The spatial pattern of the steric height from the EOF analysis is then projected back onto the entire (1960–2012) non-detrended and non-smoothed steric height fields from the source region to produce a series of each PC.
Finally, the PCs from step 2, now solely based on steric height from the open ocean, are projected onto the spatial patterns from tide-gauge residuals (SDSLTG), which produces an EOF-based SDSL reconstruction (SDSLEOF) over the entire 1960–2012 period at each site.
The SDSLEOF reconstructions are then evaluated as spatial averages over all sites in each region. By using varying barystatic corrections and by considering randomly varied parameter choices, we produce 5,000 ensemble members that are used to assess uncertainties of the EOF reconstruction. The EOF approach has two major advantages compared to simple field averages over the source region as formerly used in refs. 22,23 While a simple area average assumes that the steric signal from the open ocean is one on one transferred toward the coast, there are factors at play that may dampen or amplify the signal while propagating from the open ocean toward the coast. The regression character of the EOF approach explicitly takes this into account and scales the signal to match the variability seen at tide gauges. Furthermore, EOF analysis decomposes a field into common modes. As typical for many other atmospheric or oceanographic time series, most of the variance of the entire field is contained in the first few modes, while local effects move as noise into lower EOFs. Thus, the EOF approach is robust against local outliers. To test the stability of the EOF reconstruction against increasing data uncertainties in the steric products before the 1980s, we also calculated a second 5,000-member ensemble on the basis of PCs calculated since solely 1980 (Extended Data Fig. 7). We do not find any significant differences in the reconstructions based on PCs calculated over the entire period or only since 1980. Correlations are, in all cases, >0.98 with the only exception being the three western boundary currents where the correlation coefficients are between 0.92 and 0.94. The trends are also not significantly different between the two ensembles (Extended Data Fig. 7b).
Earth system models
To test the performance of the EOF technique, we apply it to the historical runs of 11 ESMs from the Coupled Model Intercomparison Project Phase 5 (CMIP5) [75], which are all listed in Supplementary Table 2. As CMIP5 models usually have a horizontal resolution that barely exceeds one degree, we additionally included one eddy-permitting model (CNRM-CM6-1-HR; ref. 76) from the HighResMIP initiative. CNRM-CM6-1-HR has a resolution of 25 km, providing a more realistic representation of the coastal zone than its CMIP5 counterparts and many ocean reanalyses19. We decided to use ESMs rather than ocean reanalyses because the latter suffer from (regionally varying) drifts due to sparse data assimilation, uncertainties in air–sea fluxes and mixing errors19, while ESMs are entirely consistent without data assimilation. All simulations have been linearly de-drifted using the pre-industrial control run (PiControl). ESMs have the advantage of providing predescribed components of the sea-surface height (‘zos’), ocean-bottom pressure (‘pbl’) and global mean thermosteric sea-level changes (‘zostoga’). As the ESMs are volume- rather than mass-conserving, we sum zos and zostoga and subsequently subtract pbl to derive steric height from each model (one might also estimate steric height from temperature and salinity profiles in each model, which is, however, more time consuming). We sample coastal sea-level (zos + zostoga) at the grid points closest to the tide-gauge locations and perform a similar correlation analysis with steric height to identify the source regions. Then the EOF technique is applied in a similar fashion (with 1,000 randomly varied smoothing factors and mode-selection thresholds) as for observations. We compare the reconstruction to both the available 53-yr calibration periods as well as a randomly selected and independent validation period of 53 consecutive years that have not been used for calibration. There are three factors to keep in mind when using ESMs as a testbed for the EOF approach. First, the resolution of ESMs is, even in CNRM-CM6-1-HR, still too coarse to fully resolve coastal processes. Second, the model provides ‘perfect observations’ at each location without gaps and measurement errors. Third, we do not have corrections for barotropic wind forcing available for the ESMs as we apply for real observations. While the first two factors spatially smooth the observations and therefore tend to increase the covariance compared to real observations, the missing barotropic correction may lead to a slightly degraded performance in high latitude regions compared to reality.
Statistics
All correlations and regressions were performed after removing linear trends. Statistical significance and error estimates for the correlation analyses are computed using Fisher’s z-transform77. While the test does not account for temporal correlations in the data, it is much more time-efficient and leads to barely different source regions than with repeated simulations using autoregressive processes. Uncertainties of linear trends are calculated assuming that the residuals are temporally correlated following an autoregressive process of the order one. Nonlinear trends in the different budget components have been calculated by a singular spectrum analysis with an embedding dimension of ten using the MATLAB package from ref. 78.
Extended data
Extended Data Fig. 1: Performance of ocean reanalysis in simulating SDSL variability and trends. Shown are linear trends of SDSL as simulated by the a SODAsi.378, b SODA 2.2.480, c ORA S481, d ORA20C82,83, and e GECCO214 reanalysis over the common period from 1960 to 2012. For all reanalysis systems, the model internal global average has been replaced by ref. 24. In b and d, data is only available until 2008 and 2009, respectively. Grey dots show the 89 tide-gauge locations used in this study. f Linear trends in SDSLTG from the virtual stations of the nine coastal regions (grey bars) are compared to trends calculated from SDSL as simulated by the five ocean reanalysis systems (nearest-neighbour series). d, Correlations between detrended SDSLTG and SDSL from the five ocean reanalysis systems. The grey shadings separate the different regions from each other.
Extended Data Fig. 2: Tide-gauge coherence and virtual stations for each region. a, Cross-Correlation matrix for the 89 tide-gauge records ordered by region. Black boxes mark the locations of the selected tide-gauge records for each region. b, The observed tide-gauge records (corrected for vertical land motion; coloured lines) together with the virtual station for each region (thick black line) that has been built based on gap-filled records (see Methods). The percentage of total data availability in each region is given in brackets.
Extended Data Fig. 3: Origin of coastal SDSL changes. Spatial correlation patterns between the SDSL residuals at tide gauges (SDSLTG, grey dots) and steric height24 from the open ocean used to assemble Fig. 1a but for each of the nine coastal regions separately. Only significant correlations (P ≤ 0.05) are shown. a, North Sea, b NW Atlantic north of Cape Hatteras, c NW Atlantic south of Cape Hatteras, d NE Pacific, e Hawaii, f Japan Sea, g West Australia, h New Zealand, and i West Pacific.
Extended Data Fig. 4: Reconstruction of coastal SDSL changes. Extension of Fig. 2b illustrating the reconstruction of SDSLEOF in observations (top) and in validation experiments with ESMs (here illustrated by the CNRM-CM6-1-HR, bottom) for each of the nine coastal regions. a, North Sea, b Northwest Atlantic north of Cape Hatteras, c Northwest Atlantic south of Cape Hatteras, d Northeast Pacific, e Hawaii, f Japan Sea, g West Australia, h New Zealand, and i West Pacific.
Extended Data Fig. 5: Linear trends in steric height and comparison of different observational products. Shown are the linear trends in steric height calculated over the upper 2000m for different gridded observational products. a, ref. 24 and 67, b ref. 68, c EN469 with ref. 70 corrections, and d EN4 with ref. 71 corrections. The grey dots mark the locations of tide-gauge records used in this study. e, Linear trends for the SDSLEOF reconstructions in each region using the four different data products (grey bars = ref. 24 and67; blue = ref. 68; turquoise = EN469 with ref. 70 corrections; yellow = EN469 with ref. 71 corrections. f, Same as e but showing the correlation between SDSLEOF based on the different products and SDSLTG.


Extended Data Fig. 3: Origin of coastal SDSL changes. Spatial correlation patterns between the SDSL residuals at tide gauges (SDSLTG, grey dots) and steric height24 from the open ocean used to assemble Fig. 1a but for each of the nine coastal regions separately. Only significant correlations (P ≤ 0.05) are shown. a, North Sea, b NW Atlantic north of Cape Hatteras, c NW Atlantic south of Cape Hatteras, d NE Pacific, e Hawaii, f Japan Sea, g West Australia, h New Zealand, and i West Pacific.
Extended Data Fig. 4: Reconstruction of coastal SDSL changes. Extension of Fig. 2b illustrating the reconstruction of SDSLEOF in observations (top) and in validation experiments with ESMs (here illustrated by the CNRM-CM6-1-HR, bottom) for each of the nine coastal regions. a, North Sea, b Northwest Atlantic north of Cape Hatteras, c Northwest Atlantic south of Cape Hatteras, d Northeast Pacific, e Hawaii, f Japan Sea, g West Australia, h New Zealand, and i West Pacific.
Extended Data Fig. 5: Linear trends in steric height and comparison of different observational products. Shown are the linear trends in steric height calculated over the upper 2000m for different gridded observational products. a, ref. 24 and 67, b ref. 68, c EN469 with ref. 70 corrections, and d EN4 with ref. 71 corrections. The grey dots mark the locations of tide-gauge records used in this study. e, Linear trends for the SDSLEOF reconstructions in each region using the four different data products (grey bars = ref. 24 and67; blue = ref. 68; turquoise = EN469 with ref. 70 corrections; yellow = EN469 with ref. 71 corrections. f, Same as e but showing the correlation between SDSLEOF based on the different products and SDSLTG.
Extended Data Fig. 6: Validation of the vertical land motion (VLM) correction. Comparison between observed trends (after the removal of barystatic gravitation, rotation and deformation terms) and residual VLM plus Glacial Isostatic Adjustment from the difference between tide gauges and the hybrid reconstruction from ref. 50 as well as the observed trends and residual VLM from Global Navigation Satellite System plus Glacial Isostatic Adjustment and the difference between tide-gauge and satellite altimetry as calculated by ref. 4 (Fred).
Extended Data Fig. 7: Validation of the EOF approach. a, Shown are the time series of SDSLEOF based on reconstructions using principal components that have been calculated and regressed on the steric height from the open-ocean over the entire 1960 to 2012 period (and as used in the main paper) as well as those based on principal components that have been calculated (and regressed on the steric height from the open-ocean) over the period from 1980–2012. The grey shading marks the corresponding validation period from 1960–1979. b, The corresponding linear trends of SDSLEOF over the common period from 1960–2012. Shadings and error bars represent the 95% confidence intervals.


Chen and Tung (2018)
Title:
Global surface warming enhanced by weak Atlantic overturning circulation
Key Points:
The role of AMOC has shifted its primary role from transporting surface heat northwards to storing heat in the deeper Atlantic, thereby buffering global surface warming.
During an accelerating phase from the mid-1990s to the early 2000s, the AMOC stored about half of the global excess heat, contributing to a slowdown in global warming.
Changes in the AMOC since the 1940s are best explained by multidecadal variability, rather than a trend forced by human activities.
Recently, the AMOC and oceanic heat uptake have weakened, suggesting a period of rapid global surface warming.
