# Dynamical properties of the North Atlantic atmospheric circulation in the past 150 years in CMIP5 models and the 20CRv2c Reanalysis

David Rodrigues, M Carmen Alvarez-Castro, Gabriele Messori, Pascal Yiou, Yoann Robin, Davide Faranda

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David Rodrigues, M Carmen Alvarez-Castro, Gabriele Messori, Pascal Yiou, Yoann Robin, et al. Dynamical properties of the North Atlantic atmospheric circulation in the past 150 years in CMIP5 models and the 20CRv2c Reanalysis. Journal of Climate, 2018, 10.1175/JCLI-D-17-0176.1. hal-01504478v2

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Dynamical properties of the North Atlantic atmospheric circulation in the
past 150 years in CMIP5 models and the 20CRv2c Reanalysis

David Rodrigues¹, M. Carmen Alvarez-Castro¹, Gabriele Messori²,

Pascal Yiou¹, Yoann Robin¹, Davide Faranda¹,³*

¹Laboratoire de Sciences du Climat et de l'Environnement, UMR 8212 CEA-CNRS-UVSQ,IPSL,
Université Paris-Saclay, 91191 Gif-sur-Yvette, France.

²Department of Meteorology and Bolin Centre for Climate Research, Stockholm University,
Stockholm, Sweden.

³London Mathematical Laboratory, 14 Buckingham Street, London WC2N 6DF, UK

*Corresponding author address: Laboratoire de Sciences du Climat et de l'Environnement, UMR 8212 CEA-CNRS-UVSQ,IPSL, Universite Paris-Saclay, 91191 Gif-sur-Yvette, France.

E-mail: davide.faranda@lsce.ipsl.fr
ABSTRACT

It is of fundamental importance to evaluate the ability of climate models to capture the large-scale atmospheric circulation patterns and, in the context of a rapidly increasing greenhouse forcing, the robustness of the changes simulated in these patterns over time. Here we approach this problem from an innovative point of view based on dynamical systems theory. We characterize the atmospheric circulation over the North Atlantic in the CMIP5 historical simulations (1851 to 2000) in terms of two instantaneous metrics: local dimension of the attractor and stability of phase-space trajectories. We then use these metrics to compare the models to the 20CRv2c reanalysis over the same historical period. The comparison suggests that: i) most models capture to some degree the median attractor properties and models with finer grids generally perform better; ii) in most models the extremes in the dynamical systems metrics match large-scale patterns similar to those found in the reanalysis; iii) changes in the attractor properties observed for the ensemble-mean 20CRv2c reanalysis might be artifacts due inhomogeneities in the standard deviation of ensemble over time; iv) the long-term trends in local dimension observed among the 56 members of the 20-CR ensemble have the same sign as those observed in the CMIP5 multimodel mean.
## 1. Introduction

One of the main sources of uncertainty in determining the impact of climate change on extreme events is the forced response of atmospheric dynamics (Shepherd 2014; Field 2012). While changes in observables such as surface temperature are easily diagnosed, shifts in the mid-latitude atmospheric patterns have proved very difficult to quantify. Some advances have been made by focussing on specific features such as atmospheric blocking (Kay et al. 2015; Cassou and Cattiaux 2016; Faranda et al. 2016b), which in turn influence the occurrence of European cold spells and heat waves, but the broader appreciation of circulation changes is still unsatisfactory. Here we shed some light on this knowledge gap by using a dynamical systems framework. We illustrate the power of such an approach by considering the well-known Lorenz (1963) system, a conceptual model of atmospheric convection consisting of three differential equations<fnref n="1" />

$$ \dot{x} = \sigma(y-x), \quad \dot{y} = rx - y - xz, \quad \dot{z} = xy - bz, \qquad (1) $$

where x, y, z represent respectively the convection strength, the temperature difference between the surface and the top of the troposphere and the asymmetry of the convection cells. The parameters σ, r are the Prandtl and the Rayleigh numbers, while b is a ratio of critical parameters. A trajectory of the Lorenz (1963) attractor is shown in blue in Figure 1. The figure consists of 2000 points obtained by iterating the Lorenz equations with $dt \approx 0.035$, $\sigma = 28$, $r = 10$, $b = 8/3$ with a Runge Kutta scheme of order 4. These are the classical parameters used by Lorenz (1963).

To study the effects of an external forcing, we now increase σ by 2% with respect to the classical value, such that $\sigma = 28.5$ (magenta trajectory in Fig. 1). This new trajectory on average favors higher values of z, but the changes relative to the original trajectory depend on the point being

<footnote n="1">¹a list of symbol used in the manuscript is provided in the supplementary material Table S1.</footnote>
considered: some points are not displaced, while some others are mapped elsewhere. Assuming no knowledge of the system other than the trajectories' paths, is there a way to determine that they belong to Lorenz attractors with different forcings? To answer this question we would need: i) to measure the dynamical properties of an ensemble of trajectories representing the two configurations; and ii) to estimate the distance between these trajectories and determine if the shift has changed the properties of the points in a detectable way. As a further complicating factor, we note that changes in natural systems are often gradual, and a sudden jump from one forcing to a different one is unlikely. A more realistic example discussing a Lorenz attractor with a smoothly varying forcing is presented in Section 2.

The atmospheric equivalent of a point on the Lorenz attractor is the ensemble of instantaneous fields describing the atmosphere at a time $t$. Here we study the atmospheric circulation over the North Atlantic and focus on a single field: the sea-level pressure (SLP). The SLP field reflects the major modes of variability affecting the North Atlantic (Hurrell 1995; Moore et al. 2013) and can further be used to diagnose a wealth of other atmospheric features, ranging from teleconnection patterns to storm track activity to atmospheric blocking e.g. (Rogers 1997; Comas-Bru and McDermott 2014). The trajectories of our dynamical systems are the successions of daily SLP fields from 26 CMIP5 models and the 20CRv2c reanalysis over the period 1851 to 2000 (Compo et al. 2011). The choice of the North Atlantic domain is motivated by the better observational coverage over the region in the first part of the analysis period compared to other regions of the globe (Krueger et al. 2013a). In order to measure changes in the systems, one must be able to specify at each point (each day) the local (in phase-space, daily in time) dynamical properties and track their evolution. Recent contributions to dynamical systems analysis have proven that local properties of the trajectories are characterized by two quantities: the local dimension and
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the stability of the field considered (Lucarini et al. 2016; Faranda et al. 2017). They correspond respectively to the rarity and the typical persistence of the configuration. Faranda et al. (2017) and Messori et al. (2017) have also shown that these two metrics can be connected to the predictability of a given atmospheric state and that their extremes match climate extremes.

