Molecular diffusion produced by Brownian motion is no longer the dominant diffusion mechanism when the flow velocity becomes fast enough to overcome viscous forces that tend to keep fluid elements aligned along parallel paths. When this threshold is reached, the flow becomes very irregular, and the fluid elements are entrained and transported by eddies which form either from the slowing effect of the bottom and side boundaries or from the disturbances introduced by geometrical irregularities. This type of flow is called turbulent, and it is characterized by an enhanced momentum and mass transfer across the flow field. Diffusion of mass is no longer controlled by Brownian motion, but rather by the continuous displacement of fluid elements in all directions induced by turbulence. While molecular diffusion is isotropic, turbulent diffusion is often different in all directions, as eddies are continuously stretched and deformed by the flow.

Turbulent flows are usually modeled splitting the physical quantities into time-averaged mean values and fluctuations around the mean. After manipulation of the advection-diffusion equation [11], a new mathematical

transport term appears, which is the time-averaged product of the fluctuating values of velocity and concentration. If velocity and concentration fluctuations were statistically independent, then these terms would produce no net diffusive mass fluxes. It turns out instead that velocity and concentration irregularities are correlated and that the integral effect over time of turbulent fluxes is always much higher than the fluxes induced by Brownian motion. The mass transport equation can no longer be written as in eqn [11], but it becomes

where Dx, Dy, and Dz are called eddy diffusion coefficients, are direction dependent, and so much larger than the molecular diffusion coefficient Dm to replace it in all terms of the equation. The conceptual difference between eqns [12] and [11] is that the value of the coefficients Dx, Dy, and Dz is now determined by the flow regime, that is, they are flow properties, while Dm was independent of the flow and determined only by the combination of solute and solvent. Another difference resides in the fact that eddy diffusivity is scale dependent, while molecular diffusivity is scale independent. Eddy diffusivity typically scales with a 4/3 power of the length scale of the process. This implies that as diffusion makes the substance spread in the domain, diffusivity increases due to the effect of larger eddies that come into play. This dependence is important in large and deep water bodies such as the sea or lakes, where the diffusion process involves several different scales over time. In rivers, instead, the size of the eddies is controlled by water depth and width, and diffusivity is no longer affected by the scale of the process. Typical values of eddy diffusion coefficients are m2s 1

in slow-moving, deep, and stratified water bodies such as lakes and the sea,

10-10 m s for horizontal surface flows such as rivers and channels.

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