This study analyses data from 15 models from the Coupled Model Intercomparison Project Phase 6 (CMIP6) . Historical experiments, using historical external forcing, simulating 1850–2014 and four ScenarioMIP future projections for 2015–2100 ; ssp126 (low emission), ssp245 (medium emission), ssp370 and ssp585 (high emission) are investigated. Since the focus of this study is the AMOC streamfunction, the models selected require the necessary monthly-resolution variables to compute them for both depth- and density-space: meridional velocity (vo), practical salinity (so) and potential temperature (thetao) for the historical and all the previously mentioned ScenarioMIP experiments. Additionally, where available, variable vertical thickness (thkcello) is used in density streamfunction calculations. The exceptions to this are ACCESS-CM2 and ACCESS-ESM1-5, which lack sufficient vo on JASMIN for all scenarios but have computed depth-space streamfunctions available. We also choose to use the provided density-space streamfunctions for sigma-level models. This is to avoid issues with interpolating from sigma-levels to depth-levels and back to sigma-levels. These requirements limit the number of models in this study to 15. For consistency, priority is given to offline computed streamfunctions over the provided streamfunctions. As many ensemble members were used as possible. The variables used for each CMIP6 model in this study are listed in Table and for models with multiple ensemble members, they are listed in Table . For models with multiple ensemble members, the ensemble mean values are used unless otherwise specified.

The depth-space streamfunction is computed with Eq. (), integrating the meridional velocity (“vo”).

The depth-space streamfunctions were calculated by the integral as described in , 1ψz(y,z′,t)=∫xwxedx∫-H(x,y)z′(vo(x,y,z,t)-vo^(y,t))dz where ψz(y,z′,t) is the streamfunction, vo is the meridional velocity, vo^ is the section averaged volume transport as a correction term for mass conservation, 2vo^(y,t)=1A(y)∫xwxe∫-H(x,y)zsurfvo(x,y,z,t)dxdz where A(y) is the section area at a given latitude y. H(x,y) is the bottom topography, x is longitude with xe and xw as the eastern and western boundary, y is latitude, zsurf is surface and t is time. A comparison between this product and provided online streamfunctions is found in Appendix , as well as a comment on the use of meridional mass transport (“vmo”) to compute depth-space streamfunctions. In summary, we find using “vo” to compute the streamfunction results in differences (> 1 Sv) from the online streamfunction (Fig. ) whereas using “vmo” has much smaller differences (≪ 0.1 Sv). We suggest this is due to the inclusion of the bolus velocity in “vmo”, which is absent in “vo”. For consistency with the variables used for the density-space product, we choose to use “vo” to calculate the depth-space streamfunction.

Calculating the density-space streamfunction requires the density at every grid point. Practical salinity (“so”) and potential temperature (“thetao”) are interpolated from the model T-grid points to the model V-grid points and sigma-2 density is calculated using the Gibbs SeaWater (GSW) Oceanographic Toolbox for python . Sigma-2 is density referenced to 2000 dbar using the TEOS-10 equation of state.

The density-space streamfunctions were calculated with the integral, 3ψρ(y,ρ′,t)=∫xwxedx∫ρ(x,y,-H(x,y))ρ′v(x,y,ρ,t)dzdρdρ where ψρ(y,ρ′,t) is the streamfunction, v is the meridional velocity component, ρ(x,y,-H(x,y)) is the lower bound of density, dzdρ is the isopycnal thickness, x is longitude, y is latitude, ρ′ is variable density and t is time. The density-space streamfunctions were calculated as in Eq. () using the NEMO Cookbook python package . Volume transports were computed and then binned into sigma-2 densities over the range of 28–38 kg m⁻³ in 0.125 kg m⁻³ bins. These calculations are done on monthly output. For C grids, interpolation is done by taking the average of the so and thetao data to the north and south of the desired V-grid point. For B grids, so and thetao data to the north and south of the desired V-grid line are averaged and the velocity data to the east and west are averaged onto the same point. Partial cells were included as part of the mesh files and thkcello, where available. Interpolation is done prior to the use of the NEMO Cookbook.

The four modelling centres with provided the density-space streamfunction (see Table for the label “msftmrho”) are sigma-level models and to avoid issues with interpolating from sigma-level to depth-level and back to sigma-levels, the provided streamfunctions for these models are used. A comparison between the provided and computed density streamfunctions are found in Appendix .

