4 J.R. Ristorcelli and A.C. Poje
T
vv
=
Z
∞
R
vv
η dη, 6
T
2 1
vv
= −
Z
∞
η R
vv
η dη, 7
the meridional dispersion becomes, in the diffusion limit [ t → ∞],
hx
2
tx
2
ti
∞
= 2 hvvi[T
vv
t + T
2 1
vv
] .
8 This kinematic result indicates that asymptotic meridional dispersion is determined by the zeroth and first
moments of the meridional velocity autocorrelation. As one of the objectives of this article is to understand the role that
β plays on the dispersal of fluid particles it is necessary to understand how the imposition of a mean vorticity gradient influences the integral time scales
T
vv
and T
1
vv
.
3. Moment Analysis of the Potential Vorticity Equation
In this section several consequences of a moment analysis of 2 are derived. A Lagrangian analysis of the potential vorticity equation is used to obtain the necessary statistical information of the dynamics appearing
in Taylor’s kinematic dispersion result.
3.1. Temporal Microscale of ζt
The two-point temporal correlation of the vorticity, for a stationary process, can be written as R
ζζ
τ = hζtζt + τi
hζtζti .
9 The correlation is an even function and it is customary to define a time scale, the temporal microscale, by
the curvature of R
ζζ
at the origin: for small τ ,
R
ζζ
τ = 1 − τ
2
τ
2
ζ
+ · .
10 Using the definition of
R
ζζ
a Taylor series is used to relate τ
2
ζ
to the square of the temporal derivative of the vorticity:
R
ζζ
τ = 1 − τ
2
2 hζζi
¿
d d
t ζ
d d
t ζ
À
+ · .
11 Squaring and averaging the vorticity equation, 2, indicates that
h ˙ζ ˙ζi =
¿
d d
t ζ
d d
t ζ
À
= β
2
hvvi = 2
hζζi τ
2
ζ
12 and thus the temporal microscale is known in terms of single point second-order moments:
τ
2
ζ
= 2
hζζi β
2
hvvi .
13 Note that curvature at the origin of the
ζ autocorrelation, a “small scale” quantity, has been related to “large scale” second moment quantities,
hζζi, hvvi. Note also that in the limit of inviscid two-dimensional turbulence, [
β → 0], that the microscale, for statistically homogeneous stationary, is infinite.
Lagrangian Covariance Analysis of
β
-Plane Turbulence 5
3.2. Integral Time Scale of vt
The Lagrangian solution for the vorticity, 2, can be formally written as ζt|x
, t =
ζ − β
Z
t
vt
1
|x , t
d t
1
. 14
The meridional velocity that drives 2 is related to the vorticity through a Poisson equation. Thus expression 14, while not immediately useful as a solution per se, allows several rigorous results for the statistical
properties of the solution of 2 to be deduced. The quantity
ζ is a random variable denoting the vorticity at the tagging time and location,
t ,
x
i
. Multiplying 14 by
ζt and averaging produces hζtζti = hζtζ
i − β
Z
t
hζtvt
1
i dt
1
. 15
The fluid particle identification has, for convenience, and without loss of generality, been dropped. As t → ∞, hζtζ
i → 0 and one finds that hζ
ζ i = −β
Z
∞
hζtvt − ηi dη; 16
the integral time scale of the joint process, {ζ, v}, is finite and known in terms of the initial enstrophy and
β. The fact that the variance, hζζi = hζ ζ
i, is stationary for the inviscid problem has been used. The moment of 14 with the meridional velocity produces
hvtζ i − β
Z
t
hvtvt
1
i dt
1
= 0 ,
17 where again stationarity implies
hvtζti = 0. This can be rewritten as hvtζ
i = βhvvi
Z
t
R
vv
η dη, 18
which indicates that, in the diffusion limit, the first integral time scale of the meridional velocity is given by T
vv
=
Z
∞
R
vv
η dη = 0, 19
since hvtζ
i → 0 as t → ∞. This result and its consequences are the central result of this article. The vanishing of the first integral time scale for finite
β implies that the meridional particle dispersion is bounded see [18].
The above results are a rigorous consequence of 2, hζvi = 0, stationarity, and that the statistics of the
flow forget their initial condition, i.e., hζtζ
i → 0 and hvtζ i → 0 as t → ∞.
3.3. Autocorrelation of ζt and the Second Integral Time Scale of vt
