We compiled a data set of 541 bankfull measurements of alluvial rivers (see supporting information) and used Bayesian linear regression to examine empirical and theoretical support for the hypothesis that alluvial channels adjust to a predictable condition of basal shear stress as a function of sediment transport mode. An empirical closure based on channel slope, bankfull channel depth, and median grain size is proposed and results in the scaling of bankfull Shields stress with the inverse square root of particle Reynolds number. The empirical relationship is sufficient for purposes of quantifying paleohydraulic conditions in ancient alluvial channels. However, it is not currently appropriate for application to alluvial channels on extraterrestrial bodies because it depends on constant‐valued, Earth‐based coefficients.

Fishes that swim upstream in rivers to spawn must navigate complex fluvial velocity fields to arrive at their ultimate locations. One hypothesis with substantial implications is that fish traverse pathways that minimize their energy expenditure during migration. Here we present the methodological and theoretical developments necessary to test this and similar hypotheses. First, a cost function is derived for upstream migration that relates work done by a fish to swimming drag. The energetic cost scales with the cube of a fish's relative velocity integrated along its path. By normalizing to the energy requirements of holding a position in the slowest waters at the path's origin, a cost function is derived that depends only on the physical environment and not on specifics of individual fish. Then, as an example, we demonstrate the analysis of a migration pathway of a telemetrically tracked pallid sturgeon (Scaphirhynchus albus) in the Missouri River (USA). The actual pathway cost is lower than 10⁵ random paths through the surveyed reach and is consistent with the optimization hypothesis. The implication—subject to more extensive validation—is that reproductive success in managed rivers could be increased through manipulation of reservoir releases or channel morphology to increase abundance of lower‐cost migration pathways.