The European Consortium Earth System Model Version-3 (EC-Earth) scenarios analyzed in this study were: PI (100 years run B405 and 200 years run B400), MH_PMIP (100 years run Z6KA and 200 years run B6KA), MH_GS (100 years run G105 and 50 years run G100, simulations with prescribed vegetation in the Sahara region) and MH_GSdr (100-year run G506, and 200-year run G501, simulations with prescribed vegetation in the Sahara region and dust reduction).

EC-Earth standard configuration consists of the atmosphere model IFS, including the land surface module HTESSEL and the ocean model NEMO3.6 with the sea ice module LIM3. Coupling variables are communicated between the different component models via the OASIS3-MCT coupler . The EC-Earth model is used to contribute to CMIP6 in several configurations, for example, the EC-Earth3-Veg configuration, which couples the LPJ-Guess dynamic vegetation model to the atmosphere and ocean model. However, the performance of EC-Earth3 and EC-Earth3-Veg is very similar .

The Community Earth System Model (CESM) outputs from different scenarios used were from CCSM-Toronto: PI, MH_PMIP, MH_GS, and MH_GSsl (with prescribed soil and lake inputs), 100 years run each; and iCESM: PI, MH_PMIP, MH_GS (100 years run each).

The CESM models used here are from the CMIP6 multi-model ensemble. The CCSM-Toronto simulations are a PMIP experiment for the mid-Holocene with Green Sahara and mid-Holocene with soil and lake inputs made by the University of Toronto (UofT), Canada. The model configuration was made by UofT-CCSM4 (2014), atmosphere from CAM4 (finite-volume dynamical core; 288 × 192 longitude/latitude; 26 levels; top level 2 hPa) ; ocean: POP2; sea ice: CICE4; land: CLM4. The iCESM simulations used in this study were first presented in . iCESM is configured with CAM5, POP2, CLM4, CICE4, and RTM . The atmosphere and land have a 1.9 × 2.5° horizontal resolution, and the ocean and sea ice have a nominal 1° horizontal resolution. Model configurations include a preindustrial simulation (1850 CE), a mid-Holocene simulation with a 6 ka orbit and greenhouse gases and preindustrial vegetation, and a mid-Holocene Green Sahara simulation with a 6 ka orbit and greenhouse gases and a vegetated Sahara. Dust emissions from the Sahara are reduced in the mid-Holocene Green Sahara simulation. For additional model configuration details, see .

The scenarios from the Goddard Institute for Space Studies Model E2 coupled with the Russel ocean model (GISS-E2-R) were: PI, MH_PMIP, and MH_GSNA with North African vegetation only, MH_GSEX with Extra-Tropical vegetation only, and two runs of MH_GSALL with Full vegetation (100 years run each).

All runs – except for GS Full Vegetation Run 1 – use updated aerosol and ozone inputs for non-anthropogenic simulations and apply the Green Sahara vegetation based on Nancy Kiang’s regression on leaf area index . In contrast, GS Full Vegetation Run 1 employs a regression script based on the Köppen–Geiger classification to prescribe the leaf area index . Several experiments have been set up for the last millennium with GISS due to uncertainties in past forcings and their effects, with different combinations of solar, volcanic, and land use/vegetation .

The Global Precipitation Climatology Project (GPCP) is a precipitation dataset based on the sequential combination of microwave, infrared, and gauge data. From 1979 to 2020, and offers globally complete satellite‐only precipitation estimates . To examine the precipitation over the Atlantic in the satellite era and compare its Shannon Entropy values to the ones calculated using model simulations, we utilized the GPCP Version 2.3 Combined Precipitation dataset . This dataset provides both continental and ocean precipitation data in a monthly, 2.5° × 2.5° grid, during the 1979–2015 period.

The observational sea surface data comes used is from Met Office Hadley Centre's sea ice and sea surface temperature dataset (HadISST1), a monthly 1° × 1° dataset , which is a Reanalysis dataset that uses observational data from ship expeditions and platforms interpolated by a numerical model to recreate the Global SST. The sea surface temperature anomaly was calculated with respect to the 1979–2015 period. To separate the Atlantic SST anomaly pattern of variability from any global warming signal, the mean global temperature anomaly was also calculated and subtracted from the SST anomaly time series .

In this study, the HadISST1 dataset has been used to measure the values of Entropy from the observational data using the phase space retrieved from the merged simulations dataset. While historical climate model simulations are best suited for comparisons with observational data, this study focuses on pre-industrial and mid-Holocene scenarios. Nevertheless, we applied our methodology to satellite data to quantify Shannon’s Entropy for the observational period.

Principal Component (PC) analysis (also known as Empirical Orthogonal Functions – EOF) is a technique used to reduce data dimensionality. When studying a high-dimensional dynamical system, such as numerical climate model simulations, finding relevant statistical information that emerges from the system's dynamics can be inefficient and overwhelming . The PC analysis derives a new set of orthogonal coordinates from your data, ordering the EOF patterns that maximize the data's variance in a decreasing fashion, enabling a drastic reduction in dimensionality of your dataset while preserving most of its variation . The PC time series is the projection of our data in the corresponding EOF pattern.

