A decline in Atlantic meridional overturning circulation (AMOC) strength has been observed between 2004 and 2012 by the RAPID-MOCHA-WBTS (RAPID – Meridional Overturning Circulation and Heatflux Array – Western Boundary Time Series, hereafter RAPID array) with this weakened state of the AMOC persisting until 2017. Climate model and paleo-oceanographic research suggests that the AMOC may have been declining for decades or even centuries before this; however direct observations are sparse prior to 2004, giving only “snapshots” of the overturning circulation. Previous studies have used linear models based on upper-layer temperature anomalies to extend AMOC estimates back in time; however these ignore changes in the deep circulation that are beginning to emerge in the observations of AMOC decline. Here we develop a higher-fidelity empirical model of AMOC variability based on RAPID data and associated physically with changes in thickness of the persistent upper, intermediate, and deep water masses at 26

In the Northern Hemisphere, the Atlantic meridional overturning circulation (AMOC) carries as much as 90 % of all the heat transported poleward by the subtropical Atlantic Ocean

The importance of the AMOC means that since 2004 it has been observed by the RAPID-MOCHA-WBTS (RAPID – Meridional Overturning Circulation and Heatflux Array – Western Boundary Time Series, hereafter RAPID array) mooring array at 26

In one proxy reconstruction,

At 26

Here, we revisit the approach of

Our regression models were trained on RAPID data from 27 May 2006 to 21 February 2017

The internal geostrophic transport,

The strength of the AMOC is then the maximum of

For use in the regression models, absolute salinity, conservative temperature, and in situ density were calculated from the gridded in situ temperature and practical salinity data created during the RAPID calculations. As all AMOC transports are filtered during the RAPID calculation, the same Butterworth 10 d, low-pass filter was also applied to the salinity, temperature, and density data; the filtered data were then averaged from a 12-hourly resolution to a monthly mean. Anomalies of these data and RAPID-estimated transports were created by subtracting the mean between 27 May 2006 and 21 February 2017, and these monthly mean anomalies were used to train all our regression models.

We revisited the linear regression made by

Linear regression of the monthly mean RAPID thermocline (0–800 m) transport anomaly on the monthly mean conservative temperature anomaly at

We expanded this one-layer model by creating multiple linear regression models using two, three, and four explanatory variables to represent two, three, and four layers respectively, reflective of the water mass and circulation depth structure. We used RAPID-defined layers: the thermocline between the surface and 800 dbar; Antarctic Intermediate Water (AAIW) between 800 and 1100 dbar; Upper North Atlantic Deep Water (UNADW) between 1100 and 3000 dbar; and Lower North Atlantic Deep Water (LNADW) between 3000 and 4820 dbar (Fig.

Initially a model with two explanatory variables representing the thermocline (

The OLS regressions were checked against a number of assumptions that should hold for a linear regression model to be fit for purpose, for example, a known issue for those models based on time series is autocorrelation of residuals. We found that all our OLS models, whether using simple or multiple linear regression, showed autocorrelation of residuals, indicated by Durbin–Watson values between 0 and 1. An alternate linear regression model was used, the generalised least-squares model with autocorrelated errors (GLSARs), which models autocorrelation of residuals for a given lag

The simplest model selected by the algorithm, regressing UMO transport anomaly on the western boundary density anomaly at 780 dbar, gives an adjusted

Comparison of UMO transport anomaly predicted by GLSAR(1) regression models using one to four layers with the UMO transport anomaly observed by RAPID. The layers represented by the regression-independent variables are shown above each plot, and the model

The selected model was cross-validated using a

Our model was trained on monthly mean density anomalies but was to be used with hydrographic data from much shorter periods of a day or two. To evaluate how well these “snapshot” profiles represented the longer periods, we simulated them by randomly selected 20 single points from the 7961 available 12-hourly values from the most recent RAPID data. These were applied to the model and the predicted UMO compared to the observed monthly mean UMO for the same time, with the model error being the difference between the two. Bootstrapping the model prediction showed that around 65 % of the observed UMO values were within the prediction interval of the corresponding model UMO. The standard deviation of the bootstrapped model errors was 2.8 Sv.

