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.
The number of locations with sea-level data available for the reconstruction is larger than in our earlier 2004 (Church et al. 2004) and 2006 (Church and White 2006) studies, particularly prior to 1900 (Figure 1). In the 1860s there are only 7-14 locations available, all north of 30˚N. In the 1870s, there is one record available South of 30˚N but still none in the southern hemisphere and it is only in the second half of the 1880s (Fort Denison, Sydney, Australia starts in January 1886) that the first southern hemisphere record becomes available. While we attempted the reconstruction back to 1860, the results showed greater sensitivity to details of the method prior to the 1880s when the first southern hemisphere record is available (see below for further discussion). As a result, while we show the reconstruction back to 1860, we restricted the subsequent analysis (computation of trends, etc.) to after 1880. The number of locations with data available increases to 38 in 1900 (from 71 individual gauges), including several in the southern hemisphere, to about 85 locations in 1940 (from 130 individual gauges but with still less than 10 in the southern hemisphere), and to about 190 in 1960 (from about 305 individual gauges with about 50 locations in the southern hemisphere). The number of locations peaks in May 1985 at 235 (from 399 individual gauges, with slightly less than one-third in the ocean-dominated southern hemisphere; Figure 1). The largest gaps are in the Southern Ocean, the South Atlantic Ocean and around Africa (Figure 1f). Through the 1990s there are at least 200 locations available from between 370 and 400 gauges. For the last few years there are fewer records available because of the unavoidable delay in the transmission by national authorities of monthly and annual mean information to the PSMSL. In December 2009, there are 135 locations available from 250 gauges.
Figure 1: The number and distribution of sea-level records available for the reconstruction. (a) The number of locations for the globe and the northern and southern hemispheres. (b-f) indicate the distribution of gauges in the 1880s, 1910s, 1930s, 1960s and 1990s. The locations indicated have at least 60 months of data in the decade and the number of records are indicated in brackets.
Sea-level measurements are affected by vertical land motion. Corrections for local land motion can come from long-term geological observations of the rate of relative local sea-level change (assuming the relative sea-level change on these longer times scales is from land motions rather than changing ocean volume), or from models of glacial isostatic adjustment, or more recently from direct measurements of land motion with respect to the centre of the Earth using Global Positioning System (GPS) observations. Here, the ongoing response of the Earth to changes in surface loading following the last glacial maximum were removed from the tide-gauge records using the same estimate of glacial isostatic adjustment (GIA; Davis and Mitrovica 1996; Milne et al. 2001) as in our earlier study (Church et al. 2004).
We completed the analysis with and without correction of the sea-level records for atmospheric pressure variations (the “inverse barometer” effect). The HadSLP2 global reconstructed atmospheric pressure data set (Allan and Ansell 2006) was used for this correction.
We tested the impact of correcting the tide-gauge measurements for terrestrial loading and gravitational changes resulting from dam storage (Fiedler and Conrad 2010). For the large number of tide gauges used in the period of major dam building after 1950 (mostly over 200), the impact on global mean sea level is only about 0.05 mm year^{-1} (smaller than the 0.2 mm year^{-1} quoted by Fiedler and Conrad, which is for a different less globally-distributed set of gauges). Tests of similar corrections for changes in the mass stored in glaciers and ice caps, and the Greenland and Antarctic Ice Sheets show that these effects have an even smaller impact on GMSL.
Satellite Altimeter Data Processing Techniques
The TOPEX/Poseidon, Jason-1 and OSTM/Jason-2 satellite altimeter missions measure sea surface height (SSH) relative to the centre of mass of the Earth along the satellite ground track. A number of instrumental and geophysical corrections must be applied. Every 10 days (one cycle) virtually global coverage of the worldÕs ocean, between 66˚N and S, is achieved. Our gridded data set as used here goes to 65˚N and S.
Our satellite altimeter data processing mostly follows the procedures, and uses the edits and tests, recommended by the providers of the satellite altimeter data sets, and are similar to those described in Leuliette et al. (2004). The documents for the three missions used are Benada (1997) for TOPEX/Poseidon, Aviso (2003) for Jason-1 and CNES (2009) for OSTM/Jason-2.
Orbits from the most recent versions of the Geophysical Data Records (GDR Þles; MGDR-B for TOPEX/Poseidon, GDR-C for Jason-1 and GDR-T for OSTM/Jason-2) are used. GDR corrections from the same Þles for tides, wet troposphere, dry troposphere, ionosphere, sea-state bias (SSB), inverse barometer correction (when required) and the mean sea surface are applied in accordance with these manuals, except for some TOPEX/ Poseidon corrections: firstly, the TOPEX/Poseidon wet troposphere correction has been corrected for drift in one of the brightness temperature channels (Ruf 2002) and offsets related to the yaw state of the satellite (Brown et al. 2002). Secondly, the inverse barometer correction (when used) has been recalculated using time-variable global-mean over-ocean atmospheric pressure, an improvement on the GDR-supplied correction which assumes a constant global-mean over-ocean atmospheric pressure. This approach makes the correction used for TOPEX/Poseidon consistent with the Jason-1 and OSTM/Jason-2 processing.


Satellite Altimeter Data Processing Techniques
The TOPEX/Poseidon, Jason-1 and OSTM/Jason-2 satellite altimeter missions measure sea surface height (SSH) relative to the centre of mass of the Earth along the satellite ground track. A number of instrumental and geophysical corrections must be applied. Every 10 days (one cycle) virtually global coverage of the worldÕs ocean, between 66˚N and S, is achieved. Our gridded data set as used here goes to 65˚N and S.
Our satellite altimeter data processing mostly follows the procedures, and uses the edits and tests, recommended by the providers of the satellite altimeter data sets, and are similar to those described in Leuliette et al. (2004). The documents for the three missions used are Benada (1997) for TOPEX/Poseidon, Aviso (2003) for Jason-1 and CNES (2009) for OSTM/Jason-2.
Orbits from the most recent versions of the Geophysical Data Records (GDR Þles; MGDR-B for TOPEX/Poseidon, GDR-C for Jason-1 and GDR-T for OSTM/Jason-2) are used. GDR corrections from the same Þles for tides, wet troposphere, dry troposphere, ionosphere, sea-state bias (SSB), inverse barometer correction (when required) and the mean sea surface are applied in accordance with these manuals, except for some TOPEX/ Poseidon corrections: firstly, the TOPEX/Poseidon wet troposphere correction has been corrected for drift in one of the brightness temperature channels (Ruf 2002) and offsets related to the yaw state of the satellite (Brown et al. 2002). Secondly, the inverse barometer correction (when used) has been recalculated using time-variable global-mean over-ocean atmospheric pressure, an improvement on the GDR-supplied correction which assumes a constant global-mean over-ocean atmospheric pressure. This approach makes the correction used for TOPEX/Poseidon consistent with the Jason-1 and OSTM/Jason-2 processing.
Calibrations of the TOPEX/Poseidon data against tide gauges have been performed by Gary Mitchum and colleagues (see, e.g., Nerem and Mitchum 2001). Here and in earlier publications, we have used the calibrations up to the end of 2001 (close to the end of the TOPEX/Poseidon mission). One of the problems these calibrations address is the changeover to the redundant “side B” altimeter electronics in February 1999 (at the end of cycle 235) due to degradation of the ÔÔside AÕÕ altimeter electronics which had been in use since the start of the mission. An alternative processing approach to address the side A to side B discontinuity is to use the separate Chambers et al. (2003) SSB models for TOPEX sides A and B without any use of the Poseidon data, as this correction does not address the substantial drift in the Poseidon SSH measurements, especially later in the mission. No tide-gauge calibrations are applied to Jason-1 or OSTM/Jason-2 data. The altimeter data sets as used here are available on our web site at http://www.cmar.csiro.au/sealevel/sl_data_cmar.html.
The Analysis Approach
The full details of our approach to estimating historical sea level were reported in Church et al. (2004). Briefly, the reconstructed sea level H^r(x, y, t) is represented as
H^r(x, y, t) = U^r(x, y) alpha(t) + epsilon
where U^r(x, y) is a matrix of the leading empirical orthogonal functions (EOFs) calculated from monthly satellite altimeter data mapped (using a Gaussian filter with a length scale of 300 km applied over a square with sides of 800 km) to a one degree by one degree grid for the ice free oceans between 65ûS and 65ûN, epsilon is the uncertainty, x and y are latitude and longitude and t is time. This matrix is augmented by an additional “mode” that is constant in space and used to represent any global average sea-level rise. In the reduced space optimal interpolation, the amplitude of the constant mode and these EOFs are calculated by minimising the cost function
S(alpha) = (K U^r alpha - H^0)^T M^{-1} (K U^r alpha - H^0) + alpha^T Lambda alpha
This cost function minimises the difference between the reconstructed sea levels and the observed coastal and island sea levels H^0, allowing for a weighting related to the observational uncertainties, omitted EOFs and also down-weights higher order EOFs. K is a sampling operator equal to 1 when there is observed sea-level data available and 0 otherwise, Lambda is the diagonal matrix of the eigenvalues of the covariance matrix of the altimeter data and M is the error covariance matrix given by
M = R + KU’ Lambda’ U’^{T}K^T
where R is the matrix of the covariance of the instrumental errors (assumed diagonal here) and the primes indicate the higher order EOFs not included in the reconstruction.
