42
SI 4hl.b.Y ('ORWSIN
Ciaussian within the accuracy of measurement (see, e.g.. Batchelor, 1953).
and this leads to Gaussian concentration distributions about a single raggiiig point. Sirice the elementary solution of the simple dimusion equation is a
Gaussian fiinciion, it is essential to remind ourselves that the occurrence of
Geussian distributions does nor ensure that the process can be described by
a simple diffusion equation. On the other hand, by allowing the diffusivity to
be ti function of time (see footnote 7 to Section 7). we should be able to
dcvisc a diffusion equation whose solution is a spreading Gaussian function
with hirly general growth history.
As anothcr example, consider a random walk with a Gaussian p.d.f. of
step length
(52)
(52)
r(2,o) -= ii(:),
p,(l) = (2n) ' 0 , I expl - P / ~ u : ; .
If the initial concentration field is H Dirac function.
thcn it cpn hc shown that after
(54)
f+(z. 11) = ( 2 ~ ) - 'u*- exp{ -z2.i2a,2j,
with
tinies steps
( 5 5 )
0, = n%, .
(56)
Ic 1 ' ! 2 ,
With equal time steps, Eq. ( 5 5 ) is equivalent to
This behavior is appropriatc to solutions of the simple diffusion equation.
Yet the system certainly does not fulfill a condition like Eq. (13). which is
ncccssary if a process is in fact dcscriblible by the simple diffusion equation.
Onc of the qualitative differences hetween transport processes whose scale
is nor much smaller than the characteristic length of the mean concentration
profilc and classical transport processes, which fulfill such a condition. is
that t hc former can apparently transport properties against a local concentration gradient.' In turbulent transport. this possibility has led to such
apparent paradoxes as "reversed ' * kinetic energy transformation. wherein
energy is fed locally from turbulence into mean flow, instead of the other
way around (Erian and Eskinazi, 1969).
" There hits not been time to calculate an illustration. Any concentration distrihution with
an unsynimetriul maximum or minimum should serve IS an appropriate exempie. Paren1hcticiilly. this constant density phcnomenon shoiild not he confiised with the "countergradiciit
vertical hoat flux" observed in thc planetary atmospheric boundary layer (see, e.g.. Dcardorfl,
1972).
SI 4hl.b.Y ('ORWSIN
Ciaussian within the accuracy of measurement (see, e.g.. Batchelor, 1953).
and this leads to Gaussian concentration distributions about a single raggiiig point. Sirice the elementary solution of the simple dimusion equation is a
Gaussian fiinciion, it is essential to remind ourselves that the occurrence of
Geussian distributions does nor ensure that the process can be described by
a simple diffusion equation. On the other hand, by allowing the diffusivity to
be ti function of time (see footnote 7 to Section 7). we should be able to
dcvisc a diffusion equation whose solution is a spreading Gaussian function
with hirly general growth history.
As anothcr example, consider a random walk with a Gaussian p.d.f. of
step length
(52)
(52)
r(2,o) -= ii(:),
p,(l) = (2n) ' 0 , I expl - P / ~ u : ; .
If the initial concentration field is H Dirac function.
thcn it cpn hc shown that after
(54)
f+(z. 11) = ( 2 ~ ) - 'u*- exp{ -z2.i2a,2j,
with
tinies steps
( 5 5 )
0, = n%, .
(56)
Ic 1 ' ! 2 ,
With equal time steps, Eq. ( 5 5 ) is equivalent to
This behavior is appropriatc to solutions of the simple diffusion equation.
Yet the system certainly does not fulfill a condition like Eq. (13). which is
ncccssary if a process is in fact dcscriblible by the simple diffusion equation.
Onc of the qualitative differences hetween transport processes whose scale
is nor much smaller than the characteristic length of the mean concentration
profilc and classical transport processes, which fulfill such a condition. is
that t hc former can apparently transport properties against a local concentration gradient.' In turbulent transport. this possibility has led to such
apparent paradoxes as "reversed ' * kinetic energy transformation. wherein
energy is fed locally from turbulence into mean flow, instead of the other
way around (Erian and Eskinazi, 1969).
" There hits not been time to calculate an illustration. Any concentration distrihution with
an unsynimetriul maximum or minimum should serve IS an appropriate exempie. Paren1hcticiilly. this constant density phcnomenon shoiild not he confiised with the "countergradiciit
vertical hoat flux" observed in thc planetary atmospheric boundary layer (see, e.g.. Dcardorfl,
1972).