On the stability of a class of shoreline planform models

The evolution of beaches in response to the incident wave conditions has long attracted the attention of researchers and engineers. A popular mathematical model describing the change in the position of a single height contour on the coastline assumes that the beach profile is stable and the plan shape evolves due to wave-driven long-shore transport. Extensions of this model include more contours and allow for beach profile alteration through cross-shore transport of sediment. Despite this advantage, models with multiple contours remain relatively underused. In this paper we examine the stability of this class of model for the cases of one to three contours. Unstable modes may exist when there is more than one contour. These include short waves whose growth rate is strongly dependent upon wavenumber. For the case of three contours an additional long wave instability is possible. A necessary, but not sufficient, condition for instability is found. It requires a reversal of transport direction amongst the contours. The existence of these instabilities provides a possible explanation for the difficulties found in implementing computational multi-line models, particularly where structures alter the natural longshore transport rates so they satisfy, locally, the condition for instability.