The term attractor dimension usually refers to a global measure . D indicates the average number of degrees of freedom of a dynamical system. Several methods to measure D were developed in the 1980s . These techniques have a certain number of adjustable parameters and require the system to be embedded in a subspace of the phase space. They provide good estimates of D only when the trajectories are sufficiently long to estimate the embedding parameters. Such computations are therefore problematic in systems with large numbers of degrees of freedom and give biased results when applied to atmospheric flows .

The technique we exploit here results from the application of extreme value theory to Poincaré recurrences in dynamical systems . In this approach, the returns to points on chaotic attractors are fully characterized by extreme value laws. In practice, one needs long trajectories x(t) that approximate sequences of states on the attractor. One then fixes a point ζ on the trajectory and computes the probability P that x(t) will return in a ball of radius ϵ centered on point ζ. The theorem, modified in , states that logarithmic returns g(x(t))=-log⁡(dist(x(t),ζ)) are distributed as P(g(x(t))>s(q),ζ)≃exp⁡-x(t)-μ(ζ)σ(ζ). Here s is a high threshold associated with a quantile q of the series g(x(t)) itself, linked to the radius ϵ via s=g-1(ϵ). In other words, requiring that the orbit falls within a ball of radius ϵ around the point ζ is equivalent to asking that the series g(x(t)) is over the threshold s. Here we adopt q=0.98 to determine s. The resulting distribution is the exponential member of the generalized Pareto distribution family. The parameters μ and σ depend on the point ζ chosen on the attractor. σ(ζ) then provides the local dimensions d(ζ) via the simple relation σ=1/d(ζ). This result is very powerful because it provides a new way to compute local dimensions on the attractor and to recover D as the average of d on all the ζs without the need for embedding, as is required in most dimension computation algorithms . We want to stress that this procedure is not just a statistical fitting. The reason why it provides good estimates of d that were impossible to obtain with previous techniques derives from the universality of the extreme value statistics for Poincaré recurrences: one knows a priori the statistics of such recurrences and can then check that they are achieved for the numerical trajectory examined.

The stability of the state ζ is measured by θ(ζ), namely the inverse of the average residence time of trajectories around ζ. For discrete maps θ is uniquely defined (see , for details): if ζ is a fixed point of the dynamics, θ(ζ)=0. For a point that leaves the neighbourhood of ζ immediately, θ=1. For continuous flows, the definition of θ depends on the Poincaré map chosen and precisely on the Δt chosen to discretize the flow. Since θ is the inverse of the average residence time, it is measured in units of 1/Δt. In general, the higher the persistence of the point ζ, the longer the previous and subsequent states of the system will resemble ζ. The residence time can be computed by introducing a further parameter in the previous law. This parameter, known as the extremal index, is such that P(g(x(t))>s(q))≃exp⁡-θx-μ(ζ)σ(ζ). To estimate θ, we adopt the Süveges estimator . For a fixed quantile q, the estimator is defined as θ=∑iNc(1-q)Si+N+Nc-∑iNc(1-q)Si+N+Nc2-8Nc∑iNc(1-q)Si2∑iNc(1-q)Si, where N is the number of recurrences above the chosen quantile, Nc the number of observations which form a cluster of at least two consecutive recurrences, and Si is the length of each cluster i. This length is the number of consecutive time steps during which the trajectory remains within a radius ϵ of ζ. For further details on the derivation of this estimator, the reader is referred to .

We use daily fields from the NCEP/NCAR reanalysis , with a horizontal resolution of 2∘. The analysis is carried out over the whole Northern Hemisphere for all days of the year over the period 1948–2013. The observables of interest are sea-level pressure, 2 m temperature, and precipitation frequency. A previous study has shown that the results obtained are largely independent of the dataset used and of its spatial resolution.

Anomalies are defined as deviations from the long-term daily mean. So, for example, the anomaly of t2m at a given location on 5 December 2000 is computed relative to the mean value of all 5 Decembers in the dataset at that location.

