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