Dérivation microscopique des équations de la MHD

Ce document utilise le livre: « Fundamentals of Physics », P.M. Bellan, Cambridge University Press (2006) Debye shielding

Originates from theory of liquid electrolytes (Debye and Huckel 1923).

Assume and densities are initially equal and spatially uniform. They do not need to be in thermal equilibrium with each other and so the and will be allowed to have separate temperatures denoted by T i, T e.

We assume the following:

1. The plasma is assumed to be nearly collisionless so that collisions between particles may be neglected to first approximation.

2. Each species, denoted as σ, may be considered as a ‘fluid’ having a density n, a temperature T, a pressure P = nκT(κ is Boltzmann’s constant), and a mean velocity u so that the collisionless equation of motion for each fluid is

Where m is the particle , q is the charge of a particle, and E is the electric . Now consider a perturbation with a sufficiently slow time dependence to allow the following assumptions: 1. The inertial term ~ d/dt on the left hand side is negligible and may be dropped. 2. Inductive electric fields are negligible so the is almost entirely electrostatic, i.e., E ~ −grad φ. 3. All temperature gradients are smeared out by thermal particle motion so that the temperature of each species is spatially uniform. 4. The plasma remains in thermal equilibrium throughout the perturbation (i.e., can always be characterized by a temperature). Invoking these approximations, the previous equation reduces to:

leading to: Let us now slowly introduce a test particle of charge qT at the orgin of coordinates. This perturbation will result in a small but finite potential, superposition of the potential of the test particle and that of the other particles that have moved in response to the presence of the test particle: this slight displacement is called shielding. Poisson’s equation becomes:

We can assume and thus, assuming that the plasma response remains Boltzmann-like:

Using neutrality, we get with and

Since ions cannot move fast enough to keep up with an electron test charge which would be moving at the nominal electron thermal velocity, the shielding of electrons is only by other electrons, whereas the shielding of ions is by both ions and electrons.

One gets:

This test-particle/shielding-cloud analysis makes sense only if there is a macroscopically large number of plasma particles in the shielding cloud; i.e., the analysis makes sense only if

This will be seen later to be the condition for the plasma to be nearly collisionless. A conventional plasma does not have sufficient to become substantially non-neutral for distances greater than a Debye length. Small v. large angle collisions in plasmas

What happens to the and energy of a test particle of charge qT and mass mT that is injected with velocity vT into a plasma. This test particle will make a sequence of random collisions with the plasma particles (called “field” particles and denoted by subscript F); these collisions will alter both the momentum and energy of the test particle. The scattering angle θ is given by Grazing (small angle) collisions occur when the test particle impinges outside the shaded circle and so occur much more frequently than large angle collisions. Although each grazing collision does not scatter the test particle by much, there are far more grazing collisions than large angle collisions and so it is important to compare the cumulative effect of grazing collisions with the cumulative effect of large angle collisions. Since the sum of angles of grazing collisions will be zero, we must use the square of the angle and use random walk statistics: scattering is a diffusive process. What N must be in order to have

But: where

Thus

or Since the condition for the cumulative effect

of grazing collisions to exceed the large collisions is equivalent to

We conclude that the criterion for an ionized gas to behave as a plasma (i.e., Debye shielding is important and grazing collisions dominate large 3 angle collisions) is the condition that nλ D >> 1.

A useful way to decide whether Coulomb collisions are important is to compare the collision frequency with the frequency of other effects, or equivalently the mean free path of collisions with the characteristic length of other effects. Electron and ion collision frequencies

Momentum scattering is characterized by the time required for collisions to deflect the incident particle by an angle π/2 from its initial direction, or more commonly, by the inverse of this time, called the collision frequency.

We assume the particle incident velocity to be the species thermal 1/2 velocity vT = (2 κT/m) . We normalize all collision frequencies to νee , and for further simplification assume that the ion and electron temperatures are of the same order of magnitude.

The reduced mass for ei collisions (electrons scattering from ions) is the same as for ee collisions (except for a factor of 2), the relative velocity is the same — hence, we conclude that νei ~ νee .

Because the temperatures were assumed equal, σ∗ ii ≈ σ∗ ee and so the collision frequencies will differ only because of the different velocities in the expression ν = nσv. The ion thermal velocity is lower by an amount (me/mi) 1/2 giving 1/2 νii ≈ (me/mi) νee . Momentum is conserved in a collision so that in the lab frame mivi = −meve

Hence

Electric resistivity:

(Spitzer and Harm 1953).

Derivation of fluid equations: Vlasov, 2-fluid, MHD

At any given time, each particle has a specific position and velocity. We can therefore characterize the instantaneous configuration of a large number of particles by specifying the density of particles at each point x,v in phase-space. The function prescribing the instantaneous density of particles in phase-space is called the distribution function and is denoted by f(x,v,t).

Rate of change of the number of particles in a small box dxdv of phase space Denoting by a(x,v,t) the of a particle, it is seen that the particle flux in the horizontal direction is fv and the particle flux in the vertical direction is fa.:

1D Vlasov equation: Generalizing to 3D:

With the Lorentz

It can also be written:

Thus, the distribution function as measured when moving along a particle trajectory (orbit) is a constant.