The EOFs are constructed from the covariances of the altimeter sea-level data after removal of the mean. Any overall increase in sea level as a result of ocean thermal expansion or the addition of mass to the ocean is difficult to represent by a finite number of EOFs. We therefore include an additional “mode” which is constant in space to represent this change in GMSL.
Because the sea-level measurements are not related to a common datum, we actually work with the change in sea level between time steps and then integrate over time to get the solution. The least squares solution provides an estimate of the amplitude of the leading EOFs, global average sea-level and error estimates.
Christiansen et al. (2010) tested the robustness of various reconstruction techniques, including an approach similar to that developed by Church et al. (2004) using thermosteric sea level calculated from climate model results. They used an ensemble of model results (derived by randomising the phase of the principal components of the model sea level, see Christiansen et al. (2010) for details). For a method similar to that used here (including the additional “constant” mode and for a 20 year period for determining the EOFs), the trend in the ensemble mean reconstruction was within a few percent of the true value when 200 gauges were available (with about a 10% variation for the inter.quartile range of individual estimates, decreasing to about 5% when a 50 year period for determining the EOFs was available). When only 40 gauges were used, the ensemble mean trend was biased low by a little under 10% (with an interquartile range of about 15%). They further showed that the reconstructions tend to overestimate the interannual variability and that a longer period for determining the EOFs is important in increasing the correlation between the reconstructed and model year to year variability. Reconstructions that do not use the constant mode perform poorly compared to those that do. These results are similar to our own tests with climate model simulations, with the reconstruction tending to have a slightly smaller trend. Christiansen et al. also found a simple mean of the tide gauges reproduces the trend with little bias in the ensemble mean and about a 10% variation in the interquartile range. However, the simple mean has larger interannual variations and correlates less well with the model interannual variability.
The GMSL estimates are not sensitive to the number of EOFs (over the range 4Ð20 plus the constant mode) used in the reconstruction, although the average correlation between the observed and reconstructed signal increases and the residual variance decreases when a larger number of EOFs is used. For the long periods considered here and with only a small number of records available at the start of the reconstruction period, we used only four EOFs which explain 45% of the variance, after removal of the trend.
Computation of EOFs


Christiansen et al. (2010) tested the robustness of various reconstruction techniques, including an approach similar to that developed by Church et al. (2004) using thermosteric sea level calculated from climate model results. They used an ensemble of model results (derived by randomising the phase of the principal components of the model sea level, see Christiansen et al. (2010) for details). For a method similar to that used here (including the additional “constant” mode and for a 20 year period for determining the EOFs), the trend in the ensemble mean reconstruction was within a few percent of the true value when 200 gauges were available (with about a 10% variation for the inter.quartile range of individual estimates, decreasing to about 5% when a 50 year period for determining the EOFs was available). When only 40 gauges were used, the ensemble mean trend was biased low by a little under 10% (with an interquartile range of about 15%). They further showed that the reconstructions tend to overestimate the interannual variability and that a longer period for determining the EOFs is important in increasing the correlation between the reconstructed and model year to year variability. Reconstructions that do not use the constant mode perform poorly compared to those that do. These results are similar to our own tests with climate model simulations, with the reconstruction tending to have a slightly smaller trend. Christiansen et al. also found a simple mean of the tide gauges reproduces the trend with little bias in the ensemble mean and about a 10% variation in the interquartile range. However, the simple mean has larger interannual variations and correlates less well with the model interannual variability.
The GMSL estimates are not sensitive to the number of EOFs (over the range 4Ð20 plus the constant mode) used in the reconstruction, although the average correlation between the observed and reconstructed signal increases and the residual variance decreases when a larger number of EOFs is used. For the long periods considered here and with only a small number of records available at the start of the reconstruction period, we used only four EOFs which explain 45% of the variance, after removal of the trend.
Computation of EOFs
For each altimeter mission the along-track data described above are smoothed onto a 1˚-by-1˚-by-1 month grid for the permanently ice-free ocean from 65˚S to 65˚N. The smoothing uses an e-folding length of 300 km and covers 90% of the global oceans. The three data sets are combined by matching means at each grid point (rather than just the global average) over the common periods between TOPEX/Poseidon and Jason-1 and between Jason-1 and OSTM/Jason-2. This is an attempt to overcome the problem of different geographically correlated errors in the missions, for example due to different sea-state bias corrections. The overlap between TOPEX/Poseidon and Jason-1 was from 15-January-2002 to 21-August-2002 (T/P cycles 344-365, J-1 cycles 1-22) or, effectively, February to July 2002 in our monthly data sets. The overlap between Jason-1 and OSTM/ Jason-2 was from 12-July-2008 to 26-January-2009 (J-1 cycles 240-259 and J-2 cycles 1-20) or, effectively, August to December 2008 in our monthly data sets.
Separate versions of the altimeter data sets with and without the inverse barometer correction and with and without the seasonal signal are produced, as follows:
Only whole years (in this case 17 years) are used.
Grid points with gaps in the time series (e.g. due to seasonal sea ice) are ignored.
The data are area (cos(latitude)) weighted.
The global-mean trend is removed.
The GIA correction appropriate for this data is applied (Mark Tamisiea, NOC Liverpool, private communication).
In the original (Church et al. 2004; Church and White 2006) reconstructions, the EOFs were defined with the 9 and 12 years (respectively) of TOPEX/Poseidon and Jason-1 satellite altimeter data available at those times. There are now 17 years of monthly satellite altimeter data available, almost twice as long as the original series. This longer time series should be able to better represent the variability and result in an improved reconstruction of global average sea level, as found by Christiansen et al. (2010). After removing the global average trend and the seasonal (annual plus semi-annual) signal, the first four EOFs account for 29, 8, 5 and 4% of the variance (Figure 2). If the seasonal signal is not removed, the first four EOFs account for 24, 18, 14 and 4% of the variance. These EOFs characterise the large-scale interannual variability, particularly that associated with the El Nino-Southern Oscillation phenomenon, and for the case where the seasonal signal has not been removed, also include the seasonal north/south oscillation of sea level.
Figure 2: The EOFs used in the sea-level construction. The four EOFs on the left include the seasonal signal and represent a combination of the seasonal signal and interannual variability. The corresponding four EOFs on the right are after the seasonal signal has been removed from the altimeter data. The EOFs are dimensionless and of unit length.
Sensitivity of the Results
To complete the reconstruction, we need to specify two parameters: the instrumental error covariance matrix R and the relative weighting of the “constant” mode to the EOFs. Church and White (2006) used the first differences between sets of nearby sea-level records to compute an average error estimate of the first differences of 50 mm and assumed errors were independent of and between locations (i.e. the error covariance matrix was diagonal). When the seasonal signal was removed, tests indicated the residual variance increased when a smaller error estimate was used but was not sensitive to the selection of larger values. Similarly, the residual variance increased when the weighting of the ÔÔconstantÕÕ mode was less than 1.5 times the first EOF but was not sensitive to larger values. The computed trends for the 1880Ð2009 increased slightly (0.06 mm year -1 or about 4%) when the relative weighting was increased by 33% from 1.5 to 2.0 or the error estimate was decreased by 40% to 30 mm. Prior to 1880 when there were less than 15 locations available and none in the southern hemisphere, there was considerably greater sensitivity to the parameter choice than for the rest of the record and hence we focus on results after 1880. When the seasonal signal was retained in the solution, a larger error estimate of 70 mm was appropriate. This solution also had a larger residual variance and a slightly greater sensitivity in the trend to the parameter choice and hence we focus on the solution with the seasonal signal removed, as in our earlier studies.
As a further test of the effectiveness of the EOFs to represent the interannual variability in GMSL, we computed EOFs using shorter periods of 9 and 12 years, similar to our earlier analyses (Church et al. 2004; Church and White 2006). The resulting estimates are well within the uncertainties.
The atmospheric pressure correction makes essentially no difference to the GMSL time series for the computations with the seasonal signal removed and no difference to the computations including the seasonal signal after about 1940. However, prior to 1940, the correction does make a significant difference to the GMSL calculated with the seasonal signal included. These results suggests some problem with the atmospheric correction prior to 1940 and as a result we decided not to include this correction in the results. This issue seems to be related to the HadSLP2 data set not resolving the annual cycle and, possibly, the spatial patterns well for the Southern Hemisphere south of 30 S for the 1920s and 1930s, presumably because of sparse and changing patterns of input data at this time and in this region. This is being investigated further.
Results
We present results for two periods: from 1880 to 2009 and the satellite altimeter period from January 1993 to December 2009. The latter is only a partial test of the reconstruction technique because the EOFs used were actually determined for this period.


