The actual values of the scaling exponents can be directly extracted
from the statistics of the passive scalar, which is not spoiled by
large-scale objects. In Figure 2.8 we plot the
exponents
as obtained by looking at the local
slopes of the structure functions
,
in comparison with the exponents predicted by
the Lagrangian exit-time statistics, according to
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The exit-time statistics has been obtained with the following procedure:
Exit-times provide an excellent tool
for estimating the scaling exponent of the field.
Since the thresholds can be chosen exactly within the inertial range,
the scaling of Lagrangian structure function
is not spoiled by contamination of the viscous range,
and the excellent scaling allows for a precise measurement
of the scaling exponents, which are in good agreement with those directly
observed from the structure function of the passive field.
Moreover, while the finite-time Lyapunov exponents
are achievable only in numerical simulations, the measure of exit-times
can be performed also in experiments by means of couple of Lagrangian tracers.
From the exit-time statistics it is also possible to recover
the right tail of the Cramér function as the
inverse Legendre transform [41]
of the scaling exponents
:
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