To test for mediation or suppression, three regression equations are defined (ignoring intercepts and residuals for simplicity). Prior to estimating the coefficients, X, Y, and Z were standardised. The first equation describes the total effect τ of the predictor X on the predictand Y: 1Y=τX.

Here, τ represents the correlation between the standardised November SST index and the standardised winter NAO index. The second regression describes X→Z→Y by accounting for the standardised mediator variables. The effect of X on Y changes to τ′, known as the direct effect (not through the mediator), and the effect of Z on Y when accounting for X is denoted as β: 2Y=τ′X+βZ.

The total effect of X on the mediator Z is labelled here as α in the second equation: 3Z=αX.

An important thing to note is that α encapsulates not just the direct forcing X→Z, but also all the indirect forcing through intermediate variables, crucially including via the pathway X→Y→Z.

A central concept is the product αβ, known as the indirect or mediated effect (of X on Y through Z). The total effect is the sum of the direct and mediated effects: τ=τ′+αβ. This also follows from Eqs. (1)–(3). Scaling the mediated effect by the total effect yields: 4αβτ=1-τ′τ.

According to the standard criteria for mediation laid out by , τ, α, and β must all be significantly different from zero. If τ′=0 (or is not significantly different from zero), it follows from Eq. (4) that the total and mediated effects are identical. In this case, the pathway X→Y is fully accounted for by Z, indicating that X→Z→Y represents a valid causal pathway – though not necessarily the only one.

Mediation often represents forward-directed pathways where X changes Z, and Z subsequently affects Y. In climate dynamics, however, feedback mechanisms are common, and ambiguities may arise because Y and Z are evaluated contemporaneously. This implies that both the hypothesised X→Z→Y and the alternative X→Y→Z pathways may be active. It is nevertheless possible to assess the degree to which Z responds directly to X rather than indirectly through Y by regressing out the concurrent variability of Y: 5Z=α′X+γY.

Here, α′ represents the NAO-independent SST-to-mediator influence, to be compared with α from Eq. (), which includes all routes from X to Z (including those via Y). The product α′β is then interpreted as the SST-forced component of the mediated effect; that is, the influence that would arise if the mediator responded only to direct SST forcing, while the NAO retained its full sensitivity to the mediator through β. This is a complementary diagnostic to the full mediated effect αβ, not a replacement; it helps distinguish SST-forced mediation from mediation that is predominantly atmospheric in origin.

To quantify the SST forcing onto the mediator, the sign consistency between α′ and α is assessed. In regions where the SST forcing aligns with the full forcing (which includes NAO feedbacks on Z), the ratio α′/α should be positive. Values near zero indicate that the SST forcing is weak. Because the mediators (Z) are inherently noisy, Ordinary Least Squares (OLS) estimates of α and α′ are both subject to attenuation bias , which biases coefficients towards zero and increases the chance of sign flips across different sample sets.

Accordingly, I use a conservative hypothesis test: the null hypothesis is that the SST-forced component (α′) is zero or has the opposite sign to the total forcing onto the mediator (α). Rejecting this null hypothesis indicates that the SST forcing is sufficiently robust to maintain a consistent physical direction despite attenuation. I emphasise that it does not imply that the SST forcing dominates over NAO → mediator feedbacks.

Accounting for the NAO's autocorrelation is another prudent step to prevent potential confounding of the results. showed that in ERA5, this autocorrelation was only significant from November to December and not from November to DJF; this was confirmed to be valid for the shorter period examined here for both ERA5 and SEAS5. Labelling the NAO index in November as Y0, a new regression equation could be defined as: Z=α′′X+γ0Y0+γ′Y.

However, as the coefficient γ0 was found to be negligible for both mediators, which was expected in light of the missing NAO autocorrelation, Eq. () is used in the analysis.

