We learn dynamical system models using the European Centre for Medium-Range Weather Forecasts (ECMWF) Reanalysis 5 (ERA5) dataset spanning the period from January 1950 to December 2022, which we treat as our observational “truth”. We use this data at a temporal resolution of one month, and on a spatial grid of 0.5° × 0.5°. To assess how well the family of potential discovered equations predicts data it has not been compared against, we construct an additional validation and test set.

Selecting a model based on the validation set rather than the training data avoids the possibility of choosing a model that has overfitted to the training data. The test data is then used to assess the ability of the best model to predict unseen data. For testing and validation, we use different segments of the ECMWF Reanalysis of the 20th Century (ERA20C) dataset over the period from January 1900 to December 1949. Specifically, we make data from January 1940 to December 1949 to be our validation set, and from January 1900 to December 1939 as our test set.

Precipitation in the ERA5 reanalysis data is not an assimilated variable before 2010, up to which point it is only a modelled field. The Global Precipitation Climatology Project (GPCP) Monthly Analysis Product spans 1979 to the present and integrates data from rain gauge stations and various satellite datasets over land and ocean. Thus, for 1979 onward, the GPCP dataset provides one of the most complete and accurate rainfall analyses over the oceans. The correlation between GPCP and ERA5 monthly precipitation anomalies over the North Atlantic ocean (see Sect.  for area of interest) is 0.4 for the period 1982–2020. However, unlike GPCP, ERA5 provides a consistent dataset between 1950 and the present, which is sufficiently long to learn low-order models that capture decadal variability.

We favour the later ERA5 dataset for model training, as improved data assimilation techniques and the abundance of satellite and ground observations result in a more accurate and trustworthy representation of historical Atlantic variability than in the ERA20C data . This additional accuracy is likely especially valid regarding precipitation over the Atlantic Ocean. The ERA20C dataset is longer than ERA5 but only assimilates surface observations. We choose our validation set to be a relatively short 20 % of the independent period (1900 to 1949), as we favour longer periods to evaluate the ability of a low-order model to predict the decadal variability of the AMV, NAO, and precipitation indices for various different states of the system. The correlation coefficient between monthly North Atlantic precipitation anomalies (see Sect.  for area of interest) derived from ERA5 and ERA20C datasets is strong at 0.89 for the shared period 1950–2010.

To compute the AMV index, the mean monthly SST anomalies are calculated for the region (80–0° W, 0–60° N), from which we subtract the global average (60° S–60° N) following . We then normalise the index by dividing by its standard deviation. The NAO index is defined as the difference between the mean monthly sea level pressure anomalies for a region around the Azores (28–20° W, 36–40° N) and around Iceland (25–16° W, 63–70° N) following . The NAO time series is then normalised by dividing by its standard deviation. We compute North Atlantic precipitation time series as the mean monthly total precipitation anomalies for the region (60–0° W, 15–75° N). We denote these anomalies with P, which have units of m. The precipitation index is then normalised by dividing by its standard deviation, as performed for the NAO and AMV indices. All three indices, AMV, NAO and P have zero mean across the period 1950–2022, as they were computed using monthly anomalies of SST, sea-level pressure, and total precipitation based on the climatology of the full period. Since all time series have been normalised, they are therefore all unitless.

Sparse Identification of Nonlinear Dynamics (SINDy) is a sparse regression algorithm that learns dynamical system equations from data time series, and here in the form of ordinary differential equations. A general time-evolving dynamical systems model takes the form of x˙(t)=f(x(t)) where the vector x=[x1(t)x2(t)⋅⋅⋅xd(t)]T∈Rd represents the state of the system at time, t. The number of system variables is represented by d and we have m observations of the state x. The SINDy algorithm learns the analytical form of f from data, by performing a regression analysis using a library of candidate functions. SINDy assumes the function, f, is sparse, and therefore operates to deliberately contain only a few terms from these candidate functions.