For this study the AMOC strength is defined at each latitude in the Atlantic as the maximum of the overturning streamfunction measured in Sverdrups (1 Sv ≡106 m³ s⁻¹) for both the depth- and density-space AMOC. AMOC strength can alternatively be defined using a fixed depth or density level (e.g. ). However, the difference in definition of AMOC strength does not significantly impact the conclusions of this paper (not shown). Using the AMOC maxima to define AMOC strength avoids having to take into consideration of the shoaling behaviour of the AMOC under future climate forcing. The latitudinal response of the AMOC is investigated over the Atlantic basin (30° S to 65° N). To avoid the wind driven cell, AMOC strength computation excludes the top 500 m in depth-space and excludes densities less than 35.5 kg m⁻³ in density-space.

The response of the AMOC to future climate change scenarios is calculated as the percentage reduction from a historical reference period (1850–1899). The multi-model mean (MMM) AMOC strength for the historical reference period in depth-space has a peak AMOC strength of 18.2 Sv (at 35° N) and a minimum strength south of the peak of 14.8 Sv (at the equator) (Fig. a) while in density-space the peak AMOC strength is 18.0 Sv (at 55° N) and the minimum strength south of the peak of 14.0 Sv (at 30° S) (Fig. b). A 50-year period has been chosen for the reference period to reduce the impacts of interannual and multi-decadal variability.

AMOC strength is calculated as a 10-year running mean from 1900–2100 for the historical and four future scenarios labelled at the centre year of the decade, providing “annual” AMOC strengths from 1905–2095. The percentage weakening is computed relative to the reference period at each latitude. Latitudes are resampled onto a regular 1° resolution grid from 30° S to 65° N for consistency across the models. Changes to the streamfunctions north of 65° N are not considered due to the weak mean streamfunction and curved model grids at these latitudes.

AMOC weakening refers to the reduction in AMOC strength relative to the reference period. Thus, a negative AMOC weakening indicates a strengthening of the AMOC relative to the reference period. The use of percentage weakening rather than absolute reduction (in Sverdrups) is to allow the direct comparison of weakening between models that have a wide range of AMOC strengths, as is evident in the quantile ranges seen in Fig. . The use of percentage normalisation does not remove potential method-dependent structural differences between models of underlying AMOC diagnostics.

Taken at the multi-model mean peak (35° N for depth-space, 55° N for density-space), AMOC strength ranges from 10.7 Sv (IPSL-CM6A-LR) to 23.8 Sv (TaiESM1) and from 11.5 (CanESM5) Sv to 24.7 Sv (TaiESM1) for depth- and density-space respectively. To provide context on our model set in relation to the wider literature, we calculate the AMOC decline at 26° N for ssp585 using the method described in to compare mean weakening of the two different model sets. That paper takes the historical reference over the entire historical period (1850–2014). The model set for this paper has a mean percentage weakening of 45 %, where MRI-ESM2-0 has the most weakening at 67 % and EC-Earth3 is at the low end with 27 % weakening (Fig. ). This is larger than the mean weakening of 39 % found in the aforementioned study. Our model set skews slightly towards models with greater AMOC weakening.

To connect AMOC weakening across latitudes, contours are computed at specific percentages of weakening, henceforth referred to as weakening thresholds or threshold. The first year a given weakening threshold is exceeded is recorded at each latitude. The threshold line is defined for a specific weakening threshold, e.g. 15 %, as the contour of years as it changes across latitudes. The latitude at which the threshold is first exceeded indicates the year where AMOC weakening originates. The weakening anomaly of a given threshold can be followed throughout the basin to reveal its propagation. The threshold lines are calculated at 2.5 % intervals over the range 2.5 %–60 % between 2000-2095.

The gradient of the threshold line (here-on referred to as the “weakening gradient”) represents the speed of propagation of the AMOC weakening signal at a specific percentage reduction. The larger the weakening gradient, the faster the signal is transported southward. Assuming the threshold lines are approximately linear, the weakening gradient is found by performing a linear least-squares regression between 40° N and 30° S. This latitude range is chosen to capture the propagation south of the weakening onset. To ensure confidence in the validity of the gradient for a given threshold there must be at least a continuous line of thresholds from 40° N to the equator. This prevents erroneous gradients arising from limited data points. With these criteria, the linear fits of all models have a mean p-value < 0.05 (Fig. ). All models, with the exception of ACCESS-ESM1-5 and EC-Earth3, have a mean absolute r-value > 0.8 for all four scenarios (Fig. ).