The CTD profiles to be applied to the regression model to estimate UMO transport occur at irregular intervals, so to allow comparisons between periods, we calculated the mean transport anomalies for a given time window. Additionally, we calculated the weighted rolling mean for the transport anomalies with the RAPID annual cycle removed, using a Gaussian distribution over the same time window. Since

To evaluate the co-variability of the density anomaly selected to represent each water mass transport and the observed UMO transport anomaly, we determined the coherence between them using a multi-taper spectrum following

Multi-taper spectrum coherence

The importance of the LNADW in the AMOC decline compared to the UNADW

For the LNADW linear regression, the algorithm selected the western boundary density anomaly at 3040 dbar, close to the boundary between the Upper and Lower North Atlantic Deep Water layers at 3000 dbar. The resulting linear regression Eq. (

This model was also tested using the RAPID data between February 2017 to November 2018, and the model-predicted LNADW transport anomaly shown in Fig.

As

Historical hydrographic data were obtained from the World Ocean Database 2018 (WOD2018)

Initially, we used the western boundary density anomalies derived from the transatlantic sections at 24.5

UMO transport anomaly estimated by the empirical model using density anomalies from six transatlantic hydrographic sections, compared to estimates from

When the model is applied to western boundary density anomalies from the selected hydrographic profiles, together with the eastern boundary climatology, the resolution of the UMO time series in Fig.

The model-estimated AMOC transports shown in Fig.

Compared to the LNADW estimates from transatlantic sections made by

UMO and LNADW transports estimated by empirical models using density anomalies from hydrographic CTD profiles, compared to estimates from RAPID and

The 4-year mean transports in Fig.

The 4-year mean AMOC transports show slightly more variability than the UMO, and agree slightly less well with the RAPID 4-year mean values, differing by 0.7 Sv for 2004–2008 and 2008–2012 and 1.5 Sv for 2012–2016. The 2000–2004 mean reflects the UMO downturn, with the lowest 4-year mean value of 14.8 Sv, again lower than the 2008–2012 mean AMOC transport for both model and RAPID by 0.4 and 1.1 Sv respectively.

The 4-year LNADW rolling mean suggests a non-monotonic weakening trend in the southward deep return flow between 1985 and 1999, from 8.5 Sv southwards in 1985 decreasing to 3.8 Sv southwards in 1999. The rolling 4-year mean then varies by less than 0.6 Sv between 2000 and 2008, then weakens again to between 4.6 and 4.3 Sv southwards in 2009 and 2010 respectively before increasing in strength again to a maximum southward transport of 7.7 Sv in 2013. RAPID mean southward values for 2004–2008 and 2008–2012 are stronger than the model by 0.9 and 0.3 Sv respectively, but for 2012–2016 they are 1.8 Sv weaker. The greater disagreements between model and RAPID mean values for AMOC and LNADW transports may be due to the additional smoothing caused by using monthly mean Ekman transport to estimate them rather than the 10 d filtered values used by RAPID. None of the model-estimated UMO, AMOC, or LNADW transports show an overall trend.

Although the AMOC has been well-observed since 2004 by RAPID, before this, estimates of AMOC transport were restricted to approximately decadal transatlantic sections. It has been estimated that a time series of at least 60 years is necessary to detect long-term change in the AMOC due to anthropogenic global warming

To develop this empirical model of the AMOC, we regressed UMO transport on western boundary density anomalies within each of the thermocline, UNADW, and LNADW layers and an eastern boundary climatology within the AAIW layer, using an algorithm to select the best depth for each density anomaly. The selected model was then applied to historical hydrographic CTD profiles to predict UMO and hence AMOC transport strength between 1981 and 2016, at approximately annual resolution. This resolution is sufficient to show pentadal to decadal variability, with a model uncertainty of around

Four-year means (dashed lines) from 1984 to 2016 and the Gaussian-weighted rolling mean with a 4-year window (solid line), with the markers showing the mid-point, for AMOC, LNADW, and UMO transports estimated by the relevant regression models (orange) and from RAPID observations (dark blue). The 4-year and rolling means for Florida Current (light blue) and Ekman (dark grey) transports are also shown.