Keywords:
Atlantic meridional overturning circulation, Global surface warming, Ocean heat storage, Multidecadal variability, Climate change, Oceanic heat uptake
Corresponding author:
Ka-Kit Tung
Citation:
Chen, X., & Tung, K.-K. (2018). Global surface warming enhanced by weak Atlantic overturning circulation. Nature, 559(7714), 1–14. doi:10.1038/s41586-018-0320-y
URL:
https://www.nature.com/articles/s41586-018-0320-y
Abstract
Evidence from palaeoclimatology suggests that abrupt Northern Hemisphere cold events are linked to weakening of the Atlantic Meridional Overturning Circulation (AMOC) [1], potentially by excess inputs of fresh water [2]. But these insights — often derived from model runs under preindustrial conditions — may not apply to the modern era with our rapid emissions of greenhouse gases. If they do, then a weakened AMOC, as in 1975–1998, should have led to Northern Hemisphere cooling. Here we show that, instead, the AMOC minimum was a period of rapid surface warming. More generally, in the presence of greenhouse-gas heating, the AMOC’s dominant role changed from transporting surface heat northwards, warming Europe and North America, to storing heat in the deeper Atlantic, buffering surface warming for the planet as a whole. During an accelerating phase from the mid-1990s to the early 2000s, the AMOC stored about half of excess heat globally, contributing to the global-warming slowdown. By contrast, since mooring observations began [3,4,5] in 2004, the AMOC and oceanic heat uptake have weakened. Our results, based on several independent indices, show that AMOC changes since the 1940s are best explained by multidecadal variability [6], rather than an anthropogenically forced trend. Leading indicators in the subpolar North Atlantic today suggest that the current AMOC decline is ending. We expect a prolonged AMOC minimum, probably lasting about two decades. If prior patterns hold, the resulting low levels of oceanic heat uptake will manifest as a period of rapid global surface warming.
Main
As an analogy of the flow of energy in our climate system, consider the filling of a bucket of water from a tap at the top. The feed rate of the tap is an analogue of the top-of-atmosphere radiative imbalance—the net heating—of our planet, with the water level in the bucket analogous to surface warming. The sink at the bucket bottom drains into a larger bucket below (the deeper oceans). If the drain rate is the same as the feed rate from the tap at the top, the water level in the bucket does not rise (hiatus of surface warming). If the drain is plugged, the water level will rise rapidly in the bucket (rapid surface warming). AMOC controls about half of the variation of this ‘drain rate’.
Figure 1 quantifies the energy budget of our climate system, using the subsurface ocean heat content (OHC) measured mostly by a system of autonomous profiling Argo floats, during a period, 2000–2014, when the ‘drain rate’ was large. The total OHC, as approximated by that in the upper 1,500 m of the oceans, is increasing at a rate of about 0.42 ± 0.02 W m−2, consistent with radiative imbalance7. The upper 200 m roughly corresponds to the mixed layer globally. Through wind and turbulent mixing, variations of sea surface temperature (SST) and mixed-layer OHC are highly statistically correlated (r = 0.82 in 13-month running mean). Figure 1 shows that both were in a warming slowdown for this period. Why the upper 200 m OHC was in a warming slowdown is clear: the increase in heat storage below 200 m, about 89 zettajoules (1 ZJ = 1021 J). This amount of heat is equivalent to 180 years of the world’s energy consumption at the current rate, and any future variation even within this observed range will have important consequences for the surface temperature.
Figure 1: Quantifying the global heat budget and the partition among ocean basins in the two periods 2000–2004 and 2005–2014. The SST from ERSST.v4 is shown as a black curve and the 0−200-m OHC from the ISHII and Scripps datasets (see Methods) is shown as an orange curve, showing that they co-vary and that both are in a warming slowdown, while the total OHC, as approximated by the 0−1,500-m OHC (red curve), is increasing at the regressed linear rate of 0.42 W m−2 (red dashed straight line). This excess heat from forcing is sequestered below 200 m. The orange-shaded region represents the additional amount of heat stored in the 200−1,500 m layer since 2000, about 89 ZJ. One zettajoule is equivalent to twice the world’s annual energy consumption. If this additional storage were absent, the upper 200 m would have increased at the rapid rate of the red curve. We adjusted the data for the Southern Ocean to remove a possible artefact due to the rapid transition from no-Argo to the Argo observing platform around 2002−200328. The inset shows the division of the 89 ZJ of global ocean increase in heat storage in the 200−1,500 m layer into the four ocean basins and two periods. 35° S marks the northern boundary of the Southern Ocean and the southern boundary of the Atlantic, Pacific and Indian oceans. The error bars are one-standard-deviation errors of the linear regression.
If the radiative imbalance and the heat storage below 200 m were to remain the same, the 0–1,500 m OHC would still increase at the same rate as the radiative imbalance, but the 0–200 m OHC curve would lie on the 0–1,500 m curve, increasing at the same rate, or about 0.23 °C per decade. Our best estimate for the next two decades, allowing for some increase in ocean storage, is 70% of that rate, at 0.16 °C per decade (see Methods), close to the 25-year trend of 0.177 °C per decade of the last rapid warming period in the twentieth century [8].
The inset of Figure 1 shows how the global increase in OHC storage between 200 m and 1,500 m are partitioned among the various oceans. The Pacific and the Indian oceans dominate the horizontal exchanges of heat in the upper 300 m [9,10], and the Atlantic and the Southern oceans dominate the vertical redistribution11. They accounted for about 70% of the global heat storage increase in the 200–1,500 m layer during 2000–2014, divided between the North Atlantic, which is dominant before 2005, and the Southern Ocean after 2005. The subsurface warming in the Southern Ocean started in 1993 according to the data available (see below), and was attributed to the southward displacement and intensification of the circumpolar jet [8], caused in large part by the Antarctic ozone hole [12]. The North Atlantic’s role appears to be cyclic on decadal timescales, with AMOC in an accelerating phase before 2005.
AMOC transports warm saline surface water found in the subtropical Atlantic to the subpolar Atlantic, where heat loss to the cold atmosphere increases its density. Aided by its high salinity it sinks and returns southward at depth. When AMOC is stronger (weaker), more (less) of the warm and saline water is found in the subpolar Atlantic, and subsequent sinking subducts more (less) heat there, as demonstrated in Figure 2. The contrast is dramatic between periods when AMOC is increasing and when it is decreasing. Why AMOC sometimes accelerates or declines is more complicated. It could be responding to external forcing, for example, such as the freshening of the subpolar waters from melting ice at the end of the Little Ice Age [13]. Or, AMOC could be part of a natural, multidecadal variability involving feedbacks between the density effect of salinity on deep convection in Labrador and the Nordic Seas, and the subsequent induced northward transport of surface salinity reinforcing the deep convection [14].


The inset of Figure 1 shows how the global increase in OHC storage between 200 m and 1,500 m are partitioned among the various oceans. The Pacific and the Indian oceans dominate the horizontal exchanges of heat in the upper 300 m [9,10], and the Atlantic and the Southern oceans dominate the vertical redistribution11. They accounted for about 70% of the global heat storage increase in the 200–1,500 m layer during 2000–2014, divided between the North Atlantic, which is dominant before 2005, and the Southern Ocean after 2005. The subsurface warming in the Southern Ocean started in 1993 according to the data available (see below), and was attributed to the southward displacement and intensification of the circumpolar jet [8], caused in large part by the Antarctic ozone hole [12]. The North Atlantic’s role appears to be cyclic on decadal timescales, with AMOC in an accelerating phase before 2005.
AMOC transports warm saline surface water found in the subtropical Atlantic to the subpolar Atlantic, where heat loss to the cold atmosphere increases its density. Aided by its high salinity it sinks and returns southward at depth. When AMOC is stronger (weaker), more (less) of the warm and saline water is found in the subpolar Atlantic, and subsequent sinking subducts more (less) heat there, as demonstrated in Figure 2. The contrast is dramatic between periods when AMOC is increasing and when it is decreasing. Why AMOC sometimes accelerates or declines is more complicated. It could be responding to external forcing, for example, such as the freshening of the subpolar waters from melting ice at the end of the Little Ice Age [13]. Or, AMOC could be part of a natural, multidecadal variability involving feedbacks between the density effect of salinity on deep convection in Labrador and the Nordic Seas, and the subsequent induced northward transport of surface salinity reinforcing the deep convection [14].
Fig. 2: The OHC linear trend in the Atlantic basin. The trend is zonally averaged over two periods, when AMOC is increasing (a) and decreasing (b). The two periods are chosen according to the observed AMOC trends in Fig. 3a. ISHII data are used in the first period and Scripps data are used in the second period. Stippling indicates areas of statistical significance at the 95% confidence level. The linear trend is unreliable in the Southern Ocean prior to 2005, and so that region is masked.
AMOC is commonly believed to be slowing on centennial timescales owing to global warming. The RAPID/MOCHA mooring array, deployed in 20043 off the coast of Florida to monitor AMOC, soon afterwards recorded its weakening4. The decadal decline, however, is ten times larger than the predicted forced response5, causing concerns about its long-term trend and possible deficiencies of the models used. Figure 3a, constructed from various independent proxies from 1945 to the present (see Extended Data Fig. 1 for unfiltered time series and Extended Data Fig. 2 for error bars), shows that it is dominated instead by reversing phases. The weakening AMOC, by 3.7 Sverdrups (Sv) since 2005 measured by the RAPID/MOCHA array, was actually preceded by an acceleration15,16. Altimetry data of sea-surface heights (SSH) available since 199317 were used to deduce18 via geostrophic balance that at 41° N AMOC sped up by 4 Sv from the early 1990s to 2005, consistent with Zhang’s subsurface fingerprint proxy6. We use multiple independent proxies to infer subpolar AMOC strength back in time to 1945. Many of the proxies used here have been validated by models: Zhang’s subsurface temperature fingerprint was highly coherent with AMOC strength6,19,20 at low frequencies in the model (GFDL CM2.1) at mid-latitudes. The subpolar gyre SST proxy21, and the upper ocean subpolar salinity proxy20 were also model-validated. Along with the long record of tide gauges along the east coast of the USA22, these proxies consistently indicate a period of low AMOC from the mid-1970s to the 1990s. The shading in Fig. 3 shows that this period coincided with a period of rapid surface warming. See also Extended Data Fig. 3 for the coincidence of Atlantic OHC change and global surface warming. See Methods for model–observation reconciliation.
Figure 3: AMOC and GSTA variations. a, Mid and subpolar latitude AMOC strength, as calculated at 41° N using altimetry measurements, from ref. 18 (red, two-year running mean, Sverdrup scale shown on the right); inferred from integrated subpolar salinity in 0–1,500 m and 45–65° N in the Atlantic as a proxy, using the ISHII (dark blue) and Scripps (purple) datasets, with a two-year running mean. The green curve is the subpolar salinity, similarly calculated but using EN4. The AMOC fingerprint6 (dark blue) and the accumulated sea-level index (turquoise) calculated from historical tide gauge measurements22 were smoothed with 10-year and 7-year low-pass filters, respectively, from their sources. The subpolar gyre SST index21 in orange is also a two-year running mean. See Methods for details. The inset shows RAPID-measured AMOC at 26° N. b, Shown are GSTA from HadCRUT4.6 (black), the nonlinear secular trend (close to the 100-year linear trend) (brown) and variation about the trend for timescales longer than decadal (multidecadal variability (MDV), red). The inset shows the SST spatial pattern associated with MDV obtained by regressing SST onto its time series. The blue curve is the smoothed version of GSTA obtained as the sum of the secular trend and MDV. The faint lines around the solid lines are from 100 ensemble members of the HadCRUT4.6, which assess the range of uncertainty of the data used in the solid lines.