The use of a rolling decadal mean is essential to the functioning of the first exceedance framework used to calculate thresholds. Without it, interannual variability would dominate the threshold lines and present a very noisy dataset, significantly lowering confidence in the linear fit and onset years. As we are interested in the mean state change of the AMOC and not in the propagation of annual and sub-decadal anomalies, our use of a rolling decadal mean is justified.

When comparing differences between future scenarios, the significances are computed using a two sided student t-test and a p-value of 0.05.

The MMM reference streamfunctions for depth-space (Figs. a and a) and density-space (Figs. b and b) respectively show peak overturning at 35 and 55° N. By the end of the 21st century both streamfunctions have shoaled and weakened in amplitude (Fig. c and d). The streamfunction maxima are now at 30° S for both representations and there is no longer a clearly defined latitudinal maxima in the North Atlantic. Taking the difference between the reference streamfunction and the ssp585 streamfunction confirms that the greatest reduction in the AMOC occurs where the AMOC is strongest in the reference (Fig. e and f). Figure f also shows an increase in the density-space streamfunction at the lightest densities north of 40° N. This increase is due the surface waters in the SPG becoming less dense due to subsurface warming , so it is not that the streamfunction has strengthened but instead the density classification has shifted. The standard deviations of the differences in the model ensemble reveal that the largest model spread is not aligned with maximum weakening (Fig. g and h). Instead, the greatest standard deviations are on the contours above and below the peak. This could indicate the model spread is most strongly determined by the intensity of shoaling and change in vertical position of the AMOC streamfunction as it weakens. This relates to a recent study that found models with deeper AMOCs exhibit greater shoaling than with shallower AMOCs and these models with deeper AMOCs may be more sensitive to warming than observations imply .

By representing AMOC weakening as a function of latitude and time, the connectivity of AMOC weakening across the Atlantic basin is revealed and the timescales of AMOC weakening can be inferred. Figure shows the percentage weakening Hovmöllers for depth-space and density-space plots from the first decade after the reference period (1905) to the end of the ssp585 experiment (2095). Changes from 1905–2000 are not of particular interest to this study but are included for completeness. This time period does show AMOC strengthening between 1960–2000 of approximately 10 % in both depth and density space, attributed to aerosol emissions , before the AMOC starts declining from 2015.

AMOC weakening begins at high latitudes, north of 40° N, with a weakening signal propagating to the southern boundary of the Atlantic basin in both depth-space and density-space (Fig. a and b). This is clearest when following the threshold lines. At the 5 % threshold, the time to traverse from the latitude of onset to 30° S is approximately three decades (2015–2045). At the 25 % threshold, this takes ∼ 45 years for depth-space and ∼ 55 years for density-space. Density-space has a longer signal than depth-space because the latitude of onset is further north in density-space than depth-space. This might be expected when looking at the streamfunctions , as the density streamfunction takes into account changes in the zonal structure in higher latitude North Atlantic, which are suppressed in the depth-space streamfunction and the maximum of the density-space streamfunction is likewise further north than the depth-space streamfunction. In both depth and density space representations of the AMOC streamfunction, there is a southward shift of the onset latitude as the threshold increases, 51 to 45° N in depth-space and 61 to 51° N in density-space from 5 % threshold to 45 % threshold. The difference between onset latitudes means density-space shows AMOC weakening earlier than in depth-space. For example, the 25 % threshold reached in 2035 in density-space compared to 2047 in depth-space. South of 40° N the timings of the threshold lines realign between the two representations. That AMOC weakening begins in the north suggests methods for early warning signals should focus more on high latitude monitoring.

