The dynamics of a plasma has a great complexity and can be described using different approximations depending on the physical aspect to analyze. The most complete descrip- tion is provided by the kinetic theory, which considers the microscopic velocities of the particles over the distribution function of each species.
The distribution function (fα) indicates the number of particles of the species α per
volume in the phase space. A phase space is a mathematical space in which all possible states of a system are represented, and where each possible state corresponds to one unique point in this space. Every axis of the phase space is a parameter or a degree of freedom of the system, i.e. position and velocity.
When the ionisation of the atoms balances the recombination of the electrons and ions, the average number of plasma particle species is constant in time. In these conditions, full kinetic description of a plasma can be expressed with the equation of conservation of
the particle distribution function (Boltzmann equation): ∂fα ∂t + ~v· ∂fα ∂~x + ~ Ftot mα ·∂fα ∂~v = Cα (2.4)
where mα is the particle mass, t is the time, v is the speed vector of the particle, x is
the position vector of the particle and Cα represents collisions with all species. The
force ( ~Ftot) exerts on plasma particles is the addition of two kind of forces: the external
ones ( ~Fext) provided by external fields and the internal ones ( ~Fint) provided by the fields
generated from the particles far away of the Debye sphere. These mentioned fields are macroscopic.
The Debye sphere is a characteristic plasma parameter which divides the space in two regions: one with the short-range interaction inside the sphere and the other region with the long-range interaction outside it. The number of particles inside the Debye sphere will determine much of the physics involved in plasma [24,25]. The radius of this sphere is the Debye length which may be expressed as:
λD = s
ε0kBT
n2e (2.5)
here ε0 is the permittivity of free space, kB is Boltzmann’s constant, T is the absolute
temperature and ne is the density of electrons.
The ~Fextis modelled by the Lorentz equation (Eq.2.6), which calculates the movement
of a particle of mass m, velocity v and charge q moving in an electric field E and a magnetic field B. ~ Fext= md~v dt = q h ~E + ~v × ~Bi (2.6) Using the conservation of number of particles (Eq.2.4) (without particle recombination or escape from the system), the full kinetic description of a plasma can be expressed using the Vlasov equation (Eq. 2.7) where particles are assumed to be uncorrelated.
df dt ≡ ∂f ∂t + ~v· ∂f ∂~x + ~Fext mα + Z ~ Fint m f (~x, ~v, t)d~xd~v ! ·∂f ∂~v = 0 (2.7)
The Vlasov equation is a differential equation that describes the time evolution of the distribution function of plasma consisting of charged particles with long-range interaction. It is said that particles are uncorrelated when there are no collisions among them, so Ci is considered to be equal to 0. This situation is accomplished when the number of particles into the Debye sphere is large, typically nλ3
D ≈ 10 9.
In addition, the electromagnetic fields (external and internally) are modeled by the Maxwell’s equations (Eq. 2.8).
∇· ~E = 1 ε0 X α qα Z fαd3v ∇· ~B = 0 ∇ × ~E = −∂ ~B ∂t ∇ × ~B = µ0X α qα Z ~ vfαd3v + 1 c2 ∂ ~E ∂t (2.8)
where ~E is the electric field, ~B is the magnetic field, µ0 is the permeability of free space
and c is the speed of light in free space.
To compute the field ~E, or equivalently −∇φ, we solve the Poisson’s equation (Eq.2.9), ∇2φ = −q
ε0 Z
f (~x, ~v, t) d~v (2.9)
where φ is the electrostatic potential arising from ensemble-averaged charge density distribution of all the species in plasma, i.e.
ρ =X
α
qα Z
fα(~x, ~v, t) d~v (2.10)
The high electric conductivity of plasmas, due to electrons are very mobile, implies that any charge that is developed is readily neutralized, and thus plasmas can be considered as electrically neutral. The quasi-neutrality equation (Eq. 2.11) describes the apparent charge neutrality of a plasma overall, or in other words it describes that the densities of positive and negative charges in any sizeable region are equal. When the region is less
than the Debye sphere, the quasi-neutrality is not guaranteed. X
α
qαnα = 0 (2.11)
Finally, it is interesting to note an approximation used for investigating low frequency turbulence in fusion plasmas. As it was explained in Section 2.3.1, the trajectory of charged particles in a magnetic field is a helix that winds around the field line. This trajectory can be decomposed into a fast gyration about the magnetic field lines (called gyromotion) and a slow drift of the guiding center along the field line (Fig. 2.13).
-
B
-B
- - - - - gyromotionB
guiding centerB
guiding centerFigure 2.13: Gyrokinetic model (using 4-point average)
For most plasma behavior, this gyromotion is irrelevant. So the gyromotion of a particle is approximated by a charged ring moving along the field line and the gyro-angle dependence is removed in the equations. From Vlasov equation, by averaging over this gyromotion, one arrives at the gyrokinetic equation which describes the evolution of the guiding center in a phase space with one less dimension (3 spatial, 2 velocity and time) than the full Vlasov equation (3 spatial, 3 velocity and time):
∂f ∂t + dvk dt ∂f ∂vk +d ~R dt ∂f ∂ ~R = 0, (2.12)
where vk is the velocity parallel to the magnetic field line and ~R is the position vector of