The exact solution of the problem of river morphodynamics derived in Part 1 is
employed to formulate and solve the problem of planimetric evolution of river
meanders. A nonlinear integrodifferential evolution equation in intrinsic coordinates
is derived. An exact periodic solution of such an equation is then obtained in terms
of a modified Fourier series expansion such that the wavenumbers of the various
Fourier modes are time dependent. The amplitudes of the Fourier modes and their
wavenumbers satisfy a nonlinear system of coupled ordinary differential equations
of the Landau type. Solutions of this system display the occurrence of two possible
scenarios. In the sub-resonant regime, i.e. when the aspect ratio of the channel is
smaller than the resonant value, meandering evolves according to the classical picture:
a periodic train of small-amplitude sine-generated meanders migrating downstream
evolve into the classical, upstream skewed, train of meanders of Kinoshita type.
Evolution displays all the experimentally observed features: the meander growth rate
increases up to a maximum and then decreases, while the migration speed decreases
monotonically. No equilibrium solutions are found. In the super-resonant regime the
picture is essentially reversed: downstream skewing develops while meanders migrate
upstream.Numerical solutions of the planimetric evolution equation are obtained for the
case when the initial channel pattern exhibits random small perturbations of the
straight configuration. Under these conditions, the evolution displays the typical
features of solutions of the Ginzburg–Landau equation, in particular, the occurrence
of spatial modulations of the meandering pattern which organizes itself in the form of
wavegroups. Furthermore, multiple loops develop in the advanced stage of meander
growth.
A large deviation principle for small perturbations of random evolution equations in Hölder norm
