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The amplitude of the El Niño-Southern Oscillation (ENSO) would be normally distributed if the coupled Pacific ocean-atmosphere were a linear system forced by Gaussian weather noise. Moment estimates of skewness and kurtosis demonstrate that this is not the case for monthly mean anomalies in Pacific sea surface temperatures during 1950–97. The noted predominance of El Niño events compared to La Niña events is related to the high skewness in the eastern Pacific. Skewness and kurtosis both exhibit an intriguing geographical variation from positive in the eastern to negative in the western Pacific. We have also examined Niño-3 indices generated by three climate models having widely different complexity. These exhibit a wide range of skewness and kurtosis values rather different from those found for the observations. Skewness and kurtosis can be used to diagnose non-linear processes and provide powerful tools for validating models, and for testing observed sea-surface temperatures for the presence of possible climate change.