Introduction - Turbulence in the Atmosphere - page 15

of three-dimensional turbulence. Without viscous dissipation the kinetic energy of
turbulence could grow without bound (Problem 1.5) . This kinetic-energy cascade
is a statistical concept, but it has direct implications for instantaneous turbulence
fields: it says they not only have large, prominent, energetic eddies that we can see
in clouds and smoke plumes, but they also have very much smaller eddies whose
viscous forces dissipate kinetic energy at the required rate. It is generally accepted
that vortex stretching is one physical process responsible for the generation of this
wide range of smaller eddies in three-dimensional turbulence.

1.6.2 Random, stochastic
Imagine generating turbulent flow of a certain geometry (in the laboratory, say)
any number of times. Because of the sensitivity of turbulence to inevitable small
differences in its initial state, each resulting flow, called a realization , is unique. We
call such a flow random , by which we mean different in every realization .
A property at any point in a turbulent flow has a mean value and fluctuations
about that mean. As we shall discuss in Chapter 2 , the mean can in principle be
an ensemble mean, the average of values at the point over many realizations of the
flow; a time mean, which we can use in statistically steady cases to approximate
the ensemble mean; or a spatial mean, which we can use in spatially homogeneous
cases. The fluctuations about this mean are stochastic , which we define as varying
irregularly in space or time in a given realization. †
Numerical models of turbulent flows generally provide only such mean values.
The most common atmospheric-diffusion models estimate the ensemble-mean con-
centration downwind of an effluent release (although some users' manuals call it a
time mean). But an effluent plume in the daytime atmospheric boundary layer will
fluctuate in time and space about that ensemble mean, and will behave differently
in each realization. Such concentration excursions can induce dangerously high
concentrations of toxic effluents over short periods.

1.6.3 The effective diffusivity
In turbulent flows the advective fluxes in the Navier-Stokes (1.26) and conserved
scalar constituent (1.31) equations are spatially and temporally chaotic, three
dimensional, and generally (except at solid surfaces) much larger than themolecular
fluxes. This gives turbulent flow a much greater “mixing power” or “effective diffu-
sivity” than laminar flow. But these are, to use G. I. Taylor's term, virtual properties
of turbulence - properties of themean flow, not of a realization. Effluents do disperse

†

This distinction between random and stochastic is not always made in turbulence, but it is useful.