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) \[ ∂_tc̄ + ū∂_xc̄ = κ̄∂^2_xc̄ + ∫_0∞ ∂τD∂^2_xc̄(x-X,t-τ)\,dτ, \] where ū(t) 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.