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Accueil > Recherche > Plasmas Spatiaux > Thématiques scientifiques > Theoretical modelling of collisionless plasmas

Theoretical modelling of collisionless plasmas

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Most space plasmas studied at LPP can be called "collisionless". This property has important consequences on the phenomena that occur in these plasmas as well as how to understand and model them.
The study of these questions is naturally one of the specialties of the researchers of the space plasma team. A book, "Collisionless Plasmas in Astrophysics", presents the state of the art in this field. Two of its co-authors belong to the space plasma team of LPP. A course for PhD Students is also provided annually at the Meudon Observatory, by the authors and their colleagues from Observatory, on this subject.

  LPP team

N. Aunai, G. Belmont, T. Chust, C. Krafft, L. Rezeau, R. Smets

 Selected publications

 Collisionless ?

What is a plasma without collisions ? First of all, what do we mean by "collisions" in a plasma ? Is it comparable to the usual notion of collisions between neutral particles ? The answer is no.

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Collision dans un gaz neutre ou dans un plasma. Les collisions entre particules neutres sont binaires (figure de gauche), c’est-à-dire qu’une particule ne subit l’influence que d’une autre particule à la fois, et seulement lorsqu’elle en passe très près. Entre ces "collisions", la particule va en ligne droite. Dans un plasma (figure de droite), chaque particule a une trajectoire sans point anguleux, régie à chaque instant par le champ électrique dû à toutes les autres. Cette trajectoire serait celle en pointillé si les particules environnantes formaient un fluide continu, mais les petits écarts au caractère "fluide continu" font que la particule s’éloigne progressivement de cette trajectoire idéale. C’est ce qu’on appelle "collisions" en physique des plasmas.

In a neutral medium (left figure), each particle goes straight on, except at a few points where it "meets" another particle (that is to say at an extremely close pass) and where it is deflected suddenly. The trajectory is thus a succession of line segments, with some turning points (the number of turning points tends to zero when the density decreases). The distance between two turning points is called "mean free path".
Conversly in a plasma (right picture), interactions between charged particles are not made at very short distance, but at long range via the electrostatic interaction, in 1 / r2. We are therefore in the opposite limit to the neutral gas : each particle is at every moment influenced by a very large number of other particles which surround it. Its trajectory is always essentially governed by a collective field due to all the other particles that are considered as a continuous fluid. To this "ideal" field are superimposed fluctuations due to the non-continuous nature of the sources of the field. It follows disturbances - "soft" and never angular- which make it diverge substantially from its ideal path only after some time. The distance traveled during this time is also called the "mean free path", but its physical significance is therefore substantially different from the previous one, in neutral gas.

A plasma can be told "collisionless" when one can completely ignore the non-continuous nature of the sources of the field, that is to say, by definition, whenever we study phenomena at smaller scales than the mean free path.

  kinetic and fluid

How to describe a collisionless plasma ? The most comprehensive statistical way to do this is to consider the velocity distribution function f (v), which is the probability density of observing a particle of velocity v at a place r and at a time t. All classical quantities of fluids theories, the density n, the fluid velocity u and the pressure P are deduced from it, because they are integrals of this function (moments of order 0, 1 and 2).

How to model the evolution of a collisionless plasma ? The evolution of the distribution function is calculable because it is governed by a deterministic equation : the Vlasov equation. As soon as one is able to determine the evolution of f (v), the evolution of macroscopic momentum n, u and P is deduced naturally. Instead, when you know those moments in an initial condition without knowing f (v), their subsequent evolution cannot be uniquely determined because multiple distribution functions, evolving differently, can share these same momentums (Figure 2) . So it’s a situation very different from thermodynamics, where the knowledge of the first few initial moments allows determining unambiguously the subsequent evolution of these moments.

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La fonction de distribution de gauche, qui est Maxwellienne, partage les mêmes premiers moments, densité, vitesse moyenne et pression, que la fonction de droite, qui est constituée de deux faisceaux séparés se propageant en sens inverse. Il est peu probable que si l’on part, dans une portion limitée de l’espace, de l’une ou l’autre de ces distributions, l’évolution ultérieure des distributions et de ses moments, soit la même. Ceci montre la difficulté de modéliser les plasmas sans collisions en ne considérant que leurs moments.

If one wants (or can) only use the moments, as is done in thermodynamics, that is to say, without introducing the function f (v), how to do it ?
By integrating it, one can deduce from Vlasov equation a set of equations that link the evolution of the different moments. The firsts of these equations are then essentially identical to usual fluid equations, continuity equation, Navier-Stokes, energy, etc ... One can that way build an infinite number of equations, but it will never form a closed system as long as all the moments until an infinite order are not introduced. If we want to stick to moments of orders 0, 1 and 2, as in thermodynamics, the system includes the moment of order 3 (heat flux). If we want to describe the evolution of moment of order 3, the system includes the moment of order 4, etc. This is consistent with the fact that in a collisionless plasma, the knowledge of only the first few moments in the initial condition does not allow to determine uniquely the subsequent evolution of these given moments.

