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The effect of cross-stream depth variations upon contaminant dispersion in a vertically well-mixed current

The effects of two depth topography configurations on the dispersion of contaminant from an outfall in shallow water are investigated theoretically. Firstly, a sharp cross-stream depth change, with uniform water depth and current velocity on either side of it, is considered; the second configuration is that of a uniformly sloping beach with a longshore current. The dispersion of contaminant is modelled by a two-dimensional advection-diffusion equation. Current velocities scale as the square root of the water depth, and eddy diffusivities as the 3/2 power of the water depth, though these scalings are not crucial to the solution of the problem. By neglecting longitudinal diffusion, it is possible to obtain solutions for both configurations: the method of images is used for the sharp depth change, and Laplace transforms for the sloping beach. The solutions are represented graphically by drawing contours of contaminant concentration, and also by plotting the contaminant flux. These plots show that, while deeper regions have greater contaminant flux because of the higher current velocity and eddy diffusivity, contaminant tends to accumulate to higher concentrations in shallower water; the direction of motion of contaminant should not be confused with the direction of the central axis of the contaminant plume. Finally, for the sloping beach, the concentration of contaminant at the shoreline is calculated, and it is shown that putting the outfall further out to sea.


Anthony Kay - Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge CB3 9EW, U.K

Author Keywords: pollutant dispersal; diffusion-advection models; horizontal variations; mathematical analysis; outfalls; depth profiles; beach pollution; isopleths can drastically reduce this shoreline concentration


A two part paper, the first part of which is motivated by the behaviour observed when diffusion occurred cooling water leaving a power station outfall over a sharp depth change and the effects of topography upon this. The second part of the paper models a sloping topography supposed to represent a shoreline. The argument for this approach was that by putting outfalls much further out to sea, shoreline concentrations of pollutants can be greatly reduced.

The first example is modelled using a laboratory setup with dye, in order to observe the spreading of the plume. The results are used to compare predictions of a two dimensional mathematical model to the problems presented. The second example of a uniformly sloping topography, which is much more common in reality, is modelled using mathematics and the results are plotted graphically. The overall conclusion is that transport processes are much more efficient in deeper water, so that contaminant fluxes are much greater there. As a result, high concentrations of contaminant are retained in shallower regions.

The paper makes a general acknowledgement as to the accuracy of the experiments including that while flow speeds and depths may scale up, turbulence does not, and these effects may contribute to particularly incorrect validation of the mathematical model.

As a paper on topographical effects upon contaminant flow, it is well written and easy and understand. However being only a two dimensional mathematical model, it has slightly limited scope and while it may prove useful for the design of outfalls, issues such as vertical mixing are not addressed. Therefore more technical analysis is not possible. Overall, an informative paper for reading around contaminant flow and gaining a good background.