In this approach, we extracted two distinct phase spaces (one for SST and another for precipitation) composed of EOFs derived from the combined simulations of all models and scenarios. This process yields a unified phase space with a shared spatial structure for the entire ensemble, providing a consistent framework for analyzing and comparing variability across different simulations . This procedure is discussed and applied by Chandler et al. (2024), who construct a merged dataset across models and extract dominant modes of variability using principal component analysis. Their approach implicitly defines a shared low-dimensional representation for inter-model comparison, which is conceptually related to our phase-space construction.

To investigate decadal variability in the Atlantic Ocean, decadal filters were applied across all datasets, calculated as simple decadal means from the original monthly time series. These filters serve two primary purposes: first, to examine the decadal coupling between precipitation and SST variables; and second, to extract consistent multi-model patterns that serve as the foundation for a shared phase space. Decadal filtering is applied to the precipitation and SST datasets before pattern extraction to effectively smooth out fine-scale structures and disparities arising from different model architectures. Since we define a single phase space to encompass the merged dataset, this filtering process is essential to ensure that the leading climatic patterns capture substantial variance across the integrated multi-model ensemble.

A more classical approach to reducing the dimensionality of Atlantic Ocean SST and identifying its modes of variability is through the use of SST indices from regional boxes . Together with pressure gradients and wind anomalies, the ocean modes arising from these SST anomalies induce precipitation in the ocean and adjacent continents . These indices are constructed from pre-defined spatial boxes and provide a reduced representation of the system by capturing key patterns of tropical and South Atlantic SST decadal variability. In particular, the Atlantic Meridional Mode (AMM: the difference between 15–5° N, 50–20° W and 15–5° S, 20° W–10° E regional boxes), the Atlantic Equatorial Mode (AEM: 3° N–3° S, 20–0° W regional box), and the South Atlantic Subtropical Dipole (SASD: the difference between 30–40° S, 10–30° W and 15–25° S, 0–20° W regional boxes) can be used to define a phase space analogous to that obtained via PC analysis. However, unlike the PC analysis, these indices can be highly correlated, and they have no reciprocal precipitation modes. In our study, this framework is used to compare the system’s decadal SST variability using different techniques to define the phase-space.

Projecting a simulation series onto the “n” dimensions of our phase space generates a trajectory in that phase space. To elaborate more, we present a simplified 2-dimensional phase space for the SST decadal anomalies of the EC-Earth – PI experiment, using the two leading EOFs from the merged dataset. First, we apply the dimensionality reduction using the PC analysis to find the merged dataset two leading EOFs (Fig. a and b). Then, we project the EOFs into the simulation's Atlantic Ocean time series, generating its PCs (Fig. c and d). The trajectory of our system in the 2-D continuous phase space is shown in Fig. f. Defining negative, neutral, and positive phases for each index, using a threshold linked to Shannon’s Entropy (as described in Sect. 2.3), we coarse-grain this space into n3 states (9 possible states for this 2-D problem). The discrete state evolution of our system (Fig. e) is defined by quadrants in the 2-D map (States from Fig. f). Finally, the trajectory can also be seen as a discrete system evolving in time, from one possible state to the next, as depicted in Fig. g.

Since the continuous trajectory in a high-dimensional phase space can be challenging to plot, the trajectories of a system in a higher-dimensional phase space can be depicted with directed graphs (Figs. , h and h, and –). These graphs can hold different information regarding the system's trajectory in phase space. Each node in these graphs represents a state of the system at a particular time step, with the node's size indicating the number of months the system remained in that state. Nodes that appear darker have a higher degree, meaning they are connected to more transitions to and from other states. This indicates that the system frequently returns to the same state, which results in a darker color for that node. The distance between nodes is irrelevant; it was adjusted for the graph. The information in a directed graph can be overwhelming to analyze for every simulation run; therefore, we utilize its information to calculate a macro-property that reflects its variability in phase space.

Shannon's Entropy (Eq. ) is used as a measure of variability in unfied the phase space. 1H=-∑k=0NP(xk)ln⁡(P(xk))

The entropy of a dynamic system is a measure of its organization in a coarse-grained space. For example, the systems from Fig. a and b have the same initial and final states (0 and 4, respectively); however, system “a” evolves directly from the initial to the final state, while system “b” varies across different states until arriving at the final state. This means that system “a” is less chaotic, its trajectory is more organized, and hence its entropy is smaller.

Looking at Eq. (), the probability (P(xk)) of our system x (the simulated Atlantic Ocean SST and precipitation) being found in each possible state (k) can be calculated empirically from the simulation time series. For example, if a specific simulation has been in only one state during its whole time series, that state (xk) has a probability equal to one to be found in that specific state and zero in the others. Considering that ∑k=1NP(xk)=1, the entropy from that time series is the lowest possible, zero. In our study, defining a 3-dimensional phase space and discretizing each dimension in 3 possible phases (positive, negative, or neutral) creates 27 possible states of our system. The entropy from each time series will reflect its variability in that discrete phase space. Since we are using the same space for all the models and scenarios, we can compare their variability.

