Statistical inference of causal interactions and synchronization between dynamical phenomena evolving on different temporal scales is of vital importance for better understanding and prediction of natural complex systems such as the Earth's climate. This article introduces and applies information theory diagnostics to phase and amplitude time series of different oscillatory components of observed data that characterizes El Niño/Southern Oscillation. A suite of significant interactions between

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... ocesses operating on different time scales is detected and shown to be important for emergence of extreme events. The mechanisms of these nonlinear interactions are further studied in conceptual low-order and state-of-the-art dynamical, as well as statistical climate models. Observed and simulated interactions exhibit substantial discrepancies, whose understanding may be the key to an improved prediction of ENSO. Moreover, the statistical framework applied here is suitable for inference of cross-scale interactions in human brain dynamics and other complex systems.