A perturbation method for nonlinear dispersive wave problems

Abstract

A Simple purely empirical ecxpression has been found which closely correlates the existing laboratory data on the measured rates of unsuspended, bedload transport of sediment in flumes with recently published data on the like transport rates in a wide variety of natural rivers. Let $\omega$ be the stream power $\overline{\tau}\overline{u}$ per unit bed area, and let $\omega_0$ be the threshold value of $\omega$ at which sediment begins to be moved. Then if the flow depth is Y and the peak size, or mode, of the sediment is D, the transport rate i$_b$, by immersed weight, varies empirically as $i_b \propto (\omega - \omega_0)^\frac{3}{2} Y^{-\frac{2}{3}} D^{-\frac{1}{2}}.$ This is dimensionally incomplete for it omits, for instance, the excess density of the solids, which happens to be virtually constant for all the data examined. The relation can however be made dimensionless by writing $\frac{i_b}{(i_b)_*} = \big\lbrack\frac{\omega - \omega_0}{(\omega -\omega_0)_*}\big\rbrack^\frac{3}{2} (\frac{Y}{Y_*})^{-\frac{2}{3}} (\frac{D}{D_*})^{-\frac{1}{2}},$ where the starred symbols refer to any standard set of values chosen from a reliable experimental plot. The above expression has so far been found to predict the average actual values within the limits of measuremental error over 10$^3$-fold ranges of flow depth Y and mode size D, and over a 10$^4$-fold range of stream power $\omega$. When the bed material of a natural river is bimodal, the measured transport rates are found to occur, seemingly at random, between the two limiting values computed respectively for the larger and the smaller values of mode size D. A physical explanation of such an apparently general correlation poses an intriguing problem in fluid mechanics. Some qualitative clues are discussed and a tentative outline explanation is given of the general tendency of streams that flow over mobile material towards a quasi-rhythmic fluctuation of channel cross-section.