The relevance of the composite anomaly maps for the different variables is evaluated using a sign test. This identifies geographical areas where at least 2/3 of the composite maps have the same-sign anomaly. Assuming a binomial process with the same number of draws as the composite maps and equal chances of positive or negative outcomes (binomial distribution with success rate 0.5), a 2/3 threshold is beyond the 99.99th percentile of the distribution. In Figs. , , and , such regions are marked by the thick black lines.

The local dimension, d, of the slp field shows a marked variability throughout the analysis period, with values ranging from as low as 8.6 to as high as 33.2 (Fig. a). The average dimension D, which in this case is roughly 19.4, therefore provides incomplete information concerning the field of interest, since the number of locally active degrees of freedom (identified by d) can vary by a factor of almost 4. The autocorrelation function (ACF) of d (Fig. c) highlights a robust variability pattern which is not immediately evident from the raw time series. There is a clear semi-annual cycle, with peak autocorrelation values in excess of 0.21. Over a full year there are therefore two positive and two negative peaks in autocorrelation, with the second positive peak typically displaying a larger magnitude than the first. This is consistent with previous analyses which have identified a strong seasonal dependence in d . The presence of a semi-annual cycle leads us to interpret the ACF as being modulated by the four seasons, with the first positive peak corresponding to cross-season correlation and the second, larger, peak corresponding to correlation between the same seasons in successive years. This periodicity could be linked to semi-annual slp variability features at the mid-latitudes .

The inverse persistence, θ, shows a marked variability, with values ranging from 0.28 to 0.65 (i.e. 1.6 to 3.6 days in terms of 1/θ) (Fig. b). We note that these values should not be compared directly to the persistence of the traditional weather regimes defined using clustering algorithms as (i) here we consider a full hemispheric domain, while weather regimes are typically computed for specific regions, and (ii) the requirement that the flow does not leave the neighbourhood of the state ζ is a more restrictive condition than continued permanence within a given cluster. Indeed, if one considers the typical partition of the atmospheric patterns over the North Atlantic into four weather regimes, the probability of being in one of them is of order 0.25, whereas the probability of being close to ζ is set by the threshold s – in our case 0.02 (see Sect. 2.1). Concerning the ACF (Fig. d), θ shows a very different pattern to d. The year-to-year correlation between the same seasons is still large and positive, but the semi-annual oscillation seen in d is almost entirely absent. Indeed, the winter values appear to be anticorrelated with those of the other three seasons.

An analysis of monthly-mean values confirms the strong seasonal control on the dynamical characteristics of the field. In the summer months, both the magnitude (Fig. a, b) and variability of the two metrics reaches a minimum (Fig. c, d). With the autumn season, the local dimension and θ increase rapidly while the variability remains low. As winter progresses the variability increases, θ remains roughly constant while the local dimension shows a marked decrease, albeit remaining well above the summertime values. In spring d grows back to values similar to those seen in autumn, while its variability peaks and θ starts decreasing. This picture is consistent with the ACFs described above. The annual cycle of θ can be explained as follows: stability peaks in summer when the mid-latitude storm tracks and wave activity are comparatively weak, decreases in autumn and winter and starts increasing again during spring. For the annual cycle of d, the maxima occurring in the intermediate season can be explained as follows: assuming that there is a winter and a summer attractor, the transitional seasons are more unstable because the atmospheric flow can explore both the summer and winter configurations. In dynamical systems terminology, the spring/autumn atmospheric flow sits on a saddle-like point of the dynamics.

