0 Dynamical analysis of a reduced model for the North Atlantic Oscillation 1

We apply a regularized vector autoregressive clustering technique to identify recurrent and persistent states of atmospheric circulation patterns in theNorthAtlantic sector (110◦W-0◦E, 20◦N-90◦N) associated with the Atlantic Ridge (AR) and the North Atlantic Oscillation (NAO). The technique additionally provides the temporal behavior in terms of a time-dependent switching between the respective cluster states. Using the resulting cluster affiliations for each day, we set the switchingsequence a priori to define a non-smooth linear delayed map that we use to analyze the dynamics associated with the resulting cluster-based model. We compute the time-dependent covariant Lyapunov vectors (CLVs) and their associated finite-time covariant Lyapunov exponents (FTCLEs), with a particular focus on indicators of transitions between the states. We find that the window chosen to compute the CLVs acts as a filter on the dynamics. For short windows, CLV alignment and changes in FTCLE growth rates are indicative of individual transitions between persistent states. For long windows, we observe an emergent annual signal manifest in the alignment of the CLVs characteristic of the observed seasonality in the respective NAO and AR indices. Analysis of the average finite-time dimension reveals the NAO− as the most unstable state relative to the NAO+, with persistent AR states largely stable. 10


Introduction
the atmospheric circulation in the Atlantic sector, the FEM-BV-VAR method yields a set of states 143 consistent with differing phases of the NAO. By treating the clustering as a non-smooth linear 144 delay system, it is possible to directly compute the Lyapunov spectrum and CLVs of the model, as 145 well as dynamical indicators of transitions such as increased finite-time instability (Norwood et al. 146 2013) and alignment of CLVs (Beims and Gallas 2016;Sharafi et al. 2017;Kuptsov and Kuznetsov 147 2018). The relationship between these dynamical quantities and the particular regime transitions 148 can then be compared to assess whether the reduced-order model exhibits non-trivial dynamics. 149 In this study we analyze the optimal model for the NAO resulting from applying the FEM-BV-150 VAR method to atmospheric reanalysis data. The remainder of this article is structured as follows.

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In section 2 the data and clustering methods used to derive a reduced order model for circulation 152 regimes is described. We introduce the general properties of the optimal model and validate it 153 against an observed NAO index. In section 3 we define the corresponding discrete time dynamical 154 system through construction of a delay-embedded non-smooth linear map that corresponds to the 155 time-dependent dynamics of the optimal model from the fit. Through this novel interpretation of 156 the system we calculate the corresponding CLVs and their properties as they evolve in time. We therefore choose to keep the leading = 20 PCs, accounting for approximately 91% of the total variance; the corresponding EOFs are shown in appendix A. Additionally, to assess the qualitative 185 behavior of the regimes identified by the clustering analysis, we make use of the daily NAO index Given the daily timeseries of = 20 PCs between 1 January 1979 and 31 December 2018, 191 corresponding to a sample of length = 14610 days, we next extract a set of persistent states by 192 applying the FEM-BV-VAR clustering method (Horenko 2010b;Metzner et al. 2012). 193 In this approach, the behavior of the system is taken to be described by an underlying model The resulting model is interpreted as representing the observed fields in terms of a set of recurrent 207 circulation regimes that govern the local, short-term (e.g., day-to-day) variability, which the system 208 repeatedly transitions between.

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To determine both an assignment of individual days to a state as well as the parameters Θ 210 characterizing each state, we minimize a loss function of the form such that the loss function is a convex combination of the individual losses and the complete set 217 of affiliations = [γ 1 , . . . , γ ] ∈ R × may be interpreted as providing a soft clustering of the 218 data into the states. The observed persistence of large-scale coherent features in the mid-latitude 219 troposphere implies that the switching process described by the affiliations should also exhibit 220 some degree of persistence, yielding regimes that are metastable. To enforce this behavior, the 221 affiliation sequence is required to satisfy a constraint on the total variation norm of the sequence , 222 In the usual formulation of FEM-BV clustering, it is further assumed that the affiliations can be expressed in terms of a set of compactly supported basis functions. When each basis function is non-zero over more than one time step, this essentially imposes a minimum length of time that must be spent in a given state. We choose triangular basis functions that are non-vanishing at only a single time point, allowing state transitions between adjacent time points. of the form for some constant . Each term in this sum is non-zero only if the affiliations differ between 224 times and + 1, corresponding to a transition between states, so that this constraint imposes an 225 upper bound on the total number of transitions between states. It is more convenient to express this 226 constraint in terms of a "typical" state length ≥ 0 that is independent of the time series length, in 227 terms of which we define as The form of the loss functions ℓ (x , Θ ) is governed by the assumed dynamics within the hidden 229 states. For the FEM-BV-VAR clustering method, the time evolution of the system within a given A numerical method for finding the minimum of the resulting loss function with respect to and 237 is summarized in appendix B.

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The number of clusters , VAR order , and state length constitute the set of hyperparameters 239 that must be chosen beforehand when applying the above procedure. data, and so we select as our optimal model the set of hyperparameters that minimize this metric.

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The results of this cross-validation procedure, using fold = 10 cross-validation folds, are summa-248 rized in F .
where I( ) is an indicator function equal to one if is true and zero otherwise, or by applying

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Based on the above analysis we have some confidence that the optimal FEM-BV-VAR model 302 extracts a set of metastable states that can be related to coherent features in the North Atlantic.