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In this work we will first assess whether the models and reanalysis present similar attractor properties over the full time period considered. To do this, we compute daily values for the dimension and stability of the SLP fields and study their average and extreme properties. Next, we quantify their changes over the analysis period. We then compare the changes seen in the models to those observed in the reanalysis.

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## 2. Data & Methods

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We use daily SLP model output from the historical simulations of 26 CMIP5 models (Table 1) available from the CMIP5 archive (Taylor et al. 2012). We then compare these simulations to the 20th Century Reanalysis (20CRv2c) dataset (Compo et al. 2011), studying both the ensemble mean and the 56 individual members. The analysis focuses on the region $22.5°N – 70°N$ and $80°W – 50°E$ and the period 1850-2000. SLP anomalies are defined as deviations from a daily climatology. For example, the SLP anomaly at a given geographical point on the 5th December 2000 is computed relative to the mean SLP value at that same location for all 5th Decembers over the analysis period.

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In order to compute the dynamical systems metrics, we combine the statistical tools of extreme value theory with the results obtained by Freitas et al. (2010) for Poincaré recurrences. The parameters mentioned in the introduction (local dimension *d* and stability $1/\theta$) are computed for the

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points $\zeta$ on the attractor obtained as a sequence of states of the system. The dynamical indicators are linked to the probability $p$ that a trajectory $x(t)$ explores a ball centered on $\zeta$ with diameter $2\epsilon$, i.e. the recurrence rate of the configuration $\zeta$. We briefly outline the physical meaning of these quantities and the way they are computed below.

(i) Local Dimension: The Freitas et al. (2010) theorem and its modification in Lucarini et al. (2012) states that the probability $p$ of entering a ball centred on $\zeta$ with a radius $\epsilon$ for chaotic attractors obeys a generalized Pareto distribution (Pickands III 1975). In order to compute such probability, we first calculate the series of distances $\delta(x(t), \zeta)$ between the point on the attractor $\zeta$ and all other points $(x(t))$ on the trajectory. We then put a logarithmic weight on the time series of the distance:

$$g(x(t)) = -\log(\delta(x(t), \zeta)).$$

The reason for taking the logarithm is explained by Collet and Eckmann (2009): in the dynamical system set-up the negative logarithm increases the discrimination of small values of $\delta(x,y)$ which correspond to large values of $g(x(t))$. The probability of entering a ball of radius $\epsilon$ centered on $\zeta$ is now translated in the probability $p$ of exceeding a threshold $q$. In the limit of an infinitely long trajectory, such probability is the exponential member of the generalized Pareto distribution:

$$p = \Pr(g(x(t)) > q, \zeta) \approx \exp(-[x - \mu(\zeta)]/\beta(\zeta))$$

whose parameters $\mu$ and $\beta$ depend on the point $\zeta$ chosen on the attractor. Remarkably, $\beta(\zeta) = 1/d(\zeta)$, where $d(\zeta)$ is the local dimension around the point $\zeta$. This result has recently been applied to SLP fields in Faranda et al. (2017). In this paper we use the quantile 0.975 of the series $g(x(t))$ to determine $q$. We have further checked the stability of the results against
reasonable changes in the quantile.

(ii) *Local Stability*: The probability $p$ contains information about the geometry of the ball around $\zeta$ but provides no insight on the temporal evolution of the dynamics around $\zeta$. In particular, it is interesting to know the mean residence time of the trajectory within the neighborhood of $\zeta$. To measure this quantity, we employ the extremal index $\theta$ (Freitas et al. 2012; Faranda et al. 2016a), namely the inverse of this mean residence time. Heuristically, if the trajectory enters $N$ times the neighborhood of $\zeta$ and at each time $i$ the length of the cluster (that is, the number of successive timesteps when the trajectory is within the neighborhood) is $\tau_i$, a simple estimate is:
$$ \theta^{-1} = (1/N) \sum_i \tau_i, $$
such that $\theta$ varies between 0 and 1. The value $\theta = 0$ corresponds to a stable fixed point of the dynamics where the observation $\zeta$ is repeated infinite times (as for a pendulum left in its equilibrium position). A value of $\theta = 1$ indicates a point immediately leaving the neighborhood of $\zeta$. Since $\theta$ is the inverse of a persistence time, it depends on the $dt$ used. If $dt$ is too large, the time dependence structure is hidden and $\theta$ will fail to be close to 1. If $dt$ is too small, $\theta$ is close to zero. In Faranda et al. (2017) it has been observed that $\theta$ for SLP fields over the North Altantic varies between 0.3 and 0.5, when $dt = 1$ day. In this work we use the same $dt$. The extremal index is estimated with the likelihood estimator by Süveges (2007).

Figure 2 illustrates the meaning of the indicators for atmospheric flows: the local dimension $d$ is the number of degrees of freedom needed to describe the dynamics of the system linearized around the state $\zeta$ and is therefore proportional to the number of degrees of freedom of $\zeta$. In the pictogram, $d = 2$ since we only consider two possible origins and evolutions of the state $\zeta$. The inverse of the persistence time $\theta$ is linked to the probability that the trajectory follows a path
where each field resembles those of the previous and subsequent days. In the figure we present two possibilities: i) along the green trajectory, the pattern changes every day so that $\theta(t) = 1$; ii) along the red trajectory, the same pattern is observed for three consecutive days so that $\theta = 1/3$.

In order to evaluate how close the $d$ and $\theta$ of the different models are to those of the reanalysis, we adopt a number of distance metrics. The simplest metrics that can be defined are: $R(d) = \Delta(d)/\max(\Delta(d))$ and $R(\theta) = \Delta(\theta)/\max(\Delta(\theta))$, where $\Delta$ represents the difference between the median values of $d$ or $\theta$ of each model and the 20CRv2c values. We further compute a global score:

$$R_{tot} = (R(d) + R(\theta))/2. \qquad (2)$$

To check the validity of the global score, we compare $R_{tot}$ with the Wasserstein distance $\mathcal{W}$, computed as described in Robin et al. (2017).

