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date: 27 October 2020

Multi-Fluid Effects in Magnetohydrodynamicsfree

  • Elena KhomenkoElena KhomenkoInstituto de Astrofisica de Canarias

Summary

Multi-fluid magnetohydrodynamics is an extension of classical magnetohydrodynamics that allows a simplified treatment plasmas with complex chemical mixtures. The types of plasma susceptible to multi-fluid effects are those containing particles with properties significantly different from those of the rest of the plasma in either mass, or electric charge, such as neutral particles, molecules, or dust grains. In astrophysics, multi-fluid magnetohydrodynamics is relevant for planetary ionospheres and magnetospheres, the interstellar medium, and the formation of stars and planets, as well as in the atmospheres of cool stars such as the Sun. Traditionally, magnetohydrodynamics has been a classical approximation in many astrophysical and physical applications. Magnetohydrodynamics works well in dense plasmas where the typical plasma scales (e.g., cyclotron frequencies, Larmor radius) are significantly smaller than the scales of the processes under study. Nevertheless, when plasma components are not well coupled by collisions it is necessary to replace single-fluid magnetohydrodynamics by multi-fluid theory. The present article provides a description of environments in which a multi-fluid treatment is necessary and describes modifications to the magnetohydrodynamic equations that are necessary to treat non-ideal plasmas. It also summarizes the physical consequences of major multi-fluid non-ideal magnetohydrodynamic effects including ambipolar diffusion, the Hall effect, the battery effect, and other intrinsically multi-fluid effects. Multi-fluid theory is an intermediate step between magnetohydrodynamics dealing with the collective behaviour of an ensemble of particles, and a kinetic approach where the statistics of particle distributions are studied. The main assumption of multi-fluid theory is that each individual ensemble of particles behaves like a fluid, interacting via collisions with other particle ensembles, such as those belonging to different chemical species or ionization states. Collisional interaction creates a relative macroscopic motion between different plasma components, which, on larger scales, results in the non-ideal behaviour of such plasmas. The non-ideal effects discussed here manifest themselves in plasmas at relatively low temperatures and low densities.

Subjects

  • Fluid Mechanics

1. Historical Development: General Remarks

Multi-fluid extensions of magnetohydrodynamics (MHD) were developed in parallel with classical MHD at the beginning of the 20th century. Classical MHD theory works under the assumption of the collective behavior of fluid particles and is strictly valid in the case of strong collisionally coupled plasma. The MHD approximation allows us greatly to simplify the mathematical description of the processes in plasmas. Even if the chemical composition of a plasma is complex, and positively and negatively charged particles of different species and masses are present, as long as the collisional coupling is strong, the MHD approximation can be safely applied. In situations of intermediate collisional coupling, individual plasma components start to have net motions with respect to one another. Such motions can create currents, modify the topology of magnetic structures, and produce additional friction, dissipation, and heating. The multi-fluid extension of MHD is a fluid theory allowing us to treat non-ideal plasma effects due to incomplete collisional coupling in complex plasmas in an approximate way. Multi-fluid theory preserves the classical form of MHD equations, except for the appearance of additional terms describing interactions between the ensembles of particles of different kinds.

In the wider field of physics, interest in the study of multi-fluid effects lies primarily in industrial applications aimed at understanding the behavior of magnetically confined fusion plasmas in tokamaks. In astrophysics, the first studies of multi-fluid effects addressed the dynamics of molecular clouds and interstellar shock waves (Draine, 1986; Spitzer, 1962). Plasma in the interstellar medium contains dust grains, molecules, and neutral particles. In ionospheric studies, as with tokamaks, the choice of a multi-fluid framework is a natural one, dictated by the fixed global field configuration and low density plasma having a complex chemical mixture of atoms and molecules with a varying composition in space. Interest in multi-fluid effects has also recently arisen in the field of solar physics, since cool temperatures in the photosphere and chromosphere of the Sun, and solar prominences, lead to plasmas with a very weak ionization fraction, as in the interstellar medium.

