Plasma Physics I

Il numero di Reynolds magnetico in presenza di advezione dominante

MHDRe = ( )ηm 2 advection dominating≫1,cQuesto numero può essere correlato ai cosiddetti numeri di Peclet, una classe di numeri adimensionali utilizzati per studiare i fenomeni di trasporto in un continuo. È importante notare che, in certi sistemi, possiamo avere più di una scala spaziale di interesse e, quindi, possiamo avere variazioni locali del numero di Reynolds magnetico. Come esempio, possiamo considerare una situazione in cui abbiamo un campo magnetico le cui linee di campo sono sempre parallele tra loro ma, fino a un certo punto lungo l'asse z, sono dirette in una direzione e, dopo quel punto, sono dirette nella direzione opposta.

In questo caso, data la configurazione rappresentata nello schema sopra, per lunghe scale di lunghezza, possiamo considerare che il numero di Reynolds magnetico sia molto maggiore di uno e quindi la convezione è dominante; tuttavia, nella regione vicina al punto in cui la direzione del campo magnetico si inverte, abbiamo un forte gradiente del campo magnetico e, quindi, una

much smaller characteristic length; in this case, the magnetic Reynolds number is much smaller than one and the diffusion process is not negligible. Locally, therefore, we can have effects related to the diffusion, leading to a reorganization of the magnetic field and to the phenomenon of magnetic reconnection, in which the magnetic topology is rearranged and the magnetic energy is converted to kinetic energy, thermal energy and particle acceleration. This phenomenon is particularly important in solar flares, that can be described using magnetohydrodynamics; this effect can also be obtained in laboratories under suitable conditions.

We can now remember that, in the section related to dimensional analysis, the Alfven velocity, c , related to the presence of a magnetic field and to the mass density of the fluid, was introduced as

²⁄₁⁄₂Bc = (πρm)

Substituting now the expression of this velocity, it is possible to obtain another non-dimensional number that is called Lundquist number

c LA MDS = 4c η4 πA

At this point, to complete the magnetohydrodynamics model, we need to write an energy balance equation. Working exactly as in the multiple fluid description, thus summing energy balance equations for all the a-th populations, we can obtain an equation in which the pressure is present. Since, then, we need to find a closure for this system, one possibility could be to adopt a polytropic closure by writing

d −γP( ρ )=0mdt

where γ is the polytropic index considered. Alternatively, we can start from the following equation

3 dT un +P ⋅u=(E+ ×B)⋅J −⋅Q+S2 dt cP u

in which: is the so called PdV work, neglecting viscous effects; is the total heat flux, that will⋅ ⋅Q need to be treated and modeled in a suitable way; the electromagnetic term is associated to the Joule effect; last, the term S takes into account the presence of energy losses due to

Non-elastic collisions, such as chemical reactions, nuclear reactions and emission of electromagnetic radiation. This term will be particularly useful in any case in which we want to exploit a plasma, for example, for energy production through nuclear fusion.

Limits of validity

In magnetohydrodynamics, we have thus reduced the system to the following system of equations:

{ ρ m u)=0+⋅(ρ mt d u JP+ρ =− ×Bm dt c 2B c 2 B= ×(u×B)+η 4t π

To which we have to add a closing equation, for example a polytropic closure:

d −γP( ρ )=0mdt

We can now investigate the conditions under which this assumption is valid. Considering τ the typical time MHD scale for the magnetohydrodynamics phenomena, we can associate its inverse to a characteristic magnetohydrodynamic frequency ω that must be, according to the assumption we made when developing MHD this model, much smaller than the cyclotron frequency associated to the

In an analogous way, we can define a typical magnetohydrodynamic length L associated to this model that must be much larger than the typical length associated to the variations, such as the Debye length λ associated to the ions and the Larmor radius: L ≫ λ ρ_MHD L i

We can thus define a characteristic speed for magnetohydrodynamic phenomena by computing the ratio between the characteristic length and the characteristic variation time: L_MHD ≈ uτ_MHD

The simplest way of describing a plasma is to exploit magnetohydrodynamics if the associated assumptions are satisfied, both with respect to the time scale and the length scale. In particular, this means that we have to consider scales that are much larger than the Larmor radius and the Debye length and times that are much longer than the inverse of the cyclotron frequency. The ratio of a characteristic length and a characteristic time, then, can be recognized to be a

The characteristic velocity of this system is similar to what we had in the guiding center approximation.

Often, when we derive the multiple fluid and the single fluid descriptions, we assume that the system is not far from an equilibrium condition and this assumption allows us to justify the appearance of these average quantities. In a plasma, even a collisionless description can be meaningful, therefore it is not always straightforward to give a justification to this fluid description, since collisions in plasma are not important but they are fundamental for reaching an equilibrium situation.

In the case of magnetohydrodynamics, the assumptions suggest that we can give a related but different justification of the development of this approach. Instead of starting from the momentum equation, we can describe the dynamic of the system using the guiding center approximation and, developing it, we can reach a magnetohydrodynamic description, neglecting from the beginning the terms that we have.

fluid description given by magnetohydrodynamics. Different phenomena will be studied using different models. Equilibrium configurations, for example, are usually studied using magnetohydrodynamics. Waves and collective modes, on the other hand, can be suitably described using all these three models, depending on what we are trying to describe. The stability and instability properties are usually studied in magnetohydrodynamics, while for transport properties we have to resort either to a multiple fluid description or, even, to a kinetic one.

Eventually, therefore, we can associate a plasma to a model and, from this, we can study the applications of what we have understood: this is the logical structure beneath plasma physics once we have developed suitable tools. We will now start describing the presence of waves and collective modes in plasma, where we can consider that electromagnetic waves will be related to the plasma dynamics, thus leading to the presence of collective modes and to the

Requirement of self-consistency. This is one of the most important aspects of plasma physics since any dynamic behavior will necessarily involve it; moreover, it is a starting point for understanding other aspects of this branch of physics. However, since this argument is actually very vast a careful topic selection was needed.

CHAPTER 6: WAVES IN PLASMA I

As we have seen in the previous chapters, a physical system is usually characterized by its characteristic time and its characteristic frequencies. In terms of frequencies, we have seen in a plasma the presence of various possibilities: the plasma frequency ω (in general, one for each plasma population), the cyclotron frequency p (for example in terms of electrons Ω and ions Ω ) and the collision frequency in the relaxation timee i approximation, ν ~1/t .ei r

In terms of lengths, which we can consider to be the inverse of characteristic wave vectors, we have: the Debyelength, λ ~v /ω , the Larmor radius ρ , the

mean free path in the relaxation time approximation and other typical lengths associated to the non-homogeneity of the medium considered (in which we can have gradients and a non-uniform spatial profile).

According to the values of these characteristic quantities, different studies of plasma physics can be done. In this section, we will start studying the simplest possible system, where we have an infinitely extended homogeneous plasma without any external field and in which collisions are negligible. Under these assumptions, therefore, the only remaining characteristic quantities will be the plasma frequency and the Debye length.

Considering now s