From a broader perspective, the probability distribution from the Atlantic SST or precipitation patterns (micro-states) in a simulation (trajectory) is used to determine the system's decadal variability (macro-properties) in our unified phase space (the 3 leading EOFs from our merged dataset).

Each simulation run can be represented as two trajectories in the PC phase spaces, one for the SST and one for precipitation. These trajectories of the Tropical and South Atlantic system (fully depicted for EC-Earth in the System State Classification “g” from Figs.  and ) are formed by all the states a system occupies during its time series. However, it can be challenging to present this information due to the high number of states and transistions. Even in a low 3-dimension phase space, it can be overwhelming to represent the full trajectory in a plain figure. Therefore, we chose to represent the most persistent states a system occupies (depicted in bold numbers in “g” from Figs.  and ) in the directed graph form for each simulation experiment (Figs. –). In Figs.  and , the direct graph can be seen in its ciclycal and linear forms (“h” and “i”, respectively). These graphs qualitative ilustrate the ciclicity and organization of a simulation, furthermore, their construction uses quantitative measures of the system in the phase-space. Each node in these graphs represents a system state at a given time step, with its size indicating the duration spent in that state, and its color intensity reflecting the frequency of transitions to and from other states (higher degree). Darker nodes indicate states the system revisits more often. The same properties used to design each graph were employed to calculate macroproperties such as the entropy of the time series.

In this study, Shannon's Entropy (Hsst and Hppt) is used to assess the level of organization of the tropical and South Atlantic given its possible states (as described in Eq. ).

Our choice to study decadal variability stems from the coupling between SST and precipitation at this time scale . Since these two variables are correlated and exhibit a strong feedback interaction controlling the energy balance along the Equator , we expect them to vary together, creating a distinguishable climate variability response to external forces. The first two columns of Table  show the entropy calculated in the PC phase space and its 95 % confidence interval for each model experiment. Despite differences in parameterizations and physics among models, each entropy was computed in a unified space (the states of every simulation were calculated using the EOFs extracted from the merged dataset). All model experiments were simulated over the same duration (100 years), enabling comparisons across the different models.

As seen in Fig.  and Table , the entropy values are approximately 3. This is due to the structure of our phase space: with three defined phases (positive, neutral, and negative) for each of the three principal components, the system has 3³ = 27 possible states. The maximum entropy (Eq. ) of a discrete space such as this would result in ln(27) ≈ 3.296. Since the simulation’s thresholds in phase space are individually tuned to yield its maximum possible entropy (see Methods), it naturally approaches ln(27).

EC-Earth's SST variability significantly drops from PI to the different MH scenarios (it reduces 5% in MH_PMIP and 7% in MH_GS in comparison to the PI), when dust reduction is applied its entropy recovers back to PI levels. The lowest SST entropy of all models and scenarios is measured during EC-Earth's MH_GS (blue triangles in Fig. a), indicating that in this scenario the EC-Earth model simulates a more organized Atlantic SST system. The precipitation variability shows no significant changes through all the different scenarios (blue triangles in Fig. b).

Compared to the PI experiment, GISS exhibits significant changes in both SST and precipitation variability in the MH_GSall2 experiment (5% in comparison to the PI). Besides this simulation, only MH_PMIP shown significant precipitation variability decrease when compared to PI. In this model, the largest entropy difference comes from SST in MH_GSall2 and MH_GSex (2.96 and 3.14, respectively – Table 2).

For this model, the lowest decadal SST variability happens in the MH_PMIP experiment (black hyfen in Fig. ), although without significant difference between the scenarios. The precipitation entropy shows significant decrese when vegetation is considered in the Sahara, MH_GS is the lowest of all simulations and scenarios (4% lower than the PI experiment and 6% lower than the MH_PMIP).

The decadal SST variability is higher in the PI experiment, but only significantly different from the MH_PMIP experiment (4% lower than the PI). Conversely, the MH_PMIP, MH_GS and MH_GSsl present 5% lower than PI precipitation variability.

The EC-Earth PI experiment (Fig. ) best illustrates how high correlation results in low entropy, showing the lowest entropy of all simulations within the Atlantic Ocean Modes phase space. In this specific setup, the AEM and AMM are correlated at 88 %, while the AMM and SASD show a 76 % correlation. Conversely, all MH scenarios exhibit higher entropy than the PI (6 % for MH_PMIP and 19 % for GS with/without dust) as these Atlantic modes decouple.

Compared with the PC phase space, GISS PI entropy reduces using the Atlantic ocean modes indices (Table ). The entropy from the remaining MH scenarios mostly decreases; however, they show no significant changes to the entropies calculated in the PCs phase space.

For this model, the SST entropies show a general decrease, while holding the same internal biases seen in the PC phase space entropies (Table ).

The only significant difference is still between PI and MH_PMIP; however, in this phase space, MH_PMIP holds the largest entropy amongst this model's different experiments (7 % higher than PI).