The mathematical basis for multi-fluid theory can be found in many standard books on plasma physics, as well as in specific monographs dedicated to particular areas of physics and astrophysics. Several such generic monographs may be referenced in this context, including Krall and Trivelpiece (1973), Kadomtsev (1976), Lifschitz (1989), and Goedbloed and Poedts (2004), who deal mainly with the general subject of MHD. Among the more specific ones, one may refer to Cowling (1957), Braginskii (1965), Chapman & Cowling (1970), Bittencourt (1986), Balescu (1988), Rozhansky & Tsedin (2001), and Zhdanov (2002). These monographs deal with equations for multiple species and multi-fluid effects. In the field of astrophysics and space plasma, the following monographs may be referred to for more information: Spitzer (1962), Bauer (1973), Shu (1992), Schunk & Nagy (2000), Kelley (2009), and Prölss & Bird (2012).

2. Basic Theory: Transport Equations

Plasmas susceptible to multi-fluid effects are not entirely coupled by collisions. An important assumption is that each individual component can be described by the fluid approximation. In such a case, equations governing multi-fluid magnetohydrodynamics are obtained by taking moments of the Boltzmann equation for the distribution function of an ensemble of particles of type “a”, fa,

fat+vfa+aνfa=(fat)coll.(1)

The Boltzmann equation describes the evolution of the particles’ distribution function in a unit volume as a consequence of intrinsic changes ( fa/t), the presence of a velocity field, v, acceleration due to external forces, a, and due to collisions with particles of a different kind ( fa/t|coll). “Particles of a different kind” refers to atoms of different chemical species (i.e., different atomic weights), different ionization and excitation states, molecules, and dust grains. The conservation of particles plays an important role in such an environment. There are negatively charged particles (such as electrons, some ions, and dust grains), and positively charged particles (typically ions).

Forces acting on particles in a plasma can be divided into surface and volume forces. Examples of surface forces are pressure and the viscous force; examples of volume forces are the Lorentz force and gravity. Due to the difference in particle mass and charge, the particles will have a different response under the action of the same force with the result that the average center of mass velocity of an ensemble of particles is different for particles of different kinds.

The system of transport equations for particles of type “a” has the following form:

ρat+(ρaua)=Sa,(2)
(ρaua)t+(ρauaua+p^a)=ρara(E+ua×B)+ρag+Ra,(3)
t(ea+ρaua22)+(ua(ea+ρaua22)+p^aua+qa)=Ma+EamaSa+ρauag+ρarauaE.(4)

In these equations, ρa is the mass density, ua the center of mass velocity, E and B the electric and magnetic fields, g gravity, ea the internal energy, p^a is the pressure tensor, and qa is the heat flow vector. Also ra=qa/ma is defined as the charge over mass ratio.

Unlike the classical MHD equations, multi-fluid equations contain a number of new terms (Sa, Ra, and Ma) on the right-hand side. These terms are responsible for the loss/gain of mass, momentum, and energy through different types of collision. The generic description of these terms is very complex, and approximations are needed, depending on the particular plasma conditions. The collisional terms can be divided into elastic and inelastic ones. More details are provided in Section 3.2.

The velocity of each particle is composed of its center of mass velocity and a random velocity, ca, v=ua+ca. The pressure tensor, the scalar pressure, and the heat flow vector are defined through these random velocities according to,

p^a=ρacaca;pa=13ρaca2;qa=12ρaca2ca.(5)

Here, triangular brackets mean averaging over the velocity space, υ=υfad3v/fad3v, the number density being fad3v=na, as in Lifschitz (1989). It should be kept in mind that such a definition causes the system of reference to be different for each component. The internal energy of each particle is composed of the energy of its thermal motions and the potential energy of ionization, Ea,

ea=12ρaca2+naEa=32pa+naEa.(6)

It is possible to write the same system of transport equations for photons as for any other type of particle. However, since photons are massless, the only equation that makes sense is the energy conservation equation. After some manipulation, the energy conservation equation for photons takes the form of a radiative transfer equation (Mihalas, 1986). The incorporation of the effects of radiation into the plasma transport equations is discussed in Khomenko, Collados, Díaz, & Vitas (2014a). The solar atmosphere is one of the examples of a multi-fluid plasma environment where the interaction between plasma and radiation plays a crucial role in the energy transport.

