Abstract
In this paper, we explore the mega riverbed-patterns, whose longitudinal and vertical length dimensions scale with a few channel widths and the flow depth, respectively. We perform the stability analyses from both linear and weakly nonlinear perspectives by considering a steady-uniform flow in an erodible straight channel comprising a uniform sediment size. The mathematical framework stands on the dynamic coupling between the depth-averaged flow model and the particle transport model including both bedload and suspended load via the Exner equation, which drives the pattern formation. From the linear perspective, we employ the standard linearization technique by superimposing the periodic perturbations on the undisturbed system to find the dispersion relationship. From the weakly nonlinear perspective, we apply the centre–manifold-projection technique, where the fast dynamics of stable modes is projected on the slow dynamics of weakly unstable modes to obtain the Stuart–Landau equation for the amplitude dynamics. We examine the marginal stability, growth rate and amplitude of patterns for a given quintet formed by the channel aspect ratio, wavenumber of patterns, shear Reynolds number, Shields number and relative roughness number. This study highlights the sensitivity of pattern formation to the key parameters and shows how the classical results can be reconstructed on the parameter space.
Highlights
The formation of incipient alternate bars explains the possible origin of the onset of meandering of rivers [7], while the central and braided bars cause the braiding of rivers [4]
The riverbed-patterns are thought of as double-harmonic waves that develop as a consequence of the so-called instability phenomenon, which arises as a response of the fluid–riverbed interface to the flow and the particle transport [8]
In the unstable zone, the incipient riverbed-patterns with a given transverse mode grow with time
Summary
For a given flow and particle transport conditions, the mega riverbed-patterns are expected to form above a threshold aspect ratio (e.g. channel half-width to flow depth ratio) [3]. Most of the theoretical studies have analysed the pattern formation in bedload-dominated channels [3,28,29,31], some studies have taken into account the effects of sediment suspension in addition to the bedload transport [9,30,32]. We analyse the mega riverbed-patterns, being free from an external forcing, from both linear and weakly nonlinear perspectives. The main objective of the linear stability analysis is to explore how the mega riverbed-patterns are sensitive to the characteristic parameters, e.g. channel aspect ratio, shear.
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