Modeling of annual oscillations and 1/ f-noise of daily river discharges

Abstract

Summary A frequency dependence of river runoff power spectrum is deduced from the integrated runoff and storage dynamics, derived from the mass and momentum balance equations for the water on a river catchment. The power spectrum consists of three multipliers: a 1/ f β noise factor with β ∼ 1 reflecting a stochastic behavior of precipitation; a Lorentzian factor 1 / ( f c 2 + f 2 ) damping high frequency runoff oscillations; and the sum of spectral modes of annual and intra-annual periodicity. This spectrum has two different power-law trends, separated by the crossover frequency f c , with the exponents β and β + 2 at low and high frequencies, respectively. The theoretical results are tested on four discharge time series (3580–6575 daily records) of the river Volga at Staritsa and three rivers of the Volga basin, catchment areas varied from 1850 to 21,100 km 2 . The time series are used to calculate empirical power spectra of river discharges and to construct the theoretical power spectra by fitting model parameters to the data. The calculations indicate that the spectral exponent β is approximately equal to 1 for three rivers and to 0.67 for one river. The crossover frequency f c corresponds to a period of 12 days for large catchment areas and ∼3 days for small one. The comparison shows that the model and empirical spectra by their spectral trends at low and high frequencies, crossover frequency, and positions of first several spectral modes are in satisfactory agreement. To isolate an effect of fluctuations, we consider a transformed runoff time series for one of the rivers, obtained by subtraction of the multiyear average daily hydrograph from the raw time series. The transformed time series interpreted as discharge fluctuations is used to calculate the fluctuation power spectrum. This spectrum has two different power-law trends with a spectral exponent of 0.6 at low frequencies and 2.7 at high frequencies. The difference is about 2 like for the power spectrum of raw time series, but the crossover frequency slightly moves to a value corresponding to a period of 15 days.