As a further test of the effectiveness of the EOFs to represent the interannual variability in GMSL, we computed EOFs using shorter periods of 9 and 12 years, similar to our earlier analyses (Church et al. 2004; Church and White 2006). The resulting estimates are well within the uncertainties.
The atmospheric pressure correction makes essentially no difference to the GMSL time series for the computations with the seasonal signal removed and no difference to the computations including the seasonal signal after about 1940. However, prior to 1940, the correction does make a significant difference to the GMSL calculated with the seasonal signal included. These results suggests some problem with the atmospheric correction prior to 1940 and as a result we decided not to include this correction in the results. This issue seems to be related to the HadSLP2 data set not resolving the annual cycle and, possibly, the spatial patterns well for the Southern Hemisphere south of 30 S for the 1920s and 1930s, presumably because of sparse and changing patterns of input data at this time and in this region. This is being investigated further.
Results
We present results for two periods: from 1880 to 2009 and the satellite altimeter period from January 1993 to December 2009. The latter is only a partial test of the reconstruction technique because the EOFs used were actually determined for this period.
The reconstructed and satellite estimates of GMSL have somewhat different error sources. The two largest uncertainties for the reconstructed sea level are the incomplete global coverage of sea-level measurements (particularly in the southern hemisphere), and uncertainties in land motions used to correct the sea-level records. The former contributes directly to the formal uncertainty estimates that are calculated on the basis that the sea-level records are independent. In estimating uncertainties on linear trends and accelerations, we recognise the series are autocorrelated and the number of effective degrees of freedom is only a quarter of the number of years of data. Previous tests using various GIA models suggest an additional uncertainty in trends of about ±0.1 mm year^{-1} (Church et al. 2004) that should be added in quadrature to the uncertainty in the trend estimate from the time series (but not for estimates of the acceleration in the rate of rise). The annual time series of GMSL and the estimated uncertainty estimates are available at http://www.cmar.csiro.au/sealevel/sl_data_cmar.html.
1993-2009
The global mean sea level can be computed cycle-by-cycle (every 10 days) directly from the along track satellite data or from the mapped (monthly) satellite data. After averaging the cycle-by-cycle data set over 3 cycles these two estimates for the Jason-1 mission (February 2002 to January 2009; Figure 3) are very similar; the differences have a standard deviation of 1.0 mm. The trends are different by only 0.2 mm year^{-1}, with the trend from the gridded data set being higher numerically, but not statistically different from the trend from the along-track data set.
Figure 3: Comparison of the satellite-altimeter estimates of GMSL from the along-track data (including all ocean areas where valid data are available) and the mapped data (for a fixed grid) for the duration of the Jason-1 altimeter mission.
The reconstructed and altimeter GMSL both increase from 1993 to the end of the record (Figure 4). The larger year-to-year variability of the reconstructed signal (compared with the altimeter record) of ~4-5 mm is less than the one standard deviation uncertainty estimates of about ±7 mm. For almost all of the record, the reconstruction is within the one standard deviation uncertainty estimate of the altimeter record. The uncertainty of the reconstruction increases slightly in the last couple of years because of the smaller number of tide gauge records available through the PSMSL.
Figure 4: Global average sea level from 1990 to 2009 as estimated from the coastal and island sea-level data (blue with one standard deviation uncertainty estimates) and as estimated from the satellite altimeter data from 1993 (red). The satellite and the in situ yearly averaged estimates have the same value in 1993 and the in situ data are zeroed in 1990. The dashed vertical lines indicate the transition from TOPEX Side A to TOPEX Side B, and the commencement of the Jason-1 and OSTM/Jason-2 records.
After correcting for the GIA, the linear trend from the altimeter data from January 1993 to December 2009 is 3.2 ± 0.4 mm year^{-1} (note the GIA values appropriate for correcting the altimeter data are different to that necessary for the in situ data). The uncertainty range (1 standard deviation) comes from fitting a linear trend to the data using uncertainties on the annual averages of 5 mm and is consistent with an updated error budget of altimeter sea-level trend uncertainties (Ablain et al. 2009). They estimate the largest uncertainties are related to the wet tropospheric (atmospheric water vapour) correction, the bias uncertainty of successive missions, orbit uncertainty and the sea-state bias correction. These total to about 0.4 mm year -1, similar to our uncertainty estimate. The reconstructed global average sea-level change over the same period is almost the same as for the altimeter data. However, as a result of different interannual variability, the trend of 2.8 ± 0.8 mm year^{-1} is smaller but not significantly different to the altimeter estimate after correction for glacial isostatic adjustment.
1880-2009
The GMSL time series (Figure 5) are not significantly different from our earlier 2006 result (Church and White 2006). The total GMSL rise (Figure 5) from January 1880 to December 2009 is about 210 mm over the 130 years. The trend over this period, not weighted by the uncertainty estimates, is 1.5 mm year^{-1} (1.6 mm year^{-1} when weighted by the uncertainty estimates). Although the period starts 10 years later in 1880 (rather than 1870), the total rise (Figure 5) is larger than our 2006 estimate of 195 mm mostly because the series extends 8 years longer to 2009 (compared with 2001).
Figure 5: Global average sea level from 1860 to 2009 as estimated from the coastal and island sea-level data (blue). The one standard deviation uncertainty estimates plotted about the low passed sea level are indicated by the shading. The Church and White (2006) estimates for 1870-2001 are shown by the red solid line and dashed magenta lines for the 1 standard deviation errors. The series are set to have the same average value over 1960-1990 and the new reconstruction is set to zero in 1990. The satellite altimeter data since 1993 is also shown in black.
The interannual variability is mostly less than the one standard deviation uncertainty estimates, which range from ~25 mm in 1880 to a minimum of ~6 mm in 1988 (as shown in Figure 5, where the yearly GMSL time series is plotted over the envelope of smoothed (±3 year boxcar) 1 standard deviation limits). However, there are a number of features which are comparable to/larger than the uncertainty estimates. Firstly, there is a clear increase in the trend from the first to the second half of the record; the linear trend from 1880 to 1935 is 1.1 ± 0.7 mm year^{-1} and from 1936 to the end of the record the trend is 1.8 ± 0.3 mm year^{-1}. The period of relatively rapid sea-level rise commencing in the 1930s ceases abruptly in about 1962 after which there is a fall in sea level of over 10 mm over 5 years. Starting in the late 1960s, sea level rises at a rate of almost 2.4 mm year^{-1} for 15 years from 1967 and at a rate of 2.8 ± 0.8 mm year^{-1} from 1993 to the end of the record. There are brief interruptions in the rise in the mid 1980s and the early 1990s.


Figure 5: Global average sea level from 1860 to 2009 as estimated from the coastal and island sea-level data (blue). The one standard deviation uncertainty estimates plotted about the low passed sea level are indicated by the shading. The Church and White (2006) estimates for 1870-2001 are shown by the red solid line and dashed magenta lines for the 1 standard deviation errors. The series are set to have the same average value over 1960-1990 and the new reconstruction is set to zero in 1990. The satellite altimeter data since 1993 is also shown in black.
The interannual variability is mostly less than the one standard deviation uncertainty estimates, which range from ~25 mm in 1880 to a minimum of ~6 mm in 1988 (as shown in Figure 5, where the yearly GMSL time series is plotted over the envelope of smoothed (±3 year boxcar) 1 standard deviation limits). However, there are a number of features which are comparable to/larger than the uncertainty estimates. Firstly, there is a clear increase in the trend from the first to the second half of the record; the linear trend from 1880 to 1935 is 1.1 ± 0.7 mm year^{-1} and from 1936 to the end of the record the trend is 1.8 ± 0.3 mm year^{-1}. The period of relatively rapid sea-level rise commencing in the 1930s ceases abruptly in about 1962 after which there is a fall in sea level of over 10 mm over 5 years. Starting in the late 1960s, sea level rises at a rate of almost 2.4 mm year^{-1} for 15 years from 1967 and at a rate of 2.8 ± 0.8 mm year^{-1} from 1993 to the end of the record. There are brief interruptions in the rise in the mid 1980s and the early 1990s.
The linear trend from 1900 to 2009 is 1.7 ± 0.2 mm year^{-1} and from 1961 to 2009 is 1.9 ± 0.4 mm year^{-1}. However, there are significant departures from a linear trend. We estimate an acceleration in GMSL by fitting a quadratic to the time series, taking account of the time variable uncertainty estimates. From 1880 to 2009, the acceleration (twice the quadratic coefficient) is 0.009 ± 0.003 mm year^{-2} (one standard deviation). This estimate is slightly less than but not significantly different from the (one standard deviation) estimate of Church and White (2006) of 0.013 ± 0.003 mm year^{-2}, but still significantly different from zero at the 95% level. From 1900 to 2009, the acceleration is also 0.009 ± 0.004 mm year^{-2}. If the variable uncertainty estimates are ignored the equivalent accelerations are 0.010 and 0.012 mm year^{-2}.