Relating Fig. to the latitudes of the observational arrays RAPID (26° N, ) and OSNAP (53–60° N, , where we use 57° N to represent the whole array) reveals that for thresholds over 10 %, weakening at OSNAP leads weakening at RAPID by at least a decade (10–22 years, the range increases as threshold increases) in density-space while only precedes by 4–11 years in depth-space. As an illustrative example, we consider the difference in detection time of 25 % weakening at OSNAP and RAPID in depth-space and density-space. In density-space, OSNAP precedes RAPID by 17 years for ssp585 (ranging over 17–21 years, depending on scenario). In depth-space, OSNAP precedes RAPID by only 5 years for ssp585 (ranging over 4–8 years, depending on scenario). Since the onset latitudes of AMOC weakening in depth-space are further south than the OSNAP array, there are delays in detection at OSNAP when compared to the detection of weakening in density-space, where the onset latitudes broadly overlap the OSNAP latitudes. Depth-space performs better for early warning when measured at the latitude of most intense weakening (∼ 45° N) to 26° N. For 25 % weakening, depth-space weakening at 45° N precedes 26° N by 8 years for ssp585 (ranging over 8–15 years, depending on scenario). Density-space still generally outperforms along this latitude range, with a time difference of 11 years for 25 % weakening in ssp585 (ranging over 11–16 years, depending on scenario) and still has the additional signal from further north (8 years for 25 % weakening in ssp585). For less than 10 % weakening, depth-space signal precedes the density-space signal by 0–7 years. This is likely due to the high latitude (40–60° N) weakening between 2000–2015 that is seen in depth-space and not density-space.

The timescales associated with the weakening gradients found are longer than the propagation timescales of density anomalies (e.g. ), however they do align with timescale of some studies on AMOC adjustment to high latitude forcing . This will be discussed further in Sect. .

For the multi-model set, weakening gradients are calculated for each model and each scenario and are calculated in the unit degrees per year (° yr⁻¹). There are two approaches for considering the MMM weakening gradients.

Method 1: Calculating weakening gradients of the MMM Hovmöller threshold lines (seen on Fig. ). We call this the explicit multi-model mean (EMMM) (Fig. a and b, solid black line).

Method 2: At each threshold, computing the weakening gradient of each model individually before taking the mean across all the models. This does not include the MMM Hovmöller threshold lines. We call this the statistical multi-model mean (SMMM) (Fig. a and b, dotted black line).

All the models in the model set contribute to all the values of the EMMM weakening gradients whereas since the SMMM weakening gradients are calculated separately at each threshold, the number of models involved can vary (Fig. c, d). Analysis will primarily focus on the EMMM weakening gradients with the SMMM weakening gradients and associated 5 %–95 % quantiles acting as a reference for the behaviour of the model set.

For both depth-space and density-space, there is a gradual reduction of the EMMM weakening gradient ending in a plateau starting at around the 25 % threshold as the threshold increases. The EMMM weakening gradient reduces from 3.01 to 1.39° yr⁻¹ for depth-space and 3.04 to 1.45° yr⁻¹ for density-space, i.e. in both representations the weakening gradient has halved and, by proxy, we infer the propagation speed of weakening has halved. The story is similar for the SMMM weakening gradient, although the gradients are consistently slightly higher (by an average of 0.28° yr⁻¹) and due to a handful of models with weakening gradients at high thresholds, the SMMM weakening gradients persist along higher thresholds. The SMMM weakening gradients at high threshold should be considered cautiously, since the number of models contributing to statistical calculations are diminishing (Fig. c, d). The model missing at the 2.5 % and 5 % threshold is EC-Earth3 as this model starts the century more than 5 % weaker than its historical reference, and weakening gradients cannot be calculated at these thresholds.

The interpretation of changes to weakening gradients seen in Fig. is that as the AMOC weakens there is a gradual slowdown in the southward speed of propagation of AMOC weakening, down to a minimum, constant propagation speed. This is consistent in both depth and density space analysis.

Figure shows an intriguing reduction of the weakening gradient as the AMOC weakens but also poses three questions: (1) Do all future projection scenarios have similar weakening gradients? For example, does a 25 % reduction in the AMOC propagate at the same speed across the Atlantic basin regardless of specific forcing? (2) Do individual models follow the same gradient curve as the EMMM? (3) Why does the gradient slow at lower thresholds and plateau at higher thresholds?

The next section (Sect. ) will address the first question and Sect. will address the second question while the third question will be a topic of discussion.

The differences between depth and density-space have been explored in this section. While there are differences between the high latitude response to AMOC weakening in depth-space and density-space, the behaviour in weakening gradients south of 40° N is comparable between the two representations. As density-space reveals more details of high latitude processes than depth-space and is more descriptive for AMOC propagation in connection to those high latitudes , the remainder of the analysis will be presented using the density-space representation. This will capture a more complete representation of AMOC weakening than depth-space. There is very little difference between depth and density space and the equivalent figures for depth-space have been included in Appendix .