An interesting special case occurs when τ′/τ>1, which means that the mediated effect αβ has the opposite sign to the total effect τ (Eq. ). In these cases, Z is referred to as a suppressor because the regression coefficient linking X and Y is inflated when Z is accounted for . In the context of this study, this could mean that X (November SST anomalies) drives changes in Z (e.g. flux anomalies), but the response of Y (the NAO) to those fluxes is of opposite sign to the direct X→Y pathway. This can occur because Y itself feeds back onto Z, helping to make β negative. In other words, Z acts as a negative feedback, transmitting a damping influence on Y that partly cancels (suppresses) the predictive signal from X. In the raw correlation, this feedback reduces the apparent strength of X as a predictor of Y, but once Z is controlled for, the hidden strength of the X→Y link is revealed. Put differently, had it not been for the negative feedback through Z, X would have exerted stronger predictive power on Y.

The mediation framework is applied to X, Z and Y as defined over the North Atlantic sector. Controls for remote precursors such as ENSO or stratospheric anomalies are not included. Consequently, any shared influence of such processes on both pre-winter SST and the winter NAO can appear implicitly in the estimated relationships; the results should therefore be interpreted as structural diagnostics rather than formal causal attribution across the full suite of teleconnections.

Throughout the paper, sample coefficients in Eqs. (1)–(5) (i.e. coefficients estimated through OLS fitting) are denoted by carets; for instance, τ^ is the estimated τ value.

Reanalysis and seasonal forecast data are used. The reanalysis reference is ERA5 , produced by the European Centre for Medium-range Weather Prediction (ECMWF), and the forecast system is SEAS5, the ECMWF's seasonal prediction system . The reason only one model is investigated here is that its reforecast period extends back to 1981, while reforecasts are only available from 1993 and onwards for comparable systems – this shorter period would render the mediation analysis less robust. The analysis covers the winters from 1981/82 to 2023/24 (hereafter referred to as 1981–2023).

The atmospheric component of SEAS5 is the Integrated Forecast System (IFS) atmosphere model. The grid spacing for the ocean model in SEAS5 is 0.25 degrees, which has been shown to yield a decent representation of air–sea interaction along the Gulf Stream front compared to lower-resolution models . It seems the resolution will not change in the new SEAS6 system due to be released soon, but the new ocean model nevertheless appears to yield multiple improvements, including large reductions in SST errors along the Gulf Stream their Fig. 2a.

A well-documented feature of SEAS5 relevant for this study is a warm SST bias in the western North Atlantic up to the mid-1990s . Inherited from issues with the ocean reanalysis, this bias allowed SST errors to grow rapidly and produced a local warm anomaly that affected near-surface temperature and surface heat fluxes. To verify that it did not affect the conclusions of this paper, the core analysis was repeated using only the period 2001–2023; the results were practically unchanged.

SEAS5 reforecasts were used from 1981 to 2016 with 25 ensemble members, and real-time forecasts from 2017 to 2023 with only the first 25 of 51 members used to ensure consistency with the reforecasts. The analysis was based on individual ensemble members (i.e. not ensemble means) unless otherwise specified.

SEAS5 forecasts are issued once per month. In this study, the November forecasts and reforecasts are used, corresponding by convention to lead times of 1–4 months for November through February. A potential drawback of using the November initialisations is that the SST fields are similar across ensemble members due to oceanic inertia. However, repeating the complete analysis with October initialisations (for which the November SST fields are more diverse) produced qualitatively similar results. I chose to base the analysis on the November runs, as this allows evaluation of the model's skill for the set of forecasts used operationally for predicting winter conditions.

The variables considered are SST, mean sea level pressure (SLP), and the sum of sensible and latent heat flux, hereafter referred to as surface heat flux, or SHF (positive upwards). Baroclinicity is quantified by the Eady growth rate maximum e.g., defined for the 700–850 hPa layer as σE=cf∂v∂z/N, where the unit is day⁻¹, c=86 400×0.3098, f is the Coriolis parameter, v is the wind vector, z is the geopotential height, and N is the Brunt-Väisälä frequency, given by N=gθ∂θ∂z, with θ the potential temperature and g the gravitational acceleration.

Anomalies were calculated by subtracting the overall mean and dividing by the overall standard deviation, spanning all years and ensemble members.