Discussion
There are other recent estimates of changes in GMSL for this period widely available (Jevrejeva et al. 2006; Holgate and Woodworth, 2004; Fig. 6). They all agree approximately with our updated GMSL time series and the longer of these estimates (Jevrejeva et al. 2006) also has an acceleration in the 1930s and a pause in the rise commencing in the 1960s. These changes are also present in a number of individual sea-level records (Woodworth et al. 2009). However, note that the interannual variability in the Jevrejeva et al. series is unrealistically large in the early part of the record and larger than their uncertainty estimates. The Jevrejeva et al. estimate of sea level prior to 1850 (Jevrejeva et al. 2008) indicates an acceleration in the rate of rise commencing at the end of the eighteenth century. Note that their pre-1850 estimate uses only three-sea level records. We do not attempt to extend our construction back prior to 1860. If instead of the recon.struction technique, we employed a straight average of tide gauges, the overall trend back to 1910 is very similar but there is larger interannual variability (Figure 6). Prior to 1910, the variability is even larger (consistent with the results of Christiansen et al. (2010), with unrealistic decadal trends of ±10 mm year^{-1}.
One source of error is the poor corrections for land motion. Bouin and Wooppelmann (2010) used GPS time series for correcting tide-gauge records for land motion from all sources and estimated a global average sea-level rise of 1.8 mm year^{-1} for the twentieth century, consistent with the present results and early studies (e.g. Douglas 1991). These GPS series are just now beginning to be long enough to provide useful constraints on land motion from all sources (not just GIA).
A significant non-climatic influence on sea level is the storage of water in dams and the depletion of ground water from aquifers, some of which makes it into the ocean. Chao et al. (2008) estimated that about 30 mm of sea-level equivalent is now stored in man-made dams and the surrounding soils; most of this storage occurred since the 1950s. Globally, the rate of dam entrapment has slowed significantly in the last decade or two. The depletion of ground water (Konikow et al. personal communication; Church et al. in preparation) offsets perhaps a third of this terrestrial storage over the last two decades and the rate of depletion has accelerated over the last two decades.
Figure 6: Global average sea level from 1860 to 2009 as estimated from the coastal and island sea-level data (blue) compared with estimates of Jevrejeva et al. (2006, brown), Holgate and Woodworth (2004, red) and from a simple average of the gauges (yellow). All series are set to have the same average value over 1960-1990 and the reconstructions are set to zero in 1990. The satellite altimeter data since 1993 is also shown in black.
We remove this direct (non-climate) anthropogenic change in terrestrial water storage (both dam storage and aquifer depletion) from our observations to focus on the sea-level change related to climatic influences. The resulting time series (Figure 7) shows a slightly faster rate of sea-level rise since about 1960 and a slightly larger acceleration for the periods since 1880 and 1900. Terrestrial storage contributed to the sea level fall in the 1960s but does not fully explain it. The volcanic eruptions of Mt Agung in 1963, El Chichon in 1982 and Mt Pinatubo in 1991 probably contribute to the small sea level falls in the few years following these eruptions (Church et al. 2005; Gregory et al. 2006; Domingues et al. 2008) but it has not yet been possible to quantitatively explain the mid 1960s fall in sea level (Church et al. in preparation).
Figure 7: Global average sea level from 1860 to 2009 as estimated from the coastal and island sea-level data (grey) and after correcting for the changes in terrestrial storage associated with the building of dams and the deletion of aquifers (blue). Note these series are virtually identical before 1950.
The acceleration in the rate of sea-level rise since 1880 is in qualitative agreement with the few available long (mostly northern hemisphere) sea-level records and longer term estimates of sea level from geological (e.g. salt-marsh) data (for example Donnelly et al. 2004; Gehrels et al. 2006). These data mostly indicate an acceleration at the end of the nineteenth or start of the twentieth century (see Woodworth et al. 2011, this volume, for a summary and references).


Figure 6: Global average sea level from 1860 to 2009 as estimated from the coastal and island sea-level data (blue) compared with estimates of Jevrejeva et al. (2006, brown), Holgate and Woodworth (2004, red) and from a simple average of the gauges (yellow). All series are set to have the same average value over 1960-1990 and the reconstructions are set to zero in 1990. The satellite altimeter data since 1993 is also shown in black.
We remove this direct (non-climate) anthropogenic change in terrestrial water storage (both dam storage and aquifer depletion) from our observations to focus on the sea-level change related to climatic influences. The resulting time series (Figure 7) shows a slightly faster rate of sea-level rise since about 1960 and a slightly larger acceleration for the periods since 1880 and 1900. Terrestrial storage contributed to the sea level fall in the 1960s but does not fully explain it. The volcanic eruptions of Mt Agung in 1963, El Chichon in 1982 and Mt Pinatubo in 1991 probably contribute to the small sea level falls in the few years following these eruptions (Church et al. 2005; Gregory et al. 2006; Domingues et al. 2008) but it has not yet been possible to quantitatively explain the mid 1960s fall in sea level (Church et al. in preparation).
Figure 7: Global average sea level from 1860 to 2009 as estimated from the coastal and island sea-level data (grey) and after correcting for the changes in terrestrial storage associated with the building of dams and the deletion of aquifers (blue). Note these series are virtually identical before 1950.
The acceleration in the rate of sea-level rise since 1880 is in qualitative agreement with the few available long (mostly northern hemisphere) sea-level records and longer term estimates of sea level from geological (e.g. salt-marsh) data (for example Donnelly et al. 2004; Gehrels et al. 2006). These data mostly indicate an acceleration at the end of the nineteenth or start of the twentieth century (see Woodworth et al. 2011, this volume, for a summary and references).
In addition to the overall increase in the rate of sea-level rise, there is also considerable variability in the rate. Using the yearly average data, we computed trends for successive 16 year periods (close to the length of the altimeter data set) from 1880 to the present (Figure 8). We find maxima in the rates of sea-level rise of over 2 mm year^{-1} in the 1940s and 1970s and nearly 3 mm year^{-1} in the 1990s (Figure 8). As in earlier studies (using 10 and 20 year windows; Church and White 2006; Church et al. 2008), the most recent rate of rise over these short 16 year windows is at the upper end of a histogram of trends but is not statistically higher than the peaks during the 1940s and 1970s. Consistent with the findings of Christiansen et al. (2010), our computed variability in the rates of rise are almost a factor of two less than those where an average of tide gauges (Holgate and Woodworth 2004; Holgate 2007) is used to estimate GMSL. The rate of sea-level rise since 1970 has now been quantitatively explained (Church et al. in preparation) by a gradual increase in ocean thermal expansion, with fluctuations at least partly related to volcanic eruptions, and an increasing cryospheric contribution. The contribution from glaciers and ice caps (Cogley 2009), and the Greenland Ice Sheet (Rignot et al. 2008, 2011) both increased in the 1990s. There are also recent indications of an increasing contribution from the West Antarctic Ice Sheet (Rignot et al. 2011). The larger rate of rise in the 1940s may be related to larger glacier and ice-cap contributions (Oerlemanns et al. 2007) and higher temperatures over Greenland resulting in larger sea-level contributions at that time.
Figure 8: Linear trends in sea level over successive 16 year periods for the yearly averaged reconstructed sea-level data. The trend from the satellite altimeter data are shown at the end of the time series.
The rate of sea-level rise as measured both by the satellite altimeter record and the in situ reconstruction of about 3 mm year^{-1} since 1993 is near the upper end of the sea-level projections for both the Intergovernmental Panel on Climate Change’s Third (Church et al. 2001) and Fourth (Meehl et al. 2007 - see also Hunter 2010) Assessment Reports. However, note that the in situ data also indicates that there was little net change in GMSL from 1990 to 1993, most likely as a result of the volcanic eruption of Mount Pinatubo in 1991 (Domingues et al. 2008; Church et al. in preparation).
Significant progress has been made during the last decade in estimating and understanding historical sea-level rise. However, much remains to be done. Of particular importance is the maintenance and continuation of the observing network and associated infrastructure such as the PSMSL archive. The in situ sea-level data set continues to provide a very valuable contribution to our understanding of late nineteenth, twentieth and early twenty first century sea-level rise. Data archaeology and paleo observations to extend the spatial and temporal coverage of in situ sea-level observations need to be vigorously pursued. Modern GPS measurements at tide-gauge locations, which are now beginning to provide valuable information on vertical land motion (e.g., Bouin and Woppelmann 2010) should be continued and expanded. This applies in particular to the use of in situ data to monitor the accuracy of satellite altimeter measurement systems. Increasing the number and geographical distribution of these GPS observations is a priority. Of course a major priority is maintaining a continuous record of high-quality satellite-altimeter observations of the oceans and continuing to improve the International Terrestrial Reference Frame and maintaining and expanding the associated geodetic networks. These improved observations need to be combined with more elegant analysis of the observations, including, for example, considering changes in the gravitational field associated with evolving mass distributions on the Earth and using observations of sea-level rise, ocean thermal expansion and changes in the cryosphere in combined solutions.
Acknowledgments:
This paper is a contribution to the Commonwealth Scientific Industrial Research Organization (CSIRO) Climate Change Research Program. J. A. C. and N. J. W. were partly funded by the Australian Climate Change Science Program. NASA & CNES provided the satellite altimeter data, PSMSL the tide-gauge data.