We use the SINDy algorithm to fit dynamical models evolving in time to the AMV index (AMV), NAO index (NAO) and North Atlantic precipitation anomalies (P). Since we are interested in the relationships between state variables on longer decadal timescales, a twelve-month running average is applied to the three variable time series spanning the training set from January 1950 to December 2022, after which the model is fitted. This approach removes higher frequency signals and improves the performance of fitting, consequently enhancing the quality and stability of the resulting dynamical models. Our chosen library of candidate functions consists of an offset term and linear and nonlinear polynomial terms of the three state variables up to order two. This gives a total of ten terms in each of the three equations for each state variable, resulting in 30 terms overall, describing the time evolution of the system variables. Given that the data set is limited in length, we choose polynomial terms only up to order two to reduce the risk of overfitting. The general form for the model with our candidate functions is: 1dxidt=ai+∑j=13bijxj+∑1≤j≤k≤3cijkxjxk,i=1,2,3, where x1, x2 and x3 are the state variables AMV, NAO and P respectively. In Eq. (), a constant term in each equation is represented by ai, the coefficients of the linear terms are represented by bij, and the coefficients of unique quadratic cross-products and squares are represented by cijk. The 30 coefficients are solved for simultaneously in the equation discovery process. To solve for these coefficients, we require observations of the instantaneous changes, dxidt. These are computed numerically from the data. The candidate functions are evaluated at every point in time using observations of the state variables. The SINDy algorithm then regresses dxidt against the observed values of the candidate functions to find the unknown coefficients ai, bij, and cijk.

To perform the regression, we use the LASSO optimiser, which introduces sparsity into the regression procedure. It does this by adding a penalty to the sum of the absolute values of the regression coefficients, resulting in some coefficients becoming exactly zero, thereby excluding less important variables and preventing overfitting – this is called “regularisation” in the ML community. The threshold parameter, λ, determines the strength of the penalty: coefficients are set to zero if their magnitude is less than λ.

We use two approaches to improve the robustness of the fitted model. Firstly, we use an “ensembling” approach. We randomly sample the data with replacement to create 1000 “bootstrapped” datasets before fitting a model to each one. We then take the median coefficients across the ensemble of 1000 models to produce a final averaged model. The sizes of the random samples are taken to be 25 %, 50 %, 75 % of the total number of data points in the data matrix X. Since we have three monthly time series then d=3, and these span January 1950 to December 2022 inclusive, meaning m=876. Therefore, we have a total of d×m=2628 data points. We also fit models using 100 % of the training data without bootstrapping.

Secondly, we apply “sparsity thresholding” in the learning process. This repeats the ensembling process for a range of threshold values, λ. Here we choose lambda to range from 0 to 1 with a step of 0.005. By using the four different bootstrap sample sizes and 200 different possible values of lambda, we learn a total of 800 candidate models when both ensembling and sparsity thresholding are applied. We finally discard any duplicate models, such as those which contain no terms due to the sparsity threshold being too high, or those with the exact same coefficients. The remaining set of candidate models is then evaluated on the validation data set.

The statistical significance of the low-order model's predictive performance in Sect.  (quantified using anomaly correlation coefficients) accounts for autocorrelation by performing block bootstrapping on model forecasts and observations, followed by a one-sided Student's t test against the null hypothesis of non-positive (negative) correlation, applied at the 0.05 significance level. We use a one-sided Student's t test as we are only concerned with the “positive skill” of the low-order model, since negative correlations are evidence of “anti-skill”. When computing regressions between indices and other variables in Sect. , significance levels are computed based on a random phase test . Here, we generate 10 000 phase-randomised surrogates of the index that preserve its power spectrum (and autocorrelation), and compute the regression coefficients between the surrogate distribution and the chosen variable time series. The null hypothesis is that a correlation arises purely from internal serial correlation. We apply this test at the 95 % significance level.

