2.2 Equations for a multi-component plasma
2.2.1 Five-moment transport equations and collisional terms
The dynamics of a given fluid is described by the set of so-called transport equations that can be obtained by taking velocity moments of the Boltzmann equation, that is multiplying Equation (2.1) by powers of vs and integrating over velocity space. Due to the complexity
of this process, the computations are not shown here, only the resulting equations. They do not constitute a closed set and, hence, the adoption of an approximate expression for the fluid velocity distribution is required (Schunk [1977]). In this work, a Maxwellian distribution is chosen, i.e., fs = ns ms 2πkBTs 3/2 exp −msevs2 2kBTs , (2.5)
where ns, ms, and Ts are the number density, mass and temperature of the species s. The
variableves is the random part of the velocity and is defined asevs≡ vs−us, where usis the av-
erage part of the velocity. This choice, which is valid when the fluid is close to thermodynamic equilibrium, leads to the five-moment transport approximation, where the dynamics of each species is given by the conservation equations for mass, the three components of momentum and energy, and effects like thermal diffusion, thermal conduction or anisotropy of pressure are neglected. The influence of stress and heat flows can be considered by assuming a more compli- cated function for the velocity distribution and taking a higher-order moment approximation, at the expense of more complex computations. Nevertheless, the five-moment approximation is accurate enough for the investigation developed in this Thesis.
The equation for the conservation of mass, also known as mass continuity equation, is given by
D(msns)
Dt + msns∇ · Vs = 0, (2.6)
where Vs is the velocity of species s and DtD ≡ ∂t∂ + Vs· ∇ is the material derivative for time
variations following the plasma motion.
The equation of motion, which expresses the conservation of momentum, is given by msns
DVs
Dt = −∇Ps+ F , (2.7)
where Ps is the thermodynamic pressure, which has been assumed to be isotropic, and the
term F includes all the forces (actually, force densities) that may act on the fluid, such as the Lorentz force felt by charged particles, gravity or the variation of momentum due to collisions with the other components of the plasma.
The Lorentz and gravity force densities are given by
and
Fg = msnsg, (2.9)
respectively, where qs = Zse is the electrical charge, with Zs the signed charged number and
e the elementary electric charge. E, B and g are the electric and magnetic fields and the acceleration of gravity, respectively. The collisional interaction is represented by the term P
t6=sRst, whose precise definition is given later.
For numerical purposes, it is useful to write the mass continuity and the momentum equa- tions in conservative form. In this way, the left-hand side of the equation consists of the temporal evolution of the quantity that is conserved plus the divergence of a certain flux. The right-hand side corresponds to a source term (which may be equal to zero). Therefore, due to the definition of the material derivative, Equation (2.6) is equivalent to
∂(msns)
∂t + Vs· ∇ (msns) + msns∇ · Vs= 0, (2.10) which can be rewritten in conservative form as
∂(msns)
∂t + ∇ · (msnsVs) = 0. (2.11)
Likewise, the left-hand side of Equation (2.7) can be expressed as msns ∂Vs ∂t + msnsVs· ∇Vs = ∂ (msnsVs) ∂t − Vs ∂ (msns) ∂t + msnsVs· ∇Vs. (2.12) Taking into account that, according to Equation (2.11), ∂(msns)
∂t = −∇ · (msnsVs) and using
vector identities, the expression above is equivalent to ∂ (msnsVs)
∂t + ∇ · (msnsVsVs) , (2.13)
where VsVs is a dyadic product, which can be written in matrix form as
VsVs =
V
2
s,x Vs,xVs,y Vs,xVs,z
Vs,yVs,x Vs,y2 Vs,yVs,z
Vs,zVs,x Vs,zVs,y Vs,z2
. (2.14)
Hence, the conservation of momentum can be finally expressed as ∂ (msnsVs)
∂t + ∇ · (msnsVsVs+ PsI) = qsns(E + Vs× B) + msnsg+ X
t6=s
Rst, (2.15)
where the pressure term in the right-hand side of Equation (2.7) has been moved to the left-hand side and the relation ∇Ps = ∇ · (PsI), with I the identity matrix, has been used.
The last equation of the five-moment approximation concerns the conservation of energy. For the purposes of this Thesis, it will be expressed in terms of the pressure, in the following way: ∂Ps ∂t = − (Vs· ∇) Ps− γPs∇ · Vs+ (γ − 1) X t6=s Qst, (2.16)
where γ = 5/3 is the adiabatic constant for mono-atomic ideal gases and Qst is the term that
takes into account the variation of energy caused by the collisional interaction with the rest of the species. Since pressure is not a conserved quantity, this equation cannot be written in conservative form.
There are several reasons to solve this equation in terms of the pressure instead of a related quantity. For instance, it is straightforward to compute the internal energy density of each species s, denoted by eP,s, from the pressure since those variables are related by the following
formula:
eP,s=
Ps
γ − 1. (2.17)
An alternative approach commonly used in single-fluid models is to solve one equation for the total energy density, which is the sum of the kinetic, internal and magnetic energy densities. In this approach, numerical issues may appear when computing the pressure from the total energy. Pressure must have always positive values but, in the process of subtracting the kinetic and the magnetic energy from the total energy, negative values may be obtained due to unwanted numerical effects. In the multi-fluid formalism used in this Thesis, there is not a unique equation that accounts for the conservation of the total energy of the plasma but several equations connected to the temporal evolution of the internal energy of the different species. Nevertheless, the total energy is a conserved magnitude, which is given by
ET = Z V " X s eP,s+ 1 2msnsV 2 s + B 2 2µ0 # dV, (2.18)
where V is the volume of the plasma.
