Geoscience Reference
In-Depth Information
value of the vertical displacement of diffusing effluent particles.
z
p
(t)
is related to
another Lagrangian quantity, the particle vertical velocity
w
p
(t)
,by
t
w
p
(t
)dt
.
z
p
(t)
=
(4.17)
0
Multiplying by 2
w
p
=
2
dz
p
/dt
, ensemble averaging, and using the commutativity
of time differentiation and ensemble averaging yields
2
t
0
dz
p
dt
=
dz
p
dt
=
2
z
p
dz
p
w
p
(t)w
p
(t
)dt
.
dt
=
(4.18)
This can be rewritten as
2
w
p
t
0
dz
p
dt
=
R(τ) dτ,
(4.19)
with
R
the vertical velocity autocorrelation function of a particle,
+
w
p
(t)
w
p
(
t
τ)
R(τ)
=
.
(4.20)
w
p
We can diagnose the behavior of
R
in the short-time and long-time limits. The first
is simple:
R(
0
)
1. At long times we would expect a moving particle to “forget”
its initial vertical velocity, so that
R
=
0.
In hom
og
eneous, stationary turbulence in a c
ons
tant-density fluid the Lagrangian
variance
w
p
is equal to the Eulerian variance
w
2
(
Lumley
,
1962
;
Corrsin
,
1963
).
Thus,
Eq. (4.19)
can be written
→
2
w
2
t
0
dz
p
dt
=
R(τ)dτ.
(4.21)
This result is due to G. I.
Taylor
(
1921
) and is known as “Taylor's theorem.” The
short-time limit
R
=
1 implies
dz
p
dt
→
2
w
2
t,
t
→
0
.
(4.22)
=
The long-time limit
R
0 yields
dz
p
dt
→
2
w
2
τ
L
,t
→∞
,
(4.23)