Two scalar indices are central to the analysis: the DJF NAO index, and an SST-based index representing the November SST anomaly pattern in the extratropical North Atlantic most strongly correlated with the following winter's NAO index.

To construct the NAO index, the first Empirical Orthogonal Function (EOF) of interannual DJF mean ERA5 SLP anomalies was computed over the domain 20–80° N, 90° W–40° E, using the eofs Python package and applying cos⁡ϕ latitude weighting. For both ERA5 and SEAS5, the corresponding NAO index time series were obtained by projecting their respective gridded SLP anomalies onto the ERA5-based spatial EOF pattern. It was a deliberate choice to use the ERA5 loading pattern for both datasets, as the purpose of this study is to assess how SEAS5 represents the real-world NAO pattern.

The SST index was calculated in a similar way. November SST anomalies from ERA5 were first regressed onto the interannual ERA5 NAO index to obtain a spatial regression pattern. SST anomalies were then projected onto this pattern within a reference domain extending from the 20 to 70° N and from 100° W to 20° E , after which the resulting series was standardised to form the SST index. As with the NAO index, the SST index for SEAS5 was computed by projection onto the ERA5-based pattern, not a model-specific optimal pattern, to maintain consistency across the datasets.

No masking for sea ice was applied. In both ERA5 and SEAS5, grid cells covered by sea ice are not missing values but contain subzero SSTs, which remain valid anomalies in this framework. Masking would risk introducing artificial discontinuities in space and time, since the ice edge varies between months and years.

Bootstrapping was used to estimate statistical significance by creating 10 000 randomised series through sampling with replacement. To ensure comparability between the two datasets, the bootstrap sample length was set equal to the number of years in the study period for both datasets (i.e. 43). This avoids giving SEAS5 an artificial advantage with respect to ERA5 due to its larger sample size (25 members per year). When assessing the significance of a metric (e.g. a correlation) at a significance level of 5 % (used throughout this study), the 2.5th and 97.5th percentiles of the correlation coefficient across those 10 000 randomised series were computed, and if the interval between these percentiles did not include zero, the correlation was deemed significant.

The ERA5 SST anomaly regression pattern (i.e. November SST anomalies regressed onto the DJF NAO index) is shown with shading in Fig. a. As expected, it is similar to the pattern in Fig. 1f in , which was also computed based on ERA5 but for a longer period (1940–2022). The contours in Fig. a display the regression of DJF SLP anomalies onto the NAO index.

Figure b shows the interannual November SST index, obtained by projecting the SST anomalies onto the regression pattern in Fig. a, together with the winter NAO index, both from ERA5 data. Although only a few of the local SST coefficients in Fig. a are significant, the sample correlation between the two indices is relatively high (τ^=0.49, p≈0.001), underscoring the strong link between late-autumn SSTs and the subsequent winter NAO. The SST index captures well the two exceptionally negative NAO winters of 2009/10 and 2010/11, as well as the extended positive NAO phase around 1990, though there are also seasons with weak correspondence, such as 2000/01. It is emphasised that τ^ does not represent a skill score, as no independent training and evaluation periods were defined.

Turning to the (ensemble mean) SEAS5 indices shown in Fig. c, it is evident that the SST index covaries with the SST index in ERA5 (r=0.91). However, several differences between the datasets are also apparent. Most important, the SST–NAO correlation is substantially lower (τ‾=0.21, p=0.18, with the overbar signifying that the ensemble mean was used) than in ERA5, demonstrating a discrepancy in the linkages between the observed “NAO-optimal” SST pattern and the winter NAO. Here it is acknowledged that the SST–NAO correlation is weaker than in ERA5 at least partly because both indices were deliberately derived from ERA5-based spatial patterns.

A second point is the non-significant NAO skill: the anomaly correlation coefficient between the NAO index in SEAS5 and ERA5, denoted henceforth as ρ, is not significant for the ensemble mean (ρ‾=0.29, p=0.06). This low predictive power is consistent with .