An example of consequences: If the energy E of particles is constant along their orbits then f = f(E) is a solution to the Vlasov equation. Moments of the distribution function:

density

velocity

etc… Treatment of collisions in the Vlasov equation

In order to see how collisions affect the Vlasov equation, let us now temporarily imagine that the grazing collisions are replaced by an equivalent sequence of abrupt large scattering angle encounters as shown beow: The details of these individual jumps in phase-space are complicated and yet of little interest since all we really want to know is the cumulative effect of many collisions. It is therefore both efficient and sufficient to follow the trajectories on the slow time scale while accounting for the apparent “creation” or “annihilation” of particles by inserting a collision operator on the right hand side of the Vlasov equation.

where Cσα (f σ) is the rate of change of f σ due to collisions of species σ with species α

(a) Conservation of particles – Collisions cannot change the total number of particles at a particular location so (b) Conservation of momentum – Collisions between particles of the same species cannot change the total momentum of that species so

(c) collisions between different species must conserve the total momentum of both species together so

(d) Conservation of energy –Collisions between particles of the same species cannot change the total energy of that species so

(e) collisions between different species must conserve the total energy of both species together so Two- fluid equations

Multiplying the Vlasov equation by unity and then integrating over velocity gives

because fσ → 0 as v → ∞, the third term on the r.h.s. cancels and we are left with

After multiplying by v we get: writing v = v ′(x,t) +u(x,t), one gets

where the frictional terms have the form

It can be rewritten

where the pressure tensor is given by It f σ is an isotropic function of v’ (which is the case is there is enough collisionality):

In dimension N, 3 should be replaced by N

Another form:

With the convective derivative:

This chain of equations needs to be closed. Pressure equation (assume isotropy and space dimension N):

Multiplying the Vlasov equation by mv 2/2 and integrating over velocity gives

One has

where is the heat flux and The collision term becomes

where dW/dt is the rate at which species σ collisionally transfers energy to species α.

Combining, one gets

Or, using the velocity equation to calculate the rate of change of the kinetic energy: Two limiting cases:

1. The isothermal case: The heat flux term dominates all other terms in which case the temperature becomes spatially uniform. This occurs if the typical wave phase velocities are smaller than the thermal velocity of the considered particle species (take the ratio of l.h.s with the heat flux term) and if collisions can be neglected.

2. The adiabatic limit: The heat flux terms and the collisional terms are small enough to be ignored compared to the left hand side terms. This occurs when the typical wave phase velocities are larger than the thermal velocity of the considered particle species.

In both limits a « closure » is possible.

In the adiabatic case:

where One thus gets the :

Effect of collisions

In the presence of collisions, the maximisation of under the constraints of mass and energy conservation leads to

Leading to

which corresponds to the Maxwellian or

The pressure is given by the P=n k T:

When collisions are not important, the core of the distribution function can reach a Maxwellian state while the tails are still non-Maxwellian Two-fluid equations Reduction to the MHD equations: 1 fluid model suited for large scales and long time scales

New variables: Total current density

Center of mass velocity

where

MHD continuity equation :

MHD equation of motion :

one has Defining the pressure tensor by

where v’ is the deviation with respect to the barycentric velocity, one gets, after summation over particle species:

One can easily assume neutrality

and get

MHD Ohm’s law:

Use electron equation Neglecting electron inertia and approximating u_i with U

Hall term: Negligible at low frequency or if electron ion frequency is large

If the term is also negligible

then

If Vph << c the displacement current term can be dropped from Ampere’s law resulting in the so-called “pre-Maxwell” form Ideal MHD and frozen-in fl ux

If the resistive term ηJ is small compared to the other terms in Ohm’s law, then the plasma is said to be ideal or perfectly conducting .

E + U× B = E ′ where E’ is the electric field in the plasma frame.

For an ideal plasma E’=0.

Using Faraday’s law, this implies that the magnetic flux is time invariant in the frame moving with velocity U.

Thus, the magnetic field lines must convect with the velocity U, i.e., the magnetic field lines are frozen into the plasma and move with the plasma . Calculate the rate of change of the magnetic flux through a surface S(t) bounded by a material line C(t), i.e., a closed contour which moves with the plasma.

The displacement of a segment dl of the bounding contour C is Uδt where U is the velocity of this segment. The incremental change in surface area due to this displacement is S = Uδt ×dl MHD equations of state

A tube of plasma initially occupying the same volume as a magnetic flux tube is constrained to evolve in such a way that ∫B·ds stays constant over the plasma tube cross-section. Thus B A =const.

If collisions are infrequent, temperature anisotropy can persist for a long time. Parallel pressure

Using we have along the flux tube (N=1) :

with

But : and using

We have: Perpendicular pressure

Similarly: with

Single adiabatic law

If collisions are sufficiently frequent to equilibrate the perpendicular and parallel temperatures, then the pressure tensor becomes fully isotropic and the dimensionality of the system is N = 3 so that γ = 5/3. There is now just one pressure and temperature and the adiabatic relation becomes Summary:

or