP is particularly evident in the western subtropical North Atlantic, showing a basin-scale drop corresponding to about 10 cm.
 Common to all scenarios is a net redistribution of water masses from the southern to the northern hemisphere as most of the continental shelves are located in the northern hemisphere. For RCP2.6 (RCP8.5), the change in the mean sea surface height due to ocean mass redistribution is + 1.2 ( + 2.1) cm for the northern and − 0.9 ( − 1.5) cm for the southern hemisphere.
 Figure 2 shows the SAL fingerprint induced by the redistribution of ocean mass defined as the relative motion between the ocean surface and the solid earth (relative sea level, RSL). For all scenarios, SAL effects cause a moderate sea level increase over the shelf areas of about one order of magnitude smaller than the forcing. Mean RSL over the shelves, defined as those areas with a bathymetry shallower than 700 m, is 0.8, 1.0 and 1.5 cm for scenarios RCP2.6, RCP4.5 and RCP8.5, respectively. The regional RSL increase can be up to 1–3 cm on high northern latitude shelves, depending on the scenario. Outside the shelves, the largest SAL effects are found over the Atlantic, where negative OBP changes drive an additional sea level fall of up to 2 cm in RCP8.5.
 The SAL fingerprint is smoother than the input OBP fields for two reasons: (1) the sea surface represents an equipotential surface of the gravity field, which is smoother than the originating mass distribution as a result of Newton's gravitational law; (2) flexural properties control the deformation of the solid earth that acts as a low-pass filter. The same flexural properties also cause the peak RSL signal to be located towards the central areas of the shelves, even if the largest OBP changes occur close to the coast (as clearly visible in the Barents Sea for scenario RCP8.5).
 Due to the smoothness of the SAL fingerprints and the fact that most of the shelves are located in the northern hemisphere, the SAL effects are not always correlated to the local OBP fields. This is, for example, visible in the Arctic, where the resulting RSL changes are positive even in areas of negative OBP changes. Conversely, the northward migration of water further enhances the negative OBP changes in the Southern Ocean. The meridional mass displacement induces a motion of the geocenter of 4.9, 5.7 and 8.0 mm respectively (century-averaged rates are smaller than 0.1 mm/yr, or well below today's accuracy in the determination of the Earth's reference frame).
4 Summary and Conclusion
 Global steric sea level rise in NorESM1-M is 13, 18 and 28 cm for RCP2.6, RCP4.5 and RCP8.5, respectively. However, modeled sea level rise on the shallow shelf areas off most of the coastlines is mostly due to redistribution of water masses and only to a minor degree caused by local steric expansion. The right column of Figure 1 shows the change in sea surface height taking into account steric changes and sea water redistribution. The sharp cross-shelf gradients originating from steric changes only (Figure 1, left) are eliminated. The remaining sea surface height gradients are related to localized (but still large-scale) heat and fresh water anomalies in the ocean, and to changes in ocean circulation.
 The additional change in sea level due to SAL effects is relatively small compared to the combined regional changes due to steric changes and sea water redistribution. The fractional magnitude and pattern for RCP4.5 is depicted in Figure 3. The shown distribution is also representative for the other scenarios, and may therefore be taken as scalable contribution of the SAL effects examined here. The large negative and positive anomalies in the Southern Ocean result from the relatively small changes in sea surface height in this area (Figure 1, right column).
 The SAL effect is negligible over the Pacific Ocean away from the East Asian shelves, but reduces the sea level rise in the entire interior Atlantic Ocean Atlantic south of 30 °N by up to 10%. The SAL effect increases sea level rise by up to 15% in the Barents Sea and between 5 and 10% on the shallow shelves in high northern latitudes (Figure 3).
White et al.  found no difference between coastal and global sea level rise during the period 1950–2000. Our results suggest that the sea level rise on shallow shelf areas in the 21st century may be 5–6% larger (3–5% if the average is taken over coastal grid boxes only) than the globally averaged sea level rise due to thermal expansion. This is consistent with Gregory et al.  who found a coastal sea level rise 3% stronger than the global mean.
 The sea surface at rest represents an equipotential surface and thus an ocean dynamic equilibrium state. Modifications due to the SAL effect induce new equipotential surfaces without lateral gravity gradients and therefore no anomalous currents. However, to equilibrate to the new equipotential surface, adjustment processes have to take place through redistribution of ocean mass. This adjustment process will, in general, redistribute ocean heat and salt and may therefore impact the baroclinic circulation. The full impact of SAL effects on the ocean circulation therefore requires the SAL effect to be online (or dynamically) incorporated into an ocean general circulation model [Kuhlmann et al., 2011].
 The results presented here are based on one climate model. Testing the robustness would require an analysis of several, preferably many, models. It is nevertheless likely that the SAL effects caused by accumulation of mass on the shelves represent a minor contribution to projected increases in sea level caused by global warming. One possible exception to this conclusion is the high northern latitude, where the SAL effects caused by accumulation of water on the shelves may add 5–15% to the conventional sea level rise due to steric expansion and ocean mass redistribution.
 The authors are grateful for the valuable comments from two anonymous reviewers. This work has received financial support from the EU FP7 MONARCH-A project (grant 242446) and the Centre for Climate Dynamics at the Bjerknes Centre for Climate Research. This is publication no. A415 from the Bjerknes Centre for Climate Research.