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Couldrey et al. (2021)
Title:
What causes the spread of model projections of ocean dynamic sea-level change in response to greenhouse gas forcing?
Key Points:
Ocean model diversity in AOGCMs is a key source of uncertainty in sea-level projections under greenhouse gas forcing.
Increased air-sea heat flux sets the broad pattern of dynamic sea-level change; wind stress and freshwater flux have localized effects.
Nonlinear dynamic sea-level responses occur with simultaneous flux perturbations, especially in the Arctic and North Atlantic
Keywords:
sea-level rise, ocean heat uptake, climate change, climate modeling, ocean model diversity, air-sea flux perturbations, FAFMIP
Corresponding author:
Matthew P. Couldrey
Citation:
Couldrey, M. P., Gregory, J. M., Dias, F. B., Dobrohotoff, P., Domingues, C. M., Garuba, O., et al. (2021). What causes the spread of model projections of ocean dynamic sea-level change in response to greenhouse gas forcing? Climate Dynamics, 56(1–2), 155–187. doi:10.1007/s00382-020-05471-4
URL:
https://link.springer.com/article/10.1007/s00382-020-05471-4
Abstract
Sea levels of different atmosphere–ocean general circulation models (AOGCMs) respond to climate change forcing in different ways, representing a crucial uncertainty in climate change research. We isolate the role of the ocean dynamics in setting the spatial pattern of dynamic sea-level (ζ) change by forcing several AOGCMs with prescribed identical heat, momentum (wind) and freshwater flux perturbations. This method produces a ζ projection spread comparable in magnitude to the spread that results from greenhouse gas forcing, indicating that the differences in ocean model formulation are the cause, rather than diversity in surface flux change. The heat flux change drives most of the global pattern of ζ change, while the momentum and water flux changes cause locally confined features. North Atlantic heat uptake causes large temperature and salinity driven density changes, altering local ocean transport and ζ. The spread between AOGCMs here is caused largely by differences in their regional transport adjustment, which redistributes heat that was already in the ocean prior to perturbation. The geographic details of the ζ change in the North Atlantic are diverse across models, but the underlying dynamic change is similar. In contrast, the heat absorbed by the Southern Ocean does not strongly alter the vertically coherent circulation. The Arctic ζ change is dissimilar across models, owing to differences in passive heat uptake and circulation change. Only the Arctic is strongly affected by nonlinear interactions between the three air-sea flux changes, and these are model specific.
Introduction
Sea-level rise presently is and will continue to be an important consequence of anthropogenically forced climate change. Global mean thermosteric sea-level rise, due to thermal expansion of a warming ocean, accounts for 30–55% of the global mean sea-level rise (GMSLR) projected for the years 2081–2100 (Church et al. 2013). The remainder results mostly from ocean mass gain due to melting land ice (glaciers and ice sheets). Regional sea-level changes are much more complicated, involving ocean and climate dynamics as well as solid-Earth processes, typically not included in coupled climate models. The latter can make up 50% or more of regional sea-level change pattern by the end of the 21st Century (Slangen et al. 2012; Stammer et al. 2013).
As part of the Coupled Model Intercomparison Project (CMIP), atmosphere–ocean general circulation models (AOGCMs) simulate anthropogenic sea-level change related to changes in climate dynamics by starting from a near-equilibrium (i.e. well spun-up) preindustrial control state (piControl) and running forward with time-varying forcing agents (greenhouse gases, anthropogenic aerosol, etc.). In these models, ocean dynamic sea level, ζ, is defined at each location and time as
ζ = η − η̅ (1)
where η is the local sea-surface height relative to a surface on which the geopotential has a uniform constant value, and η̅ is its global mean over the ocean area. η is termed ‘sterodynamic sea level’ according to recent terminology conventions (Gregory et al. 2019). By definition, the global mean of ζ is zero, but locally it is not zero due to ocean circulation and horizontal density gradients. Using CMIP terminology, ζ is the variable ‘zos’ (Griffies et al. 2016). The sterodynamic sea-level change (Δη) that an individual location experiences may differ substantially from the global mean thermosteric sea-level rise (GMSLR, Δ𝜂−), because of changes in ocean circulation and density. This present work focuses on the spatial pattern in ocean dynamic sea-level change
∆ζ(x,y,t) = Δ𝜂(𝑥,𝑦,𝑡) − ∆η̅(𝑡),(2)
projected to occur over the coming century due to anthropogenic greenhouse gas forced climate change. Accordingly, Δζ is calculated from CMIP output as the difference between the ‘zos’ fields in a forcing experiment relative to a control state (see Sects. 2.2 and 2.3 for details). In results from AOGCMs from CMIP’s fifth phase (CMIP5), the spatial standard deviation of the multi-model mean Δζ projected under scenario RCP4.5 by 2081–2100 is 0.06 m, which is 30% of the multi-model mean GMSLR due to thermal expansion (Gregory et al. 2016). Thus, in some locations, Δη is more than twice its global mean, because of ocean dynamic sea-level change. Moreover, different AOGCMs predict diverse spatial patterns and magnitudes of sea-level change (Slangen et al. 2014). The global mean of the inter-model standard deviation of Δζ in the same projections is also about 0.06 m; in other words, the systematic uncertainty in predicting the pattern of dynamic sea-level change is of first order, being about the same magnitude as the pattern itself (see also Fig. 3). It is this spread among AOGCMs that we seek to investigate: one of the largest uncertainties affecting regional impacts of anthropogenic sea-level and climate change this century and beyond.
Part of the spread among AOGCMs comes from their different representations of forcing agents, especially anthropogenic aerosol (Melet and Meyssignac 2015). The idealized scenario of increasing the concentration of CO2 by 1% per year, called ‘1pctCO2’ is simpler to interpret than more complex experiments with differing time profiles of numerous types of climate forcing (Eyring et al. 2016). After seven decades, 1pctCO2 reaches a similar magnitude of radiative forcing to RCP 4.5 by the end of the twenty-first century.
Developing a more complete understanding of the climate response to idealized 1pctCO2 forcing provides insight into how we expect the climate to respond to moderate greenhouse gas and aerosol emissions by the end of this century. However, even when AOGCMs are forced with this simple, idealized setup, they produce a range of climate response, primarily because of their differing climate sensitivities (i.e. the degree of surface warming that results per radioactive forcing). Differences in the representation of cloud feedback mechanisms in atmosphere models accounts for the greatest uncertainty in climate sensitivity (Ceppi et al. 2017). The control climate, which is different for each model, also contributes to the spread (Bouttes and Gregory 2014), e.g. via the strength of the ice coverage and water vapour feedbacks (Hu et al. 2017), and sea-surface temperature (SST) biases (He and Soden 2016). These and other factors affect the spatial patterns and magnitudes of the air-sea fluxes of heat, freshwater, and momentum, which are the drivers of Δζ. Unpacking the influence of oceanic processes from atmospheric processes is therefore difficult in experiments like 1pctCO2.
Previous work has investigated how the diversity in the changes of air-sea fluxes of heat, freshwater and momentum contribute to the spread in projections of sea-level change. A typical approach is to force a single model with ensembles of boundary conditions like SST (Huber and Zanna 2017) or air-sea flux change (Bouttes and Gregory 2014) to mimic the spread of fully coupled simulations. These studies find that forcing individual models with a variety of boundary conditions produces a large spread of ocean responses, in terms of sea-level change (Bouttes and Gregory 2014), ocean heat uptake (OHU) and circulation change (Huber and Zanna 2017). While these studies demonstrate that models are sensitive to surface fluxes, the uncertainty that results from the diversity of ocean model structure in coupled models has not yet been assessed.


Developing a more complete understanding of the climate response to idealized 1pctCO2 forcing provides insight into how we expect the climate to respond to moderate greenhouse gas and aerosol emissions by the end of this century. However, even when AOGCMs are forced with this simple, idealized setup, they produce a range of climate response, primarily because of their differing climate sensitivities (i.e. the degree of surface warming that results per radioactive forcing). Differences in the representation of cloud feedback mechanisms in atmosphere models accounts for the greatest uncertainty in climate sensitivity (Ceppi et al. 2017). The control climate, which is different for each model, also contributes to the spread (Bouttes and Gregory 2014), e.g. via the strength of the ice coverage and water vapour feedbacks (Hu et al. 2017), and sea-surface temperature (SST) biases (He and Soden 2016). These and other factors affect the spatial patterns and magnitudes of the air-sea fluxes of heat, freshwater, and momentum, which are the drivers of Δζ. Unpacking the influence of oceanic processes from atmospheric processes is therefore difficult in experiments like 1pctCO2.
