An analytical model for the morphodynamics of a trench in the presence of tidal currents

Equations are derived for the displacement and deformation of the cross-section of a trench in a sandy bottom. The trench is subject to tidal currents and waves. Tidal currents are rectilinear and perpendicular to the longitudinal axis of the trench. The tidal velocity comprises the residual velocity M0 and the harmonic constituents M2 and M4. The effect of waves on the sediment transport is incorporated as stirring. Separate equations are derived for bed-load and suspended-load transport. The equations are averaged over the tidal time scale assuming that the morphological time scale is large compared to the tidal period. To determine the leading order terms, variables are scaled and the equations are written in dimensionless form. The relative magnitude of the terms is determined by a small parameter ε. ε is the ratio of trench depth and undisturbed water depth. Retaining only leading order terms, the equations reduce to advection–diffusion equations with constant coefficients. The cross-section migrates, widens and shallows. The velocity of migration is a function of the residual velocity and the amplitudes and phases of the M2 and M4 constituents. Widening and shallowing is a result of diffusion. For bed-load transport, diffusion derives from the effect of the bed slope on the sediment transport. For suspended-load transport, diffusion is assumed to be the result of velocity shear and vertical turbulent mass exchange (shear dispersion). When accounting for higher order nonlinear terms an initially symmetric cross-section becomes asymmetric. Application to the access channel to the Port of Amsterdam yields displacement velocities of 1.3 m yr−1 for bed-load transport and 1.8 m yr−1 for suspended-load transport. The initial rates of increase of the half-width of the channel are, respectively, 0.33 and 1.02 m yr−1.