### 1.2 Dynamics vs. statistics

In Figure 9 a typical sample of turbulence as observed in a fluid flow in the Earth’s atmosphere can be
observed. Time evolution of both the longitudinal velocity component and the temperature is shown.
Measurements in the solar wind show the same typical behavior. A typical sample of turbulence as
measured by Helios 2 spacecraft is shown in Figure 10. A further sample of turbulence, namely the radial
component of the magnetic field measured at the external wall of an experiment in a plasma device realized
for thermonuclear fusion, is shown in Figure 11.
As it is well documented in these figures, the main feature of fully developed turbulence is the chaotic
character of the time behavior. Said differently, this means that the behavior of the flow is unpredictable.
While the details of fully developed turbulent motions are extremely sensitive to triggering
disturbances, average properties are not. If this was not the case, there would be little significance in
the averaging process. Predictability in turbulence can be recast at a statistical level. In other
words, when we look at two different samples of turbulence, even collected within the same
medium, we can see that details look very different. What is actually common is a generic
stochastic behavior. This means that the global statistical behavior does not change going from one
sample to the other. The idea that fully developed turbulent flows are extremely sensitive to
small perturbations but have statistical properties that are insensitive to perturbations is of
central importance throughout this review. Fluctuations of a certain stochastic variable are
defined here as the difference from the average value , where brackets means some
averages. Actually, the method of taking averages in a turbulent flow requires some care. We
would like to remind that there are, at least, three different kinds of averaging procedures that
may be used to obtain statistically-averaged properties of turbulence. The space averaging is
limited to flows that are statistically homogeneous or, at least, approximately homogeneous over
scales larger than those of fluctuations. The ensemble averages are the most versatile, where
average is taken over an ensemble of turbulent flows prepared under nearly identical external
conditions. Of course, these flows are not nearly identical because of the large fluctuations
present in turbulence. Each member of the ensemble is called a realization. The third kind of
averaging procedure is the time average, which is useful only if the turbulence is statistically
stationary over time scales much larger than the time scale of fluctuations. In practice, because of
the convenience offered by locating a probe at a fixed point in space and integrating in time,
experimental results are usually obtained as time averages. The ergodic theorem (Halmos, 1956)
assures that time averages coincide with ensemble averages under some standard conditions (see
Appendix 13).
A different property of turbulence is that all dynamically interesting scales are excited, that is, energy is
spread over all scales. This can be seen in Figure 12 where we show the magnetic field intensity
within a typical solar wind stream (see top panel). In the middle and bottom panels we show
fluctuations at two different detailed scales. A kind of self-similarity (say a similarity at all scales) is
observed.
Since fully developed turbulence involves a hierarchy of scales, a large number of interacting degrees of
freedom are involved. Then, there should be an asymptotic statistical state of turbulence that is
independent on the details of the flow. Hopefully, this asymptotic state depends, perhaps in a critical way,
only on simple statistical properties like energy spectra, as much as in statistical mechanics equilibrium
where the statistical state is determined by the energy spectrum (Huang, 1987). Of course, we cannot
expect that the statistical state would determine the details of individual realizations, because realizations
need not to be given the same weight in different ensembles with the same low-order statistical
properties.
It should be emphasized that there are no firm mathematical arguments for the existence of an
asymptotic statistical state. As we have just seen, reproducible statistical results are obtained from
observations, that is, it is suggested experimentally and from physical plausibility. Apart from physical
plausibility, it is embarrassing that such an important feature of fully developed turbulence, as the
existence of a statistical stability, should remain unsolved. However, such is the complex nature of
turbulence.