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We next assess whether a simplified dynamical model derived from this fit can be used to study 304 the dynamics associated with regime transitions between those states. To do so, the optimal FEM-system based on Eq. (1) in which the time evolution is given by where is the fitted state assignment given by Eq. (7). The cluster means µ (1) , µ (2) , µ (3) and 308 parameter matrices A ( ) for , ∈ {1, 2, 3} are constant. Note that, by constructing the model in 309 such a way, the dynamics will change in the time step prior to a transition in the affiliation sequence.

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We are interested in whether the dynamical properties of the resulting model from the FEM-

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BV-VAR framework can show any insight on the mechanisms characterizing transitions between 312 states and whether the reduced dynamical model exhibits properties that are physically plausible.

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In order to study the dynamics we use the resulting affiliation sequences and parameter matrices 314 from the optimal FEM-BV-VAR model to construct the following system: Eq. (10) describes a discrete non-smooth linear mapping system governing the tangent dynamics where A ( , 0) is the linear propagator defined by We use the cocycle A ( , ) to calculate the CLVs of the system in Eq. (10). The CLVs φ satisfy 326 the Multiplicative Ergodic Theorem (Oseledets 1968), where is the asymptotic growth rate of vectors in subspace Φ . Although the linear maps are not  In the following sections we investigate the growth rates and alignment of the leading CLVs. 336 We compare the behavior for the different push forward steps and analyze how changes in either 337 property relates to transitions between the states. The first property of the CLVs that we analyze is their finite-time growth rates, i.e., finite-340 time covariant Lyapunov exponents (FTCLEs). Due to the rapid transitioning between states, 341 we consider the growth rates over the course of one day. We define the FTCLEs as in Wolfe 342 and Samelson (2007), here Eq. (14a). To calculate the FTCLEs we use a forward difference 343 approximation to the derivative, which in our case simplifies to applying the linear propagator to 344 the CLV calculated for a given day and taking the difference of the L 2 -norms: Note that φ ( ) = 1 for CLVs computed using the Froyland et al. (2013) algorithm and therefore 346 the scaling factor is omitted from Eq. (14b).

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We compare the FTCLEs computed using Eq. (14b) to the asymptotic growth rates computed 348 from the QR decomposition method (appendix C). For the computation we use the full matrix 349 cocycle over the period of the FEM-BV-VAR fit and an orthonormalization time step of 1 day. 350 We find that asymptotically the model is stable and there is little evidence of a spectral gap in 351 the leading exponents. F . 4 plots the asymptotic exponents compared to the statistics of the 352 FTCLEs calculated for each push forward step. It can be seen that as the push forward step is 353 increased, the mean FTCLEs approach the asymptotic values and the standard deviation decreases 354 for the leading growth rates. Since the finite-time and asymptotic growth rates are computed using 355 different methods, this agreement provides confidence in the accuracy of the CLV calculation.

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To quantify the total transient growth at each time step in an asymptotically stable system, we Then the finite-time dimension measure is given as   calculate the alignment using the following: We first consider the alignment of the CLVs calculated for the short push forward step ( = 3). which remains close to Λ 2 and both oscillate around zero. We see that for long enough residency 444 in the NAO + state the instability is driven by Λ 2 overtaking Λ 1 .

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In order to obtain a more complete understanding of the alignment behavior around transitions, The peak in 1,2 occurs on the last day the affiliation sequence is in the preceding state. We also 457 observe that there is a spike in 2,3 following both transitions from the NAO − state; for NAO − 458 to AR it occurs on the day following the peak in 1,2 and for NAO − to NAO + it occurs two days 459 following. For both transitions from the NAO + state there is an increase in 1,2 , 2,3 , and 1,3 ,  Next we consider the behavior of the alignment of the leading two CLVs, 1,2 ( ), across the 466 varying push forward lengths. This is displayed in the panels of F . 10a. The first difference 467 we notice is in the timescale of variability of the alignment. For shorter push forward lengths we 468 observe that large changes in alignment occur more often than for longer push forward lengths. We 469 also observe the emergence of a low-frequency signal within the variability as the push forward is seen more through the total number of days spent in a given state and average residency times.

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As mentioned in section 2c, the NAO − state accounts for 46.5% of the total number of model days.

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The largest contribution to that comes from JJA (41%) compared to DJF which only accounts for 500 11% of NAO − days.This seasonality is similar to, but much more pronounced than, that observed 501 for the CPC NAO index; over the same period as the model fit, 45% of days had a negative daily 502 mean index, and 20% of these days occurred during DJF compared to 29% accounted for by JJA.

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The average residency length also has a seasonal signal (T 1), with its maximum in JJA (9.3 504 days) and minimum in DJF (2.5 days). We observe as expected a seasonal signal in the transition to the overall increase in DJF variability compared to JJA. 514 We now turn to the average behavior of alignment by season. F . 12 shows the alignment 515 averaged over each season of the indicated pairs of CLVs. We see a clear seasonal behavior of 1,2 516 with a maximum in summer and a minimum in autumn and winter. Interestingly, there is also a 517 seasonal signal in 2,3 , 2,4 and 3,4 (although weaker for 2,4 and 3,4 ). We do not see a seasonal 518 cycle in the alignments with the more asymptotically stable CLVs (5-7) as their dominant signals 519 have a cycle length of less than a year.

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We have presented here a dynamical analysis of a reduced model for the NAO teleconnection.

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The preferred model has been constructed through application of the FEM-BV-VAR method which 523 has been previously used to identify atmospheric pressure states consistent with known coherent the estimated parameters for state at fixed may be compactly written as where Tr[A] denotes the trace of a matrix A. This coordinate descent method finds a local minimum 597 of the loss function for a given initial guess at the optimal parameters and not necessarily a globally 598 optimal solution. In order to reduce the degree to which this occurs, in all of the results presented