When the SST–NAO correlation and the NAO skill are evaluated for all the ensemble members instead of for the ensemble mean, both metrics deteriorate, revealing higher internal noise. This behaviour is consistent with the “signal-to-noise paradox” e.g.. The sample parameter based on all members, which is used in the remainder of the analysis, is τ^=0.06, which is not significantly positive and much lower than τ‾=0.21. The member-level NAO skill also decreases from ρ‾=0.29 to the non-siginificant value of ρ^=0.07.

One of the three main purposes of this paper is to assess whether the total SST–NAO relationship (τ in Eq. 1) in the model is proportional to its NAO skill. That skill is defined here as ρ: the correlation between the DJF NAO index in SEAS5 and the corresponding DJF NAO index in ERA5, with ERA5 years selected to match the year of each SEAS5 member. If there had been no relationship between τ and ρ, it would be of limited interest to scrutinise the causal pathways through which the SSTs influence the NAO.

It is important to emphasise that ρ is an external quantity: it depends solely on the correlation between the SEAS5 and ERA5 DJF NAO indices and contains no information about the model’s internal relationships among X, Z, and Y. By contrast, the total effect τ and the mediated effect αβ derive entirely from SEAS5’s internal covariance structure. There is therefore no algebraic or definitional link between skill and any aspect of the model’s SST–NAO relationship. Any association between the two reflects actual co-variation between an external validation measure and internal model dynamics, rather than an outcome expected by construction.

To investigate the association between τ and ρ, bootstrapping was used to generate an ensemble of 10 000 SEAS5 series, each with the same length as the number of years in the study period (43). The results are not sensitive to this choice of length, and the same bootstrap ensemble is analysed further in Sect. .

For each bootstrap series, two quantities were computed: (1) τ^, the sample SEAS5-internal correlation between the November SST index and the DJF NAO index; and (2) ρ^, the sample NAO skill. Figure  shows a scatterplot of these parameters for a subset of the series. The correlation across all the 10 000 series is positive (r=0.33) and significant at the 5 % level. This does not imply that a strong SST–NAO relationship is sufficient or strictly required for high NAO skill, since the NAO is influenced by many processes unrelated to North Atlantic SST variability. Rather, the result provides support for a central premise of this study: the extent to which the model reproduces the observed influence of November SST anomalies on the winter NAO contributes meaningfully to its overall NAO skill.

Before examining the role of SHF and baroclinicity in mediating the SST–NAO relationship, it is useful to consider the climatological context. In Fig. , ERA5 climatologies and SEAS5 biases are therefore shown, starting with the mean November SSTs in the North Atlantic in Fig. a. A prominent feature is the strong SST gradient along the boundary between the warm Gulf Stream waters and the much colder waters along the North American coastline. These gradients give rise to intense SHF on the warm side of the front (Fig. b), and they also coincide with strong low-level baroclinicity (Fig. c). The last panel in the top row, Fig. d, shows the climatological SLP pattern, which is characterised by a dipole between the Icelandic Low and the Azores High.

The aforementioned warm SST bias in the western North Atlantic is visible as a tongue-like feature in the east–west direction south of Greenland in Fig. e. This is also linked to a clearly defined positive DJF SHF bias in Fig. f. The poor SEAS5 representation of the SST gradient along the Gulf Stream seen in Fig. a is also of interest. Figure e reveals a pronounced warm bias on the cold side of the front and a weaker cold bias on the warm side, resulting in an overall weakened gradient. The SHF biases in Fig. f reflect these SST errors, generally showing fluxes that are too strong in warm-biased regions and too weak in cold-biased sectors. Although not shown here, these SHF biases project strongly onto the DJF SST bias in the western part of the basin; these are larger in magnitude than the ones for November in Fig. e, but for the most part they have the same sign, suggesting a growth of the model's SST bias with lead time. Figure g indicates that the underestimated SST gradients along the Gulf Stream are associated with too weak baroclinicity in the storm track entrance region, while σE is too high to the south.