The Wasserstein distance $\mathcal{W}$ measures how two multivariate probability density functions $\mu$ and $\eta$ differ from each other. Probability distributions are like distributions of mass (normalized to 1) across space. The Wasserstein distance is proportional to the minimal work that is needed to transport one distribution of mass into another. It is therefore rooted in optimal transport theory (Villani 2008; Santambrogio 2015). In practice, the two measures $\mu$ and $\eta$ are discretized into multivariate histograms, which are sums of Dirac masses:

$$\mu = \sum_{i} \mu_{i} \delta_{x_{i}}, \quad \eta = \sum_{j} \eta_{j} \delta_{x_{j}}.$$
The distribution $\mu$ is transformed into $\eta$ by the transport plan $\gamma$ such that $\gamma_{ij}$ transports the mass $\mu_i$ at $x_i$ to $\eta_j$ at $x_j$. Therefore the $\gamma_{ij} \ge 0$ verify:

$$ \mu_i = \sum_j \gamma_{ij}, \quad \eta_j = \sum_i \gamma_{ij}, \quad \sum_{ij} \gamma_{ij} = 1. $$

If $\Gamma$ is the set of all possible transport plans of $\mu$ to $\eta$, then the Wassertein distance is:

$$ \mathcal{W}(\mu, \eta) = \sqrt{\inf_{\gamma \in \Gamma} \sum_{ij} \gamma_{ij} \|x_i - x_j\|^2}, $$

where $\|\cdot\|$ is the usual Euclidean distance.

Before beginning the analysis of the model and reanalysis data, it is necessary to outline if the dynamical properties are sensitive to the changes of the attractor due to continuous modifications of the underlying forcing and if these changes are statistically detectable. There are few theoretical results on non-stationary statistics of dynamical systems, as well as on non-stationary extreme value theory. Luckily, the recurrence approach we use here to estimate $d$ and $\theta$ allows to bypass most of the technical difficulties linked to non-stationarity, because the dynamical properties are measured with respect to each individual state $\zeta$ of the attractor. If the change affects the neighborhood of a state, it will change its dynamical properties. If most of the states are affected by the changes in the dynamics, then the average dimension of the attractor and the average persistence will change accordingly.

In order to test this idea, we consider the Lorenz (1963) systems discussed in the Introduction and perform a set of 30 realizations (trajectories), where $\sigma$ varies continuously over $28 < \sigma < 28.54$ according to: $\sigma(t) = \sigma(t - dt) + \delta\sigma$, with $\delta\sigma = 10^{-5}$. Each realization con-
sists of about 70000 iterations with time-step $dt = 0.02$. The number of time-steps and the $dt$ are chosen to mimic the persistence properties of the SLP field over the North Atlantic, which displays a median $\theta$ of around 0.5 (see Section 3). If our methodology can indeed detect the gradual change from $\sigma = 28$ to $\sigma = 28.54$, then the $d$ and $\theta$ distributions for the first and second half of the simulations should be significantly different. In order to provide a visually immediate picture, we show distributions of the medians of $(d, \theta)$ for each half of each simulation (Figure 3). It is straightforward to verify that the data forms two clouds of well-separated median centroids. This analysis therefore confirms that the indicators are sensitive to small changes in the attractor properties and that we can attempt to use them to detect long term changes in the dynamical properties of reanalysis and CMIP5 data.

### 3. Aggregate analysis of model and reanalysis attractors

We begin the analysis of the daily SLP fields from 1851 to 2000 by presenting the scatterplot of $d$ versus $\theta$ for the ensemble mean of the 20CRv2c reanalysis (Figure 4). Hereafter we will call this run 20CR-EM. With respect to the computations done for the Lorenz (1963) system, $\zeta$ is now a daily SLP map and distances are computed using an euclidean metrics at each grid point as in Faranda et al. (2017). The average of $d$ is proportional to the number of degrees of freedom needed to represent the systems' dynamics (this quantity is called attractor dimension in dynamical systems theory) while the average of $\theta$ is the inverse of the mean persistence time of a given SLP configuration. Maxima (minima) of $d$ therefore correspond to the most complex (simple) SLP configurations. Maxima (minima) of $\theta$ correspond to the most unstable (stable) configurations (Messori et al. 2017; Faranda et al. 2017). Panels A-D in Figure 4 show the composite SLP anomalies for dynamical extremes - namely days beyond the 0.98 and 0.02 quantiles of the $d$ and $\theta$ distributions. These closely – albeit not exactly – resemble the canonical
North Atlantic weather regimes (Vautard 1990). In particular, maxima of $\theta$ (A) reproduce an Atlantic Ridge pattern, while minima of $\theta$ (B) correspond to a negative North Atlantic Oscillation (NAO) phase. Similarly, maxima of $d$ (C) correspond to a blocking pattern and minima of $d$ (D) to a positive NAO. This is in agreement with previous results from Faranda et al. (2017), who further found that dynamical extremes occur mostly in the winter season. The patterns are stable if the definition of dynamical extremes is stable up to the 20th and the 80th percentiles of the relevant distributions, although the magnitude of the composite anomalies reduces when including days corresponding to lower percentiles (not shown). We note that the values of $\theta$ shown in the Figure should not be compared directly to the persistence of the traditional weather regimes defined using clustering algorithms, as the requirement that the flow does not leave the neighborhood of the state $\zeta$ is a more restrictive condition than continued permanence within a given cluster. For example, intense or frequent mobile synoptic systems leading to substantial day-to-day fluctuations in sea-level pressure could cause the dynamics to leave the ball centered on the state $\zeta$ in phase space while remaining within a weather regime cluster. Indeed, if one considers the typical partition of the North Atlantic atmospheric variability into 4 weather regimes, the probability of being in one of them is of order 0.25, whereas the probability of being close to $\zeta$ is set by the threshold $q$ in our case 0.0225 (see Section 2).

