On the time development of meander bends

According to the theory of Ikeda, Parker & Sawai (1981), meander migration rate at a point depends on a convolution integral of channel curvature from that point upstream. The problem can be quantified in terms of the bend equation. The time development of periodic bend trains of finite amplitude is analysed using the method of two-timing. The results apply near the critical wavenumber for the growth of bends of infinitesimal amplitude. A finite-amplitude equilibrium state bifurcating from the null state at the critical wavenumber was delineated by Parker, Diplas & Akiyama (1983), they called the resulting solution the Kinoshita curve. It is found herein that this equilibrium state is unstable. Bends of longer Cartesian wavelength grow to cutoff. Shorter bends are obliterated. Nevertheless, in either case, the bend train tends towards the shape of the Kinoshita curve. The theory suggests that some growing bends may be stabilized by local obstructions to downstream migration. The obstructions would cause an effective reduction in Cartesian wavelength, moving the bend from the unstable regime to the stable regime. A rather crude check of bend shape and rates of deformation generally lends support to the analysis.