Statistical properties of dune profiles

Abstract

Properties of sand waves formed by subcritical unidirectional water currents are investigated by statistical analyses of records of streambed profiles. Records of bed elevation y as a function of distance x along the channel, y = y(x), and time records at a fixed point of the channel, y = y(t), were collected in three laboratory flumes that were 8 inches, 2 ft and 8 ft wide and in a straight alluvial channel that was 55 ft wide. For all cases, the bed material was fine sand. The continuous analogue records were converted to discrete data points and were analyzed by digital computer. The analyses show that both types of records, y(x) and y(t), can be approximately represented as stationary Gaussian processes. When the data are standardized and the length or distance are expressed as ratios of the mean duration between zero-crossings of y, the statistical properties of all the flume data are similar, with no distinguishing characteristics that can be attributed to size of flume or to whether the bed forms were ripples or dunes. The field data, however, reflect the influence of large alternate bars that were not present in the flumes. The Gaussian assumption, together with the spectral properties of the records as expressed by a dimensionless parameter, 6, permit predicting the distributions of maximum and minimum values of y between successive zeros of y. These distributions represent the probability distributions of the depth of local scour and fill due to the formation and migration of sand waves, and the parameters that specify the distributions relate approximately to flow velocity and depth. Observed values of the number of zero and h-level crossings, the mean duration between zero crossings, and the mean duration of upward excursions of the process y(t) above the fixed level h compared reasonably well with theoretical values for the Gaussian model. The distribution of the duration of upward excursions is the conditional probability distribution of the rest period of a particle, given that it is deposited on the downstream face of a ripple or dune at the level h. Observed distributions of these durations can be approximated by a gamma distribution with parameters that relate to h, where h is measured in units of standard deviation from the mean bed level. These distributions and other probability distributions that enter into stochastic models of sediment transport can be determined either from the theoretical model or empirically from the observed data. The results of the study ,-show that even though the bed elevation deviates somewhat from the postulated normal distribution, reasonable estimates of many properties of the bed profiles can be derived from fairly simple statistical models.