The above transport equations (2–4) can be further adjusted, depending on the particular type of plasma, such as those in the interstellar medium, planetary ionospheres and magnetospheres, the atmospheres of cool stars, or tokamak plasma. Sections 5, 6, and 7 give an overview of these environments and of the main multi-fluid effects encountered, depending on the application. The simplest case is a hydrogen plasma containing only one type of chemical species, hydrogen, which can be either neutral or singly ionized (see Leake et al., 2014).

The strength of collisional coupling also regulates the level of approximations to be applied. For example, if the plasma is dense enough, the collision frequency can become significantly higher than any cyclotron frequency of an individual species. It makes collisional momentum transfer between the different types of particles very efficient. In such a case, particles of different kinds are expected to have the same velocity and energy, and solving individual equations for different kinds of particles therefore provides unnecessarily detailed information. Equations for different “a” components can then be added together to yield single-fluid MHD equations.

Another usual approach consists of neglecting electron inertia, since the electron mass is three orders of magnitude lower than that of a typical ion. In such a case, electron transport equations are not solved explicitly, but combined with other equations for charged species. This leads to a two-fluid formalism.

In all of the simplified cases given here, not explicitly solving the electron transport equations results in a loss of information about the behavior of the electric current. This information has to be supplied in an approximate way by adding a generalized Ohm’s law to the system. Different forms of Ohm’s law are discussed in Section 3.5.

2.1 n -fluid and Two-fluid Descriptions

Particularly useful forms of multi-fluid formalism are the so-called single-fluid and two-fluid descriptions. The two-fluid approach is valid in the situation of intermediate collisional coupling and is necessary when typical scales of processes under study are similar to ion–neutral collisional scales. The equations provided in this section are derived under the assumption that charged and neutral components behave in a sufficiently different way thanks to the Lorentz force acting only on the charges. In principle, since the inertia of plasma components with the same charge but different mass is not the same, they can also behave differently. Taking this into account would lead to a slightly more complex description, with more components to be evolved separately, that is, generically speaking it would lead to the n-fluid approach. However, the common part is that electrons are not evolved separately, but together with equations for charges.

Two-fluid transport equations take the following form,

ρnt+(ρnun)=Sn,(7)
ρct+(ρcuc)=Sn,(8)
(ρnun)t+(ρnunun+p^n)=ρng+Rn,(9)
(ρcuc)t+(ρcucuc+p^c)=[J×B]+ρcgRn,(10)
t(en+12ρnun2)+(un(en+12ρnun2)+p^nun+qn+FRn)=ρnung+Mn,(11)
t(ec+12ρcuc2)+(uc(ec+12ρcuc2)+p^cuc+qc+FRc)=ρcucg+JEMn,(12)

where the definitions of the pressure tensors ( p^n and p^c), the heat flux vectors ( qn and qc), the internal energies ( en and ec), and the radiative energy fluxes ( FRn and FRc) all involve summation over the species present in the plasma. An example of their definitions for the solar atmosphere can be found in Khomenko et al. (2014a). This form of the two-fluid equations is used in solar physics and ionospheric physics. Radiative energy exchange and heat conduction gain importance under certain plasma conditions. In the solar atmosphere, energy exchange by radiation plays a major role. In the photosphere and corona it can be treated in a simplified way by assuming local thermodynamic equilibrium or correspondingly optically thin energy losses. In the chromosphere, a complex treatment has to be applied involving the solution of a non-local equilibrium radiative transfer equation. At the same time, the chromosphere is the region expected to be susceptible to multi-fluid MHD effects. Radiation plays an important role in interstellar medium plasmas as well. In ionospheric and tokamak plasmas, energy transport by radiation is less important, while heat conduction adopts an important role, as in the solar corona. Under strong plasma magnetization, heat conduction acts differently in the directions parallel and perpendicular to the magnetic field (see Braginskii, 1965).