We call AMOC+ (AMOC–) the phase when the AMOC strength is above (below) climatology (based on the subpolar salinity, which has a long record with no trend). The high (+) phase consists of two rapid subphases. The increasing subphase (AMOCup) started in 1993, from the low point in AMOC–, first slowly and then rapidly, peaking in 2005. It is then followed by a rapid decreasing subphase (AMOCdown) (2005 to the present) (Fig. 3a). At low values of overturning (AMOC–) the strength is relatively level even though there are short-term fluctuations, because a slower poleward transport of saline water from the tropical Atlantic makes it difficult to speed up the sinking in the subpolar North Atlantic except through slower processes: The surface water could slowly become more saline through the reduction of fresh water outflow from land glaciers and from the Arctic Ocean23. The northward transport of warm and saline water increased more rapidly since 1999, and started a negative feedback as the warm surface water increased glacier melt and freshwater outflow. The previous AMOCdown subphase of 1965–1974 started with the gradual freshening of the north Atlantic waters, as can be inferred from the decreasing salinity in the subpolar region, braking the AMOC. Incidentally, both SSH at 41° N and RAPID at 26° N showed a simultaneous, short-lived 30% drop in AMOC strength in 2009–20105, partially caused by an extreme negative episode of atmospheric North Atlantic Oscillation that affected the wind field5 over both areas.
Water masses in the subpolar and subtropical gyres are different and transports across gyre boundaries need not be continuous14. For vertical heat subduction, it is mainly the subpolar AMOC that is our focus in Fig. 3a. Signals from salinity proxies at the subpolar Atlantic have almost reached the previous low. The subpolar gyre SST has started to warm. The deep Labrador Sea density, which is known to lead by 7–10 years changes in wider basin AMOC15,16, has stopped declining since 2014 (Extended Data Fig. 4). The subtropical region is more prone to higher-frequency perturbations14, and the RAPID time series is experiencing its short-term oscillations (two so far) after the recovery from the large dip in 2010 so the decadal trend may be difficult to see. Nevertheless, it appears to have stabilized at that latitude. Previously, when AMOC reached its lowest AMOC– value after 1975, that level phase lasted two and a half decades. Although we have data only for one cycle, its observed non-sinusoidal pattern characterized by a prolonged flat minimum separated by steep peaks is as expected from the physical arguments presented above.


Water masses in the subpolar and subtropical gyres are different and transports across gyre boundaries need not be continuous14. For vertical heat subduction, it is mainly the subpolar AMOC that is our focus in Fig. 3a. Signals from salinity proxies at the subpolar Atlantic have almost reached the previous low. The subpolar gyre SST has started to warm. The deep Labrador Sea density, which is known to lead by 7–10 years changes in wider basin AMOC15,16, has stopped declining since 2014 (Extended Data Fig. 4). The subtropical region is more prone to higher-frequency perturbations14, and the RAPID time series is experiencing its short-term oscillations (two so far) after the recovery from the large dip in 2010 so the decadal trend may be difficult to see. Nevertheless, it appears to have stabilized at that latitude. Previously, when AMOC reached its lowest AMOC– value after 1975, that level phase lasted two and a half decades. Although we have data only for one cycle, its observed non-sinusoidal pattern characterized by a prolonged flat minimum separated by steep peaks is as expected from the physical arguments presented above.
The longer Global-mean Surface Temperature Anomaly (GSTA) record shown in Fig. 3b, together with its low-frequency variation24,25, consists of a secular trend and a multidecadal variability (MDV), defined to be on timescales that are decadal or longer. The spatial pattern associated with MDV (inset to Fig. 3b) has the pattern of an interhemispheric seesaw in the Atlantic, with the North Atlantic being the centre of action, consistent with model results26. When the MDV is increasing it doubles the GSTA warming rate over the 100-year trend of 0.08 K per decade, and is associated with a period of rapid warming in the late and also the early twentieth century. That secular trend of 0.08 K per decade, statistically significant at over 95% confidence level against a second-order autoregressive (AR(2)) red noise, has been attributed to the underlying anthropogenic global warming trend27. The regressed spatial pattern associated with the secular trend resembles the model-predicted response from greenhouse warming24,25. The MDV in the GSTA is related to the Atlantic Multidecadal Oscillation (AMO) (see Methods), the latter having a record extending back several hundred years.
The previous period of low overturning in the AMOC− phase, from 1975 to the 1990s, coincided with a period of rapid global warming at the surface. This is more than a coincidence because the energy budget involved can be quantified. We do not have reliable subsurface data for the period when the surface warming was rapid. However, the change from that period can be quantified so that an estimate can be made for what would happen if that change were absent. During 2000–2005, in the AMOCup subphase, 52% of the global increase between 200 m and 1,500 m is sequestered in the Atlantic. Together with the heat sequestrated in the Southern Ocean, it contributed to a period of global warming slowdown. When this additional heat storage is absent, a period of rapid surface warming is expected to reoccur.
Although the Argo programme was launched around 2000, its coverage in the Southern Ocean did not become adequate until 2005. To validate the data on OHC we compare satellite SSH* (the asterisk indicates the deviation of SSH from its global mean) available since 1993 (Fig. 4a and b) to the thermosteric sea level rise (due to thermal expansion of the water column) (Fig. 4c and d) calculated using OHC above 1,500 m. The comparison is surprisingly good north of 35° S. Notable exceptions are as expected; they include areas with no Argo measurements: shallow maritime areas west of the Caribbean islands, and the deep mid-Atlantic Ocean below 1,500 m, which was not included in our OHC. South of 35° S the linear trend in the Argo data is not reliable across 2003 during the transition from no-Argo to Argo measurements28. The two datasets consistently show that in the subpolar Atlantic there is increasing (decreasing) heat storage when AMOC is increasing (decreasing). The southward (northward) displacement of the Gulf Stream at mid-latitudes created some compensating cooling (warming)21. In the AMOC’s rapidly decreasing subphase, some heat is entrained in the subtropical gyre. The Southern Hemisphere north of 35° S is mostly featureless. South of 35° S, mesoscale patterns of warming can be seen in SSH*, which is also reflected in the OHC after 2004, but not before, owing to data quality. These mesoscale eddies in the linear trend occurring south of the Antarctic Circumpolar Current may be due to its recent strengthening, and its increased baroclinic instability [29].
Figure 4: Contrasting thermosteric SSH* patterns for increasing and decreasing AMOC. a, c, Linear SSH* trend when AMOC is increasing; b, d, Linear SSH* trend when AMOC is decreasing. a and b show SSH* from remote sensing, compared with the steric sea level calculated using OHC in c and d. SSH* is SSH with its global mean subtracted, reflecting mostly the thermosteric part of SSH (see Methods).
The increased sea level (Fig. 4b) and warmer SST (Extended Data Fig. 5d) in the western subtropical Atlantic may have led to strong hurricanes and their destructive power, and the surprising string of category-5 hurricanes making landfall towards the end of the decreasing phase of the AMOC, instead of at the peak of the AMOC, when the mean SST of the entire North Atlantic is the warmest and the basin-wide hurricane number is the highest30.
Climate-model runs under preindustrial conditions demonstrated the existence of multidecadal variation in AMOC, and its associated Atlantic SST variation: the AMOC+ (AMOC−) phase corresponds to warm (cold) SST and Northern Hemisphere mean surface temperature6,19. This prevailing paradigm has permeated popular perceptions about the future climate consequence of an AMOC weakened by global warming, similar to the abrupt switch back into icy conditions of the Younger Dryas during the last deglaciation2. Over the past few decades, however, there is a positive trend of warmer subsurface water in the subpolar Atlantic (Extended Data Fig. 6), rendering the mean state lighter (see the temperature–salinity diagram in Extended Data Fig. 7). Deep convections can now carry more heat downward. In the presence of greenhouse heating from above and warmer SSTs, AMOC’s role in sequestering heat becomes important in the current global surface energy budget (Fig. 1). When AMOC is more constant, as in the AMOC− phase, little additional heat is sequestered in the Atlantic, contributing to a more rapid surface warming as more heat from radiative imbalance remains on the surface and the upper 200 m of the global oceans. We note, however, that we have discussed here only one component of a complex system: global heat balance is maintained by the combined ocean and atmosphere systems and a change in the transport of one regional component may affect the partitioning of change between other parts of the ocean or of the atmosphere, depending on the timescales involved.
Methods
Updated AMOC indices
We reproduced the unfiltered monthly AMOC indices (Extended Data Fig. 1). Their correlation coefficient with Zhang’s unfiltered AMOC fingerprint is listed on the right. All correlations are statistically significant at over 95% confidence level.
AMOC indices in Fig.3a
Extended Data Fig. 1 shows that all unfiltered AMOC proxies used in Fig. 3a are correlated with Zhang’s fingerprint AMOC proxy at over 95% confidence level. Zhang showed20 that in the Geophysical Fluid Dynamics Laboratory model the fingerprint proxy is highly coherent with the model AMOC Index, defined as the zonal integrated maximum Atlantic overturning at 40° N, at decadal and multidecadal scales. This is the reason that the fingerprint is shown smoothed with a 10-year low-pass filter. This fingerprint is calculated using the detrended 400-m subsurface temperature. (It was updated to 2017 by the author with permission to use.)
Our subpolar upper ocean salinity index is defined as the average over 45°–65° N in the Atlantic basin and integrated over 0–1,500 m. The two undetrended salinity indices shown in Fig. 3 and Extended Data Fig. 1 are from three data sources. The first index is based on ISHII and Scripps. ISHII data have not been updated since 2012 and Scripps data are only available since 2004; they are connected at 2012 when calculating the correlation coefficient with Zhang’s fingerprint AMOC proxy. The data source for the second salinity index is from EN4 (version 4.2.1).


Methods
Updated AMOC indices
We reproduced the unfiltered monthly AMOC indices (Extended Data Fig. 1). Their correlation coefficient with Zhang’s unfiltered AMOC fingerprint is listed on the right. All correlations are statistically significant at over 95% confidence level.
AMOC indices in Fig.3a
Extended Data Fig. 1 shows that all unfiltered AMOC proxies used in Fig. 3a are correlated with Zhang’s fingerprint AMOC proxy at over 95% confidence level. Zhang showed20 that in the Geophysical Fluid Dynamics Laboratory model the fingerprint proxy is highly coherent with the model AMOC Index, defined as the zonal integrated maximum Atlantic overturning at 40° N, at decadal and multidecadal scales. This is the reason that the fingerprint is shown smoothed with a 10-year low-pass filter. This fingerprint is calculated using the detrended 400-m subsurface temperature. (It was updated to 2017 by the author with permission to use.)
