Contaminant dispersion in oscillatory flows

If the time scale for cross-sectional mixing is comparable with or larger than the flow period, then, after each flow reversal, there can be a substantial time span in which the contaminant cloud is contracting. Thus, the apparent longitudinal-diffusion coefficient is negative. This means that the contaminant dispersion cannot be modelled by a diffusion equation, because negative diffusivities imply the spontaneous development of infinite concentrations. Here it is shown how this periodic contracting and expanding can be modelled by a delay-diffusion equation (Smith 1981) \[ \partial_t\overline{c} + \overline{u}\partial_x\overline{c} = \overline{\kappa}\partial^2_x\overline{c} + \int_0^{\infty} \partial_{\tau}D\partial^2_x\overline{c}(x-X,t-\tau)\,d\tau, \] where is the bulk velocity, X(t, τ) a coordinate displacement, and D(t, τ) the diffusion coefficient at time τ after discharge. The recent memory ∂τD is always positive and diffusive in character, so singularities cannot arise. However, when τ is large this memory function can be negative because of reversed flow at earlier times. Particular attention is given to estuarial flows and results are derived for the dependence of D upon the water depth and upon the width of the estuary.