disp_1998

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 terms

$\begin{displaymath} {{\rm D} C \over {\rm D}t}={\partial C\over \partial t}+{\bf... ... C\over \partial y}+w{\partial C\over \partial z}=0.\eqno{(1)} \end{displaymath}$

It will be sufficient for our purposes to take the air to be incompressible so that

$\begin{displaymath} \nabla .{\bf u}={\partial u\over \partial x}+{\partial v\over \partial y}+ {\partial w\over \partial z}=0.\eqno{(2)} \end{displaymath}$

Hence an alternative form of Eq. (1) is

$\begin{displaymath} {\partial C\over \partial t}+\nabla .({\bf u}C)= {\partial C... ...er \partial y}(vC)+{\partial\over \partial z}(wC)=0.\eqno{(3)} \end{displaymath}$

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

$\begin{displaymath} {{\rm D} C \over {\rm D}t}=S(x,y,z,t),\eqno{(4)} \end{displaymath}$

where

represents the source term (in kgm $^{-3}$ s $^{-1}$ ). Examples could be:
(i) $\displaystyle S=-\alpha C\quad (\alpha\hbox{~constant}).$ (5)
This represents decay of

by, for example, chemical decomposition or radioactive decay.
(ii) $\displaystyle S=S_0\delta ({\bf r}-{\bf r}_0)\quad (S_0\hbox{~constant})$ , (6)
where $\delta ({\bf r}-{\bf r}_0)$ is a delta function at ${\bf r}={\bf r}_0$ . This represents a continuous point source (e.g. emission from a chimney).
(iii) $\displaystyle S=S_0\delta ({\bf r}-{\bf r}_0)\delta (t-t_0)\quad (S_0\hbox{~constant})$ . (7)
This represents an instantaneous point source occurring at time

at ${\bf r}={\bf r}_0$ (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

$\begin{displaymath} {{\rm D} C \over {\rm D}t}=D\left ({\partial^2C\over \partia... ...artial^2C\over \partial z^2}\right ) =D\nabla ^2 C. \eqno{(8)} \end{displaymath}$

is the molecular diffusivity.

has dimensions ${\rm L}^2{\rm T}^{-1}$ . Typical values for pollutants in the atmosphere are 5-50 $\times 10^{-6}$ m

s $^{-1}$ . We are usually concerned with dispersion on the scale of hundreds of metres to hundreds of kilometres. Consider ${\rm L}=100$ m. Then the time-scale is

$\begin{displaymath} {\rm T}={{\rm L}^2\over D}={10^4\over 50\times 10^{-6}} = 2\times 10^8 \hbox{~s} \end{displaymath}$

which is infinite for all practical purposes. Hence molecular diffusion is never directly relevant to atmospheric dispersion.

1.2 Turbulence

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:

$\begin{displaymath} \hbox{Average of $\phi$~}=\overline \phi={1\over T}\int_t^{t+T}\phi (x,y,z,t)\, {\rm d}t .\eqno{(9)} \end{displaymath}$

The average period

should be chosen to be long compared to the turbulence time-scales. Then we put

$\begin{displaymath} \phi =\overline \phi + \phi '.\eqno{(10)} \end{displaymath}$

$\phi '$ represents the turbulent part of $\phi$ . It follows (to a reasonable approximation at least) that

$\begin{displaymath} \overline {\phi '}=0.\eqno{(11)} \end{displaymath}$

Let us perform this separation for all the variables in the concentration equation, Eq. (3).

$\begin{displaymath} \eqalign{C&=\overline C+C'\cr u&=\overline u+u'\cr v&=\overline v+v'\cr w&=\overline w+w'\cr} \end{displaymath}$

Therefore

$\begin{displaymath} {\partial\over \partial t}(\overline C+C')+ {\partial\over \... ...\partial\over \partial z}[(\overline w+w')(\overline C+C')]=0. \end{displaymath}$

We now average this equation. Consider, for example

$\begin{displaymath} \eqalign{\overline {(\overline C+C')(\overline u+u')}&= \ov... ...rline {u'C'}\cr &=\overline u\overline C+\overline {u'C'}\cr} \end{displaymath}$

since $\overline {u'}=\overline {C'}=0$ . Similar results hold for

and

, giving

$\begin{displaymath} {\partial\overline C\over \partial t}+{\partial\over \partia... ...C'}) -{\partial\over \partial z}(\overline {w'C'}).\eqno{(12)} \end{displaymath}$

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

$\begin{displaymath} \left .\eqalign{ \overline {u'C'}=&-\varepsilon _x{\partial\... ..._z{\partial\overline C\over \partial z}\cr}\right\}\eqno{(13)} \end{displaymath}$

$\varepsilon _x$ , $\varepsilon _y$ and $\varepsilon _z$ are analogous to the molecular diffusivity

. They must be measured experimentally. They differ from

in that (i) $\varepsilon _x$ , $\varepsilon _y$ and $\varepsilon _z$ need not be equal, (ii) In general, $\varepsilon _x$ , $\varepsilon _y$ and $\varepsilon _z$ are not constant, (iii) $\varepsilon _x, \varepsilon _y, \varepsilon _z\gg D$ .

Using Eq. (13),

$\begin{displaymath} {\partial\overline C\over \partial t}+{\partial\over \partia... ...n _z{\partial\overline C\over \partial z}\right ). \eqno{(14)} \end{displaymath}$

This equation is the basis of much of the modelling of atmospheric dispersion.

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