Our subpolar upper ocean salinity index is defined as the average over 45°–65° N in the Atlantic basin and integrated over 0–1,500 m. The two undetrended salinity indices shown in Fig. 3 and Extended Data Fig. 1 are from three data sources. The first index is based on ISHII and Scripps. ISHII data have not been updated since 2012 and Scripps data are only available since 2004; they are connected at 2012 when calculating the correlation coefficient with Zhang’s fingerprint AMOC proxy. The data source for the second salinity index is from EN4 (version 4.2.1).
The sea-level index was obtained as in ref. 22 by calculating the sea-level difference between the average of a group of linearly detrended, deseasonalized tide-gauge measurements south of 35° N and that to the north. It is accumulated in time, shifted to the right by 4.8 years and smoothed with a 7-year lowpass filter.
The subpolar gyre SST index was obtained by ‘detrending’ the subpolar gyre SST by the subtraction of the global mean SST. It is averaged over the subpolar gyre region, defined by ref. 21.
Willis’ AMOC strength at 41° N was calculated [18] using altimetry SSH measurements and geostrophic approximation for the zonal-mean northward velocity vertically integrated above 1,130 m. It is not detrended or accumulated.
Error bars for data used in Fig. 3
The error bars for the salinity time series used in Fig. 3a are plotted in Extended Data Fig. 2. The uncertainty at each gridpoint is provided by each data source: ISHII, Scripps and EN4. The error bar of the salinity time series at each time is computed as the combination of the gridpoint uncertainty and one standard deviation due to the averaging in space. The uncertainty of the SSH-deduced AMOC strength was given by ref. 18. The measurement and sampling errors at each time gridpoint were ±12%. The uncertainty of tide-gauge data was discussed by ref. 22, and that of Zhang’s fingerprint proxy by ref. 30. The uncertainty of the global surface temperature data from HadCRUT4.6 was assessed by the data source using 100 ensemble members that span the uncertainty range of the data.
Calculation of warming scenarios
We emphasize that this is not a prediction, but a scenario calculation. In our current climate system, the OHC in the upper 1,500 m of the global oceans increases at the rate of 0.42 W m−2, which is approximately the top-of-atmosphere radiative imbalance. Apart from short-term variations of radiative imbalance such as those due to volcanic eruptions, it is reasonable to assume that for the next two decades there will not be an appreciable change in radiative imbalance, barring an unexpected development of carbon sequestration technology.
Scenario 1
If the OHC storage below 200 m remains the same (no increases), then the radiative imbalance of the 0.42 W m−2 heats only the top 200 m of the global oceans. That is, the increase of OHC in the top 200 m of the oceans is responsible for the increase in the entire 1,500 m of the column. The top 200 m of the global ocean then warms at the rate calculated as: 0.42 W m−2 divided by the heat capacity of 200 m of the ocean = 0.23 °C per decade. This is equivalent to that obtained for a ‘slab’ ocean of 200 m thick.
Scenario 2
As for Scenario 1 except that only the Atlantic and the Southern oceans’ heat content below 200 m remain the same for the next two decades. The Pacific and the Indian oceans continue to increase their OHC at the current rate. The warming rate is 70% of that for Scenario 1 because at present the Atlantic and the Southern oceans together are responsible for 70% of the OHC increase in the upper 1,500 m of the oceans. This is probably the more likely scenario because we have argued in the main text that AMOC is likely to remain relatively constant during the next two decades. The subsurface Southern Ocean has been warming since at least 1993, caused by the southward displacement and intensification of the westerly jet, which cannot continue much longer, first because the proposed cause (the ozone hole) has diminished in importance as the ozone hole heals, and second because there is not much more room for the jet’s southward displacement. So the increase in warming will probably stop.
Model AMOC and reconciliation with recent observations
Observational results in Fig. 3a show that there was a positive trend from 1993 to 1999, with a small peak in 1996. The rapid rising trend from 1999 to 2005 is statistically significant at the over 95% confidence level. This is seen in all proxies, most clearly in the less smoothed data (SSH and subpolar salinity). This claim is supported by observation of SSH-deduced AMOC strength, tide-gauges, the subpolar salinity proxy, and also the Zhang fingerprint proxy. (The last proxy, because of 10-year smoothing, does not show the smaller peak in the mid-1990s). A model reanalysis also showed an acceleration prior to 2005 followed by a decline at 26° N, and a peak in the mid-1990s as well as one in 2005 at 45° N16. AMOC in models is sensitive to resolution and subgrid parameterization31, resulting in little consensus among reanalysis (and hindcast) products. With one exception16 these products do not agree with the RAPID observation at 26° N. The exception is the GloSea5 model, which has a higher, eddy-permitting resolution than previous reanalyses. Supplementary figure 1 of ref. 16 shows two peaks, one at 1995 and one at 2005. The 1995 peak is slightly higher than the 2005 peak, and is referred to thus in the main text of ref. 16: “The AMOC at 45° N is representative of the changes in the subpolar gyre, with the AMOC decreasing from a maximum in the mid-1990s, followed by a slight increase (Fig. 1d)”. The peak in 2005 was not mentioned. However, the result on the 1995 peak should be treated with care, as the authors themselves stated in the supplementary information of ref. 16: “It is likely that there will be a period of spinup, where the deep ocean (where there are few observational constraints) adjusts, which may explain the divergence in trend. Hence we disregard the first few years of each experiment. There is also a shock in 1992 when the altimeter data is introduced, which may contribute to the increase in AMOC strength between 1989 and 1995. Hence we choose the period to analyse starting from January 1995, and join the two analyses in January 2002.” The relative magnitude of the 1995 peak and the 2005 peak may be unreliable as it was obtained by joining two reanalyses, one starting from 1989 and one from 1995 with “divergence in trend”16.
The observed SSH data since 1992 can be used to deduce AMOC strength using geostrophic approximation, bypassing the problems of shock and subsequent adjustment when the same SSH data were introduced in model assimilation.
SST changes during different phases of AMOC


The observed SSH data since 1992 can be used to deduce AMOC strength using geostrophic approximation, bypassing the problems of shock and subsequent adjustment when the same SSH data were introduced in model assimilation.
SST changes during different phases of AMOC
The upper branch of the climatological AMOC brings warm and saline surface water from the subtropical North Atlantic to its subpolar latitudes. When the overturning is stronger, more of this warm water is found in the subpolar northern latitudes. In the Southern Hemisphere, more of the cold water from the region of the Antarctic Circumpolar Current is brought northward into the Southern subtropics. Consequently a characteristic signature in the Atlantic SST is an opposite-signed multidecadal anomaly, with warming to the north and smaller cooling to the south when the overturning is stronger (AMOC+), and the reverse pattern when it is weaker (AMOC–) (Extended Data Fig. 5a, b). This ocean-induced SST variability is centered in the subpolar North Atlantic20. The observed tendency during the last two subphases of the AMOC is as expected (Extended Data Fig. 5c, d): As AMOC slows after 2005, the SST tends towards a cooler North Atlantic and warmer subtropics. Accompanying the strong cooling in the subpolar gyre is an interesting intense warming after 2005 in the northwest Atlantic, centered in the Gulf of Main, which was recently simulated in a high-resolution climate model32 as due to the northward displacement of the Gulf Stream when AMOC slows. The inverse relationship between Gulf Stream’s northward displacement and AMOC strength was found6 to be caused by the Labrador Current retreat and the bottom vortex stretching33.
AMO
In long coupled atmosphere–ocean model runs under preindustrial conditions (without increasing greenhouse gases) the AMO is the SST manifestation of AMOC variations, and the two time series are approximately in phase19. The definition of AMO in ref. 19 is the mean of Atlantic SST north of 45° N, which may lead the subtropical SST anomaly by two years. A more traditional definition of AMO is the mean Atlantic SST north of the Equator34, with an approximately one-year phase difference. It has been shown24, using the space-time perspective of rotated empirical orthogonal function analysis, that the AMO is mainly responsible for the observed global mean surface temperature variation on multidecadal timescales. The two are in phase during the industrial era. Since the AMOC and the global mean surface temperature variation are not in phase (as shown in Fig. 3), it follows that during the industrial era, AMOC and AMO are off phase, possibly by a quarter cycle, although AMOC’s time series is too short for an accurate determination of the phase information.
During the positive phase of AMO, SST is warm in the North Atlantic and surrounding continents. Therefore, Northern Hemisphere mean surface temperature is warm during the positive phase and cool during the negative phase of the AMO. Using multiproxy data in the Northern Hemisphere the AMO time series can be extended back several hundred years35. The longest instrumental temperature record exists in central England, and it was used27 to reconstruct the AMO timeseries back to the Little Ice Ages. An even longer record of ice cores in Greenland, in the northern Atlantic, exists, and a statistically significant at the over 95% confidence level AMO signal can be found36 extending back to 800 AD that is coherent with the instrumental record of central England27 during their overlapping period. It appears that AMO is a recurrent phenomenon of period around 65–70 years and that it is robust in the preindustrial era, with the Atlantic and the surrounding areas warm during the positive phase and cold during the negative phase. From climate model preindustrial control runs, it seems that AMO is a surface manifestation of AMOC variation. Furthermore, based on palaeoclimate evidence of cold events when AMOC slows down abruptly, a common perception is that a slowdown in AMOC would lead to a cold Northern Hemisphere. The mechanism relies on the dominant role of AMOC (and its Gulf Stream) in horizontally transporting surface heat from the tropics to the mid- and high-latitude Atlantic, where it releases some heat to the cold atmosphere before sinking in the subpolar Atlantic. The heat released to the atmosphere makes Europe warmer (when wind blows in that direction) than it should be for its latitude.
Calculating SSH* from altimetry data
SSH* is SSH with its global mean subtracted. SSH contains both the thermosteric part (due to thermal expansion of the entire water column) and the ocean water mass addition that is due to melting land ice. It is known that the ocean will adjust to any change in ocean mass rapidly through the propagation of gravity waves, and will reach a new equilibrium globally within a couple of months37. Therefore, the subtraction of the global mean largely removes the mass contribution from SSH.
Extended data figures and tables
Extended Data Fig. 1: Unfiltered AMOC proxy time series in monthly resolution. The thick solid lines are 13-month running means. The numbers to the right of each time series show the correlation coefficient with the unfiltered AMOC subsurface temperature fingerprint of Zhang. Data are taken from refs 20,21,22. All of the correlation coefficients are above 95% confidence level. The accumulated sea-level index is shifted to the right by 4.8 years in this figure. Without the time shift, its correlation with the AMOC proxy is practically zero (r = 0.06).
Extended Data Fig. 2: Error bars for the three salinity time series shown in Fig. 1. The colour lines are monthly values of uncertainty, superimposed on the 13-month means of the time series. psu, practical salinity units.
