Spatial autoregressive structure in meander evolution
The migration of a river bend depends in part on the high flow velocities that characteristically impinge on its outside bank. Recent models have treated this in terms of a spatial convolution, whereby local bend migration is mathematically a weighted aggregate of up-stream curvature and bed topography. The convolution model can be tested using river migration data after it is discretized and recast into a finite autoregressive form. Published isochrones marking former positions of bends on the Beatton River, Canada, support the hypothesis that rates of bend migration follow a convolutional relation. In addition, a comparison of the underlying flow model with published flume experiments involving constant-curvature bends illustrates how it predicts the near-bank depth-averaged velocity associated with a forced-bar topography in absence of free bars. The autoregressive form of the model is equivalent to a stochastic linear-difference equation; this allows bend curvature to be treated as a random process. Cast in the frequency domain, the convolution model predicts that big bends tend to grow at the expense of little bends and curvature irregularities in complex trains; there exists no tendency for preferential growth of an intermediate bend size. The model also predicts the well-known shift of maximum migration rates to positions down-stream of curvature apexes and implies that the magnitude of this shift increases with decreasing bend size. Predicted shifts compare well with published, measured shifts on the Nishnabotna River, Iowa. The sensitivity of the meandering process to initial bend geometries and entrance flow conditions ensures that diverse bend shapes arise along freely migrating rivers independently of factors such as unsteady flow and nonuniform erodibility. No single geometrical form serves as an asymptotic, evolutionary state for individual bends.