When the ionization fraction becomes extremely weak, the equations can be further simplified. In the interstellar medium, the ionization fraction drops to values as low as 107. In such a case, the inertia, energy, and pressure of charged particles play no role in the dynamics, and these terms can be safely neglected in the equation of motion and in the energy equation (see Ciolek & Roberge, 2002; Falle, 2003). The only remaining terms in the equation of motion of the charged components are then the Lorentz force and the collisional force, which balance each other. After some rearrangement, this leads to the following system of equations,

ρnt+(ρnun)=0,(13)
ρa|t+(ρa|ua|)=0,(14)
ρnunt+(ρnunun+p^n)=J×B,(15)
ra|(E+ua|×B)+ρnKa|;n(ua|un)=0,(16)
t(en+12ρnun2)+(un(en+12ρnun2)+p^nun)=JE+Mn,(17)
Ma|+ρa|ra|ua|E=0,(18)

where Ka| is the collisional rate between ions and neutrals (see Ballester et al., 2018). The system is essentially n-fluid since it includes multiple ion species, marked by the sub-index a|”. Radiation, heat conduction, and source terms in the continuity equations have been dropped. Note that the momentum equation for neutrals contains the Lorentz force. This is a consequence of the fact that in such a weakly ionized medium the Lorentz force is acting on neutrals through collisions with charges.

2.2 Bi-fluid Magnetohydrodynamics

Bi-fluid MHD is a formalism for multi-fluid MHD introduced for magnetically confined tokamak plasmas (Hazeltine & Meiss, 1985, 2003) (see also recent derivations in Février, 2014; Nicolas, 2014). This prescription is useful in strongly magnetized ion–electron plasmas with a globally imposed magnetic field and weak collisional interaction, but not influenced by gravity. The properties of such plasmas are expected to be significantly different along and across the magnetic field. Bi-fluid formalism allows the reintroduction into the MHD description of diamagnetic and polarization drifts, together with certain effects from the pressure tensor (viscosity). In very weakly collisional plasmas the fluid approximation usually remains valid for velocities across the magnetic field. This is so because, even for low collisional coupling, the magnetic field acts as a substitute for collisions, assisting the plasma collective motion in the form of cyclotron gyration around the magnetic field lines. Velocities parallel to the magnetic field are therefore very different, and may not follow a fluid behavior (it is nevertheless frequently adopted for convenience and simplicity). A good discussion of these effects is provided in Freidberg (2007).

To derive bi-fluid equations, one starts from the equations for an individual species a (Eqs. 24) and decomposes the velocity of this species into components parallel and perpendicular to the magnetic field,

ua=ua,||+ua,,(19)

where

ua,=E×BB2+B×(p^a)ρaraB2+B×(ua/t+uaua)raB2,(20)

and which can be viewed as a series of drifts perpendicular to the magnetic field: that is, drift due to the electric field, diamagnetic drift, and polarization drift. The polarization drift is usually neglected as insignificant. Applied to ion–electron plasmas, equations can be written for the total plasma density, ρ, velocity, u, and energy (pressure, p), where additional terms would appear due to the drifts, as

ρt+(ρu)piri[×BB2]=0,(21)
ρ(ut+uu+ui*u)=J×Bp,(22)
pt+up+ΓpuΓrimi(Tipi+piTipeTe)[×BB2]=q,(23)

where pi,e and Ti,e are the scalar pressures and temperatures of ions and electrons, and Γ is the ratio of specific heats. These equations assume gyroviscous cancellation for the pressure tensor (Braginskii, 1965; Hazeltine & Meiss, 1985; Ramos, 2005), tui+ρ1p^gυtu+ui*u, with an ion diamagnetic drift velocity ui*=B×(p^i)/ρaraB2. Approximations and closures for these equations can be found elsewhere (Lütjens & Luciani, 2010).

