Perturbations of channel geometry (like variations of channel curvature or channel
width) in meandering rivers give rise to morphodynamic effects which display themselves
through the development of large-scale perturbations of bottom topography in
the form of stationary bars developing in the longitudinal direction. The latter may
then drive the lateral migration of the channel by enhancing bank erosion at bar
pools: through this mechanism local perturbations of channel geometry may affect
the planimetric development of meandering rivers on large timescales. The problem
tackled herein is whether such morphodynamic influence is invariably felt downstream
as the commonly employed model of river meandering would suggest.
In order to solve this problem, we derive the exact solution of the linearized form of
the mathematical problem of river morphodynamics. Linear analysis had pointed out
the existence of a resonance phenomenon: in a linear (hence ideal) context, resonance
occurs when the meander wavenumber and the width ratio of the channel take values
(λR and βR, respectively) such as
to force free spatial modes of the system consisting
of free bars which neither grow nor decay either in time or in space. Channels
characterized by values of the width ratio β larger (smaller) than βR are called
super- (sub-)resonant. The present solution, which applies to channels with constant
width and arbitrary curvature distribution, shows that two distinct scenarios may
occur: downstream influence is associated with sub-resonant channels and vice versa
dominant upstream influence occurs in super-resonant channels. Small-amplitude
waves of bottom topography are shown to migrate downstream in the former case
and may migrate upstream in the latter, as resonance also defines the threshold
conditions below (above) which small-amplitude alternate bar perturbations (may)
migrate downstream (upstream).
These results have several implications. In the present paper we examine the
overdeepening phenomenon whereby abrupt variations of channel curvature, as in
sequences of straight and constant curvature reaches, lead to sequences of stationary
alternate bars with amplitude decaying in the longitudinal direction. We show
that, along with downstream overdeepening, an upstream overdeepening scenario is
predicted in the super-resonant regime.
Implications of the upstream influence on planimetric development of meandering
rivers are investigated in Part 2.