We consider three sets of potential equations, each with a different number of terms, selected based on the Pareto curve in Fig. . The Pareto curve illustrates how errors begin to plateau for models with ten or more terms, indicated by the “knee” point of the Pareto front (Fig. ). Thus, we could choose a ten-parameter model as our “best” model, as it achieves a good trade-off between model accuracy and complexity. The ten-term model with the lowest MSE is given by Eqs. ()–(): 2τdAdt=-1.61N-1.63NP3τdNdt=+1.21A-0.66A2+2.39AP+1.58NP4τdPdt=+0.95A2-0.62N2-1.37AP-1.41NP where “A” refers to the AMV index, “N” refers to the NAO index, and “P” refers to the precipitation anomalies of the North Atlantic. This model falls on the Pareto front, and is marked by a red dot in Fig. . The natural time step, t, is months, as this corresponds to the time interval of the data. However, as we expect the SINDy methodology to quantify the longer-term variations, we have scaled the equations by multiplying through by τ=120 months (10 years). Hence, each coefficient represents the decadal tendencies of each term. Notably, this scaling makes all terms on the right-hand side of Eqs. ()–() of order unity thus balanced in magnitude, illustrating that this timescale is the one for which the SINDy algorithm is identifying changes.

To assess the model's validity, we must also examine its long-term behaviour. Due to the limited period for which data on A, N, and P are available, long-term behaviour does not form part of the SINDy fitting process, which can only be expected to capture variations up to timescales of the order of decades. However, the current observations clearly illustrate that the coupled A–N–P system exhibits substantial variability, and arguably this allows us to eliminate equation sets that fail to account for this behaviour over the longer term. Hence, we numerically integrate Eqs. ()–() forward, using a fourth-order Runge-Kutta scheme, and find that all solutions of this model converge onto one of the five stable fixed points: (A=-1.20, N=0, P=-0.84), (A=0, N=0, P=-0.99), (A=-3.29, N=-2.15, P=-0.99), (A=-1.04, N=0.31, P=-0.99) and (A=0.83, N=-0.91, P=-0.99). Once the solutions of this model converge onto stable fixed points, they remain at a constant value for all time. Hence, this equation set does not represent self-sustaining variability in the North Atlantic and therefore is an unrealistic model. More generally, we find that all models with thirteen terms or less (red region in Fig. ) either converge on stable fixed points, limit cycles, or are unstable and lead to “blow-ups” in the simulated trajectories. A limit cycle is an isolated closed trajectory in phase space that represents a stable, self-sustained oscillation. Models that follow limit cycles may generate observed variations in A, N and P, however, they may also lack important features of the observed variability.

For this reason, in addition to requiring that our model has a low MSE concerning instantaneous tendencies, we also introduce a second requirement that a credible model has realistic variability on longer timescales. This comparison is based on the estimated power spectra derived from model integrations undertaken for 50 000 years. Considering models with more terms in the equations, we find that those which contain between fourteen and twenty terms, inclusive, are all stable when integrated forward in time and have solutions that either converge onto a limit cycle or a stable fixed point (blue region in Fig. ). One such model is the 16-term Pareto-optimal model, shown below (Eqs. –), whose solutions converge onto a limit cycle: 5τdAdt=-1.69N+0.52A2+0.59P2-1.53AP-1.93NP6τdNdt=-0.90+1.33A-0.65A2+0.71P2-0.81AN+1.94AP+1.64NP7τdPdt=+0.82A2-0.74N2-1.35AP-1.43NP and where the notation and variable names are identical to those of Eqs. ()–(). While this model is stable and converges on a limit cycle representing long-term oscillations in the North Atlantic, its behaviour is not realistic when compared to the observed North Atlantic. Upon inspection of the simulated power spectra of the candidate model ensemble, we see that models which contain 20 or fewer terms are unable to produce the broad-band physical power spectra seen in observations. These models have sparse spectra, with delta-like peaks at particular frequency bins, associated with the limit cycle. The limit cycle of Eqs. ()–() is shown in the three-dimensional phase space plots in Fig. a–c. While this captures the interconnected behaviour of the NAO, AMV, and precipitation in this region, it does not capture the irregular nature of the dynamics of the North Atlantic.

