To directly check whether small-scale vorticity can be considered as passively advected by velocity, we integrated at the same time Navier-Stokes equation for the vorticity field (Eq. (2.6)) and the equation of transport of passive scalar with a finite lifetime (Eq. (2.7)) advected by the velocity fields resulting from parallel integration of Navier-Stokes equation.
The parameters appearing in Eqs. (2.6) and (2.7)
are the same ( and
),
yet the forcings
and
are independent processes with the same statistics.
As already discussed the independence of the two forcings is crucial:
were the two fields forced with the identical random process,
the difference field
would be free-decaying,
and after a short time the passive field would became
identical to the active one.
According to the picture drawn in previous section
we expect to observe the same statistics for the
fluctuations of vorticity
and passive scalar
on scales
small enough to ensure that
the passive condition (2.37)
for vorticity is satisfied.
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In Figure 2.6 we show the power spectra of vorticity, , and
of passive scalar
.
The estimate of the range of wave-numbers at which the statistics
of vorticity and passive scalar are expected to be coincident, is
.
With the actual values
,
and
(see Figure 2.9) we have
.
The two curves in Figure 2.6
are indeed parallel at large
(
),
in agreement with the expectation
.
At smaller wave-numbers we observe a big bump in
around
which has no correspondent in
.
This deviation is most likely associated to the presence
of an inverse energy flux in the Navier-Stokes equation,
a phenomenon that has no equivalent in the passive scalar case.
Due to this effect, the scaling quality
of
is poorer than the
one,
and a direct comparison of scaling exponents in physical space
is even more difficult.
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However, we observe in Figure 2.7 that the probability
density functions of vorticity and passive scalar increments,
once rescaled by their root-mean-square fluctuation, collapse
remarkably well onto each other.
That is sufficient to state, along with the result
obtained from Fig. 2.6, the
equality of scaling exponents of vorticity and passive scalar at any order:
.
The excellent collapse of the probability distribution of fluctuations confirms that the underlying mechanism which generates intermittency, that is the competition between exponential separation of Lagrangian trajectories and exponential decay of fluctuations due to the linear damping, is the same both for vorticity and passive scalar.