Geoscience Reference
In-Depth Information
where
δ
is the Dirac delta function and
u
i
is the velocity at
x
i
,t
in realization
α
.
Thus the probability that in realization
α
the velocity at
(x
i
,t)
is within
dv
i
of
v
i
is
β
α
(v
i
;
u
i
, as required. Averaging this
expression over all realizations (and denoting this by angle brackets) yields
β(v
i
;
x
i
,t)
=
β
α
(v
i
;
x
i
,t)
=
δ(u
i
−
v
i
)
.
x
i
,t)dv
i
=
δ(u
i
−
v
i
)dv
i
=
1if
v
i
=
(13.36)
In equating the probability density of velocity
u
i
(x
i
,t)
to the expected value
of a delta function involving velocity,
Eq. (13.36)
provides a link between the
dynamics of the velocity field and those of its probability density. Before showing
how
Lundgren
(
1967
) used that link, we'll need to generalize
Eq. (13.36)
to joint
probability densities.
The probability density
β
v
(
1
)
,t
is the “one-point” form. Its product with
x
(
1
)
;
i
i
dv
(
1
)
is the probability that the velocity at
x
(
1
)
and time
t
is within
dv
i
of
v
(
1
)
.
i
i
i
From
Eq. (13.36)
it is written as
β
v
(
1
)
δ
u
i
x
(
1
)
v
(
1
i
δ
u
(
1
)
v
(
1
i
,t
,t
x
(
1
)
;
=
−
≡
−
≡
β(
1
).
i
i
i
i
(13.37)
The “two-point”, or joint, probability density
β
2
v
(
1
)
t
is the
,v
(
2
)
x
(
1
)
,x
(
2
)
;
;
i
i
i
i
probability that at time
t
the velocity at
x
(
1
)
is within
dv
i
of
v
(
1
)
and the velocity at
i
i
x
(
2
)
is within
dv
i
of
v
(
2
)
.
By extension of the arguments leading to
Eq. (13.37)
it is
i
i
δ
u
(
1
)
v
(
2
i
β
2
v
(
1
)
t
v
(
1
i
δ
u
(
2
)
,v
(
2
)
x
(
1
)
,x
(
2
)
;
;
=
−
−
≡
β(
1
,
2
).
i
i
i
i
i
i
(13.38)
Lundgren's derivation of the pdf evolution equation proceeds as follows. The
time derivative of
Eq. (13.37)
is
∂δ
u
(
1
)
v
(
1
i
∂
u
(
1
)
v
(
1
i
∂
v
(
1
i
∂t
δ
u
(
1
)
−
−
∂
∂t
β(
1
)
i
i
=
−
=
∂
u
(
1
)
v
(
1
i
i
∂t
−
i
∂
∂v
(
1
)
.
δ
u
(
1
)
v
(
1
i
∂
∂t
u
(
1
)
=−
−
(13.39)
i
i
i
Lundgren then evaluated the right side of
Eq. (13.39)
with the Navier-Stokes and
continuity equations through tedious calculations. The resulting evolution equation
for the probability density is