2.3 Single-fluid Description

Where collisional coupling is strong enough, a single-fluid approach can be used, with transport equations taking the following form,

ρt+(ρu)=0,(24)
(ρu)t+(ρuu+p^)=J×B+ρg,(25)
t(e+12ρu2)+(u(e+12ρu2)+p^u+q+FR)=JE+ρug.(26)

This system is formally equivalent to the MHD system of equations. The collisional forces compensate and do not appear explicitly in the equations. Note that in the case of both single-fluid and multi-fluid systems, one needs to supply the expression for the electric field E.

3. Closure of the Equations

To close the system of multi-fluid equations such as those in Section 2.1, it is necessary to provide expressions for the collisional terms, the pressure tensor (viscosity), and radiative and thermal fluxes. These expressions are not generic, but depend on particular applications. They are obtained through collisional integrals of the distribution functions fa, fb, and it is frequently assumed that the unperturbed velocity distribution is Maxwellian. A good overview of the calculation of transport coefficients is provided in Rozhansky & Tsedin (2001) and Zhdanov (2002). Braginskii (1965) provides a derivation of viscosity and other transport coefficients based on the Chapman–Enskog method and has been a basic reference for decades. However, it must be taken into account that Braginskii (1965) omits neutrals in his derivation of the viscosity coefficients. For a weakly ionized plasma, these have been presented in the book by Zhdanov (2002), who uses rather general expressions. An attempt has recently been made to produce a model that considers both elastic and inelastic collisional interactions, and is rigorously based on Grad’s method (Zhdanov & Stepanenko, 2016a, 2016b).

3.1 Collisional Terms: General Remarks

Collisions transfer mass, momentum, and energy between particles of different kinds a. Above, we defined the mass collision term, Sa, that leads to the appearance/disappearance of new particles; the term Ra that brings/removes momentum; and the term Ma that brings/removes internal and kinetic energy. Collisions can happen between particles of different kinds and with photons, that is, through interaction with the radiation field. All collisional terms can be subdivided into two categories: elastic and inelastic. By definition, if the particle identity, a, is maintained during the collision, such a collision is called “elastic,” otherwise it is called “inelastic.” For elastic collisions, the term Sa is zero and the term Ra simplifies to a large extent.

Collisions that lead to the creation/destruction of particles are called “inelastic.” The inelastic processes most relevant for the solar atmosphere are ionization, recombination, excitation, and de-excitation. Charge-transfer processes, in which two colliding species modify their ionization state by exchanging an electron, also belong to this type of interaction. In interstellar medium plasmas, ionization/recombination is often ignored owing to the extremely low ionization fraction. Chemical reactions (which are frequent in the ionosphere) also belong to the class of inelastic collisions.

3.2 Collisions in a Purely Hydrogen Plasma

Collisional terms take a particularly simple mathematical form for a purely hydrogen plasma. More information can be obtained from Leake et al. (2014) and Khomenko et al. (2014a).

The inelastic mass collisional term, as appears in the two-fluid set of equations above (Eqs. 712), can be expressed as

Sninel=ρcΓrecρnΓion,(27)

where Γion and Γrec are ionization and recombination rates. The ionization and recombination can be purely collisional or radiative. In the first case, simplified rates are provided in Voronov (1997) and Smirnov (2003) as a function of plasma electron density and temperature, ne and Te. If radiation needs to be incorporated, the expressions become significantly more complex as they involve calculation of the radiation intensity that is frequently non-local and does not depend on a local temperature of the plasma (see Carlsson, 1986; Rutten, 2003).

The momentum collisional term is composed of inelastic and elastic parts,

Rn=ρcucΓrecρnunΓion+(ρeνen+ρiνin)(ucun).(28)

The terms proportional to Γion and Γrec are inelastic collisions. The remaining terms describe either elastic processes or collisional inelastic momentum transfer (chemical reactions or charge transfer). These processes can be introduced via collisional frequencies νin and νen (Draine, 1986). Many books on plasma physics provide expressions for collisional cross-sections (e.g., Bittencourt, 1986; Braginskii, 1965; Huba, 1998; Lifschitz, 1989; Rozhansky & Tsedin, 2001). A recent derivation of charge exchange cross-sections is given in Vranjes and Krstic (2013) for a wide range of plasma parameters.

