1 Scalar Transport in the Atmosphere
1.1 Basic Principles
Consider an element of air containing a concentration of a passive
pollutant (passive means that it doesn't react and is neutrally buoyant, i.e.
it doesn't settle).
Fig 1. As an element of air is carried along by
the flow it always contains the same air and therefore contains the same
mass of pollutant.
If the flow is also incompressible, then the volume of the fluid element
remains constant and so the concentration remains constant. In mathematical
It will be sufficient for our purposes to take the air to be incompressible
Hence an alternative form of Eq. (1) is
Suppose now that there is a source or sink of the pollutant within the
element of air. This could be because the pollutant is created or destroyed
by chemical reactions or because there is an outflow from, say, a chimney.
The equation for would then become
where represents the source term (in kgms). Examples could
This represents decay of by, for example, chemical decomposition or
is a delta function at
This represents a continuous point source (e.g. emission from a chimney).
This represents an instantaneous point source occurring at time at
(e.g. accidental release of a radioactive substance).
Another type of source/sink term is caused by diffusion. The pollutant
may diffuse in or out of our element. The equation for is then
is the molecular diffusivity. has dimensions
Typical values for pollutants in the atmosphere are 5-50
ms. We are usually concerned with dispersion on the scale of
hundreds of metres to hundreds of kilometres. Consider m. Then
the time-scale is
which is infinite for all practical purposes. Hence molecular diffusion
is never directly relevant to atmospheric dispersion.
Atmospheric flows are almost always turbulent. Turbulence occurs when
the Reynolds number is high. It is characterised by eddy motions on a wide
range of scales. When describing the dispersion of pollutants, we are usually
interested in dispersion on scales much larger than many, if not all, of the
eddies. In other words, we are interested in averages over length or time
scales large compared to the turbulence.
In order to analyse dispersion in this way, we assume that it is possible to
divide the flow into a ``mean'' flow which is slowly varying in time and a
rapidly fluctuating, or ``turbulent'' part. We could perform this separation
by defining an average as follows:
The average period should be chosen to be long compared to the turbulence
time-scales. Then we put
represents the turbulent part of . It follows (to a
reasonable approximation at least) that
Let us perform this separation for all the variables in the concentration
equation, Eq. (3).
We now average this equation. Consider, for example
. Similar results hold for and , giving
The right-hand side represents the average effect of turbulent eddies on the
concentration. Molecular diffusion is caused by the random motion of
molecules, whereas the effect here is caused by the random eddy motions. By
analogy with the molecular scale, we assume that
are analogous to the molecular diffusivity
. They must be measured experimentally. They differ from in that
need not be equal,
(ii) In general,
are not constant,
Using Eq. (13),
This equation is the basis of much of the modelling of atmospheric dispersion.
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