Extended Data Fig. 3: Coincidence of the three AMOC phases with global warming slowdown and acceleration. a, Global mean surface temperature. b, OHC north of 45° N in the Atlantic. c, Salinity north of 45° N in the Atlantic.
Extended Data Fig. 4: Deep Labrador Sea density. Average density in the 1,000–1,500 m layer of the Labrador Sea, regionally averaged over the ocean area shown in the inset, from the three data sources given. A leading signal for stronger AMOC is the increased deep Labrador Sea salinity (and hence density). The signal propagates southward along the western boundary at depth, changing the cross-basin zonal gradient, and hence the geostrophic southward velocity13. The return flow then strengthens the upper branch of AMOC with a lag of 7–10 years15,16.
Extended Data Fig. 5: SST patterns during different AMOC phases. a, When AMOC is below climatology. b, When AMOC is above climatology, SST detrended. c, SST linear trend when AMOC is increasing. d, When AMOC is decreasing.
Extended Data Fig. 6: Linear trends, from 1950 to 2017, of temperature, salinity and density. a–c, Trends in temperature (a), salinity (b) and density (c) as a function of depth. Solid curves indicate where the trend is statistically significant at 95% confidence level.
Extended Data Fig. 7: Temperature–salinity diagram. The subpolar Atlantic Ocean (45°–65° N) for each depth between 300 m and 1,500 m for the two periods, with the mean of 2000–2016 in red and the mean of 1920–1940 in blue. The dots shown are the five winter month values (NDJFM). At these depths the seasonal cycle is very small [38].


Zika et al. (2021)
Title:
Recent Water Mass Changes Reveal Mechanisms of Ocean Warming
Corresponding author:
Zika
Citation:
Zika, J. D., Gregory, J. M., McDonagh, E. L., Marzocchi, A., & Clement, L. (2021). Recent Water Mass Changes Reveal Mechanisms of Ocean Warming. Journal of Climate, 34(9), 3461-3479. doi: 10.1175/jcli-d-20-0355.1
Keywords:
Ocean, Water masses/storage, Climate change, Heat budgets/fluxes, Climate variability, Trends
Abstract
Over 90% of the buildup of additional heat in the Earth system over recent decades is contained in the ocean. Since 2006, new observational programs have revealed heterogeneous patterns of ocean heat content change. It is unclear how much of this heterogeneity is due to heat being added to and mixed within the ocean leading to material changes in water mass properties or is due to changes in circulation that redistribute existing water masses. Here we present a novel diagnosis of the “material” and “redistributed” contributions to regional heat content change between 2006 and 2017 that is based on a new “minimum transformation method” informed by both water mass transformation and optimal transportation theory. We show that material warming has large spatial coherence. The material change tends to be smaller than the redistributed change at any geographical location; however, it sums globally to the net warming of the ocean, whereas the redistributed component sums, by design, to zero. Material warming is robust over the time period of this analysis, whereas the redistributed signal only emerges from the variability in a few regions. In the North Atlantic Ocean, water mass changes indicate substantial material warming while redistribution cools the subpolar region as a result of a slowdown in the meridional overturning circulation. Warming in the Southern Ocean is explained by material warming and by anomalous southward heat transport of 118 ± 50 TW through redistribution. Our results suggest that near-term projections of ocean heat content change and therefore sea level change will hinge on understanding and predicting changes in ocean redistribution.
Introduction
Over the past 50 years, as atmospheric greenhouse gas concentrations have increased, the ocean has absorbed more than 10 times as much heat as all other components of the climate system combined (Rhein et al. 2013). This warming showed substantial spatial variability between 1993 and 2005, being up to 10 times as much in some regions as the global average (Zhang and Church 2012). It is unclear whether this variability is due to geographical variation in the interior propagation of surface warming versus redistribution of existing heat within the ocean.
Ocean warming is an important issue because ocean thermal expansion is the largest projected contribution to global mean sea level rise in the twenty-first century (Church et al. 2013). Numerical climate models disagree on the pattern and amplitude of ocean heat content (OHC) change and hence on sea level rise under anthropogenic greenhouse warming (Gregory et al. 2016). Understanding how heat has been taken up and redistributed by the ocean is essential for predicting future changes in sea level.
Numerical ocean models forced with historical atmospheric conditions have proved to be useful tools in quantifying how variability in atmospheric forcing can set variability in OHC (Drijfhout et al. 2014) and sea level (Penduff et al. 2011) at interannual to decadal time scales. However, such models can be unrealistic for simulating multidecadal climate change because of model drift and inaccuracies in long-term changes in atmospheric forcing, particularly global mean heat fluxes (GrifÞes et al. 2009). On the other hand, coupled ocean atmosphere climate models are routinely used to capture the effect of long-term climate forcing. But such models only accurately simulate past unforced variability in regional OHC when, by chance, their internal variability is in phase with the observed system.
An advance in terms of numerical ocean climate modeling has come from the separation of OHC change into an “added” and a “redistributed” component in climate model simulations, where the former is due to change in the surface heat flux, and the latter due to rearrangement of existing OHC because of altered ocean heat transports (Banks and Gregory 2006). This decomposition is analogous to the “anthropogenic” and “natural” decomposition that has revolutionized our under standing of oceanic carbon records (Khatiwala et al. 2013). Here we will present a novel method to diagnose the “material” component of OHC change, which we will show is closely related to the “added” component introduced by Banks and Gregory (2006).
Recent work has aimed to reconstruct the drivers of OHC change based on observationally derived airÐsea boundary conditions. Zanna et al. (2019) for example used surface temperature anomalies combined with a tracer-based approach to reconstruct the role of anomalous surface heat fluxes in centennial heat content change. Roberts et al. (2017) estimated the contribution of airÐsea heat flux changes in setting mixed layer and full-depth-integrated OHC budgets over recent decades and inferred the role of ocean circulation as a residual. Here we aim to circumvent reliance on such boundary conditions and infer the mechanisms of ocean heat content change directly based on water mass changes.
Water mass-based methods have been used to decompose local temperature and salinity changes into a dynamic “heave” component and an apparently material component at constant density based on a one-dimensional view of the water column (Bindoff and McDougall 1994). However, their analysis did not distinguish between material processes and horizontal advection, insofar as they affect the water mass properties of an individual water column.
Here we introduce a new method based on water mass theory, called the minimum transformation method, which we use to estimate recent drivers of three-dimensional OHC change. In section 2 we will review water mass theory and establish the relationship between changes in water masses as deÞned by their temperature and salinity and material changes in seawater temperature. We will describe in section 3 how this theory is translated into a practical method to estimate material changes in water masses and map these into geographical space. We present an application of this minimum transformation method to recent data over the Argo period in section 4 and give results in section 5. We discuss the results and compare them with existing work in section 6, and we give conclusions in section 7.
Water mass theory
Water mass analysis has long been used in physical oceanography to trace the origin of waters (Montgomery 1958). In the latter half of the twentieth century a quantitative framework emerged to describe the relationship between water masses, airÐsea fluxes, and mixing [Walin 1982; see the review by Groeskamp et al. (2019)]. Recent work has seen this framework advanced in two ways specifically relevant to our work here: to multiple tracer dimensions to understand the thermodynamics of ocean circulation (Nycander et al. 2007; Zika et al. 2012; Doos et al. 2012; Groeskamp et al. 2014; Hieronymus et al. 2014) and to unsteady problems to understand the oceanÕs role in transient climate change (Palmer and Haines 2009; Evans et al. 2014; Zika et al. 2015a,b; Evans et al. 2017, 2018).
An example of the utility of the water mass transformation framework in understanding transient change is provided by Zika et al. (2015a). They demonstrate that the distribution of water in salinity coordinates is influenced by the water cycle and turbulent mixing, the latter only being able to collapse the range of salinities the ocean covers. This means that changes in the width of the salinity distribution indicate an enhancement of the water cycle and/or a reduction in that rate at which salt is mixed. In this project we extend this concept to consider how changes in the temperature-salinity distribution relate to material changes in water masses.
Material changes in Conservative Temperature Theta (hereinafter simply “temperature” or T) following the motion of an incompressible fluid are related to Eulerian changes and advection by
frac{DT}{Dt} = frac{partial T}{partial t} + bf{u} cdot nabla{T}, (1)
where bf{u} is the 3D velocity vector and frac{DT}{Dt} is the material derivative, which is related to sources and sinks of heat and irreversible mixing. Conservative Temperature is used here since it is a more accurate “heat” variable than potential temperature (McDougall 2003), although the later is still routinely used in ocean models, including the one analyzed in section a of appendix A.
Even if a perfect record of frac{partial T}{partial t} were available at a fixed location, we would not know the relative roles of advection (bf{u} cdot nabla{T}) and material processes (frac{DT}{Dt}). To separate them, we consider the water mass perspective as an alternative to the Eulerian perspective. The following theory draws directly from Hieronymus et al. (2014).


An example of the utility of the water mass transformation framework in understanding transient change is provided by Zika et al. (2015a). They demonstrate that the distribution of water in salinity coordinates is influenced by the water cycle and turbulent mixing, the latter only being able to collapse the range of salinities the ocean covers. This means that changes in the width of the salinity distribution indicate an enhancement of the water cycle and/or a reduction in that rate at which salt is mixed. In this project we extend this concept to consider how changes in the temperature-salinity distribution relate to material changes in water masses.
Material changes in Conservative Temperature Theta (hereinafter simply “temperature” or T) following the motion of an incompressible fluid are related to Eulerian changes and advection by
frac{DT}{Dt} = frac{partial T}{partial t} + bf{u} cdot nabla{T}, (1)
where bf{u} is the 3D velocity vector and frac{DT}{Dt} is the material derivative, which is related to sources and sinks of heat and irreversible mixing. Conservative Temperature is used here since it is a more accurate “heat” variable than potential temperature (McDougall 2003), although the later is still routinely used in ocean models, including the one analyzed in section a of appendix A.
Even if a perfect record of frac{partial T}{partial t} were available at a fixed location, we would not know the relative roles of advection (bf{u} cdot nabla{T}) and material processes (frac{DT}{Dt}). To separate them, we consider the water mass perspective as an alternative to the Eulerian perspective. The following theory draws directly from Hieronymus et al. (2014).
We characterize water masses by their T and Absolute Salinity S_A (IOC/SCOR/IAPSO 2010; hereinafter simply “salinity” or S). The volume v of water per unit temperature and salinity and at temperature T^* and salinity S^* is
v(T^*, S^*) = frac{partial^2}{partial T partial S} int_{T<T^*, S<S^*}dV, (2)
where the integral is over elements dV of ocean volume that are cooler than T^* and fresher than S^*. An estimate of v that is based on recent observational analysis is given in Fig. 1a. (These data are described in detail in section 4.)
Figure 1: Portrait of changing ocean water masses: (a) inventory of ocean volume in Conservative Temperature vs Absolute Salinity coordinates (mean of 2006-17 inclusive) and (b) change in water mass volume between the early half and late half of the period divided by the six years (Sv). According to water mass theory, changes in air-sea heat and freshwater fluxes and/or changes in rates of diffusion are required for these changes to occur.
