As noted above, we expect the net dispersive flux J to be dominated by the velocity distribution in the reach. In general, this is poorly known, and so resort is often made to a general relationship between the dispersive flux and the gradient in the mean concentration, called Fick's law, as originally used to described random molecular diffusion:

Thus we would expect that, if there was a steep concentration gradient (from high to low) in the downstream direction, then as a result of the turbulent mixing in the river flow, there would be a net dispersive flux in the downstream direction in addition to the net advection downstream. This leads to strong concentration differences being gradually reduced over time as the pollutant moves downstream (Fig. 14.18). Combining these last two equations gives:

d t dx2 dx

This is the advection-dispersion equation (ADE). Note that, in developing the ADE, we have used average concentrations in the cross-section of the flow. This means that the ADE applies strictly only after the pollutant is well mixed across the cross-section. This is normally some way downstream of the point of entry of the pollutant, beyond a distance called the mixing length. The mixing length will vary with the nature of the entry point and the flow characteristics but can be expected to be of the order of 10-30 times the channel width (e.g. Rutherford, 1994).

Time 1

Time 1

Fig. 14.18 Snapshots of advection and dispersion of a discrete mass of pollutant or tracer in a river reach originating from a source at the river bank in a flow of mean velocity v at three different times, and concentration curves at three different distances (Xj, X3) downstream from the input point. Darker shading represents higher concentrations.

Fig. 14.18 Snapshots of advection and dispersion of a discrete mass of pollutant or tracer in a river reach originating from a source at the river bank in a flow of mean velocity v at three different times, and concentration curves at three different distances (Xj, X3) downstream from the input point. Darker shading represents higher concentrations.

The ADE has a simple analytical solution for flows in which the velocity and dispersion characteristics of the flow stay constant downstream of the mixing length. This is clearly rare in real rivers but can sometimes be a useful approximation. For more complex cases, it is necessary to use an approximate numerical solution. For the simplest case of an instantaneous input of mass M and no losses, in a flow of cross-section A and mean velocity v, the analytical solution is given by

This is exactly the same equation as the Gaussian distribution that is commonly used in statistical problems. It predicts that, as the input of pollutant disperses in the flow, it does so as a symmetric Gaussian curve in space, i.e. with distance downstream (Fig. 14.19a). Normally, however, we do not take measurements of concentration at the same time along a river. It is much more convenient to follow a concentration curve in time at a particular point in space. In this case, as the pollutant passes that point, the tail of the concentration curve will have been subject to more dispersion than the rising limb. Thus, the concentration curve in time will not be symmetric but slightly skewed (Fig. 14.19b).

Application of (14.41) or (14.40) requires estimates of the two characteristics of the flow, the mean velocity, v, and the dispersion coefficient, D. These are not constants. We expect the mean velocity to increase with discharge and the dispersion coefficient increases approximately with the square of discharge. A variety of equations have been suggested to allow the estimation of the dispersion coefficient in relation to the characteristics of the flow (e.g. Rutherford, 1994; Young and Wallis, 1993). When a pollutant is expected to be non-conservative, losing mass as the cloud moves downstream, application of (14.40) also requires a value for the loss coefficient, a. The loss coefficient will vary with the nature of the pollutant and the flow conditions and is difficult to generalise.

When concentration curves are examined from pollution incidents or tracer experiments in real rivers, however, it is generally found that the curves are highly skewed, with much longer tails than predicted by the ADE (Fig. 14.20). This is generally interpreted in terms of 'dead zones' in the river, caused by backwater eddies, storage within

2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 12000

Fig. 14.19 Solution of (14.41) with v = 0.5ms-1 and D = 100 m2s-1 for: (a) the pattern of concentration with distance downstream for a fixed time at 9600 s; and (b) the pattern of concentration with time for a fixed distance of 2000 m.

2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 12000

Fig. 14.19 Solution of (14.41) with v = 0.5ms-1 and D = 100 m2s-1 for: (a) the pattern of concentration with distance downstream for a fixed time at 9600 s; and (b) the pattern of concentration with time for a fixed distance of 2000 m.

Time (hours since injection)

Fig. 14.20 Fitted advection-dispersion equation (ADE) and first-order aggregated dead zone (ADZ) models to bromide tracer data collected on the River Ouse in Yorkshire. The ADE model has been fitted to get the timing of the first rise and peak right but shows how it then cannot reproduce the longer term retention of tracer (without an additional 'dead zone' component).

Time (hours since injection)

Fig. 14.20 Fitted advection-dispersion equation (ADE) and first-order aggregated dead zone (ADZ) models to bromide tracer data collected on the River Ouse in Yorkshire. The ADE model has been fitted to get the timing of the first rise and peak right but shows how it then cannot reproduce the longer term retention of tracer (without an additional 'dead zone' component).

vegetation growth in the river, or other effects that lead to part of the volume of water having much lower downstream velocities than the main stream. Then, when pollutant gets into a dead zone, it takes much longer to return to the main stream than the timescale of transport in the faster flowing part of the flow. This is what leads to the long tails of concentration commonly observed.

There have been two main approaches to making better predictions of pollutant transport. The first has been to modify the ADE by adding a transient storage component in which it is assumed that transfers to and from the storage are proportional to the concentration difference between main stream and transient storage concentration (Bencala and Walters, 1983). Thus (14.41) can be modified by the addition of a simple representation of exchange between the main river and the transient storage:

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