Space and Laboratory Plasma Physics

Physics has experienced several revolutions in the twentieth century that profoundly changed our understanding of nature. Quantum mechanics and (special, general) relativity are the best known and certainly the most important, but the discovery of the fourth state of matter – the state of plasma – as the most natural form of ordinary matter in the Universe, with more than 99% of visible matter being in this form, is unquestionably a revolution in physics. This discovery has led to the emergence of a new branch of physics called plasma physics.

Plasma physics describes the coupling between electromagnetic fields and ionized matter (electrons, ions). Thus, it is based upon one of the four foundations of physics: the electromagnetic interaction whose synthetic mathematical formulation was made by the Scottish physicist J. C. Maxwell who published in 1873 two heavy volumes entitled A Treatise on Electricity and Magnetism. The discovery of the electron by J.J. Thomson in 1897 and the formulation of the theory of the atom at the beginning of the twentieth century have contributed to the first development of plasma physics. It was in 1928 that the name plasma was proposed for the first time by I. Langmuir, referring to blood plasma in which one finds a variety of corpuscles in movement. Experimental studies of plasmas first focused essentially on the phenomenon of electrical discharge in gas at low pressure with, for example, the formation of an electric arc. These studies initiated during the second half of the twentieth century were extended to problems related to the reflection and transmission of radio waves in the Earth's upper atmosphere (this was how the first transatlantic link was established by Marconi in 1901), which led to the discovery of the ionosphere, an atmospheric layer beyond 60 km altitude with a thickness of several hundred kilometers. As explained by the astronomer S. Chapman (1931), the ionosphere consists of gas partially ionized by solar ultraviolet radiation; therefore, it is the presence of ionospheric plasma which explains why low-frequency waves can be reflected or absorbed depending on the frequency used.

Turbulence is generally associated with the formation of vortices in a fluid. There are numerous experiences in daily life where one can note the presence of turbulence: the movements of a river downstream of an obstacle, the smoke escaping through a chimney, vortical motions of the air, or the turbulence zones that we sometimes cross by plane. Since it is not necessary to use powerful microscopes or telescopes to study turbulence one could conclude that it is probably not difficult to understand it. Unfortunately that is not the case! Although significant progress has been made since the middle of the twentieth century, several important questions remain unanswered and it is clear that at the beginning of the twenty-first century turbulence remains a central research topic in physics.

The first theoretical bricks of turbulence were laid from the moment physicists started to tackle the non-linearities of the hydrodynamic equations. As we will see, it is in this context that the first fundamental law of turbulence was established: this was the statistical law of Kolmogorov (1941). Nowadays the theoretical treatment of turbulence is partly based on numerical simulations which, accompanied by very powerful tools of visualization, allow us to tackle this difficult problem from a different angle and stimulate new questions. The purpose of this chapter is to present concepts and fundamental results on fully developed turbulence. This chapter is devoted to hydrodynamics, from which some foundations of the theory of turbulence have emerged. The two other chapters in this part of the book will be devoted to MHD turbulence.

What is Turbulence?

Unpredictability and Turbulence

It is not easy to define turbulence quantitatively because to do this one requires knowledge of a number of concepts that will be defined partly in this chapter.

Without going into the details, we can notice that the disordered – or chaotic – aspect seems to be the main characteristic of turbulent flows. It is often said that a system is chaotic when two points originally very close to each other in phase space separate exponentially over time. As we will see later, this definition can be extended to the case of fluids.