This section explores differences in density-space AMOC weakening between the future scenarios; ssp126, ssp245, ssp370 and ssp585.

To initially assess differences between scenarios, timeseries of AMOC weakening are shown for 3 selected latitudes of 51° N, 26° N and 30° S (Fig. ). As was inferred from Fig. , the rate of AMOC weakening is greatest at high latitudes (51° N) and is slower at lower latitudes (26 and 30° S). The latitude with the greatest divergence in AMOC weakening between scenarios is 51° N with end of century MMM weakening of 63 % in ssp585 and 41 % in ssp126 but also the largest standard deviations for those two experiments of the latitudes (Fig. a). In contrast, end of century weakening at 26° N is 47 % in ssp585 and 32 % in ssp126 (Fig. b) and at 30° S is 25 % in ssp585 and 22 % in ssp126 (Fig. c). AMOC weakening is more responsive to climate forcing in northern latitudes than latitudes further south. For example, under ssp585 forcing 25 % weakening is reached in 2036 at 51° N, 2055 at 26° N and only in 2095 weakening at 30° S. The latitudinal responsiveness can also be seen when looking at “negative” weakening (when the AMOC has elevated strength relative to the reference). The two higher latitude figures show weakening start by 2015, whereas at 30° S it takes until the mid 2030s (depending on scenario) to start experiencing positive weakening. The divergence between scenarios appears mainly driven by a slowdown of weakening in the lower emissions scenarios (ssp126 and ssp245) as the rate of weakening for ssp370 and ssp585 appears largely unchanged over each timeseries. A point of interest is that the AMOC weakening for ssp370 is consistently less than the other three experiments until the mid-2060s, contrary to the expectation that higher emissions scenarios lead to greater AMOC weakening but by the end of the century the amplitude of weakening in each scenario corresponds to the severity of the emissions in that scenario. The AMOC weakening being less for ssp370 is a consistent feature across the model set. Another point of interest with ssp370 is that it has a slower rate of weakening to the other scenarios prior to 2050, in disagreement with what was found by . This may be due to differences in the method of calculating the timeseries, such as this use of percentage difference taking the rolling decadal average rather than rolling annual average, or because our study uses a smaller number of ensemble members than their study used.

Relative to the 1850–1899 reference period, the AMOC is shown to already be weakening from the year 2000 at 51° N (Fig. a) and 26° N (Fig. b), which is model dependent and due to the model's sensitivity to aerosols . While the RAPID array has not yet detected significant weakening across 2004–2022 , statistically significant weakening has been seen in specific water masses after decomposing the AMOC (e.g. in the deep western boundary) . Observational arrays are subject to large internal variability which could obscure AMOC weakening . Longer timescales of observations are required to detect any weakening of the real-world AMOC, with proposing at least 29–67 years of RAPID observations to be at the 90 % confidence level of detection, but decomposing the AMOC as was done by could reduce this timescale. Weakening has not yet been significantly observed by OSNAP (2014–2021) .

Having assessed AMOC weakening at specific latitudes, we now consider the full basin response to AMOC weakening for the four scenarios. All scenarios exhibit positive weakening gradients, indicative of a north-to-south propagation of AMOC weakening (Fig. ) with weakening onset occurring at high latitudes similar to what was shown for ssp585 in Sect. . At a glance, Hovmöller plots are broadly similar for the first 50 years with similar threshold lines of 5 % and 15 % weakening after which the scenarios start diverging with ssp585 experiencing the greatest weakening and ssp126 experiencing the least. Detailed comparisons between scenarios are found by taking pairwise differences of the four Hovmöller plots, testing for 95 % significance of difference found with a student t-test (Fig. ).

Considering the scenarios compared to ssp585, the greatest magnitude end of century differences are with ssp126 (Fig. a), with smaller differences for ssp245 (Fig. b) and smaller again for ssp370 (Fig. c), consistent with Fig. . Interestingly, there are hardly any significant differences between ssp585 and either ssp126 or ssp245 before 2050. When the differences do become significant they do not propagate at the speed of the weakening gradients but can occur over a large range of latitudes in a single year and at least for the case with ssp245, it appears the equator may present a barrier for differences to propagate across (Fig. b). This is not seen in Fig. c, as the ssp370 and ssp585 are significantly different for the vast majority of the basin and time period because at every latitude AMOC weakening for ssp370 is less rapid than for ssp585 (Fig. ).