The energy collisional term for hydrogen plasma, appearing in the equation of conservation of the internal plus kinetic energy, takes the form,

Mn=12ρcuc2Γrec12ρnun2Γion+1γ1kBmn(ρcTcΓrecρnTnΓion)++12(uc2un2)(ρeνen+ρiνin)+1γ1kBmn(TcTn)(ρeνen+ρiνin).(29)

Again, the terms proportional to Γion and Γrec are energy change due to inelastic collisions. The term proportional to the squared velocities is due to elastic collisions, and the last term, proportional to the temperature difference between charges and neutrals, is the thermal exchange term due to elastic collisions. Notice that both the energy equations in (1112) have symmetric collisional terms, so that the total internal plus kinetic energy lost by one fluid is directly gained by another fluid.

The energy collisional terms can be written for equations of conservation of internal energy. After removing the kinetic energy part from Eqs. (1112) the conservation of internal energy takes the following form,

ent+(unen+qn+FRn)+p^nun=Qn,(30)
ect+(ucec+qc+FRc)+p^cuc=J[E+uc×B]+Qc.(31)

The expressions for collisional terms, Qn and Qc become,

Qn=12Γrecρc(ucun)2+1γ1kBmn(ρcTcΓrecρnTnΓion)+12(ucun)2(ρeνen+ρiνin)+1γ1kBmn(TcTn)(ρeνen+ρiνin),(32)
Qc=12Γionρn(ucun)21γ1kBmn(ρcTcΓrecρnTnΓion)+12(ucun)2(ρeνen+ρiνin)1γ1kBmn(TcTn)(ρeνen+ρiνin).(33)

Note that Qn and Qc do not strictly compensate each other as was the case for Mn. There are terms proportional to the square of the velocity difference, (ucun)2 that are positive and are added to both charges and neutral energy equations. The second of these terms is frictional heating. It arises from the net difference in velocities between the two fluids and allows their kinetic energy to be dissipated and converted into heat. The appearance of frictional heating is an intrinsically multi-fluid effect and does not appear in classical MHD which assumes velocities of different plasma components to be the same. Frictional heating plays a significant role in the solar chromosphere.

3.3 Pressure Tensor and Heat Conduction Vector

The pressure tensor, p^a, can be decomposed into its diagonal and non-diagonal parts. The trace of the diagonal part provides the scalar pressure, and all the elements of p^a contain components of viscosity. In a strongly magnetized plasma non-fully coupled by collisions, the diagonal components of the pressure tensor become anisotropic and depend on the direction parallel and perpendicular to the magnetic field (leading to different definitions of temperatures in both directions).

Similarly, the components of the heat flow vector, qa would depend on the temperature parallel and perpendicular to the magnetic field and in general will be different. The heat flow vector can be expressed by using a thermal conductivity tensor as,

qa=κ^aTa,(34)

Braginskii (1965) and Rozhansky and Tsedin (2001) provide the expressions and discuss the components of p^a and qa for plasmas with different degrees of ionization, and the limiting cases for the strong and weak magnetic field. Braginskii’s expressions, modified for the case of partially ionized plasmas, can be found in Khodachenko, Arber, Rucker, and Hanslmeier (2004) and Khodachenko, Rucker, Oliver, Arber, and Hanslmeier (2006). Vranjes (2014) provides viscosity coefficients for ions and neutrals based on the derivation of Zhdanov (2002), while calculations of transport coefficients specific for the lower solar atmosphere are presented in Vranjes and Krstic (2013).

