A wavelet-based approach to detect climate change on the coherent and turbulent component of the atmospheric circulation

. The modiﬁcations of atmospheric circulation induced by anthropogenic effects are difﬁcult to capture because wind ﬁelds feature a complex spectrum where the signal of large-scale coherent structures (planetary, baroclinic waves and other long-term oscillations) is mixed up with turbulence. Our purpose is to study the effects of climate changes on these two components separately by applying a wavelet analysis to the 700 hPa wind ﬁelds obtained in climate simulations for different forcing scenarios. We study the coherent component of the signal via a correlation analysis to detect the persistence of large-scale or long-lasting structures, whereas we use the theory of autoregressive moving-average stochastic processes to measure the spectral complexity of the turbulent component. Under strong anthropogenic forcing, we detect a signiﬁcant climate change signal. The analysis suggests that coherent structures will play a dominant role in future climate, whereas turbulent spectra will approach a classical Kolmogorov behaviour.

Once the separation between coherent and noisy component is done, we study the property of X(t) and Y (t) separately. For the coherent component X(t) we use as indicator the memory of the system by measuring the integral of the autocorrelation function defined as: where E[X] stands for expectation value. The ACF measures how long the system remember an initial condition. For a white noise signal, it decays to 0 as τ > 1. For a correlated signal it decay slowly to 0 for large τ . For a perfectly periodic signal, the ACF is periodic itself. The integral of the ACF in its discrete version is written us: where we sum the correlation up to a time T sufficiently large for the ACF to decay to 0. Λ measures how long coherent structures persist in time and it is therefore linked to the predictability: the higher the correlation, the higher the probability 5 that the structure will be preserved in future times (Schubert et al., 1992).
For the noisy component Y (t) we use an indicator of the spectral complexity with respect to the canonical Kolmogorov behavior. In order to introduce this indicator we will use the class of Auto Regressive Moving Average stochastic processes. In general, a stationary time series Y t of an observable with unknown underlying dynamics can be modeled by an ARMA(p, q) 10 process such that for all t: with ε t ∼ W N (0, σ 2 ) -where W N stands for white noise -and the polynomials φ(z) = 1 − φ 1 z t−1 − · · · − φ p z t−p and θ(z) = 1−θ 1 z t−1 −· · ·−θ q z t−q . Notice that, hereinafter, the noise term ε t will be assumed to be a white noise, which is a very general condition (Box and Jenkins, 1970).

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The basic model for the noisy component is the ARMA(1,0) or simply AR(1) model which is the simplest compatible with the Kolmogorov spectrum (Thomson, 1987). When the spectral complexity increases, the best ARMA model describing the velocity time series will deviate from the basic one. We can define a normalized distance between the reference ARMA(p, q) and any other ARMA(p, q) model as the normalized difference between the BIC(n,σ 2 , p + 1, q) and the ARMA(p, q) BIC(n,σ 2 , p, q): with 0 ≤ Υ ≤ 1: it goes to zero if the dataset is well described by an ARMA(p, q) model and tends to one in the opposite case.
We have already checked that such indicators perform well in different physical systems, providing more information than the usual ones, based on the critical slow down due to the increase of correlations in the systems at the transition. These analyses 25 have been recently published in (Faranda et al., 2014) where indicators similar to Υ have been used to model different physical systems: Ising and Langevin models and turbulence. A large Υ correspond to a complex spectrum with non-trivial scaleinteractions and non-constant energy transfers, a small Υ correspond to a spectrum compatible with the Kolmogorov spectrum with constant energy fluxes between scales. The predictability will decrease with an higher Υ because more structures at different scales will have to be followed to describe the behavior of the system. In other words, for high Υ the component Y (t) 5 cannot be just modelled as simple noise.

Analysis
We illustrate the potential of Λ and Υ indicators on a climate change experiment used in the CMIP5 framework for the IPCC AR5 report (Collins et al., 2013). To explore the climate change of the next century, the IPCC has developed four different scenarios, defined in terms of radiative evolution and corresponding to a concentration of greenhouse gases year by year between 10 2006 and 2100 and extended until 2300. Here we consider two scenarios: i) the low emission scenario (RCP2.6) leading to a radiative balance of 2.6 W/m2 in 2100 with a peak at 3W/m2 and a decreasing trend ii) the higher emission scenario (RCP8.5) predicting an increase up to 8.5W/m2 in 2100. The effect of such greenhouse gases perturbations are well known for some observables, e.g. the global temperature increase ranges from 1±0.4 • C to 3.7±0.7 • C in the last part of the period 2081-2100 (Collins et al., 2013). 15 We focus on the daily horizontal winds at 700 hPa (u 700 , v 700 ) obtained from the IPSLCM5-LR model. This model is devel-

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We begin the analysis by showing typical maps of Λ and Υ for the scenario RCP 2.6 and the two components: the zonal u 700 and the meridional v 700 (Fig. 2). Λ shows, for the zonal component, a rich structure with several areas where persistent coherent structures are well identified (Fig. 2a) We have devised two indicators to study the changes in the atmospheric circulation by separating the coherent structures from the turbulent part of the signals. The indicator for the coherent structures ∆Λ is a measure of the total persistence of the coherent structure, whereas ∆Υ is a measure of the residual complexity of the turbulent spectrum, once the coherent component has been removed.

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The indicators show significant changes when the climate system is subject to greenhouse gases forcing. The difference in the indicators for the RCP 8.5 scenario between the second half and the first half of the 21st century suggests that El Nino Southern Oscillation will play a major role and blocking conditions will change the typical coherent structures observed at the mid-latitude.
10 Besides the regional patterns, we believe that the most important message is contained in the global average of our indicators.
For the RCP 8.5 scenario, ∆Λ increases by 0.5 days in the second half of the century for u 700 and by 0.2 days for v 700 . On the other hand, the spectral complexity decreases in the tropical regions of about 10%. This suggests that the coherent structures will play a major role in the atmospheric dynamics and this will probably enhance the predictability of the atmosphere on