The remaining pairwise differences between emissions scenarios (ssp126, ssp245 and ssp370) are characterised by less extreme emissions scenarios experiencing greater weakening than the more extreme emissions scenarios, especially before 2070 (Fig. d–f). This is contrary to what we expect from the end of the century, and indeed after 2070 this trend is reversed, especially north of 50° N. In comparisons with ssp370, significant differences south of the equator persist until the 2090s. Comparing similar experiments (ssp370 and ssp245, Fig. e) and (ssp245 and ssp126, Fig. f), there are wavelike periods with no significant differences in AMOC weakening possibly driven by multi-decadal variability aligning AMOC strengths. Figure d and f also show an indication that the equator is a barrier for differences to cross.

The ssp370 scenario has been shown to be consistently different from the other scenarios (Fig. c–e) and with a different rate of reduction of weakening (Fig. ). This scenario is characterised by high aerosol emissions, which have been linked with strengthening the AMOC . The increased presence of aerosols in the atmosphere may explain the slower weakening of the AMOC in this scenario.

Figure explores the magnitude and spatial distribution of differences in AMOC weakening between scenarios but to understand differences in latitudinal connectivity we must return to Fig. . At high threshold (≥ 25 %) the threshold lines in the lower emissions scenarios reduce in gradient, southward propagation is slower, while in higher emissions scenarios there isn't such a noticeable change in the threshold line gradient (Fig. ). The reduction in weakening gradient for lower emissions scenarios is clear in Fig. as is the relatively constant propagation speed in the high emissions scenarios. In Fig. , both ssp126 and ssp245 show a continuous reduction in weakening gradient as threshold increases while for ssp370 and ssp585 the weakening gradient initially reduces but plateaus at high thresholds. The EMMM weakening gradient for ssp370 also starts lower than any of the other experiments with the 2.5 % threshold gradient at 2.50° yr⁻¹ compared to 2.99° yr⁻¹ (ssp585), 2.92° yr⁻¹ (ssp245) and 3.04° yr⁻¹ (ssp126), further supporting its uniqueness from the other experiments.

The plateau of high threshold weakening gradient could indicate a minimum speed of propagation in response to strong forcing that is not required for weaker forcing. Alternatively, depending on the mechanism for southward propagation of weakening is, if the driver of weakening stops or slows in lower emissions scenarios, it is plausible that the southward progression of a weakening signal would stop or slow as well. Figure shows that the rate of AMOC weakening varies as a function of latitude. The process that sets the rate of weakening at a fixed latitude is not the object of this study but we posit is strongly related to the propagation of AMOC anomalies through the basin. Exactly understanding the mechanism that propagates weakening southward would provide insight into this slowdown across the scenarios. Does weakening in the high latitudes and SPG necessitate weakening for the rest of the AMOC?

Figure c and d show a divergence between the EMMM line and the SMMM line, where the SMMM plateaus rather than continuously declining. This is a surprise, but considering the number of models involved in the high threshold statistics (Fig. e-h), this divergence only occurs between the multi-model mean and a small number of models. As will be shown in Sect. , most models show similar behaviour to the EMMM rather than the SMMM.

This section explores intermodel behaviour of AMOC weakening. We primarily focus on the subset of models with multiple ensemble members: CNRM-CM6-1 (6), CanESM5 (16), IPSL-CM6A-LR (3), MIROC6 (3) and UKESM1-0-LL (5), with numbers in the brackets indicating the number of ensemble members. We focus on these models to reduce the uncertainty associated with internal variability. We also present three models with only one ensemble member (MRI-ESM2-0, NorESM2-LM and TaiESM1) to highlight weakening gradient behaviours not seen in the ensemble analysis.

The ensemble mean Hovmöller for ssp585 generally show smoother and less variable behaviour than their individual ensemble members (Figs. a–e and ). Likewise, the weakening gradient of the ensemble means are also less variable than their individual ensemble counterparts, especially CNRM-CM6-1 (Fig. f–j, model and ensemble lines). The speed of weakening gradients is visibly model dependent, with some models faster (e.g. CNRM-CM6-1) and some slower (e.g. CanESM5), especially when compared to the EMMM and SMMM lines.

