Bed-flow instability of the longshore current

An initially uniform longshore current on a plane erodible beach is considered and a linear stability analysis of the bed-flow system is performed in order to investigate the growth of alongshore periodic topographic features such as transverse or oblique bars. γ, numerical model based on the shallow water equations and a simple sediment transport formula is used. For a wide range of parameters instability is found, leading to the growth of large-scale topographic features (lengthscale of the order of the current width) downflow progressing. The growth rates and the dominant unstable mode depend mainly on R = cd/β parameter, where cd is the bottom friction coefficient and β is the beach slope. For a small R, say less than 0.1, instability is very weak, probably negligible. For R between 0.1 and 0.7 instability increases with R, leading typically to a quite simple transverse bars pattern. A further increase in R produces a far more complicated behaviour where complex patterns with downcurrent oriented oblique bars, bumps and holes can be dominant. In this region growth rates may either decrease or increase with R depending on the beach slope and the maximum Froude number of the basic flow, F. Usually, the most complex behaviour is found for gently sloping beaches. The physical mechanism of the instability is found to lie on the disturbances of potential vorticity caused by topographically induced differences in bottom friction. In this sense it is similar to the alternate bars growth in a river rather than the dunes or antidunes occurrence for 1D channel flow. The predictions of the model compare well with the available experimental data. The alongshore wavelength, γ, typically of the order of one to four times the width of the current, is close to four times for the most common values of R. The typical growth time is proportional to γ2 and for a wavelength of 100 m can be of the order of one day, depending on the sediment transport rate. The migrational speed is inversely proportional to γ, in accordance to earlier field data reported by Sonu (1969) Collective movement of sediment in littoral environment.