3.4 Radiation

Calculation of radiative energy fluxes that appear in the energy equations in Sections 2.1 and 2.3 require the solution of radiative transfer equations for the radiation intensity,

dIνds=jνkνIν.(35)

This equation describes the evolution of the radiation intensity, Iν, at a fixed frequency ν in the direction of the ray marked s through the medium that can absorb ( kν) or emit ( kν) radiation. The interaction between plasma and the radiation field is a very important ingredient in stellar atmospheres. A complete tutorial on this subject can be found in Mihalas (1986).

The total radiative energy flux, FR, that appears in the single-fluid equations in Section 2.3 is defined through the integral of the radiation intensity over all directions and frequencies.

FR(r,t)=0nIν(r,n,ν,t)dΩdν.(36)

where n represents the unit vector in the direction of s.

When multi-fluid effects are present in partially ionized plasmas, Equations 11–12 require splitting the total radiative energy flux into the two parts corresponding to neutrals and charges (Khomenko et al., 2014a),

FR=FRn+FRc,(37)

where

FRn=0(jνnkνnIν)dΩdν,(38)

and

FRc=0(jνckνcIν)dΩdν,(39)

and the intensities Iν are obtained from the complete radiative transfer equation above with total coefficients jν and kν. The absorption and emission coefficients kνn, kνc, jνn, jνc correspond to processes related to neutrals and charges.

3.5 Generalized Ohm’s Law

The equations of energy conservation (Eqs. 12, 26) need an expression for the electric field in the Joule heating term, JE. This electric field can be calculated in the system of reference attached to the center of mass of the whole fluid, or that of the charged/neutral fluids.

Derivation of the generalized Ohm’s law can be found in many monographs on plasma physics (e.g., Bittencourt, 1986; Braginskii, 1965; Krall & Trivelpiece, 1973). In a general situation, for a plasma composed of an arbitrary number of positively and negatively charged and neutral species Ohm’s law takes the form:

a,|0N+1ρa|ra|2(E+ua|×B)=Jt+(Juc+ucJ)+a,|0N+1(ρa|ra|wa|wa|)+a,|0N+1ra|p^a|a,|0N+1ra|Ra|.(40)

This is the most general form of the generalized Ohm’s law. It has no practical use since it carries the same detailed information as the electron momentum equation, and includes the individual velocities ua| of the charged species, drift velocities of the individual species with respect to the velocity of charges (wa|=ua|uc), and collisional terms of individual charged species, Ra|.

Several approximate forms of the generalized Ohm’s law are frequently used. The term

Jt+(Juc+ucJ)DJ/Dt,(41)

can be safely neglected when assuming variation of currents on timescales longer than ion–electron collisional times.

By limiting ourselves to the case of singly ionized ions only, and a chemical composition similar to that of the solar atmosphere with the dominance of hydrogen and helium and much lower abundance of metals, the following equation for the electric field in the frame of reference of charges can be retrieved:

[E+uc×B]=ηJ+ηH[J×B]|B|ηHp^e|B|χ(ucun).(42)

The first term on the right-hand side is the Ohmic diffusion term, followed by the Hall and Biermann battery terms. The last term, proportional to the velocity difference, can be considered small, since the coefficient χ is proportional to the electron mass (Khomenko et al., 2014a). The η coefficients are diffusivities,

η=ρe(ene)2(a,|=1νe;a|+bνe;b0);ηH=|B|ene,(43)

where the summation is overall available ion and neutral species. The details of the derivation of this equation are provided in Khomenko et al. (2014a). The expressions for the diffusion coefficients are simplified for hydrogen or hydrogen–helium plasmas, as in Pandey and Wardle (2008), Zaqarashvili, Khodachenko, and Rucker (2011), and Leake et al. (2014).

The generalized Ohm’s law in the frame of reference of the whole fluid is obtained by changing the frame of reference,

[E+u×B]=[E+uc×Bξnw×B],(44)

where ξn=ρn/ρ is the neutral fraction and w=ucun is the relative charges–neutral velocity. The following Ohm’s law is obtained:

[E+u×B]=ηJ+ηH[J×B]|B|ηHp^e|B|ηA[(J×B)×B]|B|2,(45)

where the additional diffusivity coefficient is defined as

ηA=ξn2|B|2αn,(46)

with αn=bρeνe;b0+a,|=1bρa|νa|;b0 being the collisional parameter with neutrals. The summation here is carried out over all available neutral (index b0) and ionized (index “ a|”) species. The term multiplying ηA is frequently called the ambipolar or Cowling diffusivity term.