The 20CR-EM data analyzed above is constructed by averaging instantaneous fields from a 56-member ensemble of simulations. The ensemble is less constrained at the beginning of the period, when surface observations were scarce, than at the end. This implies that the 20CR-EM fields are smoother at the beginning of the period than at the end, because the latter are obtained by averaging over an ensemble with smaller differences between individual members. This may affect the effective number of degrees of freedom as measured by $d$. For this reason, we also measure $d$
and $\theta$ for the 56 individual members and then average them to obtain a single daily value. We will refer to these quantities as the 20CRv2c means of ensemble (20CR-ME). Schematically:

• 20CR-EM indicates the daily dynamical properties computed for the 20CRv2c ensemble mean.

• 20CR-ME indicates the average of daily dynamical properties computed for each single ensemble member.

We can now compare the distributions of $d, \theta$ for 20CR-EM to $d, \theta$ for 20CR-ME. The scatterplot of $d$ versus $\theta$ for the 20CR-ME is shown in Figure 5. A number of differences relative to the 20CR-EM appear (cf. Figure 4). First, the median value of $d$ is lower for 20CR-EM than for 20CR-ME (Table 2), which indicates that the averaging of SLP fields in the Ensemble Mean has suppressed some degrees of freedom. Similarly, the ensemble mean has higher persistence (lower $\theta$) because smoother fields tend to have slower variations. Although the numerical values of $d$ and $\theta$ differ, the cross-correlation coefficient between the time series for the two data-sets are 0.93 and 0.97, respectively. This suggests that features such as the seasonality and interannual variability of the $d$ and $\theta$ time series are preserved with the EM averaging. We next look at the SLP anomaly fields corresponding to the dynamical extremes of the 20CR-ME (Panels A-D in Fig. 5). We find similar patterns to those observed for 20CR-EM (Panels A-D in Fig. 5)). Indeed, 70% of $d$ maxima from 20CR-EM match those from 20CR-ME, while this percentage is 61% for the minima. For $\theta$ maxima and minima we find values of 81% and 55%, respectively. We further investigate the differences between 20CR-EM and 20CR-ME by looking at the changes of the dynamical properties over time in Section 4.
We next compare the (d, θ) bivariate histograms obtained for the 20CR-EM (Figure 6a) with those computed for the CMIP5 models (Figure 6c,e). Two different behaviors emerge: some of the models (e.g. CMCC-CMS, Figure 6c) yield a unimodal distribution resembling that obtained for the 20CR-EM; other models (e.g. the IPSL-CM5A, Figure 6e) show bimodal distributions. We find these different behaviors to be related to different seasonal cycles: in Figure 6-b,d,f), we plot the (d, θ) scatterplots for the same models by coloring each point according to the month of the year it occurs in. In the 20CR-EM and the CMCC-CMS model, the different seasons are spread across the cloud, although maxima of θ mostly occur in winter and the summer season is biased towards low d and θ. The IPSL-CM5A displays a much stronger seasonal discrimination, with two distinct (d, θ) clouds for the winter and for the summer seasons corresponding to the different modes of the bivariate histograms. This implies that both the bulk statistics and the extremes are modified by the seasonal cycle. For a more detailed discussion of the seasonality of the dynamical extremes, we refer the reader to Faranda et al. (2017).

Given the variety of the possible behaviors, we will analyze separately the mean and the extreme behavior of the CMIP5 dynamical properties. We report the aggregate analysis in Table 2 and in Figure 7. In the latter figure, the colored numbers correspond to the median values for each model (numbered as in Table 1), the red dot to the 20CR-EM median and the black dots to the medians of each of the 56 members of the 20CRv2c ensemble. For 20CR-EM and 20CR-ME we also draw the ellipses whose semiaxes correspond to the standard deviation of the mean. We note that the median values of the 20CR-ME are so close to each other as to be almost indistinguishable, meaning that the individual members provide a coherent representation of the SLP field. As noted above, the 20CR-EM *d* and *θ* median values are both smaller than those of 20CR-ME. All of the models fall within the ellipse of 20CR-EM and most of the models within
that of 20CR-ME. In Table 2 we provide the median values and standard deviations of *d* and *θ* for all the models and also present the distance metrics *R*(*d*), *R*(*θ*) and *R*<sub>tot</sub> between models and reanalysis as introduced in Section 2. In order to verify their robustness, we also compare *R*<sub>tot</sub> to the Wasserstein distance *W*<sub>2</sub>, also defined in Section 2 (Fig. 8). The two indicators provide very similar information (Pearson coefficient: *r*<sub>pear</sub> = 0.90 and Spearman coefficient *r*<sub>spear</sub> = 0.85 (Von Storch 1999)); from now on we will therefore use the simpler *R*<sub>tot</sub> when discussing our results. *R*<sub>tot</sub> further indicates the direction of the changes (larger or smaller *d*, *θ*) while *W*<sub>2</sub> only provides this information if the transport plan is computed (Villani 2008). The latter would be particularly complex to compute for the dataset analyzed here.

All the models are within one standard deviation of the 20CR-EM (Figure 7), while 5 are not withing the 20CR-ME ellipse. Both Figure 7 and Table 2 further indicate that in many cases models with a higher horizontal resolution have median values closer to those of the reanalysis. We note, however, that the median values in *d* and *θ* are statistically different from those found in 20CR-EM for all models except model 24 and 26 for *d* and model 23 for *θ*. For 20CR-ME, the medians are different for all models except 13. The statistical significance is determined using the Wilcoxon ranksum test (Von Storch 1999). The null hypothesis is that the variables are samples from continuous distributions with equal medians. A rejection of the null hypothesis therefore indicates a significant difference between the medians.

We also find that different models have different spreads in *d* and *θ* (Table 2), suggesting investigating the extremes of these quantities and their relation with the weather regimes found in 20CR-ME and 20CR-EM. Figures S1-S26 in the supplementary material display the (*d*,*θ*) histograms and the composite SLP anomalies for the *d* and *θ* extremes computed as in Figures
4,5. A quantitative analysis is reported in Table S2 using the Root Mean Square Error (RMSE) between 20CR-EM, 20CR-ME and the CMIP5 SLP composite anomalies. In general, we find the B and D (NAO-like) patterns to have a higher RMSE whereas A and C are better represented. This is counterintuitive, since one might naively expect models to better reproduce low-dimensional rather than high-dimensional patterns. At the same time, the NAO is the dominant mode of variability in the North Atlantic region, which could explain why a higher RMSE is found in that region of the phase space. Another interesting observation is that models with low $R_{tot}$ scores do not always show the best match in the spatial patterns corresponding to the dynamical extremes.