Upon further increasing the number of terms, we find that all models with 21 terms or more exhibit long-term stability and have solutions with broad-band power spectra (green region in Fig. ); these models possess chaotic attractors. The final point on the Pareto front (blue circle in Fig. ) identifies a model containing 26 terms, which generates physically consistent broad-band power spectra when compared to observations. For the remainder of this paper, we choose this model as our “best” candidate model for further exploration, and we refer to it as the “low-order model”. The analytical form of the low-order model is shown in Eqs. ()–(): 8τdAdt=-0.29-0.18A-1.75N+0.67A2-0.11N2+0.67P2+0.51AN-1.72AP-2.07NP9τdNdt=-1.05+1.45A+0.57N-0.53P-0.63A2+0.66P2-0.91AN+2.08AP+1.74NP10τdPdt=+0.51-0.68N+0.85A2-0.72N2+0.12P2+0.34AN-1.44AP-1.38NP This set of three ODEs (Eqs. –) exhibit a very broad range of behaviours, including elements of both transient chaos and strong predictability. The model attractor in phase space is shown in Fig. d–f, obtained by simulating a 50 000-year trajectory from a randomly generated initial condition with a time step of 1 month. In panels (d) and (e) we observe two “sides” of the attractor, where in (d) the trajectories diverge on the left-hand side onto one of the two wings of the attractor, similar to the model. In panel (e), we observe that points travelling around the wings are funnelled in towards the centre of the attractor. This phenomenon can also be observed in the top-left-hand side of panel (f).

Upon examining the analytical forms of the 10-term model (Eqs. –), 16-term model (Eqs. –) and the 26-term model (Eqs. –) we see common terms appearing in all three models. For instance, some of the largest polynomial terms governing the dynamics of the AMV (Eqs. , , ) include a negative linear NAO term, and for the NAO (Eqs. , , ), a positive linear AMV term. The coefficients of these terms across the three models vary slightly in magnitude but consistently maintain positive or negative signs. This contrasts with findings in past literature. For example, demonstrated that low-frequency AMV variability is proportional to the low-pass filtered NAO index, where a positive increase in the NAO leads to positive changes in the AMV. This finding has been used by , to construct a delayed oscillator model for the quasi-periodic multidecadal variability of the NAO. However, in our equations governing the AMV (Eqs. , , ), we find that positive linear NAO values cause a decrease in the AMV due to the negative coefficient. This observation aligns more closely with heat-flux forcing of the NAO on SSTs in the subpolar North Atlantic . In this case, positive NAO forcing causes positive zonal wind stress anomalies, which enhances southward Ekman transport in the SPNA, leading to immediate cooling in the ocean surface . It is plausible that the model is extracting decadal SPNA SST signals from the AMV index, since demonstrated that the traditional AMV index is an average of several different processes occurring on different timescales. Upon removing shorter time-scale processes by decadal smoothing of the AMV index, the SST signal is indeed confined to the subpolar North Atlantic (SPNA) . This will be explored further in Sect. .

Other terms with the largest coefficients across the three models (Eqs. –) are cross terms between precipitation and the modes of variability (AP and NP). These cross terms dominate the equations for all three state variables.

For the AMV and precipitation equations, these terms have negative coefficients, while for the NAO equations, they have positive coefficients. The significance of these terms across the three models highlights the important role precipitation plays in the dynamics of large-scale variability. As precipitation is a good proxy for latent heating , these cross terms can capture highly nonlinear diabatic processes that act as a bridge between the ocean and the atmospheric circulation . It is also possible that precipitation influences the AMV directly through freshwater fluxes , which then feeds back onto the atmospheric circulation. A proposed mechanism linking the AMV and NAO with precipitation will be outlined in Sect. .