These errors imply a distorted storm track: too few cyclones over the northern North Atlantic and too many further south, which is consistent with the IFS cyclone bias investigations by and . This interpretation is supported by the mean SLP bias pattern in Fig. h, which shows that SEAS5 underestimates the amplitude of the observed NAO-like dipole. Sampled at representative grid points near the two NAO centres of action (Stykkishólmur, Iceland, and Ponta Delgada, Azores), the mean SLP bias amounts to +1.0 and -0.4 hPa, respectively, giving a bias in the north–south difference of +1.4 hPa. This confirms that the model’s climatological pressure contrast is weaker than observed, implying westerlies that are too weak across the subpolar North Atlantic. The most distinct weakening of the westerlies occurs between the mid-basin negative SLP bias and the positive bias near Iceland. This corresponds to the weak negative SHF bias observed in the same region (Fig. f). In this area with suppressed westerlies, surface fluxes are likely underestimated because reduced wind speeds dampen the intensity of the cold-air advection from the west.

Figures  and show the sample parameters α^ and β^, as well as their product α^β^, for both mediators. The unit is standard devations (SD), as all the variables in the regression equations were standardised prior to estimating the coefficients. It is repeated for emphasis that α, the regression coefficient linking November SSTs to the mediator (X→Z) in Eq. (), captures all routes through which SST anomalies influence Z. This includes the indirect effect via the NAO (i.e. the pathway X→Y→Z), as well as other pathways not explicitly considered here. The NAO-independent contribution of SSTs to Z, denoted α′ in Eq. (), is examined in Sect. .

The findings in the previous section raised questions about the directionality of the mediated effects associated with both SHF and baroclinicity. Although the pathway X→Z→Y is not meant to be interpreted as strictly unidirectional, evaluating the X→Z link in isolation helps determine the extent to which November SST anomalies generate responses independently of the NAO. To that end, the leftmost panels in Fig.  show the sample parameter α′^ from Eq. (), which isolates the X→Z influence with the NAO regressed out, for SST and SLP. These variables, which are not considered as mediators, are analysed here because they indicate changes in the lower boundary (SST) and circulation (SLP).

Starting with ERA5, Fig. a shows that, once the NAO contribution is removed, November SST anomalies induce an SLP pattern dominated by positive coefficients over the south-western North Atlantic. This pattern implies anomalous northerly advection in positive phases and southerly advection in negative phases. The associated SST response resembles the November antecedent in Fig. a (as expected from oceanic inertia), but there is more pronounced mid-basin dominance with positive values.

In SEAS5, the α′^ field in Fig. d shows an SST structure broadly similar to the ERA5 pattern in panel (a), with one notable exception: significant negative values appear south of Greenland. A similar sign discrepancy was already seen for the SHF α^ coefficient (Fig. d), indicating that the SST–flux response in this region is systematically misrepresented in SEAS5. These negative α′^ values lie near the well-documented positive SST bias during the early reforecast period , and when the analysis is repeated for 2001–2023, when this bias was much smaller, the negative values largely disappear. This suggests that the sign error may be linked to compensating adjustments associated with the bias, although the spatial correspondence is not exact and the mechanism cannot be established here. However, this issue is not central to the present study – while the bias alters some spatial details in SEAS5, the mediated effect in SEAS5 is not significant in this region in either period, and the skill–mediation covariability discussed in Sect.  is unaffected.

The SST-forced mediated effect α′^β^ via surface heat fluxes in ERA5 is shown in Fig. b. In this panel the dots indicate areas where the null hypothesis – that α′ and α have opposite signs – can be rejected at the 5 % level across the 10 000 bootstrap samples. In these areas the SST-forced component contributes in the same direction as the full mediated effect, which includes contemporaneous feedbacks from the NAO onto the fluxes. Over parts of the Subpolar Gyre, where β is positive (Fig. b), α′^β^ is partly positive or near-neutral, but few areas are marked with dots. This indicates that the strong total mediation seen in Fig. c is largely attributable to the Y→Z pathway; that is, NAO feedbacks on the fluxes dominate in this region. Further south in the North Atlantic, α′^β^ is negative over a broad area. The density of dots there indicates that the NAO-independent SST-forced component contributes to suppressing the SST–NAO correlation. A similar but less extensive pattern appears in the full mediated effect α^β^ in Fig. c.