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## 4. Changes in the attractor properties

We now investigate whether the SLP's dynamical indicators have changed over time by computing 30-year moving time averages (denoted by $\langle \cdot \rangle$) for 20CR-EM, 20CR-ME (and separately for the 56 members) and the 26 CMIP5 models. These results are presented in Figure 9. The abscissa show the final year of each averaging window. In the left panels, we consider the full 150 years and subtract from each model/reanalysis the respective mean values of the indicators in 1880 (that is, averaged over 1851-1880); in the right panels we focus on the last 25 years of data and subtract the values of 1979 (that is, averaged over 1940-1979).

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20CR-EM (red) shows an increase in $d$ over time which is opposite in sign and an order of magnitude larger than the changes observed in 20CR-ME (blue), the single members (light blue), the CMIP5 multimodel mean(black) and the single CMIP5 models (gray; Figure 9a). When focusing on the last 25 years of the datasets, the curves scale similarly and most of them show a moderate decrease of the dimension (Figure 9b). We hypothesize that the rapid increase of the

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dimension for 20CR-EM during the pre-satellite era is caused by changes in the variance of the ensemble members which, as discussed in Section 3 above, become more closely constrained as we approach the present day. To test this hypothesis, we compute the daily values of the standard deviation of the 56 SLP fields and plot the moving average of such quantity against that of the dimension of 20CR-EM (Figure 10). The standard deviation of the ensemble is in turn linked to the number of observations incorporated in the analysis (e.g. Krueger et al. (2013a)). The linear relation points to the increase in $\langle d \rangle$ 20CR-EM not being a physical trend but rather an artifact due to the scarcity of observations during the first part of the reanalysis period. However, from 1979, the variance stabilizes and both the 20CR-EM and 20CR-ME show a decrease in local dimension. This decrease is also observed in most of CMIP5 models as reflected by the multi-model mean (Fig. 9b).

The inverse persistence $\theta$ of 20CR-EM shows a similar behavior (Figure 9c). $\theta$ increases up to 1950 and then levels off. In terms of persistence time this increase is small compared to the resolution of the datasets used for the analysis (1 day). 20CR-ME shows a similar, albeit weaker, trend, while the CMIP5 models mostly oscillate around zero. No systematic trends can be identified after 1970 (Figure 9d).

We conclude by noting that the changes found in CMIP5 models are mostly of the same sign as those found for 20CR-ME, but systematically smaller in magnitude. This is associated with a much larger spread between the different models than between the different members of the reanalysis, with a number of models showing trends of opposite sign to those of the multi-model mean.

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342 **5. Discussion and Conclusions**

343 We have computed the instantaneous dynamical properties of the SLP fields over the North
344 Atlantic Region for the 20CRv2c and the CMIP5 historical runs, over the period 1851-2000.

345 The goal of our analysis was to assess whether different models with different physics and
346 resolutions quantitatively represent the *same* dynamical system and therefore possess attractors
347 with similar characteristics. The metrics we use are the local dimension *d* and the inverse
348 of the persistence time θ. As described in Faranda et al. (2017), these two quantities give a
349 complete characterization of the attractor of the system. To take into account the possibility that
350 inhomogeneities in the assimilated data create artifacts in the 20CRv2c ensemble-mean SLP
351 fields, we have computed two sets of dynamical properties. The first is composed of the dynamical
352 properties of the ensemble-mean field (20CR-EM). The second is computed as the average of the
353 instantaneous dynamical properties of each of the 56 ensemble members (20CR-ME).

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355 When the whole analysis period is considered, we find that the models successfully capture
356 many of the dynamical systems features identified in the reanalysis, such as the range and
357 variability of the *d* and *θ* metrics. The SLP fields corresponding to extremes in *d* and *θ* are also
358 similar across the models and reanalysis. At the same time, some models exagerate the effects of
359 the seasonal cycle on the dynamical indicators, and the statistical agreement in the median values
360 of the metrics is generally poor. Models with higher horizontal resolutions mostly perform better.

361

362 To detect the changes in the attractor properties with time, we have analysed the 30-year moving
363 averages of *d* and *θ* for the models and reanalysis. We find a number of interesting behaviors: up
364 to 1950, the 20CR-EM shows a rapid increase in *d* and *θ* whereas 20CR-ME shows a decrease in
365 *d* and a slower increase in θ. After 1950, there is no trend in θ and a weakly decreasing trend in *d* for both datasets. The CMIP5 multi-model mean shows a weakly decreasing trend in *d* throughout the analysis period, in agreement with 20CR-ME, and no trend in θ. These results suggest that one should be very careful in using ensemble means for studying the atmospheric dynamics of the late 19th and beginning of the 20th century, even over the North Atlantic (Krueger et al. 2013b; Ferguson and Villarini 2012, 2014). The 20CRv2c ensemble members are increasingly constrained by a growing number of SLP observations as one approaches the present day. This causes a decrease of the ensemble spread with time, since the system is more closely pinned to a specific manifold (the observations) without the possibility of exploring the full phase space. We hypothesize this is the root cause of the upward trend in the 20CR-EM local dimension.

375

The next natural question is whether we can trust the results obtained for single ensemble members. We believe that the answer is affirmative, as: i) the dataset has a sufficiently high horizontal resolution to obtain a good estimate of the local dimension distribution (Faranda et al. 2017) and ii) we focus here on the North Atlantic sector, which can be expected to perform better than elsewhere since most of the observations used to constrain 20CRv2c in the first part of the dataset are located in Europe or eastern North America (Cram et al. 2015). We therefore argue that the results obtained for the 20CR-ME and the multimodel ensemble are valuable, and that the decrease in dimension with time is a real and interesting feature of the atmospheric dynamics which merits a more detailed analysis in further studies. In fact, a decreasing dimension implies a more predictable atmosphere.