One can further look at the slp anomalies corresponding to extremes in d (Fig. ), here defined as events beyond the 0.02 and 0.98 percentiles of the full distribution (see dashed lines in Fig. a). The maxima of d occur primarily during the spring and autumn months, while the minima are mainly found in summer and winter (Fig. ). This mirrors the monthly-mean values discussed above. The d maxima correspond to a complex anomaly pattern spanning the whole hemisphere, but a sign test shows that in both the spring and autumn seasons there is very little agreement between the individual events (Fig. a). This suggests that there is no single, dominant hemispheric-scale slp configuration leading to large dimensional extremes. The wintertime d minima again display very limited sign agreement, with the only significant feature being an intensification and eastward extension of the climatological Aleutian low-pressure centre (Fig. b). Such a result is very different from what has recently been observed in the North Atlantic region, where both d minima and d maxima systematically correspond to precise large-scale features . In contrast, the summer minima display an extensive and significant region of negative anomalies over the pole. The former pattern is an enhancement of the relatively low climatological slp values seen over the Arctic basin during the summer months. This results in a strengthened climatological meridional gradient (and presumably a strengthened polar vortex and a reduced air-mass exchange between the mid and high latitudes, although we recognize that slp is not the optimal field to diagnose this), and we therefore hypothesize that it matches a relatively predictable configuration.

We next analyse slp anomalies corresponding to extremes in θ. The θ maxima occur predominantly during the autumn and winter months, while the minima are mostly found in spring and summer (Fig. c, d). The high mean persistence found in the summer months therefore also corresponds to instantaneous maxima in this quantity. The maxima of θ in both autumn and winter correspond to a circum-hemispheric wave-like structure and show very little sign agreement (Fig. c). Since θ maxima are by definition unstable states, the lack of sign agreement might simply be due to the zonal propagation of the wave-like anomalies in time, although we do not explore this idea further here. The θ minima correspond to a mostly zonally symmetric pattern with a significant positive slp anomaly over the pole and locally significant negative anomalies throughout the mid and low latitudes (Fig. d). Over the North Atlantic, this results in a negative NAO-like dipole which is consistent with the anomaly pattern found for regional persistence maxima in the Euro-Atlantic domain . The reversal in the sign of the polar anomalies relative to summertime d minima is difficult to interpret. However, we note that 's regional analysis also showed opposite-sign anomalies at the high latitudes for d and θ minima. More generally, our results highlight that persistent, predictable states are primarily associated with zonally symmetric slp anomalies and therefore modulations of the zonal flow. Indeed, past studies have interpreted the hemispheric circulation as being dominated by a zonal-flow attractor with blocked or wavy states being associated with an unstable fixed point . Similarly, an enhanced zonality of the large-scale flow has been linked to increased downstream predictability on regional scales .

The seasonal control on the two dynamical systems metrics we discuss here can be further investigated through a d–θ scatter plot (Fig. a). This highlights how each season forms a distinct diagonal band of relatively well-correlated d and θ values. Figure b indeed confirms that the two metrics have high cross-correlation values, with the lag-0 correlation approaching 0.7. The lagged cross-correlation shows a semi-yearly cycle, with the peak correlation values reflecting integer year shifts and peak anticorrelation values reflecting a shift of approximately one season. This latter feature can be easily understood in terms of the above analysis. Both d and θ peak in autumn and spring. If we imagine shifting the d curve forwards or backwards by one season, the autumn θ peak will now match a dimension trough, thus leading to a negative correlation. The smaller positive and negative peaks in the cross-correlation function correspond to shifts of one and three seasons, respectively, such that the two cycles are roughly in quadrature.

The local dimension d of the t2m field shows a marked variability throughout the analysis period, with a range similar to that of d(slp): 8.9<d(t2m)<33.3 (Fig. a). The average dimension D is roughly 17.6, slightly lower than D(slp). The ACF of d (Fig. c) again displays a semi-annual cycle, albeit with larger ACF values than those seen for the slp. We note that the ACF structure should not be linked directly to the large seasonal cycle in temperature, since here we are considering d(t2m), which is not necessarily linked to the absolute value of the field. The inverse persistence θ spans a range corresponding to periods between 1.9 and 6.3 days (0.16<θ<0.54, Fig. b), indicating a higher persistence than slp. θ's ACF (Fig. d) is again different to that seen for d. The inter-year same-season correlation is still large and positive, but the semi-annual oscillation seen in d is entirely absent. Indeed, the winter values appear to be anticorrelated with those of the other three seasons, albeit with some weak modulation in the negative correlation values on seasonal scales. This difference is driven by the small offset between the seasonal cycles of d and θ, as discussed below.

