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A simple diagnostic model for the cross-channel distribution of the depth average velocity in curved open channels

ABSTRACT A simple diagnostic narrow channel shallow water model for the cross-channel distribution of the depth average velocity is developed. It is diagnostic, because the slope of the stream-wise free surface is specified and it models the cross-channel distribution of the depth average velocity without any knowledge of the velocity field. A non-dimensional parameter that encapsulates the channel dimensions and the relative importance of bottom friction to horizontal eddy viscosity is defined. Numerical solutions for the cross-channel distribution of the depth average velocity are obtained using values of this non-dimensional parameter for different cross-channel topographies. Numerical solutions for the depth average velocity are in turn used to find the cross-channel distribution for secondary flow and vertical velocity. Two-dimensional distributions of the vertical velocity in the n–z-plane show that changes in topography dominate the vertical velocity distribution. Three-dimensional tracers in a hypothetical open channel using scaling values from geophysical observations show that secondary circulation has a loose helical flow pattern of approximately one full rotation for every of curvature. It is not unreasonable to suggest that this 1:1 relationship describes the looseness of secondary circulation in geophysical data.