Considering all of the water in the ocean and retaining the incompressibility assumption, the only way v can change is via transformation - that is, by making water parcels warmer, colder, saltier, or fresher as described by the following continuity equation [derived formally in Hieronymus et al. (2014)]:
frac{partial v}{partial t} + frac{partial}{partial T} (v dot{T}) + frac{partial}{partial S} (v dot{S}) = 0, (3)
where dot{T} is the average material derivative of T within a water mass. That is,
dot{T}(T^*, S^*) = frac{1}{v} frac{partial^2}{partial T partial S}  int_{T<T^*, S<S^*}frac{DT}{Dt} dV, (4)
and likewise dot{S} is the average material derivative of S. An estimate of recent changes in v is given in Figure 1b.
In Eq. (3) the terms vdot{T} and vdot{S} are the transformation rates in the temperature direction (Sv g^{-1} kg^{-1}; 1 Sv = 10^6 m^3 s^{-1}) and salinity direction (Sv C˚^{-1}) respectively. Equation (3) states that the amount of water between two closely spaced isotherms (T and T + partial T) and isohalines (S and S + partial S) will go up if more water is made warmer at T than at T + partial T and/or more water is made saltier at S than at S + partial S.
When the system is in a statistically steady state the water mass distribution v remains constant such that
frac{partial}{partial T} overline{v dot{T}} + frac{partial}{partial S} overline{v dot{S}} = 0, (5)
where the overbar represents a sufficiently long time average. In this steady case, the vector field described by overline{v dot{T}} and overline{v dot{S}} can be characterized by a thermohaline streamfunction (Zika et al. 2012; Groeskamp et al. 2014).
Here, we will not attempt to estimate this steady-state component of water mass transformation [e.g., as Groeskamp et al. (2017) have done]. Rather we will attempt to quantify only the component required to explain changes in v. That is, we aim to quantify the anomaly in the transformation rate (v dot{T})’ such that v dot{T} = overline{v dot{T}} + (v dot{T})’, and likewise for (v dot{S})’, with
frac{partial v}{partial t} + frac{partial}{partial T} (v dot{T})’ + frac{partial}{partial S} (v dot{S})’ = 0, (6)
Note that a steady-state component like Eq. (5) can always be added to (v dot{T})’  and (v dot{S})’ such that Eq. (6) is still satisfied. Hwever, we seek only the net change in water mass transformation required to explain changes in v and therefore seek the smallest (in a root-mean-square sense) values of dot{T}’ and dot{S}’ that satisfy Eq. (6). That is, we seek the smallest change in air-sea heat and freshwater fluxes and mixing - in a net sense - that can explain changes in water masses. We call this the minimum transformation.
Here we will use changes in v to infer the minimum transformation and therefore estimate v dot{T}’. This will allow us to estimate the material processes influencing ocean temperature change.
The minimum transformation method
We now apply water mass theory to understand changes in a discrete set of water masses describing the ocean over two time periods. We will then describe the application of a minimum transformation method that exploits an “earth moverÕs distance” (EMD) algorithm to estimate the amount of material warming required to affect changes in those water masses.
Discrete water masses
Consider the set of N discrete water masses with the ith water mass defined by the limits [T_i^{min}, S_i^{min}, mathbf{x}_i^{min}] and [T_i^{max}, S_i^{max}, mathbf{x}_i^{max}]. Essentially, our water masses are hypercubes in TÐSÐxÐyÐz space (more arbitrary space-and tim-dependent regions can be defined without affecting the method described below). To indicate whether water is within the ith water mass we define a boxcar function Pi_i such that
begin{equation}
Pi_i(mathbf{x}, t) = begin{cases}1, & T_i^{min} leq T(mathbf{x}, t) < T_i^{max}, S_i^{min} leq S(mathbf{x}, t) < S_i^{max} text{and} mathbf{x}_i^{min} leq mathbf{x} < mathbf{x}_i^{max}\
0, & text{otherwise}.  end{cases}
end{equation}
The volume of water in the ith water mass at time t is then iiint{Pi_i(mathbf{x}, t) dV}.
We consider two time periods: an early period (t_0 - Delta t leq t < t_0) and a late period (t_0 leq t < t_0 + Delta t). The average volume of the ith water mass over the early period is V1_i and the average volume of the jth water mass over the late period is V2_j such that
V1_i = frac{1}{Delta t} int_{t_0 - Delta t}^{t_0} iiint Pi_i(mathbf{x}, t) dV dt
and
V2_j = frac{1}{Delta t} int_{t_0}^{t_0 + Delta t} iiint Pi_i(mathbf{x}, t) dV dt, (8)
and the average temperature and salinity of water within V1_i is
T1_i = frac{1}{Delta t V1_i} int_{t_0 - Delta t}^{t_0} iiint Pi_i(mathbf{x}, t) T(mathbf{x}, t) dV dt
and


Discrete water masses
Consider the set of N discrete water masses with the ith water mass defined by the limits [T_i^{min}, S_i^{min}, mathbf{x}_i^{min}] and [T_i^{max}, S_i^{max}, mathbf{x}_i^{max}]. Essentially, our water masses are hypercubes in TÐSÐxÐyÐz space (more arbitrary space-and tim-dependent regions can be defined without affecting the method described below). To indicate whether water is within the ith water mass we define a boxcar function Pi_i such that
begin{equation}
Pi_i(mathbf{x}, t) = begin{cases}1, & T_i^{min} leq T(mathbf{x}, t) < T_i^{max}, S_i^{min} leq S(mathbf{x}, t) < S_i^{max} text{and} mathbf{x}_i^{min} leq mathbf{x} < mathbf{x}_i^{max}\
0, & text{otherwise}.  end{cases}
end{equation}
The volume of water in the ith water mass at time t is then iiint{Pi_i(mathbf{x}, t) dV}.
We consider two time periods: an early period (t_0 - Delta t leq t < t_0) and a late period (t_0 leq t < t_0 + Delta t). The average volume of the ith water mass over the early period is V1_i and the average volume of the jth water mass over the late period is V2_j such that
V1_i = frac{1}{Delta t} int_{t_0 - Delta t}^{t_0} iiint Pi_i(mathbf{x}, t) dV dt
and
V2_j = frac{1}{Delta t} int_{t_0}^{t_0 + Delta t} iiint Pi_i(mathbf{x}, t) dV dt, (8)
and the average temperature and salinity of water within V1_i is
T1_i = frac{1}{Delta t V1_i} int_{t_0 - Delta t}^{t_0} iiint Pi_i(mathbf{x}, t) T(mathbf{x}, t) dV dt
and
S1_i = frac{1}{Delta t V1_i} int_{t_0 - Delta t}^{t_0} iiint Pi_i(mathbf{x}, t) S(mathbf{x}, t) dV dt, (9)
respectively; likewise for V2_j we have
T2_j = frac{1}{Delta t V2_j} int_{t_0}^{t_0 + Delta t} iiint Pi_j(mathbf{x}, t) T(mathbf{x}, t) dV dt
and
S2_j = frac{1}{Delta t V2_j} int_{t_0}^{t_0 + Delta t} iiint Pi_j(mathbf{x}, t) S(mathbf{x}, t) dV dt. (10)
To change the set of volumes V1_i into the set of volumes V2_j requires a transformation of water in T-S space. When water transforms, it changes its T and S and can also move geographically.
To understand how water is transformed from the physical location and physical properties of one water mass to another we use the shorthand tilde{x}(t+Delta t | mathbf{x}, t) for the position of a water parcel at time t + Delta t conditional on it previously being at position mathbf{x} at time t. That is,
tilde{x}(t+Delta t | mathbf{x}, t) = mathbf{x} + int_t^{t+Delta t} mathbf{u}[tilde{x}(t^* | mathbf{x}, t), t^*] dt^*, (11)
where, as previously, mathbf{u} is the 3D velocity vector. We describe the transformation rate between the early and late water masses with the matrix mathbf{g}. The ith column and jth row of this matrix g_{ij} correspond to the average rate of transformation of water from early water mass i to late water mass j such that
g_{ij} = frac{1}{Delta t^2} int_{t_0 - Delta t}^{t_0} iiint Pi_i(mathbf{x}, t) Pi_j[tilde{x}(t + Delta t | mathbf{x}, t),t]dV dt. (12)
In Eq. (12) the term Pi_i(mathbf{x}, t) Pi_j[tilde{x}(t + Delta t | mathbf{x}, t),t] isolates water that was in the ith water mass at time t and was subsequently in the jth water mass at some time Delta t later. The quantity g_{ij} is therefore the average rate (m^3 s^{-1}) at which water in the ith early water mass is transformed into the jth late water mass.
Since the total volume of water is conserved between the early and late periods all the water from the early water masses (V1_i) must be transformed into late water masses. Likewise, all water masses from the late period (V2_j) are made from water masses of the early period. That is,
V1_i = Delta t sum_{j=1}^N g_{ij}
and
V2_j = Delta t sum_{i=1}^N g_{ij}. (13)
The average temperature change of water that transforms from V1_i to V2_j is then
Delta T_{ij} = frac{1}{Delta t^2 g_{ij}} int_{t_0-Delta t}^{t_0} iiint Pi_i(mathbf{x}, t) Pi_j[tilde{x}(t + Delta t | mathbf{x}, t),t]{T[tilde{x}(t + Delta t | mathbf{x}, t),t]-T(mathbf{x}, t)} dV dt, (14)
where the temperature change of individual parcel is related to the Lagrangian derivative by
T[tilde{x}(t + Delta t | mathbf{x}, t),t] - T(mathbf{x}, t) = int_t^{t + Delta t} frac{DT}{Dt}[tilde{x}(t^*|mathbf{x},t),t^*]dt^*. (15)
We can write Eq. (14) as
Delta T_{ij} = mathcal{T}2_{ji} - mathcal{T}1_{ij}, (16)
where mathcal{T}2_{ji} is the volume-weighted average temperature of the water in the jth late water mass that was previously in the ith early water mass and mathcal{T}1_{ij} is the volume-weighted average temperature of the water in the ith early water mass that is later in the jth late water mass.
The transformation g_{ij} involves a range of water parcels with a range of temperatures T(mathbf{x}, t), whose mean is mathcal{T}1_{ij},in the early period moving to a range of temperatures T[tilde{x}(t + Delta t | mathbf{x}, t),t], whose mean is mathcal{T}2_{ji}, in the late period. To simplify this problem, we assume that in both periods the water masses are well mixed. This means that we expect that the mean temperature of any sample of water parcels from water mass i in the early period will equal the mean temperature of the water mass as a whole, and in particular this is true for the sample of parcels that ends up in water mass j in the late period. Thus mathcal{T}1_{ij} = T1_i with this assumption. By a similar argument, mathcal{T}2_{ji} = T2_j, and hence the average T and S change of water transforming from the ith early to the jth late water mass as the difference of the average T and S of the two water masses. That is, Delta T_{ij} = T2_j - T1_j and Delta S_{ij} = S2_j - S1_j.
