  The river bar instability is revisited, using a hydrodynamical model based on
Reynolds averaged Navier-Stokes equations. The results are contrasted with the
standard analysis based on shallow water Saint-Venant equations. We first show
that the stability of both transverse modes (ripples) and of small wavelength
inclined modes (bars) predicted by the Saint-Venant approach are artefacts of
this hydrodynamical approximation. When using a more reliable hydrodynamical
model, the dispersion relation does not present any maximum of the growth rate
when the sediment transport is assumed to be locally saturated. The analysis
therefore reveals the fundamental importance of the relaxation of sediment
transport towards equilibrium as it it is responsible for the stabilisation of
small wavelength modes. This dynamical mechanism is characterised by the
saturation number, defined as the ratio of the saturation length to the water
depth Lsat/H. This dimensionless number controls the transition from ripples
(transverse patterns) at small Lsat/H to bars (inclined patterns) at large
Lsat/H. At a given value of the saturation number, the instability presents a
threshold and a convective-absolute transition, both controlled by the channel
aspect ratio {\beta}. We have investigated the characteristics of the most
unstable mode as a function of the main parameters, Lsat/H, {\beta} and of a
subdominant parameter controlling the relative influence of drag and gravity on
sediment transport. As previously found, the transition from alternate bars to
multiple bars is mostly controlled by the river aspect ratio {\beta}. By
contrast, in the alternate bar regime (large Lsat/H), the selected wavelength
does not depend much on {\beta} and approximately scales as H^(2/3)L^(1/3)/C,
where C is the Ch\'ezy number.