In SEAS5 (Fig. e), the spatial structure of α′^β^ resembles the pattern of the full mediated effect α^β^ in Fig. c. Some areas are marked with dots, including the mid-basin region exhibiting suppression. This implies that the SST-forced component plays a role in this suppression, matching the ERA5 result in Fig. b.

For baroclinicity, Fig. c shows that the SST-forced mediated effect is mainly positive in the two bands where the total mediated effect α^β^ is positive and significant (Fig. c), albeit noticeably weaker in magnitude. Parts of these bands are marked with dots, indicating where the SST-forced component plays a limited role in the full mediation. Figure f demonstrates that SEAS5 similarly produces weak, positive α′^β^ in the two bands, but no dots appear where the mediated effect is strongest in magnitude. This suggests that the SST-driven component of the mediated effect in the crucial areas is negligible.

In summary, the directional picture is heterogeneous but broadly consistent with a dominant NAO → mediator pathway. For SHF, the SST-driven contribution mainly projects onto the mid-basin area where suppression dominates, while over parts of the Subpolar Gyre the strong total mediation appears to be largely attributable to NAO feedbacks onto the fluxes. For baroclinicity, ERA5 indicates a modest SST-forced contribution aligned with the total response in two bands – albeit noticeably weaker than the full mediation – whereas SEAS5 shows no such contribution. These results motivate the next step: to assess whether variations in these internally generated mediation patterns are associated with variations in external NAO forecast skill.

In Sect. , a modest but significant association was identified between the November-to-DJF SST–NAO correlation and the model's NAO skill ρ^ (r=0.33). This suggests that the mediated effect associated with the SST–NAO linkage may also relate to forecast skill. Although the mediation signal in SEAS5 is weak overall, it is not absent: the positive α^β^ values for baroclinicity (Fig. f) broadly overlap with those in ERA5 (Fig. c). For SHF (Fig. c, f), there is likewise some agreement, apart from the negative values south of Greenland noted earlier.

This section examines whether variations in mediation strength across subsets of SEAS5 realisations are associated with variations in NAO skill. To do so, the 10 000-member bootstrap ensemble introduced in Sect.  is revisited. For each bootstrap sample, the SEAS5 mediated effect α^β^ is estimated separately for SHF and σE, alongside the NAO skill ρ^ and the model-internal SST–NAO correlation from Sect. .

The maps in Fig.  show where, geographically, the mediated effect α^β^ co-varies with the NAO skill ρ^ across the SEAS5 bootstrap samples. The most prominent feature is that the strongest positive correlations occur in the regions where ERA5 exhibits robust positive mediation. For SHF (Fig. a), this positive covariability appears across the Subpolar Gyre, even south of Greenland, where the overall α^β^ in SEAS5 is negative (Fig. f). This indicates that, within SEAS5, bootstrap subsets in which the model produces a mediation pattern more closely resembling ERA5 are also the subsets with higher NAO skill. Conversely, samples that yield negative α^β^ in these regions tend to have lower skill. Thus, even though SEAS5 does not reproduce the magnitude or sign of the mediated effect perfectly, its internal covariability shows that more realistic mediation pathways are associated with improved NAO prediction skill.

A few regions also display negative correlations between ρ^ and α^β^, but these do not overlap with the key regions where ERA5 exhibits strong SHF-mediated effects in the Subpolar Gyre. For baroclinicity, the correspondence between ρ^ and α^β^ is more uniformly related to the mediated effect in ERA5 (Fig. b).

Taken together, these patterns reinforce the main conclusion: the clearest and most physically interpretable skill-mediation covariability occurs in the regions where ERA5 displays robust positive mediation. In these areas, SEAS5 achieves higher NAO skill when it incidentally reproduces the observed mediation pathways, underscoring the importance of representing these air–sea feedbacks realistically in seasonal prediction systems.

As noted in Sect. , the skill and mediation metrics derive from entirely different sources of information. The fact that they co-vary in physically meaningful regions therefore supports the view that the ERA5-identified pathways correspond to mechanisms that matter for NAO predictability in the model.