1.1 What does turbulence stand for?

The word turbulent is used in the everyday experience to indicate something which is not regular. In Latin the word turba means something confusing or something which does not follow an ordered plan. A turbulent boy, in all Italian schools, is a young fellow who rebels against ordered schemes. Following the same line, the behavior of a flow which rebels against the deterministic rules of classical dynamics is called turbulent. Even the opposite, namely a laminar motion, derives from the Latin word lámina, which means stream or sheet, and gives the idea of a regular streaming motion. Anyhow, even without the aid of a laboratory experiment and a Latin dictionary, we experience turbulence every day. It is relatively easy to observe turbulence and, in some sense, we generally do not pay much attention to it (apart when, sitting in an airplane, a nice lady asks us to fasten our seat belts during the flight because we are approaching some turbulence!). Turbulence appears everywhere when the velocity of the flow is high enough¹ , for example, when a flow encounters an obstacle (cf. e.g., Figures 1

and 2

) in the atmospheric flow, or during the circulation of blood, etc. Even charged fluids (plasma) can become turbulent. For example, laboratory plasmas are often in a turbulent state, as well as natural plasmas like the outer regions of stars. Living near a star, we have a big chance to directly investigate the turbulent motion inside the flow which originates from the Sun, namely the solar wind. This will be the main topic of the present review.

Figure 1:

Turbulence as observed in a river. Here we can see different turbulent wakes due to different obstacles (simple stones) emerging naturally above the water level. The photo has been taken by the authors below the dramatically famous Crooked Bridge in Mostar (Bosnia-Hercegovina), which was destroyed during the last Balcanic war, and recently re-built by Italian people.

Figure 2:

Turbulence as observed passing an obstacle in the same river of Figure 1

, allows us to look at a clear example of wake.

Turbulence that we observe in fluid flows appears as a very complicated state of motion, and at a first sight it looks (apparently!) strongly irregular and chaotic, both in space and time. The only dynamical rule seems to be the impossibility to predict any future state of the motion. However, it is interesting to recognize the fact that, when we take a picture of a turbulent flow at a given time, we see the presence of a lot of different turbulent structures of all sizes which are actively present during the motion. The presence of these structures was well recognized long time ago, as testified by the beautiful pictures of vortices observed and reproduced by the Italian genius L. da Vinci, as reported in the textbook by Frisch (1995

). Figure 3

shows, as an example, one picture from Leonardo which can be compared with Figure 4

taken from a typical experiment on a turbulent jet.

Figure 3:

Three examples of vortices taken from the pictures by Leonardo da Vinci (cf. Frisch, 1995

Figure 4:

Turbulence as observed in a turbulent water jet (Van Dyke, 1982 ) reported in the book by Frisch (1995

) (photograph by P. Dimotakis, R. Lye, and D. Papantoniu).

Turbulent features can be recognized even in natural turbulent systems like, for example, the atmosphere of Jupiter (see Figure 5

). A different example of turbulence in plasmas is reported in Figure 6

where we show the result of a typical high resolution numerical simulations of 2D MHD turbulence. In this case the turbulent field represents the current density. These basic features of mixing between order and chaos make the investigation of properties of turbulence terribly complicated, although extraordinarily fascinating.

Figure 5:

Turbulence in the atmosphere of Jupiter as observed by Voyager.

Figure 6:

High resolution numerical simulations of 2D MHD turbulence at resolution $2048 × 2048$ (courtesy by H. Politano). Here the authors show the current density $J (x, y)$ , at a given time, on the plane $(x,y)$ .

When we look at a flow at two different times, we can observe that the general aspect of the flow has not changed appreciably, say vortices are present all the time but the flow in each single point of the fluid looks different. We recognize that the gross features of the flow are reproducible but details are not predictable. We have to restore a statistical approach to turbulence, just like random or stochastic processes, even if the problem is born within the strange dynamics of a deterministic system!

Turbulence increases the properties of transport in a flow. For example, the urban pollution, without atmospheric turbulence, would not be spread (or eliminated) in a relatively short time. Results from numerical simulations of the concentration of a passive scalar transported by a turbulent flow is shown in Figure 7 . On the other hand, in laboratory plasmas inside devices designed to achieve thermo-nuclear controlled fusion, anomalous transport driven by turbulent fluctuations is the main cause for the destruction of magnetic confinement. Actually, we are far from the achievement of controlled thermo-nuclear fusion. Turbulence, then, acquires the strange feature of something to be avoided in some cases, or to be invoked in some other cases.

Figure 7:

Concentration field $c(x,y)$ , at a given time, on the plane $(x, y)$ . The field has been obtained by a numerical simulation at resolution $2048 × 2048$ . The concentration is treated as a passive scalar, transported by a turbulent field. Low concentrations are reported in blue while high concentrations are reported in yellow (courtesy by A. Noullez).

Turbulence became an experimental science since O. Reynolds who, at the end of XIX century, observed and investigated experimentally the transition from laminar to turbulent flow. He noticed that the flow inside a pipe becomes turbulent every time a single parameter, a combination of the viscosity coefficient $j$ , a characteristic velocity $U$ , and length $L$ , would increase. This parameter $Re = U Lr/j$ ( $r$ is the mass density of the fluid) is now called the Reynolds number. At lower $Re$ , say $Re < 2300$ , the flow is regular (that is the motion is laminar), but when $Re$ increases beyond a certain threshold of the order of $Re -~ 4000$ , the flow becomes turbulent. As $Re$ increases, the transition from a laminar to a turbulent state occurs over a range of values of $Re$ with different characteristics and depending on the details of the experiment. In the limit $Re --> oo$ the turbulence is said to be in a fully developed turbulent state. The original pictures by O. Reynolds is shown in Figure 8

Figure 8:

The original pictures by O. Reynolds which show the transition to a turbulent state of a flow in a pipe, as the Reynolds number increases from top to bottom (see the website Reynolds, 1883 ).