River meandering dynamics.

The Ikeda, Parker, and Sawai river meandering model is reexamined using a physical approach employing an explicit equation of motion. For periodic river shapes as seen from above, a cross-stream surface elevation gradient creates a velocity shear that is responsible for the decay of small-wavelength meander bends, whereas secondary currents in the plane perpendicular to the downstream direction are responsible for the growth of large-wavelength bends. A decay length D=H/2C(f) involving the river depth H and the friction coefficient C(f) sets the scale for meandering, giving the downstream distance required for the fluid velocity profile to recover from changes in the channel curvature. Using this length scale and a time scale T, we explicitly trace the observed length scale invariance to the equations of motion, and predict similar time and velocity scale invariances. A general time-dependent nonlinear modal analysis for periodic rivers reveals that modes higher than the third mode are needed to describe upstream migration of bend apexes just before oxbow cutoff, and are important to accurate calculations of the time and sinuosity at cutoff.