Migrating sand waves

A simple mathematical model is described, which reproduces the major features of sand waves appearance and growth and in particular predicts their migration speed. The model is based on the linear stability analysis of the flat configuration of the sea bottom subject to tidal currents. Attention is focused on the prediction of the complex growth rate G that bottom perturbations undergo because of both oscillatory fluid motions and residual currents. While the real part Gr of G controls the amplification or decay of the amplitude of the bedforms, the imaginary part Gi is related to their migration speed. Previous works on the migration of the sand waves consider a forcing tide made up by the M2 constituent (oscillatory period equal to 12 hours) plus the residual current Z0 and predict always a down-current migration of the bedforms. However field cases exist of up-current migrating sand waves (down-current/up-current migrating sand waves mean bedforms moving in the direction of the steady residual tidal current or in the opposite direction respectively). The inclusion of a tide constituent characterized by a period of 6 hours (M4) is the main novelty of the present work which allows for the prediction of the migration of sand waves against the residual current Z0. Indeed the M4 tide constituent, as the residual current Z0 does, breaks the symmetry of the problem forced only by the M2 tide constituent and induces sand wave migration. The model proposed by [1] forms the basis for the present analysis. Previous works on the subject ([10], [12], [13], [16]) are thus improved by using a new solution procedure ([1]) which allows for a more accurate evaluation of the growth rate for arbitrary values of the parameter r which is the ratio between the horizontal tidal excursion and the perturbation wavelength.