CNRM-CM6-1, MIROC6 and UKESM1-0-LL show that as the AMOC weakens, the weakening gradient also reduces (Fig. f, g and j). This is the same trend as the EMMM and SMMM. MIROC6 shows similar behaviour to the MMM lines having a non-linear decrease with increasing threshold. The final weakening gradients are 1.06° yr⁻¹ for MIROC6 at 45 % weakening, 1.45° yr⁻¹ for the EMMM at 35 % weakening and 1.51° yr⁻¹ for the SMMM at 55 % weakening.

Unlike the non-linear reduction shown by MIROC6, CNRM-CM6-1 and UKESM1-0-LL have a more linear reduction in weakening gradient, with CNRM-CM6-1 declining from 3.13° yr⁻¹ at the 2.5 % threshold to 1.58° yr⁻¹ at the 45 % threshold (Fig. f) and UKESM1-0-LL declining from 1.95° yr⁻¹ at the 2.5 % threshold to 1.18° yr⁻¹ at the 30 % threshold (Fig. j). These amount to a 50 % reduction in weakening gradient for CNRM-CM6-1 at the 45 % threshold and a 39 % reduction in weakening gradient for UKESM1-0-LL at the 30 % threshold. While the ensemble means of these two models show a linear reduction, the UKESM1-0-LL ensemble lines are similar to the ensemble mean whereas CNRM-CM6-1 shows a range of values spanning the 5 %–95 % quantiles of the model set. CNRM-CM6-1 has been shown to have greater internal variability than the other models in this set which may explain the large range of ensemble member weakening gradient behaviours.

CanESM5 is a model with particularly low weakening gradients compared to the model set (see Appendix , Fig. ) and differs from the MMM weakening gradient behaviour. It is the model with the greatest number of ensemble members in our analysis (16) and the ensemble spread in weakening gradients is small relative to other models (Fig. g). The ensemble mean weakening gradient at the 2.5 % threshold is 1.08° yr⁻¹ which gently increases to 1.29° yr⁻¹ at the 22.5 % threshold (Fig. g). This is a near constant weakening gradient in comparison to the MMMs.

Of the five models in the model subset, IPSL-CM6A-LR has the most unclear weakening gradient behaviour with a decrease in weakening gradient followed by an increase (Fig. h). This behaviour is not inherently physically implausible but it appears more likely to be a limitation of the threshold line method rather than a physical process (Fig. c). These limitations are best explored studying the 5 % and the 15 % threshold lines (Fig. c). Weakening of 5 % propagates from 40° N to 20° S on the order of 20 years (∼ 2020 to ∼ 2040), similar to MIROC6, but 5 % weakening only occurs at the southern boundary of the Atlantic (30° S) in the 2070s. There is a similar delay in the 15 % threshold line at the equator during 2050–2070. The non-linearity of the southward propagation results in a less robust linear fit, that while still statistically significant, results in a smaller weakening gradient than would be expected when considering other parts of the threshold line. This highlights that weakening gradients are most informative when capturing latitudinally coherent weakening (e.g. the other four models) and does not perform as well when fitting noisy threshold lines. IPSL-CM6A-LR is a model with only three ensemble members and the mean may be more sensitive to internal variability.

The trends of the weakening gradients can be grouped by visual inspection by whether the gradient reduces (CNRM-CM6-1, MIROC6, UKESM1-0-LL), increases (CanESM5) or oscillates (IPSL-CM6A-LR). We sort all 15 models into Table . The majority of the models (10 out of 15) have a reducing weakening gradient with increasing threshold (Fig. ). Of those ten, two of them show a plateau in weakening gradient (CESM2-WACCM and TaiESM1) not seen elsewhere in the ensemble set. Two models have an increasing weakening gradient with increasing threshold (CanESM5 and EC-Earth3). Both are models that started with a small weakening gradient. The remaining three models have neither a distinctly increasing or decreasing weakening gradient with increasing threshold (ACCESS-CM2, IPSL-CM6A-LR, MRI-ESM2-0).