There is no overall trend in either AMOC or UMO as estimated by the model, but 4-year means, following

In addition to the four-layer UMO–AMOC model, we also created a similar model regressing LNADW transport on the deep western boundary density anomaly at 3040 dbar and Ekman transport. The 4-year mean LNADW transport estimates from the same hydrographic profiles show lower-frequency variability than the UMO–AMOC, suggesting the deep southward return flow was strong throughout the late 1980s and 1990s, weakening towards 2000. The 4-year mean is also weak during the observed AMOC downturn of 2008–2012. The rolling 4-year means for all three transports reflect the changes observed by RAPID well, with decreasing northward AMOC transport and decreasing deep southward return flow balanced by an increase in southward gyre recirculation

Although this model increases the temporal resolution of AMOC estimates, the resolution is still coarse compared to RAPID and the time intervals between profiles are inconsistent. The longest period where no interval is greater than 1 year is October 1988 to July 1994. There are only two intervals longer than 2 years: September 1981 to April 1985 and February 1998 to April 2001. The longest interval is 1328 d and the mean is 210 d. Although this resolution is sufficient to show multi-year variability, as shown by the 4-year means, the length of some of the sampling intervals and their inconsistency means the model cannot show interannual variability reliably.

In conclusion, this study shows that the dynamics of the AMOC can be represented by an empirical linear regression model using boundary density anomalies as proxies for water mass layer transports. More than one layer, represented by boundary density anomalies, is required to capture lower-frequency changes to UMO transport. Deep density anomalies combined with Ekman transport are successful in reconstructing LNADW transport, the deepest limb of the AMOC in the subtropical North Atlantic. Previous proxies for AMOC or UMO at 26

Our model, applied to historical hydrographic data, has increased the resolution of the observed AMOC between 1981 and 2004 from approximately decadal to approximately annual, and in doing so we have shown decadal and 4-yearly variability of the AMOC and its associated layer transports. The result is the creation of an AMOC time series extending over 3 decades, including for the first time deep density anomalies in an AMOC reconstruction.

Our model has not revealed an AMOC decline indicative of anthropogenic climate change

Original code for this analysis was written in Python and is available on request to the corresponding author.

Data from the RAPID AMOC monitoring project are funded by the Natural Environment Research Council and are freely available from

ELW and GDM conceived and designed the study. Analysis was carried out by ELW under supervision by GDM, JVM, and RM. BIM and DAS provided software and helped with analysis. ELW prepared the paper with contributions from all co-authors.

The authors declare that they have no conflict of interest.

ELW was supported by the Natural Environmental Research Council (grant number NE/L002531/1). GM was supported by the A4 project (grant aid agreement PBA/CC/18/01) supported by the Irish Marine Institute under the Marine Research Programme funded by the Irish Government, co-financed by the ERDF. The authors thank the many officers, crews, and technicians who helped to collect these data.

The authors would like to thank Penny Holliday for her feedback and support of the study and Eleanor Frajka-Williams for the code used to produce the coherence and phase relationship plots in Fig. 5, which is based on the jLab software package

The authors would also like to thank the two anonymous reviewers whose comments and suggestions helped improve and clarify this paper.

This research has been supported by the Natural Environment Research Council (grant nos. NE/L002531/1 and NE/N018044/1), EU Horizon 2020 project Blue-Action (grant no. 727852), the National Science Foundation (grant no. 1332978), the National Oceanic and Atmospheric Administration (grant no. 100007298), the European Regional Development Fund (grant no. PBA/CC/18/01), and the Climate Program Office (grant no. 100007298).

This paper was edited by Katsuro Katsumata and reviewed by two anonymous referees.