Secondary current and river-meander formation

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Abstract
A small-perturbation stability analysis is developed for investigation of the role of the secondary current accompanying channel curvature in the initiation and early development of meanders of alluvial and ice- or rock-incised streams. A small sinusoidal perturbation in the channel alignment of an initially straight prismatic channel is introduced. The velocity of the secondary flow is calculated for uniform quasi-steady flow conditions by application of the equation of the conservation of moment of momentum. The formulation is then closed by introducing the assumption that the differential, between the outer and inner banks, rate of boundary erosion, dissolution or melting (for incised channels) or sediment discharge (for alluvial channels) is proportional to the strength of the secondary flow. This formulation leads to a linear differential equation which is solved for its orthogonal components, which give the rates of meander growth and downstream migration. It is shown that the amplitude of the meanders tends to increase and that the meanders migrate downstream. The dominant wavelength and the corresponding phase shift between channel meandering and the velocity of the spiral motion are calculated as those corresponding to the fastest growth rate. The results are found to be in good agreement with data reported by others.

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