This above approximation preserves the following equality relating the change in global volume-weighted temperature to the transformation matrix:
sum_{j=1}^N V2_j T2_j - sum_{i=1}^N V1_i T1_i = Delta t sum_{i=1}^Nsum_{j=1}^N g_{ij} (T2_j - T1_i), (17)
and likewise for the volume-weighted salinity.
We have effectively discretized the continuum of trajectories from early to late water masses into a finite set of discrete trajectories. This discretization clearly leads to some information loss; however, such losses are unavoidable in any computationally feasible inverse method.


where mathcal{T}2_{ji} is the volume-weighted average temperature of the water in the jth late water mass that was previously in the ith early water mass and mathcal{T}1_{ij} is the volume-weighted average temperature of the water in the ith early water mass that is later in the jth late water mass.
The transformation g_{ij} involves a range of water parcels with a range of temperatures T(mathbf{x}, t), whose mean is mathcal{T}1_{ij},in the early period moving to a range of temperatures T[tilde{x}(t + Delta t | mathbf{x}, t),t], whose mean is mathcal{T}2_{ji}, in the late period. To simplify this problem, we assume that in both periods the water masses are well mixed. This means that we expect that the mean temperature of any sample of water parcels from water mass i in the early period will equal the mean temperature of the water mass as a whole, and in particular this is true for the sample of parcels that ends up in water mass j in the late period. Thus mathcal{T}1_{ij} = T1_i with this assumption. By a similar argument, mathcal{T}2_{ji} = T2_j, and hence the average T and S change of water transforming from the ith early to the jth late water mass as the difference of the average T and S of the two water masses. That is, Delta T_{ij} = T2_j - T1_j and Delta S_{ij} = S2_j - S1_j.
This above approximation preserves the following equality relating the change in global volume-weighted temperature to the transformation matrix:
sum_{j=1}^N V2_j T2_j - sum_{i=1}^N V1_i T1_i = Delta t sum_{i=1}^Nsum_{j=1}^N g_{ij} (T2_j - T1_i), (17)
and likewise for the volume-weighted salinity.
We have effectively discretized the continuum of trajectories from early to late water masses into a finite set of discrete trajectories. This discretization clearly leads to some information loss; however, such losses are unavoidable in any computationally feasible inverse method.
Note that, even if the ith water mass for the early period has the same temperature and salinity bounds as the ith water mass of the late period, the distribution of properties within the water mass can change. That is, in general T1_j ≠ T2_i and S1_j ≠ S2_i, so g_{ij} is always a transformation, even with i = j. For example, assume the ith water mass has temperature bounds 1˚ and 2˚C and that the water between those bounds is on average at 1.9˚C in the early period and 1.1˚C in the late period. Groeskamp et al. (2014) called this a “local effect” and included it as a separate term in their formulation. Here, we find it convenient to consider the transformation from the ith early water mass at 1.9˚C to the ith late water mass at 1.1˚C to be yet another transformation - no different than between any other pair of water masses.
We relate the transformation rate to the average material temperature tendency required to warm the ith early water mass to form the range of destination water masses it arrives at in the late period. That is,
dot{T}_i = frac{1}{V1_i} sum_{j=1}^N (T2_j - T1_i) g_{ij}. (18)
We use dot{T} to define a 3D material temperature change field Delta T_{material} such that
Delta T_{material}(mathbf{x}) = int_{t_0-Delta t}^{t_0} sum_{i=1}^N Pi_i(mathbf{x}, t) dot{T}_i dt
approx frac{1}{Delta t} int_{t_0-Delta t}^{t_0} left{int_t^{t + Delta t}frac{DT’}{Dt}[tilde{x}(t^*|mathbf{x},t),t^*]dt^*right} dt. (19)
Note here that we are relating dot{T}_i only to the anomaly of the Lagrangian tendency (i.e., frac{DT’}{Dt} rather than frac{DT}{Dt}) as it appears in Eq. (19). This is because our dot{T}_i describes only the changes in the transformation rate required to explain changes in the water mass distribution [as in Eq. (6)]. There can be (and indeed is) an additional “mean” transformation rate that leads to cycles of water in T-S space but does not lead to any changes in water mass inventories with time (Groeskamp et al. 2014). Implicit in Eq. (19) is the assumption that the anomalous warming of a particular water mass occurred evenly (in a volume-and time-weighted sense) over the regions and times during which that water mass existed in the early period.
We will contrast the inferred material warming at one location mathbf{x} against the total warming
Delta T(mathbf{x}) = int_{t_0 - Delta t}^{t_0} T(mathbf{x}, t + Delta t) - T(mathbf{x}, t) dt/Delta t,
with the residual of the two being a redistribution component such that
Delta T_{material} = Delta T - Delta T_{redistribution}.
By construction, Delta T_{redistribution} accounts for the advective redistribution of temperature (mathbf{u} cdot nabla{T}), which does not affect the underlying water masses and therefore is not accounted for in Delta T_{material}.
Finding the minimum transformation using an EMD algorithm
Our goal now is to estimate the transformation matrix mathbf{g}. Out of the inÞnite number of choices that could satisfy Eq. (13),we will look for the smallest (in a least squares sense) possible transformation required to change the distribution. We call this the minimum transformation.
Previous studies have diagnosed transformation rates from time-dependent changes in water mass distributions by searching for a minimum least squares solution on a regular T-S (Evans et al. 2014) or density-spiciness grid (Portela et al. 2020). Because of the dramatic variations in volume per unit temperature and salinity of the World Ocean (Fig. 1b) we choose to describe the distribution in an unstructured way. Furthermore, we exploit recent advances in the area of “optimal transportation theory” - in particular, the EMD algorithm that is mentioned at the beginning of section 3 (Pele and Werman 2008, 2009).
The EMD solves the hypothetical problem of moving earth from a set of mounds, each with varying amounts of earth, into a set of holes with varying amounts of empty space to be filled, where the total volume of the mounds is equal to that of the holes. In our case the “mounds” are the early water masses and the “holes” are the late water masses. The optimization problem is to find the set of transfers (from a mound to a hole, or the early to late water masses) that gives the smallest possible total of mass-weighted distance (the product of the mass and the distance of a transfer) that needs to be traveled in order to empty the mounds and fill the holes. For the EMD algorithm, we require a distance metric mathbf{d} ,which is a matrix whose ith column and jth row d_{ij} is the cost of moving water from the ith early water mass to the jth late water mass. The EMD algorithm then estimates mathbf{g} such that Eq. (13) is satisÞed and the following total mass-weighted “distance” is minimized:
sum_{j=1}^Nsum_{i=1}^N g_{ij} d_{ij}. (21)
We use the following distance metric:
d_{ij} = (T1_i - T2_j)^2 + [a(S1_i - S2_j)]^2 + delta_{ij}, (22)
where temperature and salinity differences are squared so that the distance is positive definite and long trajectories in TÐS space are penalized more than short ones and a is a constant that scales the salinity change relative to the temperature change and whose choice is described in the next section. The intent of delta_{ij} is to permit movement between water masses that are adjacent geographically without additional penalty but at the same time to stop direct exchange between geographically disconnected water masses, for example between water masses in the Southern Ocean and the Arctic. To achieve this we set delta_{ij} = 0 where the ith and jth water masses are in the same or adjacent geographical regions and delta_{ij} >> max{(T1_i - T2_j)^2 + [a(S1_i - S2_j)]^2} otherwise (in practice we use delta_{ij} = 10^6 in the latter case). Regions that share a meridional or zonal boundary are considered to be adjacent. The Arctic and North PaciÞc Oceans are not considered to be adjacent, whereas the Indian Ocean and equatorial PaciÞc regions are considered to be adjacent.


sum_{j=1}^Nsum_{i=1}^N g_{ij} d_{ij}. (21)
We use the following distance metric:
d_{ij} = (T1_i - T2_j)^2 + [a(S1_i - S2_j)]^2 + delta_{ij}, (22)
where temperature and salinity differences are squared so that the distance is positive definite and long trajectories in TÐS space are penalized more than short ones and a is a constant that scales the salinity change relative to the temperature change and whose choice is described in the next section. The intent of delta_{ij} is to permit movement between water masses that are adjacent geographically without additional penalty but at the same time to stop direct exchange between geographically disconnected water masses, for example between water masses in the Southern Ocean and the Arctic. To achieve this we set delta_{ij} = 0 where the ith and jth water masses are in the same or adjacent geographical regions and delta_{ij} >> max{(T1_i - T2_j)^2 + [a(S1_i - S2_j)]^2} otherwise (in practice we use delta_{ij} = 10^6 in the latter case). Regions that share a meridional or zonal boundary are considered to be adjacent. The Arctic and North PaciÞc Oceans are not considered to be adjacent, whereas the Indian Ocean and equatorial PaciÞc regions are considered to be adjacent.
Our motivation for using EMD is simply to find the smallest amount of transformation (in a least squares sense) required to explain observed water mass change. If T-S changes in the ocean could be explained purely by adiabatic redistribution of existing water masses, then our method would prioritize this solution. Our initial guess is therefore this adiabatic solution (i.e., where g_{ij} = 0 for all i and j). The EMD algorithm finds the smallest deviation possible from this adiabatic case. We cannot rule out larger compensating transformations having taken place. In principle, solutions given different initial guesses (e.g., an initial guess for mathbf{g} that is based on a numerical simulation) could be explored. We leave this to future work.
Figure 2 summarizes the minimum transformation method schematically. In the schematic just four early and four late water masses are deÞned with two in one geographical area and two in another. The minimum transformation moves water from the ith early to the ith late water masses in all four cases (i.e., g_{ii} ≠ 0 for all i). In addition, a substantial amount of water is moved from the second early water mass to the first late water mass (g_{21}) and from the third early water mass to the fourth late water mass (g_{34}). The observed change in temperature is therefore explained by a material warming of 2˚ and 1˚C of the two warmer shallower water masses and of 0.5˚C for the cooler deeper water masses. The remainder of the Eulerian pattern of temperature change is explained by redistribution. This schematic representation is vastly simplified as compared to our actual implementation of the minimum transformation method, which is described in the next section.
Figure 2: Schematic describing a simplified hypothetical implementation of the minimum transformation method. (left) Between a late and an early period, surface waters warm, especially to the south, where the ocean is fresher and the upper ocean layer becomes thicker. (center) The ocean is split into a southern region containing water masses 1 and 3 and a northern region containing water masses 2 and 4. Between the early and late periods, water masses 1 and 4 increase in volume and 2 and 3 reduce in volume. Taking into account the changing temperatures, salinities, and volumes of the early and late water masses, the “minimum transformations” g_{ij} are found using the EMD algorithm. These suggest modest warming of each water mass with some of early water mass 2 transforming to become late water mass 1 (g_{21}) and some of early water mass 3 transforming to become late water mass 4 (g_{34}). (right) The total temperature change is heterogeneous. A warming of 2˚C explains changes in water mass 1, a warming of 1˚C explains changes in water mass 2, and a warming of 0.5˚C explains changes in water masses 3 and 4. This warming is projected onto the location of those water masses in the early period to show the “material change.” The residual of the total and material changes is then explained by a “redistribution” that involves intense subsurface warming in the southern region and intense subsurface cooling in the northern region.
