Abstract
The evolution of a localized disturbance imposed upon an otherwise uniform alluvial flow is considered. For small disturbances a linearized theory is developed which shows that the initial disturbance splits into two modes. One mode is stationary and purely diffusive while the other mode propagates. The propagating mode may exhibit diffusion or, for sufficiently high Froude numbers instability of the “roll-wave” type. The theory provides the relevant diffusion, propagation and instability time scales associates with the two modes.
For finite amplitude disturbances, a weakly nonlinear theory is considered. Again the disturbance separates into two modes. The stationary mode remains as a solution of the diffusion equation, but the propagating mode is now governed by a Burger's equation.
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Needham, D.J. The development of a bedform disturbance in an alluvial river or channel. Z. angew. Math. Phys. 39, 28–49 (1988). https://doi.org/10.1007/BF00945720
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DOI: https://doi.org/10.1007/BF00945720