An information-theoretic approach for detecting interactions and information transfer between two systems is extended to interactions between dynamical phenomena evolving on different time scales of a complex, multiscale process. The approach is demonstrated in the detection of an information transfer from larger to smaller time scales in a model multifractal process and applied in a study of cross-scale interactions in atmospheric dynamics. Applying a form of the conditional mutual information

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... and a statistical test based on the Fourier transform and multifractal surrogate data to about a century long records of daily mean surface air temperature from various European locations, an information transfer from larger to smaller time scales has been observed as the influence of the phase of slow oscillatory phenomena with the periods around 6-11 years on the amplitudes of the variability characterized by the smaller temporal scales from a few months to 4-5 years. These directed cross-scale interactions have a non-negligible effect on interannual air temperature variability in a large area of Europe.