Further nonlinear quadratic dynamical dependencies of the AMV and NAO were uncovered by when they considered their low-frequency variability (greater than one year). They found a significant negative dependence of the quadratic AMV variability on the NAO, which is consistent with the three low-order dynamical models in Eqs. (), (), (). In , they also found a quadratic self-dependence of the AMV (present in Eqs. , ) and a strong influence of a quadratic NAO on the AMV (only found in the 26-term model in Eq. ).

In Sect. , we explored the structure and terms in three models generated by the SINDy algorithm. However, our approach generates a total of 344 candidate models using the procedures outlined in Sect. . These models exhibit varying degrees of sparsity, from 1-term to the full 30-term models, as shown in Fig. . Examining this ensemble as a whole provides further insights into the robustness of each term in the three models, which, in turn, provides information on the dominant interactions between variables. This analysis helps to distinguish terms related to noise from those representing true dynamics in the more complex low-order models. We now present the complete set of 344 candidate models in Fig. . In panel (a), we show the presence of polynomial terms across the entire set of models. The x axis indicates which term in the equation is being considered for each of the dAMVdt (orange), dNAOdt (green), and dPdt (blue) equations, respectively. The y axis displays the total number of terms in each model, ranging from 1 to 30 (shown in bold), along with the occurrence of that specific model class as a percentage of the total number of models (in normal font). For each equation in each model, a filled rectangle in the figure indicates the presence of that term.

In panel (b), we present the statistics of the coefficients for each polynomial term across the entire set of candidate models. The coloured bar indicates the mean value, error bars show the interquartile range, while the red triangles indicate the minimum and maximum values for each coefficient. Since the models were trained using standardised indices (they are dimensionless), the coefficients do not relate to the units of the variables but instead indicate the importance of each term. Since models have been fitted on four bootstrapped subsets of the data (see Sect. ), and while simultaneously varying the threshold parameter, λ, many models with the same number of terms can exist but differ based on small deviations in their coefficients. In some cases, two models with the same number of terms can differ based on their polynomial structure, such as the three 11-term model classes in panel (a), which also contain multiple models in each class due to differences in coefficients.

As with the 10-term, 16-term, and 26-term models described earlier in Eqs. ()–(), the same dominant terms across the full spectrum of fitted models are evident in Fig. a. As we move down the y axis in (a), we observe decreases in the number of equation terms, but the cross terms AP and NP persist for even the most sparse models, as seen in the final two triplets of columns in Fig. . The fact that these cross terms are robustly detected across many models for all three equations, dAMVdt (orange), dNAOdt (green), and dPdt (blue), suggests that these terms are more likely to be physical. The persistence of cross-terms involving precipitation across all models suggests that precipitation holds important dynamical information for AMV-NAO coupling. Moreover, it may act as a state-dependent coupling parameter, modifying the effective strength and sign of interactions between the AMV and NAO. Additionally, there is a delayed appearance of the dPdt equations, which occurs when models have five terms or more. In the most sparse models, precipitation serves as a diagnostic variable, where its current state provides useful information about the coupled ocean-atmosphere system. However, its own evolution may be too noisy or weak to resolve compared to the leading-order AMV-NAO interactions. Only when the model includes sufficient coupling terms can precipitation become dynamically active, allowing the lower-frequency deterministic component of precipitation to be separated from the unresolved noise. The emergence of the dPdt equations suggests an important role for active precipitation feedbacks on the atmosphere–ocean system, involving latent heating, freshwater fluxes, and atmospheric stability. We note that other robust terms are also present, such as the linear AMV term in the NAO equations and the linear NAO term in the AMV equations. A key takeaway from this section is that the robust terms across the low-order model ensemble highlight precipitation as an important mediator between the ocean and atmosphere in the North Atlantic. This coupling will be explored further in Sect. .