Here we analyse three models (MRI-ESM2-0, NorESM2-LM and TaiESM1) with only one ensemble member that reach a high threshold and are thus visually clearer to analyse (Fig. ). Ensemble member behaviour can vary substantially (e.g. CNRM-CM6-1) or very little (e.g. CanESM5). With only one ensemble member it is not possible to know if a model will be the former or the latter. For the sake of this analysis, we assume the individual ensemble realisations are representative of the model behaviour but acknowledge that these model results may be due to internal variability.

MRI-ESM2-0 has a somewhat oscillating behaviour, similar to IPSL-CM6A-LR, and does not show a decreasing trend in weakening gradient with threshold (Fig. d). MRI-ESM2-0 has a mean weakening gradient of 2.08° yr⁻¹. NorESM2-LM, on the other hand, has a strong decreasing weakening gradient, reducing from 6.41° yr⁻¹ at 2.5 % threshold to a minimum of 1.34° yr⁻¹ (Fig. e). This is similar to the the r5i1p1f2 ensemble member of CNRM-CM6-1, meaning this behaviour in NorESM2-LM could be influenced by internal variability, either in the historical reference or the future response. In TaiESM1, instead of a continuous reduction in weakening gradient, there appears to be a two-stage response. Initially there is a rapid, linear reduction until the 15 % threshold from an initial weakening gradient of 5.34° yr⁻¹ which then plateaus to a mean weakening gradient of 2.04° yr⁻¹. If this response is not related to internal variability, it suggests that there is an adjustment period of weakening in future forcing which then settles to a constant speed.

AMOC weakening north of 40° N is not considered in the weakening gradient calculation, but there are interesting features seen in subpolar gyre (SPG) region. CanESM5, IPSL-CM6A-LR and MIROC6 experience noticeable reductions in the SPG region (49 to 63° N) that are not immediately propagated southwards, most pronounced being the delay of the 35 % threshold line (Fig. b–d). Meanwhile CNRM-CM6-1 and UKESM1-0-LL show similar behaviour to the MMM with a less clear-cut latitude break (Fig. a and e). This indicates high latitude weakening in the streamfunction does not immediately propagate further south. A recent high resolution modelling study found that there is a strong relationship between density at OSNAP latitudes with AMOC strength at 45° N but not between AMOC strength at OSNAP latitudes (53–60° N) and 45° N, which aligns with the lack of immediate propagation. However, as is seen in the trio of single ensemble models (Fig. ), this does not apply to all models. MRI-ESM2-0 and NorESM2-LM show much broader weakening over the SPG region (Fig. a and b), displaying greater connectivity between the high and lower latitudes, while TaiESM1 has no visible region of greater weakening (Fig. c). The broad weakening response in the SPG for MRI-ESM2-0 and NorESM2-LM cannot just be due to these streamfunctions being provided by modelling centres, as CAS-ESM2-0 (Fig. ) shows a similar response and is a calculated streamfunction. These models are in disagreement with the results from , but given the differences in experimental set-up and model resolution, it is difficult to assess whether this disagreement has any significance.

Figure shows the same models as in Fig. but under ssp126 forcing. All models, with the exception of CNRM-CM6-1, show fewer and/or less steep (e.g. slower) threshold lines in the ssp126 experiment (Fig. a–e). This is confirmed when looking at the weakening gradients (Fig. f–j), where all except CNRM-CM6-1 show a reduction in gradient with increased threshold similar to the EMMM, although for CanESM5 it is hard to make a definitive statement as the ensemble mean weakening gradients do not reach the 10 % threshold. Unlike its ssp585 behaviour, the weakening gradients of CNRM-CM6-1 maintain at ∼ 3° yr⁻¹ until the 15 % threshold before reducing. An interpretation of this is that for CNRM-CM6-1, the initial onset of AMOC weakening is largely resultant from the historical forcing and associated feedbacks and only after the cumulative effects of the ssp126 forcing has occurred does the weakening gradient start to change. Linking this Sect. , the SMMM weakening gradients at high percentage threshold of ssp126 are skewed by the behaviours of CNRM-CM6-1 and MRI-ESM2-0 while the EMMM weakening gradient is more consistent with most models (see Appendix , Fig. ).

As is seen with CNRM-CM6-1 (Fig. b), it is possible for weakening signals to propagate under ssp126 forcing similarly or faster to those under ssp585 forcing (Fig. b). Understanding model differences in response to different forcing scenarios will give a clearer understanding of how AMOC weakening propagates throughout the Atlantic.