An analysis of monthly-mean values (Fig. ) confirms the strong seasonal control on the dynamical characteristics of the field, but also highlights a radically different picture from that seen for the slp. In the summer months, d and its variability peak, while θ and its variability display a local maximum. With the autumn season, both the local dimension and θ reach a local minimum, only to increase again during wintertime. During spring, both metrics display a second minimum before returning to their high summertime values. The seasonal cycle in the variability of both indicators roughly matches that of the indicators themselves. The fact that the monthly-mean minima in d are broader and occur with a 1-month shift relative to those in θ accounts for why the semi-annual ACF cycle is only seen in the former variable. The general picture is therefore consistent with the ACFs described above. We hypothesize that the summertime and wintertime local maxima in d are associated with the inherent difficulty in forecasting the onset and duration of warm and cold spells . This is presumably linked to a high-dimensional atmospheric configuration, namely one with a large number of allowed preceding and future evolutions. The annual cycle of θ also displays summertime and wintertime local maxima and suggests that the winter (and to a lesser degree the summer) temperature fields are comparatively unstable, while the transitional seasons have a more sluggish dynamical evolution. This can be linked to the presence of wintertime cold spells and summertime heat waves which are usually non-stationary and locally short-lived, although notable exceptions can occur. One can therefore picture the t2m dynamics as following a single potential well configuration, with the extremes located in winter and summer. The dynamics of slp and t2m are very different, and the interaction between the two could be akin to a Langevin-like model . The slp would be the variable pushed into the winter or summer potential wells, while the temperature acts as a forcing noise term with extremes in winter and summer.

One can further look at the t2m anomalies corresponding to extremes in d and θ, again defined as events beyond the 0.02 and 0.98 percentiles of the full distribution (see dashed lines in Fig. a). The d maxima occur predominantly during summer, while the minima are mostly found in autumn (Fig. e, f), consistently with the seasonal cycle described above. The spatial anomalies corresponding to d maxima are weak and display very little sign agreement (Fig. a), suggesting that there is no single large-scale pattern matching these extremes. This is consistent with the theory that the high dimensionality of the temperature field may be associated with warm and cold spells, which are highly non-stationary and therefore will not emerge in a composite plot. Interestingly, the only region showing strong sign agreement is over eastern Africa, and also emerges in the composite anomalies for the dynamical extremes of prp (see below). The anomalies associated with minima in d are stronger, but show a similarly low sign agreement (Fig. b). The θ maxima and minima occur predominantly during winter and spring, respectively (Fig. g, h), and again show low sign agreement (Fig. c, d). While, as discussed above, the dynamical extremes elucidate a number of features of the temperature's seasonality and variability, they seem to afford relatively little insight concerning its geographical nature.

In the d–θ scatter plot for t2m, the winter months form a cluster corresponding to high θ, relatively high d values, while the spring and autumn seasons form a low θ, low d cluster (Fig. a). Summertime forms a continuation of the spring/autumn band, extending it toward higher d and θ values. The relatively broad scatter of the cloud points to a weak correspondence between d and θ. Indeed, the two metrics show lower cross-correlation values than those seen for slp, with a lag-0 correlation of just above 0.5 (Fig. b). The lagged cross-correlation shows a semi-yearly cycle, with the peak correlation values reflecting full and half-year shifts, in agreement with the synchronous double peak in both metrics shown in Fig. . Similarly, the two large negative peaks in the cross-correlation function correspond to shifts of one and three seasons, respectively, leading to situations in which the two yearly cycles are in anti-phase.

