Atmospheric Dispersion with a Large-Eddy Simulation
263
which quantifi es the average rate of separation between two particles at a distance δ.
We stress that λ(δ) is independent of r if r is close to 1, and then that for very small
separations (i.e., δ l u ), the FSLE coincides with the MLE λ, since
0
1
lim
ln r
δ→
λ =
τ(δ)
.
Note that if the trajectories refer to Lagrangian particles, as in our analysis, λ can
also be called the Lagrangian Lyapunov Exponent (LLE).
In general, for fi nite δ, the FSLE is expected to follow a power law of the type:
− γ
λ(δ) δ
∼
2
(9.36)
where the value of γ defi nes the dispersion regime at the scale δ. In fact,
( )
for
u
l
λ δ
λ
δ
∼
∼
(9.37a)
2/3
0
( )
for u
l
L
−
λ δ
δ
δ
∼
∼ ∼
(9.37b)
2
0
for
L
−
λ(δ) δ
δ
∼
∼
(9.37c)
This means that γ = 3 refers to the Richardson diffusion within the inertial range and
γ = 1 corresponds to standard diffusion, that is, large-scale uncorrelated spreading of
particles. These scaling rules can be explained by means of dimensional arguments.
In fact, as the scaling law of the relative dispersion in time is of the form R 2 (t) ∼ t γ , the
corresponding scaling law in terms of the FSLE is given considering the inverse of
time as a function of space.
Before showing our results, we introduce another quantity that can also provide
information about the existence of the inertial range.
This quantity, which is related to the FSLE, is the mean relative Lagrangian velocity at a fi xed scale that we indicate with
1 2
2
( )
( )
v
v
⎡
⎤
δ = δ δ
⎣
⎦
(9.38)
where
( )
( )
(
)
δ δ =
−
2
1
2
2
( )
v
x
x
(9.39)
is the square (Lagrangian) velocity difference between two trajectories,
(1)
x
and
(2)
x
,
on the scale
( )
( )
δ
⎪ − ⎪ =δ
1
2
(i.e.,
).
x
x
The quantity v(δ)/δ is dimensionally equivalent to
λ(δ), so a scaling law of the type:
2 3
( )
v
−
δ
δ
δ
∼
(9.40)
is compatible with the FSLE inside the inertial range.
© 2010 by Taylor and Francis Group, LLC