In our familiar environment, matter appears in solid, liquid, or gaseous form. This triptych vision of the world was shaken in the twentieth century when astronomers revealed that most of the extraterrestrial matter – namely more than 99% of the ordinary matter in the Universe – is actually in an ionized state called plasma whose physical properties differ fundamentally from those of a neutral gas. The study of this fourth state of matter was developed mainly in the second half of the twentieth century and is now considered a major branch of modern physics. A decisive step was taken in 1942 when the Swedish astrophysicist Hannes Alfvén (1908–1995) proposed the theory of magnetohydrodynamics (MHD) by connecting the Maxwell electrodynamics with the Navier–Stokes hydrodynamics. In this framework, plasmas are described macroscopically as a fluid and the corpuscular aspect of ions and electrons is ignored. Nowadays, MHD has emerged as the central theory to understand the machinery of the Sun, stars, stellar winds, accretion disks around super-massive objects such as black holes with the formation of extragalactic jets, interstellar clouds, and planetary magnetospheres. Also, when H. Alfvén was awarded the Nobel Prize in Physics in 1970, the Committee congratulated him “for fundamental work and discoveries in magnetohydrodynamics with fruitful applications in different parts of plasma physics.”

The MHD description is not limited to astrophysical plasmas, but is also widely used in the framework of laboratory experiments or industrial developments for which plasmas and conducting liquid metals are used. In the first case, the emblematic example is certainly controlled nuclear fusion with the International Thermonuclear Experimental Reactor (ITER) in Cadarache. Indeed, the control of a magnetically confined plasma requires an understanding of the large-scale equilibrium and the solution of stability problems whose theoretical framework is basically MHD. Liquid metals are also used, for example, in experiments to investigate the mechanism of magnetic field generation – the dynamo effect – that occurs naturally in the liquid outer core of our planet via turbulent motions of a mixture of liquid metals. Most of the natural MHD flows cited above are far from thermodynamic equilibrium, with highly turbulent dynamics.

When a static equilibrium has been found (see Chapter 8), the next question that we have to address concerns the stability of this equilibrium. A part of the answer is given by the linear perturbation theory, which consists of analyzing the result of a small (i.e. linear) perturbation of the equilibrium. If the equilibrium is stable, the perturbation will behave as a wave that propagates in the medium; if it is unstable, the perturbation will increase exponentially.

Instabilities

Classification

In Figure 9.1, we present some unstable and stable situations arising from the example of a sphere placed in an external potential field. In case 1 a sphere is at the bottom of a well of infinite potential. In this position the sphere can only perform oscillations around its equilibrium position. These oscillations, once generated, are damped due to friction until the sphere reaches a static equilibrium position at the bottom of the potential well. This is a situation of stable equilibrium. In case 2 a sphere is placed a the top of a potential (a hill). In this case, a small displacement of the sphere is sufficient to move it to much lower potentials: this is an unstable situation that is often associated with a linear instability. The third case is that of a metastable state where the sphere is placed initially on a locally flat potential (a plateau): a small displacement around the initial position does not change the potential of the sphere. Finally, the last case (case 4) is that of a sphere placed in a hollow. This is an example of non-linear instability: the sphere is stable against small perturbations but becomes unstable for larger disturbances.

In plasma physics, the sphere in the previous paragraph corresponds to a particular mode of a wave and the shape of the potential can be a source of free energy. There are many energy sources in space plasmas. For example, the solar wind is a continuous source of energy for the Earth's magnetospheric plasma

which is never in a static equilibrium. The consequences of this energy input are the generation of large-scale gradients and the deformation of the distribution functions of particles at small scales.

Vector Identities

• Multiple products:

(A × B) · (C × D) = (A · C)(B · D) − (A · D)(B · C)

A · (B × C) = B · (C × A) = C · (A × B)

A × (B × C) = B(A · C) − C(A · B)

• Rules for products with derivatives:

∇(fg) = f (∇g) + g(∇f)

∇(A · B) = A × (∇ ×B) + B × (∇ ×A) + (A · ∇)B + (B · ∇)A

∇ · (fA) = f (∇ · A) + A · (∇f)

∇ · (A × B) = B · (∇ ×A) − A · (∇ ×B)

∇ ×(fA) = f (∇ ×A) − A × (∇f)

∇ ×(A × B) = (B · ∇)A − (A · ∇)B + A(∇ · B) − B(∇ · A)

• Rules for second-order derivatives:

∇ · (∇ ×A) = 0

∇ ×(∇f) = 0

∇ ×(∇ ×A) = ∇(∇ · A) – A

Vectorial derivatives

We list here the vectorial derivatives in the three usual coordinate systems (see figure A2.1).