Data and application of the minimum transformation method
Observational estimates of T and S come from the objective analysis provided by the Enact Ensemble (V4.0, hereinafter EN4; Good et al. 2013). EN4 has a 1˚-by-1˚ horizontal resolution with 42 vertical levels. We analyze each month between 2006 and 2017 inclusive. We split these data into two time periods: an early period between 2006 and 2011 inclusive and a late period between 2012 and 2017 inclusive (i.e., t_0 = 0000 1 January 2007 and Delta t = 6 years).
We then define a discrete set of water masses for each time period by splitting the ocean into nine geographical regions and within each region by splitting up the ocean according to T-S bins. Our nine geographical regions are the Southern Ocean south of 35˚S, the subtropical Pacific and Atlantic Oceans between 35˚ and 10˚S, the Indian Ocean north of 35˚S, the tropical Pacific and Atlantic Oceans between 10˚S and 10˚N, the North Pacific north of 10˚N, the Atlantic Ocean between 10˚ and 40˚N, and the Atlantic and Arctic Ocean north of 40˚N. To avoid discontinuities in our resulting analysis we transition linearly from one region to another over a 10˚ band (Figure 5).
We define T and S bin boundaries ([T_{min}, T_{max}] and [S_{min}, S_{max}] respectively) using a quadtree. The quadtree starts with a single (obviously oversized) bin with T boundaries [-6.4˚,96˚C] and S boundaries [-5.2, 46 g kg^{-1}] in which the entirety of the ocean’s seawater resides. The single bin is then split into four equally sized bins with the same aspect ratio as the original bin. The same process of splitting into four is repeated for any bin whose volume change is greater than a threshold of 62 times 10^{12} m^3 (equivalent to the volume of a 5˚ longitude by 5˚ latitude region at the equator with a depth of 200 m) or until the bin size is 0.4˚C by 0.2 g kg^{-1}. Average volumes for each water mass are shown in Figure 3. In the supplementary text we show that changing the size of these bins by a factor of 2 does not substantially change our results. The quadtree is applied within each region and for the change between the late and early periods. This results in bin edges defining N = 1447 water masses. These bins are then used to dedine both the early water masses and the late water masses.
Figure 3: Gray lines show Conservative Temperature T and Absolute Salinity S bounds of each water mass (or “bin”) generated using a quadtree for each geographical region. The average T and S of the water found within each bin are shown by the location of each marker, and the volume is represented by the color scale (log10m3). Inventories and mean T and S values represent the entire period (2006-17 inclusive). Inset panels show masks associated with each geographical region.
We choose the constant a to be the ratio of a typical haline contraction coefficient to a typical thermal expansion coefficient (a = beta_0/alpha_0 = 4.28). This does not mean that transformations along density surfaces are necessarily preferred; rather, the squares in Eq. (22) mean that density-compensated changes in T and S are penalized as much as changes of the same magnitude where one of the signs is reversed. The inferred Delta T_{material} for each water mass is shown in Figure 4. We have tested the sensitivity of our method to varying a by a factor of 2 and found only negligible changes in inferred warming (see section b of appendix A).
Figure 4: Each symbol shows Delta T_{material}, the average warming required for each early water mass in order to transform them into the set of late water masses.


Figure 3: Gray lines show Conservative Temperature T and Absolute Salinity S bounds of each water mass (or “bin”) generated using a quadtree for each geographical region. The average T and S of the water found within each bin are shown by the location of each marker, and the volume is represented by the color scale (log10m3). Inventories and mean T and S values represent the entire period (2006-17 inclusive). Inset panels show masks associated with each geographical region.
We choose the constant a to be the ratio of a typical haline contraction coefficient to a typical thermal expansion coefficient (a = beta_0/alpha_0 = 4.28). This does not mean that transformations along density surfaces are necessarily preferred; rather, the squares in Eq. (22) mean that density-compensated changes in T and S are penalized as much as changes of the same magnitude where one of the signs is reversed. The inferred Delta T_{material} for each water mass is shown in Figure 4. We have tested the sensitivity of our method to varying a by a factor of 2 and found only negligible changes in inferred warming (see section b of appendix A).
Figure 4: Each symbol shows Delta T_{material}, the average warming required for each early water mass in order to transform them into the set of late water masses.
In section a of appendix A, we compare the results of our method applied to synthetic data from a climate model simulation with an added-heat variable explicitly simulated by the model. We find good agreement between added heat and our inferred Delta T_{material} and between simulated redistributed heat and our inferred Delta T_{redistributed} when ocean temperature and salinity are fed in as “data” to the method. Section b of appendix A also explores sensitivity of our results to parameter choices. The uncertainties we place on OHC change are ±2 standard deviations of a bootstrap ensemble, also described in section c of appendix A.
To produce maps of the total, material and redistributed contributions to the heat content we multiply the density and heat capacity of seawater by the respective temperature change and vertically integrate these through the entire water column. Our method also produces a material salinity change. We leave discussion of those data to future work.
Results
Patterns of total OHC change between early and late periods are heterogeneous (Figure 5a). There are basin-scale patches of decreasing heat content in the western equatorial and tropical Pacific, in the Pacific sector of the Southern Ocean, in the subtropical south Indian Ocean, and in the subpolar North Atlantic. Warming is seen most strongly in the tropical eastern Pacific, South Atlantic Ocean, and subtropical North Atlantic. These changes are highly sensitive to the specific observation years chosen and the length of the epochs reflecting the regional time scale of variability associated with the redistributed component. Uncertainty is far larger than the signal in the majority of regions (stippling in Figure 5a) and coincident with previously identified regions of large sea level anomaly variability (Penduff et al. 2011).
Figure 5: Heterogeneous pattern of total and redistributed heat content change contrast against robust material heat content change: (a) change in depth-integrated ocean heat content between 2006-11 and 2012-17 inclusive, (b) inferred redistributed heat, and (c) inferred material heat content change based on changing water masses for the same period. Regions where the magnitude of the signal is less significant (less than 2 standard deviations of a bootstrap ensemble) are stippled.
However, there are a few regions (e.g., patches of the Southern Ocean and North Atlantic) where the regional redistributed signal is robust and emerges from the uncertainty (Figure 5b). The patterns of redistributed heat observed in the Pacific are consistent with interdecadal Pacific oscillation (IPO)-driven thermosteric sea level variability (Lyu et al. 2017). The IPO was typically positive in the late period and negative in the early period (see https://psl.noaa.gov/gcos_wgsp/Timeseries/ for these data).
Material heat content change shows a smaller amplitude but more coherent signal than redistributed heat (Figures 5b,c). Material warming is seen across almost the entirety of the globe, with maxima in the Southern Hemisphere and Atlantic subtropical convergence zones (Maximenko et al. 2009), consistent with model simulations of passive ocean heat uptake due to anthropogenic greenhouse warming (Gregory et al. 2016). In such model simulations, anomalous heat fluxes into the ocean predominate at mid-to high latitudes and this heat is distributed throughout the ocean largely passively via subduction (downwelling) in the North Atlantic and the Southern Ocean (Marshall et al. 2015).
Strikingly, the uncertainty in material heat content change is far smaller than that of total OHC change (stippling in Figure 5c). This suggests that heat was added to and distributed within the ocean persistently over the Argo period and that this warming is not an artifact of a particularly warm year or years.
Zonally integrating the net OHC change reveals a signal of roughly the same magnitude as its uncertainty at all latitudes (Figure 6a). Zonally integrated redistributed heat likewise has a small signal to uncertainty ratio except in the Southern Ocean (Figure 6a). Accumulating the redistributed heat contribution from north to south gives the meridional heat transport due to redistribution. Broadly, heat is redistributed from north to south with a southward cross-equatorial transport of 73 ± 60 TW between the two epochs (Figure 6c).
Figure 6: Material heat content change is accumulating in the tropics and subtropics, whereas existing heat is being redistributed southward. (a) Total heat content change (gray), redistribution contribution (blue), and material contribution (red). (b) Contributions to material heat content change from the Indian (green), Pacific (orange), and Atlantic (yellow) Oceans. (c) Meridional heat transport due to redistribution in the Southern Ocean (blue), Atlantic (cyan), and Indian plus Pacific Oceans (magenta). Shaded areas represent ±2 standard deviations of a bootstrap ensemble.
Material heat content change (Figure 6a) is larger than its uncertainty at most latitudes and shows a peak at 35˚S and at 15˚ and 35˚N. The material heat content change peaks at 35˚S and 35˚N are collocated with climatological wind stress curl minima, where material warming due to anomalous surface heat fluxes may be accumulating due to convergence of surface Ekman transport.
Table 1 shows material, redistributed, and total heat content changes by ocean basin. Material heat content change is distributed among the Indian, South Paciﬁc, and South Atlantic basins approximately according to their area. However, the tropical and subtropical North Atlantic stores close to 20% of the global ocean’s material heat content change despite representing less than 10% of its area (Table 1). An outsized role for the North Atlantic in storing material heat content change in the climate system has also been foreseen in numerical modeling studies (Lee et al. 2011).
Table 1: Material, redistribution, and total contributions to heat content change by ocean basin in terawatts and area as fraction of global ocean area. Heat content change estimates are based on differences between the periods 2006–11 and 2012–17 inclusive. Uncertainties are ±2 standard deviations. The Southern Ocean is deﬁned as the entire ocean south of 32˚S. The South Paciﬁc, South Atlantic, and Indian Ocean estimates exclude the ocean south of 32S. The North Atlantic is split into a region south of and a region north of 44˚N. The latter includes the Arctic Ocean.
Material
Redistributed
Total
Area fraction
Southern Ocean
90 ± 18
118 ± 50
208 ± 63
0.27
South Pacific
53 ± 16
226 ± 22
28 ± 22
0.15
North Pacific
82 ± 25
261 ± 55
21 ± 54
0.23
Indian Ocean
45 ± 10
213 ± 25
32 ± 30
0.12
South Atlantic
34 ± 11
6 ± 7
40 ± 7
0.06
North Atlantic (<44˚N)
75 ± 33
20 ± 17
95 ± 46
0.10
North Atlantic (>44˚N)
19 ± 6
240 ± 13
220 ± 16
0.08
Global Ocean
398 ± 81
0
398 ± 81
1.00
We identify robust redistributed warming signals in the subtropical North Atlantic and Southern Ocean. Warming in the subtropical North Atlantic is compensated by cooling in the subpolar North Atlantic consistent with a 40 ± 13 TW southward transport of heat across 44˚N (Figure 6c). Southward heat redistribution across 32S brings 118 ± 50 TW into the Southern Ocean.
Discussion