McKay, Donald. "Front matter" Multimedia Environmental Models ...

is much greater than B M, and the molecular processes can be ignored. In stagnant
regions, such as thermoclines or in deep sediments, B T may be small or zero, and
B M dominates. As we move closer to a phase boundary, B T tends to become smaller;
thus, it is possible that much of the resistance to diffusion lies in the layer close
to the interface. The roughness of the interface plays a role in determining the
thickness of this layer. For example, grass may damp out wind eddies and retard
the rate of diffusion from soil to air. Animal fur retards both diffusion of heat and
water vapor.
A complicating factor is that we have no guarantee that B T is isotropic, i.e., that
the same value applies vertically and horizontally. In Figure 7.3, we postulate that
some eddies may be constrained to form elongated “roll cells.” The horizontal B T
will therefore exceed the vertical value. In practice, this nonisotropic situation is
common and even leads to conditions in rivers where three B T values must be
considered: vertical, upstream-downstream, and cross-stream.
To give an order of magnitude appreciation of turbulent diffusivities, it is
observed that a vertical eddy diffusivity in air is typically 3600 m 2 /h plus or minus
a factor of 3, thus the time for moving a distance of 1 m is of the order of 1 s.
Molecular diffusion is clearly negligible in comparison. In lakes, a vertical eddy
diffusivity may be 36 m 2 /h near the surface, corresponding to a velocity over a
distance of 1 m of 1 cm/s. At greater depths, diffusion is much slower, possibly by
a factor of 100. To estimate eddy diffusivities, one can watch a buoyant particle and
time its transport over a given distance. The diffusivity is then that distance squared,
divided by the time.
Turbulent processes in the environment are thus quite complex and difficult to
describe mathematically. The interested reader can consult Thibodeaux (1996) or
Csanady (1973) for a review of the mathematical approaches adopted. We sidestep
this complex issue here, but certain generalizations that emerge from the study of
turbulent diffusion are worth noting.
In the bulk of most fluid masses (air and water) that are in motion, turbulent
diffusion dominates. We can measure and correlate these diffusivities. Generally,
vertical diffusion is slower than horizontal diffusion. Often, diffusion is so fast that
near-homogeneous conditions exist, which is fortunate, because it eliminates the
need to calculate diffusion rates.
In the atmosphere and oceans, there is a spectrum of eddies of varying size and
velocity. The larger eddies move faster. Consequently, when a plume in the atmosphere
or a dye patch in an ocean expands in size, it becomes subject to dispersion
by larger, faster eddies, and the diffusivity increases. If the velocity of expansion of
the plume or patch is constant, this implies that diffusivity increases as the square
of distance.
At phase interfaces (e.g., air-water, water-bottom sediment), turbulent diffusion
is severely damped or is eliminated, thus only molecular diffusion remains. One can
even postulate the presence of a “stagnant layer” in which only molecular diffusion
occurs and calculate its diffusion resistance. This model is usually inherently wrong
in that no such layer exists. It is more honest (and less trouble) to avoid the use of
diffusivities and stagnant layer thicknesses close to the phase interfaces and invoke
mass transfer coefficients that combine the varying eddy diffusivities, the molecular
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