The smooth-filamental transition predicted for
in the mean field case, can be recast taking into account
the intermittent behavior.
According to Eq. (2.20) the
exponent of the
-th structure function is the minimum between
, which corresponds to the smooth behavior, and
that describes the rough, filamental field.
In the Gaussian approximation the crossover between the linear
and non-linear behavior for the scaling exponent happens for:
It is worthwhile to underline that
in the non-smooth case is determined
by the value of
where
is the finite-time
Lyapunov exponent which minimizes
.
Since
is a growing function of
(in the quadratic approximation
)
higher moments of passive scalar fluctuations are
determined by the tail of the distribution of the stretching rates,
where the parabolic approximation for
does
not hold in general. Thus the behavior of high order
structure functions is extremely dependent on the detail
of the velocity field.
Nevertheless the presence of a smooth-filamental transition
is determined by the asymptotic behavior of the exponents
for
where the Gaussian approximation holds thanks to the
Central Limit Theorem, thus the smooth-filamental transition
is always present for
,
while for
the exact value of the critical moment
can deviate
from Eq. (2.23).
We notice that relations obtained in the Gaussian approximation
are exact for the peculiar case of the Kraichnan-model,
that is a smooth velocity field -correlated in time,
in which the Cramér function
is exactly parabolic.