386

As a final caveat we note that our analysis does not attempt to separate the forced variability from natural low-frequency oscillations and that, especially during the first part of the analysis
period, it is unclear whether the greenhouse forcing can be clearly discerned above the background
"climate noise" (Paeth et al. 1999; Lyu et al. 2015). We must therefore take into account the
possibility that the data's internal variability dominates over the long-term forcing trends for the
time period considered.

*Acknowledgments*. P.Yiou and D. Faranda were supported by ERC grant No. 338965, M.C.
Alvarez-Castro was supported by Swedish Research Council grant No. C0629701 and G. Messori
was supported by a grant from the Department of Meteorology of Stockholm University.

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<table><caption>TABLE 1. List of CMIP5 Models analysed and 20CRv2c reanalysis from 1851 to 2001. Models are approximately ordered by increasing horizontal resolution.</caption><thead><tr><th>No.<sup>1</sup></th><th>Model</th><th>Institution/ID</th><th>Country</th><th>Resolution<sup>2</sup></th></tr></thead><tbody><tr><td>0</td><td>20CRv2c ME</td><td>NOAA-CIRES</td><td>USA</td><td>2 x 2</td></tr><tr><td>1</td><td>20CRv2c EM</td><td>NOAA-CIRES</td><td>USA</td><td>2 x 2</td></tr><tr><td>2</td><td>CMCC-CESM</td><td>Centro Euro-Mediterraneo sui Cambiamenti Climatici</td><td>Italy</td><td>3.75x3.75</td></tr><tr><td>3</td><td>CanESM2</td><td>Canadian Centre for Climate Modelling and Analysis, CCCMa</td><td>Canada</td><td>2.81x 2.79</td></tr><tr><td>4</td><td>MIROC-ESM-CHEM</td><td>MIROC<sup>4</sup></td><td>Japan</td><td>2.81x 2.79</td></tr><tr><td>5</td><td>MIROC-ESM</td><td>MIROC<sup>4</sup></td><td>Japan</td><td>2.81x 2.79</td></tr><tr><td>6</td><td>BCC-CSM1-1</td><td>Beijing Climate Center</td><td>China</td><td>2.81x 2.79</td></tr><tr><td>7</td><td>IPSL-CM5B-LR</td><td>Institute Pierre Simon Laplace, IPSL</td><td>France</td><td>3.75x1.89</td></tr><tr><td>8</td><td>NorESM1-M</td><td>Norwegian Climate Center</td><td>Norway</td><td>2.5x1.89</td></tr><tr><td>9</td><td>FGOALS-2</td><td>Institute of Atmospheric Physics, Chinese Academy of Sciences</td><td>China</td><td>2.81x2.81</td></tr><tr><td>10</td><td>MPI-ESM-P</td><td>Max Planck Institute for Meteorology, MPI</td><td>Germany</td><td>1.87x1.87</td></tr><tr><td>11</td><td>MPI-ESM-LR</td><td>Max Planck Institute for Meteorology, MPI</td><td>Germany</td><td>1.87x1.87</td></tr><tr><td>12</td><td>CSIRO-MK3-6-0</td><td>CSIRO-BOM<sup>5</sup></td><td>Australia</td><td>1.87x1.87</td></tr><tr><td>13</td><td>CMCC-CMS</td><td>Centro Euro-Mediterraneo sui Cambiamenti Climatici</td><td>Italy</td><td>1.87x1.87</td></tr><tr><td>14</td><td>MPI-ESM-MR</td><td>Max Planck Institute for Meteorology, MPI</td><td>Germany</td><td>1.87x1.87</td></tr><tr><td>15</td><td>IPSL-CM5A-MR</td><td>Institute Pierre Simon Laplace, IPSL</td><td>France</td><td>2.5x1.26</td></tr><tr><td>16</td><td>INM-CM4</td><td>Institute for Numerical Mathematics, INM</td><td>Russia</td><td>2x1.5</td></tr><tr><td>17</td><td>ACCESS 1-0</td><td>CSIRO-BOM<sup>5</sup></td><td>Australia</td><td>1.87x1.25</td></tr><tr><td>18</td><td>MIROC5</td><td>MIROC<sup>4</sup></td><td>Japan</td><td>1.40x1.40</td></tr><tr><td>19</td><td>CNRM-CM5</td><td>CNRM-CERFACS<sup>3</sup></td><td>France</td><td>1.40x1.40</td></tr><tr><td>20</td><td>MRI-ESM1</td><td>Meteorological Research Institute, MRI</td><td>Japan</td><td>1.125x1.125</td></tr><tr><td>21</td><td>BCC-CSM1-M</td><td>Beijing Climate Center</td><td>China</td><td>1.125x1.125</td></tr><tr><td>22</td><td>MRI-CGCM3</td><td>Meteorological Research Institute, MRI</td><td>Japan</td><td>1.125x1.125</td></tr><tr><td>23</td><td>EC-EARTH</td><td>Danish Meteorological Institute, DMI</td><td>Denmark</td><td>1.125x1.125</td></tr><tr><td>24</td><td>CESM1-FASTCHEM</td><td>Community Earth System Model Contributors, NCAR</td><td>USA</td><td>1.25x0.94</td></tr><tr><td>25</td><td>CESM1-CAM5</td><td>Community Earth System Model Contributors, NCAR</td><td>USA</td><td>1.25x0.94</td></tr><tr><td>26</td><td>CESM1-BGC</td><td>Community Earth System Model Contributors, NCAR</td><td>USA</td><td>1.25x0.94</td></tr><tr><td>27</td><td>CCSM4</td><td>National Center for Atmospheric Research, NCAR</td><td>USA</td><td>1.25x0.94</td></tr></tbody></table>

<footnote n="1">¹ Order by horizontal resolution (Decreasing)</footnote>

<footnote n="2">²Longitude x Latitude (°)</footnote>

<footnote n="3">³Centre National de Recherches Meteorologiques - Centre European de Recherche et de Formation Avance en Calcul Scientifique</footnote>

<footnote n="4">⁴Atmosphere and Ocean Research Institute (University of Tokyo), National Institute for Environmental Studies, and Japan Agency for Marine-Earth Science and Technology</footnote>