To validate the low-order model of the North Atlantic, we assess its ability to predict the observed AMV, NAO and North Atlantic precipitation time series up to a decade ahead. The model is initialised every six months using observed monthly mean data from ERA5 training data (126 start dates between January 1950–December 2012) and from ERA20C test data (60 start dates between January 1900–December 1939). The model is integrated forward in time from each initial condition for ten years using a fourth-order Runge-Kutta scheme with a time step of one month. We validate the model separately on the training and testing data. We use the approach outlined in to measure the skill of model forecasts at different lead times and with varying averaging windows. The anomaly correlation coefficients (ACC) (Fig. ) are calculated for all possible lead-year ranges (LYR) determined by the start lead-year and the end lead-year, with minimum and maximum lead times of 1 and 10 years, respectively. For example, for the initialisation year January 1950, the lead-year range LYR[2–8] represents the average of the simulated variable (or observed variable) that falls between January 1952 and December 1958. The pixel in Fig.  that corresponds to the lead-year range 2–8 is then the ACC between the simulated and observed variables averaged in this way, where the correlation is computed across all initialisation years. The leading diagonal of each plot in Fig.  shows the skill of 1-year averaged forecasts at different leads. As you move from the leading diagonal to the lower right corner, the averaging length increases, which can enhance predictable signals by removing noise.

The skill of the low-order model in predicting the AMV index is high at a lead time of 1 year for both training and testing data, with the skill slowly decreasing at longer lead times. We observe significant predictive capability of the low-order model in predicting the variability of the AMV out to a decade ahead, with improved skill in the test data (Fig. d). The skill in predicting the future NAO is substantially less than that of the AMV, with significant correlations up to a lead time of 3 years for 3-year averaged predictions in both testing and training datasets. The model is particularly good at predicting North Atlantic precipitation anomalies for both datasets (c, f), with noticeable improvements at greater start lead years compared to the AMV and NAO.

For the training data, the correlations at LYR[1–1] for the AMV, NAO, and precipitation are 0.86, 0.54, 0.11, respectively. For the test data, the correlations are 0.93, 0.52, 0.7, respectively. We compare the skill of the model for LYR[1–1] to a baseline forecast, which we choose here to be persistence. To calculate the persistence correlation, we correlate observations with the initial forecast states for LYR[1–1]. We find no significant differences between model forecast skill and persistence correlation for this lead year range across all variables and datasets. This suggests that, for early lead times, the model does not provide much additional predictive information beyond the memory of the initial state. However, the low-order model exhibits dominant variability on timescales of approximately 20 years and longer (see Fig. d, e, f). As such, the skill diagnosed here, which is limited to lead times of up to 10 years, targets timescales shorter than the intrinsic timescales of the model. Therefore, the finding that forecast skill is comparable to persistence in this range does not necessarily imply the absence of predictive skill at longer lead times and averaging windows. We also note that correctly initialising an ensemble of model forecasts greatly improves the low-order model's predictive performance at longer lead times, while also beating the persistence benchmark (not shown here).

We also calculate the explained variance of the model in LYR[1–1] for both training and test. For the training data, the explained variances at LYR[1–1] for the AMV, NAO and precipitation are 0.65, 0.10, and -6.62 respectively. For the test data, the explained variances are 0.81, -0.48, and 0.40, respectively. We note that AMV variability is well represented in both training and test data in this lead year range. However, the low-order model struggles to capture the variability of the NAO and precipitation. Similarly to the skill score mentioned above, this difference is likely due in part to the enhanced persistence of the AMV initial state. We note that the explained variability of the low-order models could allow for an estimation of the noise that should be added to the models to effectively represent the full variability. In addition, explained variance in a given lead year range could also support the model selection process.

We now have some confidence in the fidelity of the low-order dynamical model based on its ability to predict unseen data. We will now study the model to gain insight into the mechanisms of climate variability in the North Atlantic.