 Thermodynamics, MagnetoHydroDynamics and kinetic phenomena.

Thermodynamics, as well as fluid mechanics or MHD (MagnetoHydroDynamics used for magnetized plasmas) are "closed" theories at the level of the first three moments, n, u and P. How is this possible ?
These theories do contain three conservation equations that can be derived from Vlasov equation and represent the conservation of particle number, momentum and energy. But they also include an additional equation, called "equation of state", which closes the system without introducing the moment of order 4 of the distribution function. This "closure" equation then relies on an assumption : there are a large number of collisions that continuously redistribute impulses and energies of particles of unknown but assumed equiprobable manner. This assumption leads to the principle of "maximization of the entropy", which means that only the most probable distribution function (Gaussian in the stationary homogeneous case) can be observed. This is obviously not applicable to a collisionless plasma where the notion of "most probable state" does not make sense : if one starts for instance from any homogeneous and stationary distribution, Gaussian or not, nothing will make it evolve over time and this stationary distribution therefore has no reason to be regarded as more or less probable than another one.

Clearly one must definitively renounce to "entropic" closures when working in a collisionless plasma. But this does not mean that a fluid theory can never be used in this case. This is possible whenever a closure equation can be found. The fact that all particles have their trajectory governed by the same collective electromagnetic field often makes this possible. For example, it is not uncommon that an adiabatic (zero heat flow) hypothesis for ions and an isothermal hypothesis (constant temperature) for electrons be justified. But these properties necessarily come from the physical properties of the system, unrelated with either the entropy or any probabilistic argument.

When no simple closure equation exists allowing to model in a "fluid" way a phenomenon, this phenomenon can be described as "kinetic". The kinetic models, which consist to look at the evolution of f (v) anywhere and at any time, are naturally much heavier than fluid models. They remain even beyond the capabilities of current computers as soon as one wants to treat 3D problems at large scales.

 Some examples of « kinetic » phenomena

Some of the space physics problems studied at LPP can be considered as being “kinetic”. A non exhaustive list of examples is given below.

Landau damping
When describing a plasma as a fluid, one can determine the number and the properties of all linear waves that can propagate in it. In the absence of viscosity or other dissipative effect, they propagate without damping. Instead of the small number of fluid modes, a full kinetic calculation (but naive) would seem to indicate that there are actually an infinite number of propagation modes. But this result is misleading : from any "reasonable" initial condition for the distribution functions, there is superposition of these different modes, thus, quickly, it only remains waves that propagate almost like fluid modes, but with one key difference : they always present a small damping, known as "Landau damping". This kinetic damping even becomes strong for waves propagating at a speed close to the particles thermal velocity. The existence of a systematic damping coming from a non dissipative equation (Vlasov) has given rise to many philosophical debates since it has been highlighted (in 1958).

Magnetopause and magnetic reconnection.

In MHD theory, a fluid theory, the magnetic field lines are always "frozen" in the plasma. This means that they are as elastic wires driven by the flow : they move, deform but each keeps its identity during movement. In this context, the solar wind may drive its field lines close to the magnetosphere, but it should not be able to cross a sealed boundary, the magnetopause, beyond which the field lines are those of the Earth magnetic field. In MHD, a field line from the solar wind cannot re-connect to a geomagnetic one. The magnetopause is an example of what is called, in plasma physics, a "tangential discontinuity". In reality, the magnetopause is sometimes thin enough to violate the limits of validity of the MHD, and the resulting small penetration of solar plasma in the magnetosphere has a series of important consequences (including, ultimately, the auroras). The study of the reconnection (between those field lines of different sources) is very dependent on kinetic effects because they a priori become the dominant ones when the MHD becomes invalid. Determine an equilibrium state of the boundary that be stationary at the kinetic scale, unstable or not vis-à-vis the reconnection is in itself a delicate study in which the LPP has widely contributed.

Turbulence. Turbulence is a set of many fluctuations, at all scales, in a nonlinear equilibrium. Its theory was highly developed first in hydrodynamics, then in MHD. When one investigates experimentally the turbulence properties, for example in the solar wind, it is found that its spectrum extends well beyond the limits of validity of the MHD, that is to say up to scales smaller than the ion Larmor radius, and even than that of the electron one. The experimental knowledge of turbulence at these very small scales has increased significantly in recent years thanks to spacecraft measurements, but the corresponding theory, for which the kinetic effects are critical, is still at its beginning. LPP team contributes to both fronts, experimental and theoretical.

CNRS Ecole Polytechnique Sorbonne Université Université Paris-Saclay Observatoire de Paris
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