<footnote n="5">⁵Commonwealth Scientific and Industrial Research Organisation(CSIRO), Bureau of Meteorology(BOM)</footnote>
<table><caption>TABLE 2. List of median values and standard deviations of dimension *d* and inverse persistence *θ* for 20CR-EM, 20CR-ME and the 26 CMIP5 models. The table further presents the relative distances *R*(*d*) = *δ*(*d*)/*max*(*δ*(*d*)) and *R*(*θ*) = *δ*(*θ*)/*max*(*δ*(*θ*)) normalized with respect to the farthest model and the global score *R*<sub>tot</sub> = (*R*(*d*) + *R*(*θ*))/*2*.</caption><thead><tr><th>Model</th><th>median(d)</th><th>median(θ)</th><th>std(d)</th><th>std(θ)</th><th>R(d) EM</th><th>R(θ) EM</th><th>R<sub>tot</sub> EM</th><th>R(d) ME</th><th>R(θ) ME</th><th>R<sub>tot</sub> ME</th></tr></thead><tbody><tr><td>00-20CR-ME</td><td>12.26</td><td>0.53</td><td>1.85</td><td>0.05</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>01-20CR-EM</td><td>11.56</td><td>0.50</td><td>2.06</td><td>0.06</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>02-CMCC-CESM</td><td>12.22</td><td>0.51</td><td>1.90</td><td>0.06</td><td>0.54</td><td>0.13</td><td>0.33</td><td>0.04</td><td>0.23</td><td>0.14</td></tr><tr><td>03-CanESM2</td><td>11.99</td><td>0.51</td><td>2.06</td><td>0.07</td><td>0.35</td><td>0.02</td><td>0.19</td><td>0.34</td><td>0.30</td><td>0.32</td></tr><tr><td>04-MIROC-ESM-CHEM</td><td>12.54</td><td>0.47</td><td>1.87</td><td>0.06</td><td>0.80</td><td>0.75</td><td>0.77</td><td>0.35</td><td>0.83</td><td>0.59</td></tr><tr><td>05-MIROC-ESM</td><td>12.48</td><td>0.47</td><td>1.95</td><td>0.05</td><td>0.75</td><td>0.76</td><td>0.76</td><td>0.27</td><td>0.84</td><td>0.56</td></tr><tr><td>06-BCC-CSM1</td><td>12.12</td><td>0.51</td><td>1.95</td><td>0.05</td><td>0.46</td><td>0.12</td><td>0.29</td><td>0.16</td><td>0.24</td><td>0.20</td></tr><tr><td>07-IPSL-CM5B</td><td>12.73</td><td>0.46</td><td>1.79</td><td>0.06</td><td>0.95</td><td>0.93</td><td>0.94</td><td>0.58</td><td>0.95</td><td>0.77</td></tr><tr><td>08-NorESM1-M</td><td>12.12</td><td>0.48</td><td>2.27</td><td>0.06</td><td>0.46</td><td>0.44</td><td>0.45</td><td>0.18</td><td>0.62</td><td>0.40</td></tr><tr><td>09-FGOALS-S2</td><td>11.63</td><td>0.45</td><td>1.89</td><td>0.06</td><td>0.06</td><td>1.00</td><td>0.53</td><td>0.78</td><td>1.00</td><td>0.89</td></tr><tr><td>10-MPI-ESM-P</td><td>12.17</td><td>0.51</td><td>1.82</td><td>0.07</td><td>0.50</td><td>0.06</td><td>0.28</td><td>0.11</td><td>0.28</td><td>0.19</td></tr><tr><td>11-MPI-ESM-LR</td><td>12.13</td><td>0.51</td><td>1.91</td><td>0.05</td><td>0.47</td><td>0.14</td><td>0.30</td><td>0.16</td><td>0.23</td><td>0.19</td></tr><tr><td>12-CSIRO-MK3-6-0</td><td>12.66</td><td>0.50</td><td>1.91</td><td>0.05</td><td>0.90</td><td>0.02</td><td>0.46</td><td>0.50</td><td>0.34</td><td>0.42</td></tr><tr><td>13-CMCC-CMS</td><td>11.95</td><td>0.52</td><td>2.14</td><td>0.06</td><td>0.32</td><td>0.22</td><td>0.27</td><td>0.38</td><td>0.17</td><td>0.28</td></tr><tr><td>14-MPI-ESM-MR</td><td>12.09</td><td>0.51</td><td>1.86</td><td>0.06</td><td>0.43</td><td>0.09</td><td>0.26</td><td>0.21</td><td>0.26</td><td>0.24</td></tr><tr><td>15-IPSL-CM5A</td><td>11.86</td><td>0.48</td><td>1.90</td><td>0.06</td><td>0.25</td><td>0.51</td><td>0.38</td><td>0.49</td><td>0.67</td><td>0.58</td></tr><tr><td>16-INM-CM4</td><td>12.79</td><td>0.47</td><td>2.14</td><td>0.07</td><td>1.00</td><td>0.70</td><td>0.85</td><td>0.65</td><td>0.80</td><td>0.72</td></tr><tr><td>17-ACCESS '1-0</td><td>11.74</td><td>0.49</td><td>1.92</td><td>0.06</td><td>0.15</td><td>0.31</td><td>0.23</td><td>0.64</td><td>0.53</td><td>0.58</td></tr><tr><td>18-MIROC5</td><td>12.58</td><td>0.49</td><td>2.00</td><td>0.07</td><td>0.83</td><td>0.33</td><td>0.58</td><td>0.39</td><td>0.54</td><td>0.47</td></tr><tr><td>19-CNRM-CM5</td><td>12.36</td><td>0.47</td><td>2.08</td><td>0.05</td><td>0.65</td><td>0.59</td><td>0.62</td><td>0.12</td><td>0.72</td><td>0.42</td></tr><tr><td>20-MRI-ESM1</td><td>11.72</td><td>0.51</td><td>2.09</td><td>0.05</td><td>0.13</td><td>0.12</td><td>0.13</td><td>0.67</td><td>0.23</td><td>0.45</td></tr><tr><td>21-BCC-CSM1-M</td><td>11.45</td><td>0.55</td><td>2.00</td><td>0.07</td><td>0.09</td><td>0.91</td><td>0.50</td><td>1.00</td><td>0.30</td><td>0.65</td></tr><tr><td>22-MRI-CGCM3</td><td>11.74</td><td>0.51</td><td>1.66</td><td>0.07</td><td>0.15</td><td>0.07</td><td>0.11</td><td>0.64</td><td>0.27</td><td>0.45</td></tr><tr><td>23-EC-EARTH</td><td>11.87</td><td>0.50</td><td>1.99</td><td>0.07</td><td>0.26</td><td>0.01</td><td>0.13</td><td>0.47</td><td>0.33</td><td>0.40</td></tr><tr><td>24-CESM1-FASTCHEM</td><td>11.56</td><td>0.51</td><td>1.90</td><td>0.06</td><td>0.00</td><td>0.18</td><td>0.09</td><td>0.86</td><td>0.19</td><td>0.53</td></tr><tr><td>25-CESM1-CAM5</td><td>11.88</td><td>0.51</td><td>1.83</td><td>0.07</td><td>0.26</td><td>0.13</td><td>0.20</td><td>0.47</td><td>0.23</td><td>0.35</td></tr><tr><td>26-CESM1-BGC</td><td>11.53</td><td>0.51</td><td>2.01</td><td>0.06</td><td>0.02</td><td>0.11</td><td>0.06</td><td>0.90</td><td>0.24</td><td>0.57</td></tr><tr><td>27-CCSM4</td><td>11.57</td><td>0.51</td><td>1.84</td><td>0.07</td><td>0.01</td><td>0.12</td><td>0.07</td><td>0.85</td><td>0.24</td><td>0.54</td></tr></tbody></table>
<table><tr><td>Fig. 10.</td><td>30-year moving average of the standard deviation of SLP fields across the 20CRv2c ensemble &lt;std(SLP)&gt; versus &lt;d&gt; for 20CR-EM. The color scale shows time.</td><td>38</td></tr></table>
<figure><img src="image_2.png" /><figcaption>FIG. 1. Two realizations of the Lorenz attractor. Blue: classic attractor with: $dt \approx 0.035$, $\sigma = 28$, $r = 10$, $b = 8/3$; Violet: $\sigma = 28.5$.</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>FIG. 2. Schematic representation of the dynamical indicators on fictive SLP maps. H indicates high and L low pressure systems; the arrows mimic horizontal wind fields. The local dimension *d* is proportional to the number of possible configurations preceding and following the day analyzed (here *d* = 2). *θ* is the inverse of the persistence time of a given configuration. If the pattern persist for three days (red path), then *θ* = 1/3. If the patterns change every day (green path), then *θ* = 1.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>FIG. 3. Medians of *d* and *θ* (dots) and standard deviation (semiaxes of the ellipses) for 30 realizations of the Lorenz attractor with *σ* varying continuously over 28 < *σ* < 28.54. The blue markers correspond to the first half of each simulation *σ*₁; the red markers to the second half *σ*₂. The asterisks mark the medians of the ensemble of realizations.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>FIG. 4. Scatterplot of the daily values of instantaneous dimension *d* and inverse persistence *θ* for the 20CRv2c ensemble mean SLP (20CR-EM). The straight black lines mark the 0.02 and 0.98 quantiles of *d* and *θ*. The composite anomalies in SLP obtained averaging the days beyond the quantiles can be associated with known weather regimes: A) Atlantic Ridge (maxima of *θ*), B) NAO- (minima of *θ*), C) Blocking (maxima of *d*), D) NAO+ (minima of *d*). The black contours in panels A-D indicate regions where at least 2/3 of the composite members display sea-level pressure anomalies of the same sign.</figcaption></figure>
<figure><img src="image_6.png" /></figure>

