THA\SPOHT MODEL 1 IMIT \I IONS IS TL‘RRULENCE
39
standard deiiation of contaminatit dispersed from a point source by turbulence behaws like 0 + w’f (Taylor. 1921). which is identical with Eq. (45).
the wave equation result.
Another concept may be appended to complicate at least the semantics of
this discussion: when a simple wave beam (like a pencil of light) propagates
through a random medium. the average intensity does not spread linearly.
For example, if the inhomogcneity scale is much larger than the wavelength.
each ray (i.e.. characteristic) performs a (continuous) random walk, and the
” large t ” intensity spread is diffusionlike.”
8. C o N . r i N u o u s LIMIT OF A RANDOM WALK THAT P t ~ s s ~ s s ~ . ~
A principal shortcoming of the telegraph cquation as a turbulent diffusion
model is that it possesses no Eulerian (i.e., spatial) statistical properties,
whcreas Eulerian coordinates are those in which most experimental investigations and theoretical analysis of turbulent motion are conveniently carried
out. This is. in fact. a major deficiency of all classical random walk processes
a s simulations of turbulent diffusion.
To overcome this shortcoming, Lumley and Corrsin (1959) introduced the
random walk process we have come to call the “walk on a random field.” In
these processes. we prescribe not a stochastic algorithm for the peregrinations in time of 1 particle on a blank lattice. but rather the statistical properties of a velocity field on a field of spatial and temporal intervals. These
velocity field properties are naturally Eulerian. It is then a straightforward
matter to compute the random trajectories of “material” points on these
ficlds; thc associated statistical properties are most directly Lagrangian.
t’attcrson and Corrsin (1966) reported some computer experiments on
r;i h c r gencral binary examples, particularly comparing the Lagrangian velocity autocorrelation function with the Eulerian velocity correlation function
in cpacc time. Binary fields were used for simplicity and economy, but there
is no conceptual difficulty in doing the same thing for continuous fields when
enough computing capacity is available, and this generalization has recently
h e w carl-icd o u t by Liu and Thompson (1973).
In the special case of binary Eulerian fields generated by two Markov
chains, one for space and one for time, Patterson (1966 and this volume,
p. 61) corrected the solution of Lumley and Corrsin, and also devised a
continuous limit procedure which converted the difference equations to a
gcneralized telegraph equation. His work is presented in these proceedings,
so a brief outline will suffice here.
The Eulerian binary velocity field has uniform velocity over distances at
least b% and over times at least fit. If p is the probability that neighboring
HOTH EUIXKIAN A N D LACiHANCilAN STATISTICS
39
standard deiiation of contaminatit dispersed from a point source by turbulence behaws like 0 + w’f (Taylor. 1921). which is identical with Eq. (45).
the wave equation result.
Another concept may be appended to complicate at least the semantics of
this discussion: when a simple wave beam (like a pencil of light) propagates
through a random medium. the average intensity does not spread linearly.
For example, if the inhomogcneity scale is much larger than the wavelength.
each ray (i.e.. characteristic) performs a (continuous) random walk, and the
” large t ” intensity spread is diffusionlike.”
8. C o N . r i N u o u s LIMIT OF A RANDOM WALK THAT P t ~ s s ~ s s ~ . ~
A principal shortcoming of the telegraph cquation as a turbulent diffusion
model is that it possesses no Eulerian (i.e., spatial) statistical properties,
whcreas Eulerian coordinates are those in which most experimental investigations and theoretical analysis of turbulent motion are conveniently carried
out. This is. in fact. a major deficiency of all classical random walk processes
a s simulations of turbulent diffusion.
To overcome this shortcoming, Lumley and Corrsin (1959) introduced the
random walk process we have come to call the “walk on a random field.” In
these processes. we prescribe not a stochastic algorithm for the peregrinations in time of 1 particle on a blank lattice. but rather the statistical properties of a velocity field on a field of spatial and temporal intervals. These
velocity field properties are naturally Eulerian. It is then a straightforward
matter to compute the random trajectories of “material” points on these
ficlds; thc associated statistical properties are most directly Lagrangian.
t’attcrson and Corrsin (1966) reported some computer experiments on
r;i h c r gencral binary examples, particularly comparing the Lagrangian velocity autocorrelation function with the Eulerian velocity correlation function
in cpacc time. Binary fields were used for simplicity and economy, but there
is no conceptual difficulty in doing the same thing for continuous fields when
enough computing capacity is available, and this generalization has recently
h e w carl-icd o u t by Liu and Thompson (1973).
In the special case of binary Eulerian fields generated by two Markov
chains, one for space and one for time, Patterson (1966 and this volume,
p. 61) corrected the solution of Lumley and Corrsin, and also devised a
continuous limit procedure which converted the difference equations to a
gcneralized telegraph equation. His work is presented in these proceedings,
so a brief outline will suffice here.
The Eulerian binary velocity field has uniform velocity over distances at
least b% and over times at least fit. If p is the probability that neighboring
HOTH EUIXKIAN A N D LACiHANCilAN STATISTICS