5. Dispersion in Real Environments

The Gaussian plume equation, Eq. (47) is derived from Eq. (22) and this in turn is only a true solution of the concentration equation, Eq. (21) if the eddy diffusivities are constant. This is generally not true. For example, in the neutrally stratified surface layer, the eddy viscosity $K$ is given by $u_*kz$ and we can expect $\varepsilon _z$ to behave similarly. In order to analyse real environments we have at least 3 possibilities: (i) Estimate (for example using surface layer theory) or measure (usually indirectly by measuring wind fluctuations $\overline {v'^2}$ and $\overline {w'^2}$) $\varepsilon _y$, $\varepsilon _z$ and then solve Eq. (21) numerically. (ii) Assume that the solution (47) is still a reasonable approximation but that the variation of $\sigma_y$ and $\sigma_z$ with $x$ is different to Eq. (48). Instead, we replace (48) by empirical relationships between $\sigma_y$, $\sigma_z$ and $x$. (iii) For some special cases, take $\varepsilon _z$ to vary with $z$ and solve the concentration equation analytically.

We now consider examples of (ii) and (iii).

5.1 Pasquill-Gifford Stability Classes

It is observed that in neutral stratification, Eq. (48) does not hold. Instead, measurements suggest that $\sigma_y$ and $\sigma_z$ are proportional to $x^\alpha$ for $\alpha$ in the range 0.75 to 1. This observation changes when the atmospheric stability changes. For example, when the air is very stable, vertical mixing is inhibited and $\sigma_z$ grows only slowly with $x$. On the other hand, when there is strong solar heating of the surface, there may be strong convective activity with large vertical motions; then $\sigma_z$ increases rapidly with $x$.

Based on measurements of atmospheric turbulence over flat plains, Pasquill and Gifford produced empirical results for the variation of $\sigma_y$ and $\sigma_z$ with $x$ for six stability classes. These are shown in Table 2.

truept Guidelines are given for estimating the stability class from the wind speed, cloud cover and time of day. These are given in Table 3.

truept Figs. 9 and 10 show the variation of $\sigma_y$ and $\sigma_z$ with $x$ for the six stability classes. These variations may be approximated by

\begin{displaymath} \sigma_y=ax^{0.894}\eqno{(49)} \end{displaymath}

and

\begin{displaymath} \sigma_z=cx^d+f\eqno{(50)} \end{displaymath}

where the constants $a$, $c$, $d$ and $f$ depend on the stability class as shown in Table 4.

truept With these values of $\sigma_y$ and $\sigma_z$, the concentration can be determined directly from Eq. (47). An example is shown in Fig. 11.

\epsfbox{fig9.eps}=7.8 Fig. 9. Variation of $\sigma_y$ with downwind distance $x$ for the six Pasquill-Gifford stability classes.

Fig. 10. Variation of $\sigma_z$ with downwind distance $x$ for the six Pasquill-Gifford stability classes.

Fig. 11. The solution to the Gaussian plume equation (47) with $\sigma_y$ and $\sigma_z$ given by the Pasquill-Gifford recommendations for stability class C. Contours of concentration are shown at $z=0$ for a wind speed of 10 ms$^{-1}$.

5.2 Dispersion from a Continuous Line Source

A major application here is to the dispersion of emissions from motor vehicles travelling along a road on a cross-wind. If the wind is in the $x$-direction, perpendicular to the road, and $y$ is along the road, then we do not expect the concentration of pollutant to depend on $y$. Just as described in §4, we can neglect diffusion in the windward direction, so that the concentration equation for a steady concentration distribution is

\begin{displaymath} u{\partial C\over \partial x}={\partial\over \partial z}\lef... ...varepsilon _z {\partial C\over \partial z}\right ).\eqno{(51)} \end{displaymath}

For neutral stability, it is reasonable to assume that $\varepsilon _z$ is equal to the eddy viscosity in the neutral surface layer, so that

\begin{displaymath} \varepsilon _z=ku_*z.\eqno{(52)} \end{displaymath}

Therefore

\begin{displaymath} u{\partial C\over \partial x}={\partial\over \partial z}\left ( ku_*z {\partial C\over \partial z}\right ).\eqno{(53)} \end{displaymath}

We need also to describe the height variation of the wind, $u$. It would seem to be consistent with the assumptions on $\varepsilon _z$ to take a logarithmic velocity profile but unfortunately we cannot then solve Eq. (53) analytically. Instead, a commonly made assumption is that

\begin{displaymath} {u\over u_1}=\left ({z\over z_1}\right )^p\eqno{(54)} \end{displaymath}

where $u_1$ is the wind speed at a fixed height $z_1$. This can only be approximately equal to the logarithmic profile (41) over a relatively small height range. To get a reasonable correspondence, note that from Eq. (54)

\begin{displaymath} {1\over u}{{\rm d}u\over {\rm d}z}={p\over z} \end{displaymath}

and therefore

\begin{displaymath} {{\rm d}u\over {\rm d}z}={pu\over z}. \end{displaymath}

If we now substitute from Eq. (40) and (41) into the left and right sides of the above equation, we get

\begin{displaymath} {u_*\over kz}={p\over z}\ln\left ({z\over z_0}\right ) \end{displaymath}

and hence

\begin{displaymath} p={1\over \ln\left ({z\over z_0}\right )}.\eqno{(55)} \end{displaymath}

Of course $p$ should be a constant and so in Eq. (55) we take an average value of $z$ over the range of interest. The precise value of $p$ should really depend not only on the height range of interest but also on the atmospheric stability. Generally, values in the range 0.1-0.4 are used, but for high stability they may be even larger. Typical values are given in Table 5.

Using Eq. (54), Eq. (53) becomes

\begin{displaymath} u_1\left ({z\over z_1}\right )^p{\partial C\over \partial x}... ...}\left ( ku_*z {\partial C\over \partial z}\right )\eqno{(56)} \end{displaymath}

and the solution is

\begin{displaymath} C(x,z)={q\over (p+1)u_*kx}{\rm exp}\left (-{z_1^{-p}u_1z^{p+1}\over (p+1)^2 u_*kx}\right ).\eqno{(57)} \end{displaymath}

Note that the concentration at the ground decreases with $x$ like $1/x$, unlike the one-dimensional diffusion solution with constant diffusivity which suggests decay like $1/\sqrt{x}$. Note also that the decay rate is larger for smaller $p$, i.e. for unstable air flow. Eq. (57) has been shown to agree quite well with observations.

back to syllabus