The transport of sediment by a fluid along the surface is responsible for
dune formation, dust entrainment and for a rich diversity of patterns on the
bottom of oceans, rivers, and planetary surfaces. Most previous models of
sediment transport have focused on the equilibrium (or saturated) particle
flux. However, the morphodynamics of sediment landscapes emerging due to
surface transport of sediment is controlled by situations out-of-equilibrium.
In particular, it is controlled by the saturation length characterizing the
distance it takes for the particle flux to reach a new equilibrium after a
change in flow conditions. The saturation of mass density of particles
entrained into transport and the relaxation of particle and fluid velocities
constitute the main relevant relaxation mechanisms leading to saturation of the
sediment flux. Here we present a theoretical model for sediment transport
which, for the first time, accounts for both these relaxation mechanisms and
for the different types of sediment entrainment prevailing under different
environmental conditions. Our analytical treatment allows us to derive a closed
expression for the saturation length of sediment flux, which is general and can
thus be applied under different physical conditions