Previous work has investigated how the diversity in the changes of air-sea fluxes of heat, freshwater and momentum contribute to the spread in projections of sea-level change. A typical approach is to force a single model with ensembles of boundary conditions like SST (Huber and Zanna 2017) or air-sea flux change (Bouttes and Gregory 2014) to mimic the spread of fully coupled simulations. These studies find that forcing individual models with a variety of boundary conditions produces a large spread of ocean responses, in terms of sea-level change (Bouttes and Gregory 2014), ocean heat uptake (OHU) and circulation change (Huber and Zanna 2017). While these studies demonstrate that models are sensitive to surface fluxes, the uncertainty that results from the diversity of ocean model structure in coupled models has not yet been assessed.
The change of the Atlantic Meridional Overturning Circulation (AMOC) strength in response to climate change is model specific, and is believed to be a key factor setting the pattern of North Atlantic sea-level change (Yin et al. 2009; Hawkes 2013; Bouttes et al. 2014). Globally, ocean heat uptake has been related to the degree of AMOC weakening (Xie and Vallis 2012; Rugenstein et al. 2013; Kostov et al. 2014). However, more recently, the correlation between OHU and AMOC strength was shown to arise because both are affected by mesoscale eddy transfer (Saenko et al. 2018), where OHU is more intense and deep reaching with decreasing mesoscale eddy transfer. The ocean components of most CMIP5 and CMIP6 models are not able to resolve mesoscale eddies, so their effects are parameterized in different ways across models. It is therefore plausible that the differing representations of mesoscale and other unresolved phenomena are likely to contribute to the spread of sea-level projections among AOGCMs, through their influence on seawater properties (e.g. temperature, salinity and density) and ocean transports of heat and salt.
The Flux-Anomaly-Forced Model Intercomparison Project (FAFMIP) outlines a protocol for forcing different AOGCMs with perturbations to their air-sea fluxes—heat, freshwater, and momentum—to systematically explore the oceanic response to CO2-forced climate change (Gregory et al. 2016). The key goal of FAFMIP is to replicate the oceanic response to 1pctCO2 forcing, while excluding the model spread due to changes in air-sea fluxes. Initial FAFMIP results have highlighted the importance of the heat flux perturbation in setting much of the global pattern of Δζ, and that both the wind stress and heat flux perturbations set the Southern Ocean dipole (Gregory et al. 2016).
Building on previous findings, we present in this study: (1) new sea-level results based on AOGCM simulations forced with simultaneous rather than separate flux perturbations, (2) an intercomparison of the roles of temperature and salinity-driven density changes, and (3) further examination of the decomposition of ocean heat content (OHC) changes due to changes in temperature and transport. Our study also includes new CMIP6 simulations and paves the way for possible forthcoming FAFMIP analyses.
This paper is structured as follows: an explanation of the configuration of model experiments and analysis methods is given in Sect. 2. In Sect. 3.1, the sea-level responses to FAFMIP and 1pctCO2 forcing are compared, followed by a comparison of the 1pctCO2 sea-level response from CMIP5 and CMIP6 in Sect. 3.2. The response of AOGCMs to individually applied surface flux perturbations (heat, freshwater and momentum) is assessed in Sect. 3.3, while nonlinear interactions between these flux perturbations are described in Sect. 3.4. A decomposition of ocean heat uptake based on a subset of the AOGCMs is presented in Sect. 3.5. Results are discussed in Sect. 4 and the conclusions are laid out in Sect. 5. An appendix with further notes on the decomposition of ocean heat uptake is included in “Appendix”.
Methods
Perturbation of air-sea fluxes
The FAFMIP protocol presents a method that mimics the effect of 1pctCO2 forcing on the ocean but applies identical perturbations to each model (Gregory et al. 2016). The perturbations are the multi model mean changes in the air-sea fluxes of heat, freshwater, and momentum from 1pctCO2 simulations averaged over the 61st-80th years of forcing relative to the piControl state. This period covers the time where CO2 concentration reaches double its preindustrial values (at year 70). The suite of CMIP5 AOGCMs available at the time to derive the required surface flux perturbations for the FAFMIP protocol comprises 13 members; CNRM-CM5, CSIRO-Mk3-6-0, CanESM2, GFDL-ESM2G, HadGEM2-ES, MIROC-ESM, MIROC5, MPI-ESM-LR, MPI-ESM-MR, MPI-ESM-P, MRI-CGCM3, NorESM1-ME, and NorESM1-M. It was decided that all perturbations should be derived from a common set of models to allow for consistent comparison of model-mean sea-level change and the associated spread (Gregory et al. 2016). Further details about FAFMIP and the protocol, including the perturbation files can be found at https://www.fafmip.org.
Time-dependent CO2 and other forcing causes a varying magnitude of sea-level change, while the spatial pattern is relatively time-invariant (Hawkes 2013; Perrette et al. 2013; Slangen et al. 2014; Bilbao et al. 2015). This phenomenon of ‘pattern scaling’ means that time-dependent forcing is not necessary for our investigation of the spatial structure. Therefore, in the interest of simplicity the FAFMIP flux perturbations are applied as a constant forcing for the full 70 years of each experiment, with no time-variation except for the annual cycle.
Experiments
FAFMIP perturbations to the fluxes of heat, water and momentum (Fig. 1) were applied in five different experiments, as listed below. All perturbations were applied at the air-sea interface in direct contact with seawater surface, such that sea ice is not directly affected. However, there will be indirect effects on sea ice due to the redistribution of heat and freshwater in response to all the perturbations. The heat and freshwater fluxes are defined positive downward into the ocean, and the momentum flux perturbations are positive eastward and northward. FAFMIP experiments were run by nine modelling groups using 13 AOGCMs (Table 1).
Fig. 1: Annual means of downward flux perturbations applied in FAFMIP experiments at the ocean surface for heat, water, eastward momentum, and northward momentum, a–d respectively. Perturbations are the multi model mean surface flux anomalies from simulations forced with 1% per year rising CO2 concentrations averaged over years 61–80
Table 1 Key features of the main AOGCMs studied, where dashes in Ocean horizontal resolution indicate a spatially varying resolution


Time-dependent CO2 and other forcing causes a varying magnitude of sea-level change, while the spatial pattern is relatively time-invariant (Hawkes 2013; Perrette et al. 2013; Slangen et al. 2014; Bilbao et al. 2015). This phenomenon of ‘pattern scaling’ means that time-dependent forcing is not necessary for our investigation of the spatial structure. Therefore, in the interest of simplicity the FAFMIP flux perturbations are applied as a constant forcing for the full 70 years of each experiment, with no time-variation except for the annual cycle.
Experiments
FAFMIP perturbations to the fluxes of heat, water and momentum (Fig. 1) were applied in five different experiments, as listed below. All perturbations were applied at the air-sea interface in direct contact with seawater surface, such that sea ice is not directly affected. However, there will be indirect effects on sea ice due to the redistribution of heat and freshwater in response to all the perturbations. The heat and freshwater fluxes are defined positive downward into the ocean, and the momentum flux perturbations are positive eastward and northward. FAFMIP experiments were run by nine modelling groups using 13 AOGCMs (Table 1).
Fig. 1: Annual means of downward flux perturbations applied in FAFMIP experiments at the ocean surface for heat, water, eastward momentum, and northward momentum, a–d respectively. Perturbations are the multi model mean surface flux anomalies from simulations forced with 1% per year rising CO2 concentrations averaged over years 61–80
Table 1 Key features of the main AOGCMs studied, where dashes in Ocean horizontal resolution indicate a spatially varying resolution
Experiment 1 In FAF-passiveheat, the heat flux perturbation (Fig. 1a) is applied to a ‘passive added temperature’ tracer, Ta. FAF-passiveheat is a control (similar to piControl), since its climate is not perturbed and experiences only internal variability while the extra tracer allows for the passive uptake of the heat perturbation to be quantified. Ta is initially set to 0 everywhere and the forcing, F, is applied at the surface like a heat flux (none of it penetrates below the surface, like shortwave radiation does). It is transported within the ocean via the same schemes that each model uses to advect and diffuse temperature, T, without affecting the evolution of the ocean state at all because it is passive. Since the perturbation has positive and negative values locally, Ta can be positive and negative. While the input of the passive added heat tracer via the prescribed surface heat flux is identical across models, the geographic patterns of its distribution in the ocean will differ across models, depending on each model’s preindustrial circulation and parameterized tracer transports. The FAF-passiveheat experiment makes it possible to consistently compare the unperturbed tracer uptake across models, which is necessary for a decomposition of the heat uptake in each model when forced with transient climate change (Sect. 2.4).
