The term relative diffusion is used to describe relative motion of pairs of particles viewed as a diffusion process. Richardson explained the importance of using relative, rather than absolute, motion in turbulence studies . His main intention was to separate the turbulent variations in the velocity field from the average velocity field.
We can look at relative diffusion in a form similar to that used for absolute
but it is more common to look at the behaviour of , as Richardson did, or at the behaviour of as Kolmogorov did.
If you notice that is the average squared separation of a pair of
particles, and assume that there exists a constant, , such
that , or , it is
possible to show that both and can be written
as powers of R which only depends on .
Both Richardson's data and Kolmogorov's derivations gives the same value for in three-dimensional turbulence:
When we look at or , we have freed ourselves of the problem of tracking every pair of particles back to a moment where they were close together. We can just look at their relative motion as a function of their separation.
Even though the formal descriptions of the quantities Richardson and Kolmogorov looked at are equivalent, their selections of what data we should look at are quite different. Richardson talks about a collection of particles, which initially are very close together, whereas Kolmogorov looks at a randomly selected collection of points throughout the whole body of fluid in question. Richardson's view is equivalent to studies of chaos , where you look at the exponential growth of an infinitely small difference in the initial conditions.
In both of the two extreme types motion; Brownian and ballistic , is equal to D(t), only scaled by respectively a factor of 2 and a factor of 4.
The relative motion of two random walkers (Brownian motion) for a time, t,
is equivalent to one random walker moving for twice as long a time:
The relative motion of two particles in ballistic motion for a time, t,
is similarly equivalent to one particle in ballistic motion for twice that
period of time, but here is , and thus:
So we find that for velocity fields with absolutely no spatial correlations is independent of R and for velocity fields with strong spatial correlations is proportional to R.