We construct a daily precipitation frequency variable as follows: we assign a value of 1 to each grid point and time step for non-zero precipitation rates and a value of 0 otherwise. For this variable the statistical fit of the recurrences to the expected distribution is better than for the precipitation rate itself (not shown). Another motivation to use the precipitation frequency data can be found in the multifractal analysis performed by , where the authors warn against using the precipitation data directly to measure dimensionality. The local dimension d of the precipitation frequency (prp) shows a large variability throughout the analysis period, with markedly higher values than those of the previous variables: 48<d(prp)<132 (Fig. a). This is also reflected in the average dimension D=83.1. These high values are consistent with the very scattered, noisy nature of the precipitation field. The autocorrelation function of d displays a semi-annual cycle with large autocorrelation values at full-year lags and near-zero values at 6-month lags (Fig. c). This is very different from the positive 6-month autocorrelation values seen for d(slp) and d(t2m). The inverse persistence θ (Fig. b) spans a range corresponding to periods between 1.0 and 2.1 days (0.3<θ<0.7), indicating a lower persistence than the previous variables, compatible with the precipitation's noisy nature. θ's autocorrelation function displays features similar to those discussed in Sects. 3.1 and 3.2 (Fig. d), namely a yearly ACF peak with negative autocorrelations at intermediate lags.

An analysis of monthly-mean values (Fig. ) reveals a marked semi-seasonal cycle in d, whose absolute values and variability both peak during spring and late summer/early autumn. These are seasons with enhanced convective precipitation at mid-latitudes. θ displays a similar variability behaviour, while the magnitude has a minimum in summer and an extended period of higher values from autumn into early spring. During the summer months, the mature phase of NH monsoon systems provides comparatively persistent and predictable precipitation patterns. The high persistence might also be favoured by the predominantly dry summers in the Mediterranean and other mid-latitude regions, with long dry spells at regional scale being the norm. This picture is consistent with the autocorrelation functions described above. In particular, the low absolute ACF values of d seen at lags of 2 to 10 months can be linked to the asymmetry in the positive and negative peaks. For example, for a lag of 6 months the first peak will roughly match the second peak, but the first trough will not match the second trough.

Notwithstanding the local and noisy nature of precipitation, a number of coherent features emerge from the geographical composites corresponding to extremes in the two dynamical systems metrics, again defined as events beyond the 0.02 and 0.98 percentiles of the full distribution (see dashed lines in Fig. a). The bulk of the d and θ maxima occur during spring, while the minima are primarily associated with the summertime (Fig. i–l). d maxima show predominantly negative precipitation anomalies (Fig. a). Significant features include decreases over the East Asian Monsoon region, the Indian Ocean, and continental and eastern Europe. Weak positive anomalies are mainly found over the central Pacific Ocean, parts of the USA, and the Canadian Arctic Archipelago. The anomalies associated with θ maxima closely track the d maxima throughout the Northern Hemisphere, albeit with a stronger preference for negative anomalies (Fig. c). Since the positive extremes in both metrics occur predominantly during spring, we hypothesize that the anomalies over East Asia could be linked to the shift between the northerly flow associated with the winter monsoon and the southerly flow associated with the summer monsoon. It is reasonable to expect that this shift between very different large-scale flow configurations could contribute to the high local dimensions. Similarly, the anomalies over the Indian Ocean could be associated with the onset of the Indian Monsoon's large-scale flow. d minima (Fig. b) show strong negative anomalies over the continental USA and southern Indochina and the strong positive anomalies over eastern Africa and the mid-latitude Pacific. θ minima again track closely the patterns seen for the d minima (Fig. d). The eastern African positive anomalies, roughly corresponding to northern Ethiopia, might indicate a modulation of the late-summer rainfall peak in the region . Similarly, the widespread negative anomalies across eastern continental North America suggest a modulation of the wet season over the Great Plains and northern Mexico.

The d–θ scatter plot (Fig. a) shows two clouds of points and a clear separation between the late spring/summer and autumn/winter seasons. While both clusters span a wide range of d and θ, there is a clear vertical and horizontal offset between the two, with the former seasons corresponding to lower values than the latter. The two prp metrics show high cross-correlation features (Fig. b), with the lag-0 correlation exceeding 0.6. The lagged cross-correlation shows a yearly cycle, a negative correlation for lags of 2–4 months, and a local maximum for lags of around 7 months.