Experiment 2 In FAF-heat, the heat flux perturbation F is applied as a forcing to ocean temperature, T. (Note that we call it a “forcing” because it is an external perturbation to the climate system, but it is not a radiative forcing, such as is given by CO2 increase.) The perturbation is strongly positive in the North Atlantic and in the Southern Ocean (Fig. 1a). Relative to annual mean climatological heat fluxes, the perturbation reduces the North Atlantic (north of the equator) basin-mean upward heat flux by about 50% (the precise amount varies between models). In the Southern Ocean south of 45° S, the perturbation is roughly 160% of the climatological flux and of opposite sign, switching the basin into a region of net ocean heat uptake. To avoid the atmosphere’s tendency to eliminate the SST anomaly through an opposing air-sea flux, a further passive tracer is used, called the redistributed temperature tracer, Tr which is initialized to equal T at the start and is transported within the ocean in the just the same ways as T but is not forced by the surface heat flux perturbation. The atmosphere is decoupled from the SST of T, and instead sees the surface field of Tr (Fig. 2). The result is that the perturbation gets added to the ocean, where it accumulates and modifies seawater density, and changes ocean transport, but the atmosphere does not absorb any of the added heat and is only modified by changes to surface Tr that arise indirectly through the changing ocean circulation. In FAF-heat, fields of Tr and T quickly diverge from each other by a value approximately equal to Ta. Sub-grid scale schemes (e.g. boundary layer schemes; neutral diffusion; parameterized eddy advection) have nonlinear effects on the transport of temperature that we ignore, meaning that T is assumed equal to Tr + Ta (Gregory et al. 2016). In FAF-passiveheat, Tr (if introduced and treated in the same way) would be identical to T, because no forcing is applied to T. By comparing the distribution of Ta in FAF-heat (whose circulation changes) against FAF-passiveheat (whose circulation follows the steady state) it is possible to identify regions where the changing circulation stores added heat ‘actively’ (i.e. unlike a passive tracer).
Fig. 2: Treatment of surface heat flux perturbation in FAF-heat and FAF-all, redrawn from (Gregory et al. 2016). Q is the net surface heat from the atmosphere and sea ice into the ocean and F is the flux perturbation. The SST used to calculate the surface heat flux to atmosphere and sea ice is coupled to a redistributed temperature tracer Tr, which does not feel F
Experiment 3 The water flux perturbation is derived from the CMIP5 ‘wfo’ diagnostic, which is the sum of precipitation, evaporation, river inflow and water fluxes between floating ice and seawater. There is no perturbation applied over land. The freshwater flux perturbation applied in FAF-water has a very small global annual average, and mainly redistributes freshwater (Fig. 1b). The perturbation is broadly consistent with the “wet gets wetter, dry gets drier” pattern (Held and Soden 2006), reinforcing evaporation in the mid latitudes (by about 10%), and adding freshwater elsewhere (also about 10% reinforcement): namely the equatorial Pacific, the Southern Ocean, the Arctic Ocean, and the high latitude North Pacific and North Atlantic.
Experiment 4 The surface momentum perturbation applied in FAF-stress is mainly characterized by a reinforcement and southward shift of the Southern Ocean westerlies (Fig. 1c). Between 45° and 65° S, the perturbation strengthens the westerlies by about 10%. The perturbation has smaller effects on the zonal and meridional downward momentum fluxes in the mid latitudes (Fig. 1c, d). The perturbation is added to the momentum balance of the ocean surface, such that it does not directly affect sub-grid scale parameterization schemes (e.g. planetary boundary closures) that depend on wind stress or ice stress.
Experiment 5 All three perturbations are applied together in the FAF-all experiment. This experiment serves two purposes: to assess how well the perturbations mimic the effect of CO2 forcing as in 1pctCO2, and to determine the extent to which the perturbations counteract or amplify each other’s effects on sea level when applied simultaneously. If the flux perturbations interact with each other when applied together in FAF-all, then the FAF-all sea-level response will not equal the sum of the sea-level responses to the individual perturbations. This aspect of the FAFMIP design was not tested by Gregory et al. (2016), since no results for FAF-all were available at the time from the five pre-CMIP6 models analysed.
Calculation of change due to perturbations
∆ζ can be derived from CMIP output as the difference in the ‘zos’ field from a forced experiment relative to the ‘zos’ field in an unforced control state. Although ‘zos’ is usually defined to have a zero global area mean (Griffies et al. 2016), for some models it was necessary to subtract the nonzero area mean. This study is focused on the regional sea-level changes expected by the end of the twenty-first century. Because exponentially increasing CO2 concentration, such as in 1pctCO2, gives a radiative forcing which increases linearly in time, and the FAFMIP perturbations correspond to 1pctCO2 forcing at year 70, 70 years of time invariant FAFMIP forcing integrates to approximately the same as a 100-year time integral of 1pctCO2 forcing. ∆ζ is therefore calculated from the final decade (years 61–70) of the perturbation experiments (FAF-stress, -water etc.), and from years 91–100 of 1pctCO2 experiments for comparison. This approach means the amplitudes of ∆ζ will be approximately similar, but in any case, the spatial pattern of ∆ζ is the object of interest in this study.


Experiment 5 All three perturbations are applied together in the FAF-all experiment. This experiment serves two purposes: to assess how well the perturbations mimic the effect of CO2 forcing as in 1pctCO2, and to determine the extent to which the perturbations counteract or amplify each other’s effects on sea level when applied simultaneously. If the flux perturbations interact with each other when applied together in FAF-all, then the FAF-all sea-level response will not equal the sum of the sea-level responses to the individual perturbations. This aspect of the FAFMIP design was not tested by Gregory et al. (2016), since no results for FAF-all were available at the time from the five pre-CMIP6 models analysed.
Calculation of change due to perturbations
∆ζ can be derived from CMIP output as the difference in the ‘zos’ field from a forced experiment relative to the ‘zos’ field in an unforced control state. Although ‘zos’ is usually defined to have a zero global area mean (Griffies et al. 2016), for some models it was necessary to subtract the nonzero area mean. This study is focused on the regional sea-level changes expected by the end of the twenty-first century. Because exponentially increasing CO2 concentration, such as in 1pctCO2, gives a radiative forcing which increases linearly in time, and the FAFMIP perturbations correspond to 1pctCO2 forcing at year 70, 70 years of time invariant FAFMIP forcing integrates to approximately the same as a 100-year time integral of 1pctCO2 forcing. ∆ζ is therefore calculated from the final decade (years 61–70) of the perturbation experiments (FAF-stress, -water etc.), and from years 91–100 of 1pctCO2 experiments for comparison. This approach means the amplitudes of ∆ζ will be approximately similar, but in any case, the spatial pattern of ∆ζ is the object of interest in this study.
Decadal fields of ζ are calculated to reduce (but cannot not totally eliminate) the effect of unforced interannual variability, reflecting our interest in understanding the climate response to forcing by the end of this century. A change (i.e. ∆ζ) is deemed significant for our purposes if its magnitude is more than twice the decadal standard deviation (a 95% interval for a normal distribution) of variability determined at that location in the relevant control simulation (i.e. piControl for 1pctCO2 and FAF-passiveheat for all other experiments). Decadal mean fields of ζ are calculated for each decade of control simulation, and the threshold for significance is taken to be double the standard deviation across the decadal averages. Insignificant ∆ζ features (within ± 2 standard deviations) are set to 0 for plotting purposes.
The steric sea-level responses to each perturbation can be decomposed into the thermosteric (∆ζT) and halosteric (∆ζS) components:
Δ𝜁𝑇=∫𝜂𝐻(𝛼Δ𝑇Δ𝑧)−𝑙𝜃, (3)
Δ𝜁𝑆=−∫𝜂𝐻𝛽Δ𝑆Δ𝑧. (4)
The thermosteric sea-level change (resulting from temperature change), ∆ζT (3), is the depth integral (from the surface, η, to the full ocean depth, H, with a layer thickness Δz) of the change in temperature (∆T, °C) multiplied by the seawater thermal expansion coefficient (α, °C−1) with the global mean thermosteric sea-level change, 𝑙𝜃, removed. We focus on the thermosteric component with its global mean removed to because the spatial pattern of change is the quantity of interest for this study, not the global mean change. The temperature change is the difference in potential temperature from the forced experiment relative to the control (FAF-passiveheat) averaged over the final decade. A similar Eq. (4) can be constructed for the halosteric component of sea-level change, using the change in salinity (S) and the haline contraction coefficient of seawater (β, dimensionless). Since saline water of a given mass has a smaller volume than the same mass of freshwater, a minus sign converts contraction to expansion, which is more readily comparable with ∆ζ and ∆ζT. The halosteric change typically has a near-zero global mean because total ocean salinity changes are small, making only a very small or negligible contribution to the global mean sea-level change (Gregory et al. 2019). α and β were calculated using the mean temperature, salinity and pressure fields of the final decade of the control simulation using standard nonlinear equations of state (McDougall and Barker 2011).
A sensitivity test (not shown) found that the calculation of ∆ζT and ∆ζS is not strongly affected by the choice of decade used to derive α and β; the effect of the temperature and salinity changes on α and β are small and it is the spatial patterns of Δ𝑇 and Δ𝑆 that set ∆ζT and ∆ζS. Note that the dynamic sea-level change will differ from the steric change (the sum of ∆ζT and ∆ζS) in locations where there is a large barotropic redistribution of density such as the subpolar North Atlantic and the Arctic (Lowe and Gregory 2006; Yin et al. 2010). The dynamic and steric sea-level changes will also differ in locations such as shelf seas, where the change in mass loading of the full water column (a non-steric effect) can be large (Landerer et al. 2007; Yin et al. 2010).
