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Numerical modeling of the longitudinal profiles of rivers at grade is accomplished using the basic equations of open-channel flow, sediment transport equations, and empirical relations for downstream variation in flow discharge, sediment discharge, sediment caliber, and channel width. Only in some cases are the computed stream profiles fit exactly by any one of the commonly supported mathematical function analogs to graded profile form-exponential, logarithmic, or power function, but in most cases any of these functions can provide a fit with a degree of error smaller than would be noted in treating field data. Profiles dominated by spatial change in fluid and sediment discharge are distinctly power functions, while profiles dominated by sediment size reduction are not necessarily exponential in form. Other important controls on profile shape are the degree of downstream width change in response to increasing discharge and the general range of sediment size. A dynamic model of a river's approach to grade indicates that disequilibrium river profiles closely approximate a graded profile shape even while the general slope is relatively high, and significant erosion remains to achieve equilibrium.
Snow et al. (Thu,) studied this question.