We hypothesise that the 20-year variability in the low-order model is a damped oscillatory mode forced by the atmosphere. The timescale is set by internal ocean dynamics, but its quasi-oscillatory nature results from decadal forcing of the NAO on the ocean state. In , the existence of such a decaying oscillatory eigenmode of North Atlantic SST variability was demonstrated using an inverse linear model, with an oscillatory period of 21.1 years. However, the origin and physical mechanisms behind this mode were not specified. In this study, they also found two additional eigenmodes of variability with periods of 13.4 years and 36.8 years, which have similar timescales to the dominant peaks in the power spectra of the low-order model AMV index (15.5 years and 35 years in Fig. d). Further work in using a LIM, identified an oscillatory mode of the North Atlantic system with a period of 19.72 years.

When regressing observed SST onto the observed DNAO in Fig. , we see clear differences between patterns at zero lag and patterns when the SST lags. We argue that this indicates two different processes: a direct, fast response to NAO-related wind stress and heat flux forcing, followed by a slow response due to ocean adjustment of the gyre and overturning circulation. The instantaneous effect of the NAO in Fig.  is to cool the SPNA through anomalous wind stress, setting up a tripole pattern in SST. As we increase the lead-year of the NAO to SST, warm anomalies begin to accumulate in the SPNA and Labrador Sea regions around 6–8 years later in observations. This warming counteracts the initial cooling response of the positive NAO anomaly, thereby acting as a damping effect (negative feedback, ). Persistent NAO forcing on decadal timescales can give rise to an “oscillation” in sea surface temperature in the absence of other forcing. It is this slow process that has potential predictability due to the existence of an oscillation and its slow evolution. We have demonstrated that this type of quasi-decadal variability in the North Atlantic is predictable and is well captured by the low-order model (Fig. ). The high skill of the low-order model in predicting the AMV in Fig. a and d directly reflects these slow, predictable ocean responses to the NAO. More importantly, the reason for the high predictive skill of North Atlantic precipitation by the low-order model (Fig. c) may also be attributed to these processes, as the quasi-20-year variability captured by the DP index varies in phase with the ADV index. Anomalous atmospheric convection and precipitation are direct responses to a warmer ocean. The broad 20-year peak in the observed precipitation index is similar to that of the observed AMV index in Fig. , reflecting a damped oscillation as the ocean integrates the atmospheric forcing.

Some studies attribute decadal ocean density changes associated with a ∼20-year mode to be driven mostly by temperature rather than salinity . However, in Sect. , we found that precipitation played a key role in governing the evolution of the low-order model's AMV, NAO, and precipitation indices (see Fig. ). Here, it is plausible that the role of precipitation in the model is partly realised through freshwater fluxes (therefore modifying ocean salinity), as well as acting as a proxy for latent heating in the atmosphere. The influence of precipitation on the other variables in the low-order models arose through cross-terms of precipitation and a mode of variability. For the equations that determine the evolution of the AMV index, we found a robust dependence on the -NP terms: when the AMV is in a positive phase, and both N>0 and P>0, there is a weakening of a positive AMV towards a more neutral phase. In the case where the AMV is negative and both N>0 and P>0, the AMV tends towards a stronger negative phase. When P>0, positive rainfall anomalies emerge around the Labrador Sea (Fig. i) increasing flux of freshwater into the ocean. An increase in freshwater flux could lower surface salinity and increase stratification , leading to a reduction in deep ocean convection and a weakening of the AMOC and the AMV . These effects on the AMV are reinforced during a positive NAO state (N>0), as a positive NAO causes an immediate weakening of the subpolar overturning circulation due to anomalous Ekman transport. This initial weakening lasts for several months following the initial positive NAO event .