We quantify the part of dynamic sea-level change that is non-steric, Δ𝜁𝑁, as
Δ𝜁𝑁=Δ𝜁−Δ𝜁𝑇−Δ𝜁𝑆.(5)
In plots of Δ𝜁𝑁, we subtract the area mean to reveal the spatial pattern of non-steric sea-level change, since spatial anomalies are the quantity of interest for this work. According to recent terminology conventions (Gregory et al. 2019), Δ𝜁 is related to other components of sea-level change through
Δ𝜁=Δ𝐵+Δ𝑅𝑚+Δ𝜁𝑇+Δ𝜁𝑆−ΔΓ−𝑙𝑏,(6)
where Δ𝐵 is the change due to the inverse barometer effect, Δ𝑅𝑚 is the manometric sea-level change (due to change in ocean mass per unit area), ΔΓ is the change due gravitational, rotational and deformational (GRD) effects from the redistribution of mass over the surface of the planet, 𝑙𝑏 is the barystatic change (due to addition of water to the ocean, mostly from land ice). The difference between the dynamic sea-level change and the steric change in the real world is
Δ𝜁−Δ𝜁𝑇−Δ𝜁𝑆=Δ𝜁𝑁=Δ𝐵+Δ𝑅𝑚−ΔΓ−𝑙𝑏.(7)
However, in the AOGCMs considered in this study (Table 1) ΔΓ and 𝑙𝑏 are zero because these processes are not represented. Δ𝐵 is not readily quantifiable from these models, but is assumed to be small and average to near zero over long time periods since it is of most relevance on meteorological rather than climate timescales (Ponte 2006; Gregory et al. 2019). Therefore, maps of Δ𝜁𝑁 relative to its global area mean in these models reveals the spatial pattern of Δ𝑅𝑚, the manometric sea-level change. This sea-level component is broadly analogous to the ‘barotropic component’ of sea-level change discussed in previous literature, although they are calculated differently (e.g. see Lowe and Gregory 2006).
Decomposition of ocean heat content change


In plots of Δ𝜁𝑁, we subtract the area mean to reveal the spatial pattern of non-steric sea-level change, since spatial anomalies are the quantity of interest for this work. According to recent terminology conventions (Gregory et al. 2019), Δ𝜁 is related to other components of sea-level change through
Δ𝜁=Δ𝐵+Δ𝑅𝑚+Δ𝜁𝑇+Δ𝜁𝑆−ΔΓ−𝑙𝑏,(6)
where Δ𝐵 is the change due to the inverse barometer effect, Δ𝑅𝑚 is the manometric sea-level change (due to change in ocean mass per unit area), ΔΓ is the change due gravitational, rotational and deformational (GRD) effects from the redistribution of mass over the surface of the planet, 𝑙𝑏 is the barystatic change (due to addition of water to the ocean, mostly from land ice). The difference between the dynamic sea-level change and the steric change in the real world is
Δ𝜁−Δ𝜁𝑇−Δ𝜁𝑆=Δ𝜁𝑁=Δ𝐵+Δ𝑅𝑚−ΔΓ−𝑙𝑏.(7)
However, in the AOGCMs considered in this study (Table 1) ΔΓ and 𝑙𝑏 are zero because these processes are not represented. Δ𝐵 is not readily quantifiable from these models, but is assumed to be small and average to near zero over long time periods since it is of most relevance on meteorological rather than climate timescales (Ponte 2006; Gregory et al. 2019). Therefore, maps of Δ𝜁𝑁 relative to its global area mean in these models reveals the spatial pattern of Δ𝑅𝑚, the manometric sea-level change. This sea-level component is broadly analogous to the ‘barotropic component’ of sea-level change discussed in previous literature, although they are calculated differently (e.g. see Lowe and Gregory 2006).
Decomposition of ocean heat content change
We can decompose OHC change into components due to changes in ocean temperature and in transport. Following Gregory et al. (2016), we let Φ represent the transport operator that encompasses all processes that affect heat transport, including resolved and parameterized advection, diffusion, and convection. That is, Φ(𝑇)=−∇∙(𝑢𝑢𝑇+𝑃𝑃), the convergence of temperature due to the three dimensional resolved velocity field 𝑢𝑢 and parameterized subgrid-scale tracer transport processes 𝑃𝑃. In Φ(𝑇) parentheses around 𝑇 indicates the action of ocean tracer transport on the temperature within the parentheses. At steady state, the ocean has an unperturbed temperature, 𝑇−, and an unperturbed three-dimensional transport, Φ−, where overlines indicate a long time average over the control run. The convergence of unperturbed temperature transport, Φ−(𝑇−), is zero in the steady state, except at the surface where it balances the surface heat flux.
Forced climate change modifies the surface fluxes. It alters the ocean temperature by an amount T′, relative to the unperturbed state by the addition of heat, and the transport by Φ′ through changing both the wind driven and density driven transports. As a result, the convergence Φ(𝑇) of heat is modified as well and is no longer zero. Hence, interior temperature changes according to
∂𝑇′∂𝑡=[Φ− + Φ′](T− + 𝑇′) = Φ−(𝑇′)+Φ′(T−) + Φ′(𝑇′), (8)
where [Φ−+Φ′](𝑇−+𝑇′) is symbolically the action of both the unperturbed and perturbed transport acting on the unperturbed and perturbed temperature. Note also that Φ−(𝑇−)=0.
Thus we distinguish three different causes of temperature change that arise from the convergence of: Φ−(𝑇′), transport of the added heat by the unperturbed transport processes; Φ′(𝑇−), changes in the ocean transport redistributing unperturbed heat; and Φ′(𝑇′), perturbation in the transport that redistributes the added heat (8).
If the oceans absorbed the added heat from anthropogenic climate change like a passive tracer that does not affect ocean circulation or other transport processes, then OHU would be driven entirely by the first term, Φ−(𝑇′). This term is therefore the “passive uptake of added heat”. In reality, the ocean circulation and subgrid scale processes are affected by temperature change and other surface flux changes, so the other terms play a part. The second term is a pure redistribution, whose global volume integral is small (but not precisely zero because 𝑇− can be fluxed to the atmosphere). The final term is of second order in perturbation quantities, but it is not always negligible.
Consider the following illustrative situation in a convective zone of the Labrador Sea, where climate change causes increased heat flux into the ocean, and the ocean temperature gets warmer at around 250 m depth. Positive Φ−(𝑇′) describes the change in temperature that results from the unperturbed circulation and subgrid processes that passively transport the additional heat downwards. However, these transport processes are weakened by the input of 𝑇′ because the air-sea heat input strengthens stratification and weakens downward heat transport by convection. The weakened downward transport carries less additional heat than in the passive case; and Φ′(𝑇′) is weakly negative because Φ′ is negative. Finally, weakened downward transport means that less heat from lower latitudes gets brought northward and downward, causing strongly negative Φ′(𝑇−).
FAFMIP experiments and diagnostics allow for the three contributions to the change in the convergence of heat to be distinguished. We can express the change in ocean heat content (∆h, Jm−2) due to each contribution by converting the appropriate temperature (T) field using a reference heat capacity for seawater (cp0 = 4000 J kg−1 K−1), a reference density (ρ0 = 1026 kg m−3) and the ocean grid cell vertical thickness (Δz, m). Differencing particular experiments and temperature fields (T, Ta, or Tr) yields different components of OHC change. Further notes on the time evolution of Tr, Ta and T, describing how different temperature change terms are grouped in the decomposition are included in the “Appendix”.
The OHC change due all three convergences of temperature in (8), Δh, is
Δℎ=Δ𝑇𝑐𝑝0𝜌0Δ𝑧, (9)
where ∆T, is the difference of the model’s temperature field (T) between the final decades of FAF-heat and FAF-passiveheat (9). In FAF-heat, the heat flux changes the transport processes (Φ′ ≠ 0) and the temperature (𝑇′ ≠ 0) and, thus there is a change in heat content due to all three types of temperature convergence.
The OHC change due to passive uptake of additional heat, Δℎ[Φ⎯⎯⎯⎯⎯(𝑇′)] is
Δℎ[Φ⎯⎯⎯⎯⎯(𝑇′)]=𝑇⎯⎯⎯⎯𝑎𝑐𝑝0𝜌0Δ𝑧, (10)
where the notation Δℎ[…] symbolically represents the change in OHC due to the convergence of temperature enclosed by the square brackets. Since passive temperature is initially 0, its decadal mean change by the end of simulation is simply 𝑇−𝑎, the time mean Ta from FAF-passiveheat for the years 61–70.
The OHC change due to the redistribution of unperturbed temperature, Δℎ[Φ′(𝑇⎯⎯⎯⎯)] is
Δℎ[Φ′(𝑇⎯⎯⎯⎯)]=Δ𝑇𝑟𝑐𝑝0𝜌0Δ𝑧, (11)
where ∆Tr is the difference between FAF-heat redistributed temperature, Tr, and FAF-passiveheat T, averaged over the final decade.
The OHC change due to the perturbation in the transport redistributing the added heat, Δℎ[Φ′(𝑇′)] is