Sub-grid transport of matter in two-dimensional tidal flow over an uneven bed

Abstract

A bottom topography, the horizontal length scale of which is too small to be resolved in a given coarse-grid two-dimensional tidal computation, may cause circulations at a sub-grid scale. These sub-grid circulations may influence the coarse-grid representation of the distribution of the concentration of matter discharged into the considered tidal flow. This paper explores the possibility of treating the sub-grid transport of matter and its effect on the coarse-grid concentration field following Taylor's development for turbulent pipe flow, i.e. by approximating this transport as a gradient type sub-grid dispersive transport due to the interaction between horizontal mixing and horizontal shear, induced by the sub-grid bottom topography. It shows that this method can be applied to a bottom topography with a height which is small compared with the water depth, provided that a non-linear time dependence is taken into account. This nonlinear time dependence is due to the fixed and non-randomly distributed position of the sub-grid circulations in the horizontal plane. For a bottom topography with a horizontal length scale between one and six times the amplitude of the tidal excursion it may be expected to induce resonance with respect to the production of sub-grid variations in the concentration field and the magnitude of the sub-grid dispersion coefficient.