For the equations that determine the evolution of the NAO index, one of the most robust terms is +NP. From this term, we find that during a negative NAO (N<0) and during positive precipitation periods (P>0), the negative NAO is reinforced by an increase in precipitation in the southern node of the NAO (see Fig. i) agreeing with findings from . On seasonal timescales, demonstrated that convective precipitation in the southern NAO node is a significant mediator of the late Autumn North Atlantic SST forcing on the NAO. Diabatic heating in this region, due to enhanced convection, acts as a positive feedback on a negative NAO by lowering SLP . We also find, using the same reasoning, that a decrease in precipitation (P<0) in this region weakens a negative NAO state. We expect the response of the NAO to northern North Atlantic SST to be of opposite sign to the anomalies that induced it, in line with , , , . Here, the tripole SST pattern associated with a positive NAO (similar to a negative ADV SST pattern when A<0) in Fig. b would act as a positive feedback onto the positive NAO by enhancing the meridional temperature gradient which strengthens zonal winds and encourages a positive NAO event. If this feedback is indeed present in the low-order model, then it would have to be conditioned on the sign of North Atlantic precipitation through the +AP terms in the dNdt equations. In this case a cool SPNA and a warm subtropical gyre (associated with A<0) would reinforce the positive NAO event which induced it if the AP term is positive; which is the case when P<0. Negative precipitation anomalies in the southern node of the NAO (when P<0) are often associated with a positive NAO (Fig. h) and a negative ADV. Positive precipitation anomalies would favour a negative NAO and weaken a positive NAO . Therefore, we conclude that the positive feedback pathway of North Atlantic SST onto the NAO is strongest when the North Atlantic precipitation pattern favours a positive NAO state. This highlights the importance of diabatic processes in controlling the response of the NAO to North Atlantic SST. The exact generation of the 20-year NAO variability is partly hinted at by the low-order model, with local North Atlantic SST being a strong candidate , as well as latent heating feedback mechanisms. However, further work will be required to understand these feedbacks, and the potential role of external forcing in generating North Atlantic decadal variability. Such influences on the North Atlantic atmosphere and ocean we do not consider in this study include volcanic eruptions and lunar cycles . We note that the SINDy-derived equations in this work encode the dynamical couplings between AMV, NAO and North Atlantic precipitation, but they do not in themselves distinguish cause and effect. The mechanistic pathways we propose are supported by consistency with existing theory, but not uniquely proven by the low-order models alone. Establishing causal links will require further work.

The advance presented here lies in identifying low-order deterministic models whose nonlinear structure can reproduce key aspects of decadal variability without the need for explicitly prescribed stochastic forcing. While residual dynamics are not explicitly modelled as stochastic noise in our framework, we acknowledge that such processes are present in the real system. Some of these unresolved processes may be implicitly captured through the robust nonlinear terms identified by the low-order models. As mentioned in the discussion above, the observed coupled 20-year mode of variability is partly excited by stochastic NAO forcing and maintained by slow ocean dynamics. The “stochastic driving” of this mode of variability in the low-order models is internally generated through deterministic nonlinear interactions. A comparison of our low-order models to a Linear Inverse Model (LIM) would provide insight into whether the identified nonlinearities offer a better representation of the dynamics, allowing for improved predictive capability. We leave this as an important direction for future work. Furthermore, having these low-order dynamical models provides a tool to study the sources of decadal variability and predictability. For example, various experiments could be performed, such as suppressing terms to probe one-way couplings between equations and further parameterising the model dynamics with stochastic forcing. Further research is currently underway to study the state-dependent predictability of the low-order models and to diagnose the models' underlying physical processes by comparing them to coupled climate model simulations.

We make a final remark on the use of precipitation to represent ocean–atmosphere coupling, where we have used it as an integrated proxy for both atmospheric latent heating and freshwater fluxes. Moreover, precipitation is a useful variable from a climate impacts perspective, particularly in relation to European precipitation variability associated with the NAO and AMV. To further develop this work, one could also incorporate both sensible and latent heat fluxes, as well as freshwater fluxes, which would allow for a clearer discrimination of the role of these different processes. The SINDy algorithm can easily accommodate additional variables and higher-order nonlinearities. However, increasing the dimensionality of the models in our case introduces problems, namely the risk of overfitting on such limited observational data. Accurately disambiguating the processes linking many co-varying parameters and identifying both stable and robust models would be more challenging when working with short datasets.