Similar to two chaotic oscillators that begin to synchronize once brought into contact with each other [e.g., (15)], we also expected oscillatory climatic subsystems that are coupled to each other to exhibit this behavior. To infer the phase coherence between two observables, we first needed a two-dimensional embedding of each time series. A common approach for this purpose is to calculate the analytic signal of the time series via a Hilbert transform (15), which is defined in Eq. 7. To define a meaningful phase of the time series, this signal needs to exhibit a well-centered oscillation around a common reference point. Instead of the time series itself, Osipov et al. (36) argue that it is also possible to define a meaningful phase by using the derivative and its Hilbert transform. This results in a more concise definition of the phase, since the derivative is better centered than the time series itself, and slow variations are eliminated (16). The derivatives were calculated with the standard fourth-order finite differences formulas. Thus, denoting x(t) as any of the three time series, we defined its phase asEmbedded Image(8)

Figure 4 shows an example of a time series and its embedding. We saw that the definition of the phase in the above described way is justified, since most oscillations revolve around the origin. After each full period, 2π was added to unwrap the phase. As we were investigating a seasonal phenomenon, we were interested only in phase coherence during the NDJF season. Thus, we only considered NDJF data and performed an end point matching to concatenate the data of different seasons. The end point matching minimizes the Euclidean distance between the joint vector of all three time series, their derivatives and Hilbert transforms, as well as an additional penalty that is linear in time, and favors end points late in the season and start points early in the season.

Aside from investigating and comparing the phase difference time series, we also investigated the histogram of the phases of all observables (see Fig. 6). To assess the significance of these phase histograms, we used iAAFT surrogates. These surrogates are refined Fourier transform surrogates. Fourier transform surrogates were computed by multiplying the Fourier-transformed time series with a random phase vector and transforming it back into the original space. Therefore, the surrogates exhibit the same spectrum as the original time series but have randomized phases. For a detailed account of these surrogates, see (29).

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