The functions Rst and Qst represent the momentum and the heat transfer due to elastic
collisions between two species s and t, and are defined as (see, e.g., Schunk [1977], Draine
[1986], Leake et al.[2014],Khomenko et al. [2014])
Rst≡ αst(Vt− Vs) Φst, (2.19) and Qst≡ 2αst ms+ mt Ast 2 kB(Tt− Ts) Ψst+ 1 2mt(Vt− Vs) 2 Φst , (2.20)
respectively, where Ast= 4 for collisions between electrons and neutral species and Ast = 3 for
the remaining types of collisions. The difference in the values of Ast is a consequence of the
different velocity dependence used to calculate the scattering cross-sections of each interaction. A detailed explanation can be found in Draine[1986]. The parameter αst = αts is the friction
coefficient for collisions between species s and t. The mass, temperature and velocity of species s are denoted by ms, Ts and Vs, respectively and kB is the Boltzmann constant. From the
definition of the momentum transfer term it can be checked that Rts = −Rst, which means
that the momentum lost by one of the species involved in a binary collision is gained by the other one, and viceversa. The functions Φst and Ψst depend on the drift speed, |Vs− Vt|, and
on the reduced thermal speed, Vtherm ≡
p
2kB(mtTs+ msTt) / (msmt), and can be taken as
equal to unity when the drift speed is much smaller than the thermal speed (Schunk [1977]). Although collisions tend to heat the plasma, the precise effect on each species depends on which of the two terms of the right-hand side of Equation (2.20) dominates. If the difference
of temperature between the species s and t is large enough, the first term dominates over that associated with the velocity drifts. For instance, in a case with Tt > Ts, the heat transfer Qst
would be positive while Qts would be negative. Consequently, the temperature of species s
would rise and the temperature of species t would decrease. This would continue until they had the same temperatures. Thus, collisions tend to equalize the temperatures of the components of the plasma. On the other hand, if all the species already have the same temperature, the Qst
terms are always positive because of their quadratic dependence on the velocity drifts, which means that all the species are heated.
The friction coefficients have different expressions depending on whether the collisions in- volve only ionized species or also neutral species (Braginskii [1965],Callen[2006]). For the case of collisions between two ionized species s and t, the friction coefficient is given by
αst=
nsntZs2Zt2e4ln Λst
6π√2πǫ2
0mst(kBTs/ms+ kBTt/mt)3/2
, (2.21)
where ns is the number density of species s, ǫ0 is the vacuum electrical permittivity, and mst =
msmt/ (ms+ mt) is the reduced mass. The function ln Λst, known as Coulomb’s logarithm (see,
e.g., Spitzer [1962], Vranjes and Krstic [2013]), represents the factor by which the cumulative effect of the small-angle Coulomb collisions dominates over large-angle hard-sphere collisions and is given by ln Λst= ln " 12πǫ3/20 kB3/2(Ts+ Tt) |ZsZt|e3 TsTt Z2 snsTt+ Zt2ntTs 1/2# . (2.22)
For collisions between neutral species n and another species s, that can be either neutral or ionized, the friction coefficient is
αsn= nsnnmsn 4 3 8 π kBTs ms + kBTn mn 1/2 σsn, (2.23)
where σsn is the collisional cross-section. Values of this parameter for the collisions in hydro-
gen and helium plasmas are shown in Table 2.1. Some of the cross-sections are taken from quantum-mechanical models but to the best of our knowledge such accurate computations are not available for all types of collisions. Hence, in the case that quantum-mechanical computa- tions are not available, the cross sections have been computed using the classical hard-sphere model, in which the cross-section of a collision between two species s and t is given by
σst= π (rs+ rt)2, (2.24)
where rs is the radius of a particle of the species s.
As shown by Vranjes and Krstic [2013], the hard-sphere model may underestimate the correct values of the cross-sections by one or two orders of magnitude. However, since the dominant ion in the plasmas studied in the present work is proton, it is not expected that the use of larger cross-sections for σHHe ii, σHHe iiiand σHeHe iii would significantly modify the results
that are discussed in the present Thesis. In addition, it must be noted that the cross-sections are actually not constants but functions of the temperature. However, according toVranjes and Krstic[2013], the variation of the cross-sections in the range of temperatures considered in this Thesis is small. Consequently, the values included in Table2.1are good enough approximations.
Table 2.1: Cross-sections of collisions with neutral species Value (m−2) Model
σpH 10−18 Vranjes and Krstic [2013]
σpHe 10−19 Vranjes and Krstic [2013]
σeH 1.5 × 10−19 Vranjes and Krstic [2013]
σeHe 5 × 10−20 Vranjes and Krstic [2013]
σHHe 1.5 × 10−19 Lewkow et al. [2012]
σHHe ii 2 × 10−20 Hard sphere
σHHe iii 1 × 10−20 Hard sphere
σHeHe ii 5 × 10−19 Dickinson et al. [1999]
σHeHe iii 3 × 10−21 Hard sphere
The system given by the five-moment transport equations contains more unknowns than equations. For each species in the plasma there are five unknowns (number density, three- components of the momentum and pressure) with their respective five equations. But addi- tionally, there are other unknowns related to the three components of the magnetic field, the three components of the electric field and the temperature of each species. Hence, the system is not closed and must be completed with additional equations. Such equations are presented in the following sections.