Lagrangian Covariance Analysis of
β
-Plane Turbulence 7
The integral time scale of the vorticity is related to the second moment of the autocorrelation of the meridional velocity.
4. Application of Taylor’s Dispersion Result
The Taylor analysis is now applied to the meridional mixing on the β-plane. The meridional dispersion, in
the diffusion limit, is hx
2
tx
2
ti = 2hvvi
Z
∞
t − ηR
vv
η dη = 2hvvi[T
vv
t + T
2 1
vv
] .
32 It has been shown that
T
vv
= 0 and as a consequence the meridional dispersion is finite and independent of time:
hx
2
x
2
i = 2hvviT
2 1
vv
= 2 hζζi
β
2
. 33
The meridional dispersion is bounded and the bound is determined by the enstrophy and β. Note that the
meridional dispersion bound does not depend on the energy of the meridional velocity, hvvi, as might be
expected from Taylor’s kinematic dispersion result. The notion of a bounded meridional dispersion of a Lagrangian particle is not new; physical discussions of
the rectified circulation on a β-plane [6], [7] reflect a theoretical consideration of the possibility of bounded
particle dispersion for wave motion. Here we have shown that the particle dispersion is bounded for turbulent flow interacting with Rossby waves. The
β-plane simulations of [3], [4], or [5] present observations of the boundedness of the meridional dispersion. What we have here is, apparently, a mathematical explanation
in terms of the first two integral scales of the autocorrelation, of the bounded meridional dispersion seen in
β-plane simulations. Figure 12 of [4] shows a striking numerical example of the boundedness of the meridional particle dispersion predicted by this result.
4.1. Alternate Dispersion Bound Derivation
The relationship 33 has been arrived at by taking the diffusion limit of moments of the formal solution of 14. In that way several useful statistical results were derived. There is, however, a faster way, not using
Taylor’s dispersion expression and without any results for integral scales, of deducing 33. The result 33 is recognizable as being a direct consequence of the first integral of d
dtζ = −βv = −βddtx
2
= −βddty. Taking the first integral and squaring and averaging produces
hζtζti − 2hζtζ i + hζ
ζ i = β
2
hy − y
2
i. 34
For the inviscid unforced problem hζζi = hζ
ζ i and the equation is rewritten as
2 hζζi[1 − R
ζζ
t] = β
2
hy − y
2
i. 35
If the flow is ergodic one can assume R
ζζ
t → 0 as t → ∞. That the flow is ergodic is not a straightforward matter. Both inviscid two-dimensional turbulence or a random field of
β waves, asymptotic limits of the β-plane turbulence, have nonfinite integral time scales and are therefore not ergodic see [25] for additional
viewpoints. In Section 6 we verify that the integral time scale of the ζt process is finite which allows
R
ζζ
t → 0 as t → ∞ to obtain hy − y
2
i = 2 hζζi
β
2
, t → ∞,
36 i.e., the result 33. While the result 36 has a shorter derivation, it is important to recall that the initial
derivation leading to 33 placed this result in the context of the Taylor dispersion analysis and produced several additional results describing the statistical properties of stationary
β-plane turbulence.
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