In its primitive form the Kolmogorov theory states that the four-fifths law can be generalized to higher-order structure functions according to relation (11.33) by assuming self-similarity. Experiments and numerical simulations clearly show a discrepancy from this prediction (see Figure 11.10): this is what is commonly called intermittency. Even if intermittency remains a still poorly understood property of turbulence because it still challenges any attempt at a rigorous analytical description from first principles (i.e. the Navier–Stokes equations), several models have been proposed to reproduce the statistical measurements, of which the simplest is probably the fractal model, also called the β model, which was introduced in 1978 (Frisch et al., 1978). As we shall see, this model is based on the idea of a fractal (incompressible) cascade and is therefore inherently a self-similar model. However, because the structure-function exponents are not those predicted by the Kolmogorov theory, one speaks of intermittency and anomalous exponents. Refined models have also been proposed, and we will present in this chapter the two most famous models: the log-normal and log-Poisson models.

Intermittency

Fractals and Multi-fractals

The idea underlying the β fractal model is Richardson's cascade (Figure 11.7): at each step of the cascade the number of children vortices is chosen so that the volume (or the surface in the two-dimensional case) occupied by these eddies decreases by a factor β (0 < β < 1) compared with the volume (or surface) of the parent vortex. The β factor is a parameter less than one of the model to reflect the fact that the filling factor varies according to the scale considered: the smallest eddies occupy less space than the largest.

We define by ln the discrete scales of our system: the fractal cascade is characterized by jumps from the scale ln to the scale ln+1.We show an example of a fractal cascade in Figure 13.1: at each step of the cascade the elementary scale is divided by two.

The origin of the Earth's magnetic field constitutes one of the most fascinating problems of modern physics. Ever since the works of the physicist W. Gilbert in 1600, we have known that the magnetic field detected with a compass has a terrestrial origin, but a precise understanding of its production (together with the good regime of parameters) remains elusive. The generation of the Earth's magnetic field by electric currents inside our planet was proposed by Amp`ere just after the famous experiment done by Ørsted in 1820 (a wire carrying an electric current is able to move the needle of a compass). These currents cross the Earth's outer core, which is made of liquid metal (mainly iron) at several thousand degrees. If it were not maintained by a source these currents would disappear within several thousand years through Ohmic dissipation. Indeed, in the absence of any regenerative mechanism the Earth's magnetic field would decay in a time τdiff that can be estimated with the simple relation τdiff ~ l2/η. Since the Earth's metallic envelope is characterized by a thickness of ~ 2000 km and a magnetic diffusivity η ~ 1m2 s-1, we obtain τdiff ~ 30 000 years. Also, in order to explain the presence of a large-scale magnetic field on Earth since several million years ago, it is necessary to introduce the dynamo mechanism. It was Sir J. Larmor in 1919 who first suggested that the solar magnetic field could be maintained by what he called a self-excited dynamo, a theory explaining the formation of sunspots. Generally speaking, the dynamo effect explains the solar, stellar, and even galactic magnetic fields.

Geophysics, Astrophysics, and Experiments

Experimental Dynamos

The simplest experiment concerning a self-excited dynamo is Bullard's dynamo as shown in Figure 5.1: it is made of a conducting disk which rotates in a medium

where an axial magnetic field B0 is present. An electric wire going around the axis is connected on one side to the axis and on the other side to the disk (the electric contacts do not prevent the rotation). Because of the Lorentz force an electric potential is induced between the center and the side of the disk. The induced electromotive force e1 generates a current i1 in the loop, which in turn generates an axial induced magnetic field B1, which adds to the initial field.