- We study the daily river flow fluctuations of 30 international rivers. Using the detrended fluctuation analysis, we study the correlations in the magnitudes of river flow increments (volatilities), and find power-law correlations in volatilities for time scales less than 1 year; these correlations almost disappear for time scales larger than 1 year. Using surrogate data test for nonlinearity, we show that correlations in the magnitudes of river flow fluctuations are a measure for nonlinearity. We propose a simple nonlinear stochastic model for river flow fluctuations that reproduces the main scaling properties of the river flow series as well as the correlations and periodicities in the magnitudes of river flow increments. According to our model, the source of nonlinearity observed in the data is an interaction between a long-term correlated process and the river discharge itself.