To evaluate the moments of vorticity fluctuations
it is necessary to perform an average over the statistics
of forcing which generates the fluctuations,
and over exit-times statistics:
Since the velocity field is smooth, two-dimensional and incompressible,
particles separate exponentially fast and the statistics of exit-times
can be replaced by the statistics of finite time Lyapunov exponents,
which measure the growth rate of the logarithm of the distance between
two particle during a time . The finite-time Lyapunov exponent
and exit-times are thus related in a smooth flow by the condition:
We remark that this relation does not hold in the case of non-smooth flows,
where the relative dispersion follows a power law [40].
In this case the growth of the distance between two
trajectories is dominated by the structure
of the flow at the same scale. The scaling law for the intensity
of the eddies of size
characterized by the Hölder
exponent of the rough velocity field, gives origin to a power law for
particle separation. Thus the exit-time
value is dominated
by the eddy at large scale
and is almost independent from the small
initial separation
.
Replacing the average over the statistics of exit-times with
the statistics of finite-time Lyapunov exponents, and using
the asymptotic behavior for the distribution
(see Section (2.4)), one gets
the following estimate for moments of vorticity fluctuations
The scaling exponents are evaluated from Eq. (2.31)
by a steepest descent argument as:
In the range
the scaling
exponent of the second order structure function for vorticity
is related to the spectral slope of vorticity power spectrum