8 J.R. Ristorcelli and A.C. Poje
4.2. Approach to the Diffusion Limit
Achieving the dispersion bound predicted by 36 or 33 depends on how rapidly R
ζζ
vanishes. The result 36 can be also be written, using 26, as
hζtζ i = hζ
ζ i − β
2
hvvi
Z
t
t − ηR
vv
η dη. 37
Since R
vv
has an oscillatory character one anticipates that hζtζ
i may not vanish rapidly. If one takes the two-time correlation of
ζ in the forced β-plane simulations of Figure 3 of [4] as indicative of hζζ i one sees
that R
ζζ
t decreases slowly. That the approach of
R
ζζ
t → 0 may be slow can also be anticipated in the following way: in the absence of the
β-effect ζ is materially conserved and R
ζζ
t is constant. For the β-plane, R
ζζ
t vanishes in proportion to the size of
β. Dividing through by hζ ζ
i the approach to the diffusion limit goes as β
2
hx
2
− x
20 2
i 2
hζtζti = 1
− R
ζζ
t. 38
One might view 33 or 36 as an upper bound on the particle dispersion: R
ζζ
while positive for small t changes sign becoming oscillatory as it decreases with increasing t. There is another useful way of
understanding the intermediate time result 38. The autocorrelation of the vorticity can be computed from the meridional particle dispersion.
5. Summary of Results
The assumptions of statistical stationarity have allowed several important statistical properties of β-plane
turbulence to be deduced. In anticipation of the next section, in which numerical simulations are used to verify and extend the predictions of the mathematical development, the results are summarized:
1. The micro-time-scale of the vorticity is τ
2
ζ
= 2
hζζi β
2
hvvi .
39 Thus the curvature at the origin of the autocorrelation, ostensibly a small scale quantity, is related to
the large scale quantities of enstrophy and meridional energy. Note that for a two-dimensional β = 0
inviscid homogeneous turbulence the microscale is not finite. 2. The integral time scale of the meridional velocity is zero,
T
vv
=
Z
∞
R
vv
η dη = 0, 40
as is characteristic of wave processes. 3. The second integral time scale of the meridional velocity is related to single point second-order quantities:
T
2 1
vv
= −
Z
∞
ηR
vv
η dη = hζζi
β
2
hvvi =
1 2
τ
2
ζ
. 41
4. A relationship between the autocorrelation functions of the meridional velocity and the vorticity has been obtained:
R
ζζ
τ = 1 − β
2
hvvi hζζi
Z
τ
Z
η
R
vv
η
1
d η
1
d η
42 = 1
− β
2
hvvi hζζi
Z
τ
τ − ηR
vv
η dη. 43
The integral on the last line appears in Taylor’s [12] treatment of the dispersion process: the meridional par- ticle dispersion is diagnostically related to the vorticity autocorrelation. For an inviscid two-dimensional
turbulence, β = 0, in which vorticity is materially conserved, R
ζζ
τ = 1.
Lagrangian Covariance Analysis of
β
-Plane Turbulence 9
5. The integral time scale of the vorticity is related to the second moment of the velocity autocorrelation: T
ζζ
= β
2
hvvi 2
hζζi
Z
∞
τ
2
R
vv
τ dτ. 44
6. A diagnostic expression relating the autocorrelation of the vorticity to the time evolution of an ensemble of Lagrangian particles has been derived:
R
ζζ
t = 1 − β
2
hx
2
tx
2
ti 2
hζζi .
45 7. The diffusion limit of the diagnostic relationship above predicts a bound on the dispersion of an ensemble
of Lagrangian particles. Unlike the classic Taylor dispersal problem, in which the particle diffusion length scales as a Brownian diffusion,
∼ t
1 2
, the meridional particle dispersion in a homogeneous β-plane turbulence is bounded by
hx
2
x
2
i
∞
= 2
hζζi β
2
. 46
This is due to the vanishing of the integral time scale of the meridional velocity [18] and the fact that an explicit expression for the second integral time scale of
vt is possible. That the meridional particle dispersion does not scale as a Brownian diffusion means, from a practical point
of view, that turbulent transport is not suitably parameterized by an eddy diffusivity. This statement is only valid on time and length scales appropriate to the diffusion limit.
6. Numerical Results
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