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

Figure 9:

Turbulence as measured in the atmospheric boundary layer. Time evolution of the longitudinal velocity and temperature are shown in the upper and lower panels, respectively. The turbulent samples have been collected above a grass-covered forest clearing at $5 m$ above the ground surface and at a sampling rate of $56 Hz$ (Katul et al., 1997

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 $y$ are defined here as the difference from the average value $dy = y - <y >$ , 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).

Figure 10:

A sample of fast solar wind at distance $0.9 AU$ measured by the Helios 2 spacecraft. From top to bottom: speed, number density, temperature, and magnetic field, as a function of time.

Figure 11:

Turbulence as measured at the external wall of a device designed for thermonuclear fusion, namely the RFX in Padua (Italy). The radial component of the magnetic field as a function of time is shown in the figure (courtesy by V. Antoni).

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.

Figure 12:

Magnetic intensity fluctuations as observed by Helios 2 in the inner solar wind at $0.9 AU$ , for different blow-ups. The apparent self-similarity is evident here.

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.