  Understanding how rivers adjust to the sediment load they carry is critical
to predicting the evolution of landscapes. Presently, however, no physically
based model reliably captures the dependence of basic river properties, such as
its shape or slope, on the discharge of sediment, even in the simple case of
laboratory rivers. Here, we show how the balance between fluid stress and
gravity acting on the sediment grains, along with cross-stream diffusion of
sediment, determines the shape and sediment flux profile of laminar laboratory
rivers which carry sediment as bedload. Using this model, which reliably
reproduces the experiments without any tuning, we confirm the hypothesis,
originally proposed by Parker (1978), that rivers are restricted to exist close
to the threshold of sediment motion (within about 20%). This limit is set by
the fluid-sediment interaction and is independent of the water and sediment
load carried by the river. Thus, as the total sediment discharge increases, the
intensity of sediment flux (sediment discharge per unit width) in a river
saturates, and the river can only transport more sediment by widening. In this
large discharge regime, the cross-stream diffusion of momentum in the flow
permits sediment transport. Conversely, in the weak transport regime, the
transported sediment concentrates around the river center without significantly
altering the river shape. If this theory holds for natural rivers, the aspect
ratio of a river could become a proxy for sediment discharge - a quantity
notoriously difficult to measure in the field.
