Stability of longitudinal sediment waves formed by turbidity currents: Linear and weakly nonlinear perspectives

We perform the linear and weakly nonlinear stability analyses of longitudinal sediment waves triggered by the interaction of turbidity currents with an erodible bed. The mathematical framework is based on the two-dimensional flow equations, advection-diffusion equation for sediment concentration and Exner equation for bed evolution. Using the standard linearization, the linear analysis offers the resonant wavenumber that maximizes the growth rate of sediment waves. We study the influence of the key parameters, such as the gravitational parameter, longitudinal bed slope, Rouse number, shear Reynolds number, relative roughness number and erosion coefficient, on the growth rate, resonant wavenumber and phase velocity. The results reveal that the longitudinal sediment waves migrate both upstream and downstream. By plotting the gravitational parameter against the dimensionless wavenumber, we obtain a stability diagram that accurately captures the experimental data plots within the unstable zone. Additionally, we explore the perturbation fields of velocity components, pressure and sediment concentration. Employing the centre manifold projection, the weakly nonlinear analysis provides the equilibrium amplitude of sediment waves and its sensitivity to the key parameters. The predicted wavelength and amplitude of sediment waves are comparable with field observations in the Atlantic Ocean, the Sea of Japan and the Pacific Ocean.