A key hypothesis we intend to test in this study is that the variability mode corresponding to the South American precipitation dipole could be explained by northward propagating waves triggered by the southern hemisphere Rossby wave train. For this purpose, we first introduced the following conceptual model: Denoting the dimensionless GPH along an arbitrary direction x as h and the precipitation along the direction as p, we chose to model h with a wave equationEmbedded Image(5)with the one-dimensional d’Alembertian Embedded Image and p as its spatial derivativeEmbedded Image(6)since precipitation, on a large scale, typically occurs at the fronts between highs and subsequent lows. This equation solves to a traveling wave for p(x, t). By embedding this traveling wave in the same grid as the data and adding Gaussian damping along and perpendicular to the propagation direction, we generated the model data PM(λ, ϕ, t). Its parameters are the mean values of the Gaussian damping λ0, ϕ0, their SDs σλ, σϕ, the wavelength L, and the direction θ of the wave (see the Supplementary Materials for the full equations). The model data PM(λ, ϕ, t) could be used to calculate the first two EOFs of the conceptual model. These EOFs were then fitted, by optimizing the model parameters via least squares, to the EOFs of the precipitation data (see Fig. 2). While the parameters referring to the Gaussian damping and the direction θ roughly account for the location and orography, the wavelength or wave number is an important parameter of the modeled wave.

The CEOF analysis extends the standard EOF analysis by applying the PCA to the complexified time series, i.e., the analytical signal [e.g., (21)]. The analytical signal Embedded Image is usually computed by augmenting the time series with its Hilbert transform as its imaginary part, so that Embedded Image. The Hilbert transform Embedded Image is defined asEmbedded Image(7)with P.V. denoting the Cauchy principal value of the integral. It induces a 90° phase shift to every frequency component of the time series. Figure 4 shows an example of a Hilbert transform and the signal it was calculated from. This two-dimensional embedding of the time series enabled us to analyze oscillations in time series with methods that rely on phase information, as we have used here throughout the article. Hence, the CEOF method is especially well suited for identifying oscillatory patterns and propagating waves (20). We followed here the notation of Barnett (20): The eigenvectors Bn(x) of the covariance matrix of the spatiotemporal complexified data Embedded Image and its principal components Embedded Image are all complex valued and can therefore not be analyzed directly, as it is the case for the standard EOF analysis. Thus, we investigated the following three measures, which separate the temporal and spatial domain, as well as the phase and amplitude information:

1) Spatial phase function Embedded Image

2) Spatial amplitude function Embedded Image

3) Temporal phase function Embedded Image

More details on CEOF analysis are given in (20).

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