FIG. 5. Scatterplot of the daily values of instantaneous dimension *d* and inverse persistence *θ* for the SLP fields of the 56 individual 20CRv2c members (20CR-ME). The straight black lines mark the 0.02 and 0.98 quantiles of *d* and *θ*. The composite anomalies in SLP obtained averaging the days beyond the quantiles can be associated with known weather regimes: A) Atlantic Ridge (maxima of *θ*), B) NAO- (minima of *θ*), C) Blocking (maxima of *d*), D) NAO+ (minima of *d*). The black contours in panels A-D indicate regions where at least 2/3 of the composite members display sea-level pressure anomalies of the same sign.
<figure><img src="image_7.png" /><figcaption>FIG. 6. ($d, \theta$) bivariate histograms (a,c,e) and scatter-plots (b,d,f) for the 20CR-EM reanalysis (a,b), CMCC-CMS (c,d) and IPSL-CM5A models (e,f). The color scales in (a,c,e) indicate the frequency of observations in number of days. The color scales in (b,d,f) indicate the month of the observation and show the dependence of the ($d, \theta$) diagrams on the seasonal cycle.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>FIG. 7. Comparison between the 56 20CR-ME median values of ($d, \theta$) (black points whose average is denoted by 0), the 20CR-EM (in red and numbered by 1) and all the CMIP5 models (progressive numbers 2-27, see table 1 for details). The semiaxes of the two ellipses represent one standard deviation of $d$ and $\theta$ for 20CR-EM (red) and 20CR-ME (black).</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>FIG. 8. Comparison between $R_{tot}$ values (black) and Wasserstein distances $W$ (red) between the $(d, \theta)$ of 20CR-EM (a) and 20CR-ME (b) and of the CMIP5 models. The distances are normalized by the maximum value.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>FIG. 9. 30-year moving averages $\langle\cdot\rangle$ of instantaneous dimension $d$ and inverse persistence $\theta$ minus the respective mean values over 1851-1880 (a,c) and 1950-1979 (b,d). Note that labels on the abscissa mark the last year in each 30-year averaging window. (a,b): local dimension $d$; (c,d): inverse persistence $\theta$. Red: 20CR-EM; Blue: 20CR-ME; Light blue: single 20CRv2c members; Black: CMIP5 multimodel mean; Grey: single CMIP5 models; Green: zero line.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>FIG. 10. 30-year moving average of the standard deviation of SLP fields across the 20CRv2c ensemble $\langle \text{std}(\text{SLP}) \rangle$ versus $\langle d \rangle$ for 20CR-EM. The color scale shows time.</figcaption></figure>