In plasmas with an extremely low ionization fraction, as in the ionosphere, ions play no role in the dynamics and their velocity is irrelevant. In that case, the relevant system of reference is that of the neutrals. By performing the change,

[E+un×B]=[E+ui×Bw×B].(47)

Ohm’s law in the frame of reference of the neutral fluid becomes,

[E+un×B]=ηJ+ηH[J×B]|B|ηHp^e|B|ηA[(J×B)]×B]B|2|.(48)

This equation looks formally the same as Eq. 45, but the definition of ηA is changed to

ηA=ξn|B|2αn.(49)

Therefore, the change of the frame of reference affects the Joule heating. More information can be obtained from the discussion in Vasyliūnas and Song (2005) and Leake et al. (2014). Ohm’s law in the neutral frame of reference is used in ionospheric and interstellar medium plasma (Ciolek & Roberge, 2002; Falle, 2003).

4. New Physical Effects from the Generalized Ohm’s Law

The generalized Ohm’s law contains various terms that are not present in the ideal one: Ohmic, Hall, and Bierman battery terms, and the ambipolar (Cowling) term (as written in the neutral or total plasma reference system). Each of these terms produces variations in the electric field, and in the magnetic field and currents through Faraday’s and Ampere’s laws,

Bt=×E;J=1μ×B.(50)

By performing a simple calculation, the generalized induction equation in the frame of reference of the total fluid is obtained,

Bt=×[(u×B)ηJηH[J×B]|B|+ηHp^e|B|+ηA[(J×B)]×B]|B|2].(51)

The induction equation in the neutral system of reference can be formally recovered by substituting the total center of mass velocity u for the neutral velocity un, and the expression for the ηA coefficient from Eq. 46 to 49.

In the frame of reference attached to the ions, the induction equation has a similar form,

Bt=×[(uc×B)ηJηH[J×B]|B|+ηHp^e|B|χ(ucun)].(52)

The diffusion coefficients η, ηH, and ηA have formally the same units. The relative importance of each of the new terms in the generalized induction equation can be evaluated by comparing the values of these coefficients in a plasma. As an example, Figure 1 (taken from Khomenko & Collados (2012)) shows η, ηH, and ηA as a function of height in the solar atmosphere. In this example, three zones can be distinguished. At heights below 0 kilometers, the Ohmic term, η, is dominant so that the main non-ideal effect is classical Ohmic diffusion. At intermediate heights between 0 and 800 km, the Hall effect and battery effects, both proportional to ηH, are dominant. Above 800 km, the ambipolar effect ηA is the leading one. Note that the vertical scale in the figure is in logarithmic units, so that the difference in magnitude between different η-coefficients can be significant.

Figure 1. Coefficients η, ηH, and ηa calculated in the model atmosphere representing a solar magnetic fluxtube from Khomenko and Collados (2012).

4.1 Ohmic Diffusion

Ohmic diffusion is a classical non-ideal plasma effect present in pure ion–electron plasmas. The reason for Ohmic diffusion is the presence of collisions. Collisions disrupt the trajectories of gyration of charged particles around the magnetic field, thereby breaking the frozen-in condition and producing magnetic field diffusion. A description of the effects of Ohmic diffusion can be found in almost any plasma physics tutorial (e.g., Bittencourt, 1986; Krall & Trivelpiece, 1973).

Ohmic diffusion assists the process of relaxing the magnetic field configurations by dissipating currents; it therefore allows us to convert magnetic energy into heat. Ohmic diffusion produces magnetic reconnection and also affects waves and instabilities, and its influence is important for phenomena on very small spatial/temporal scales compared with typical plasma scales.

In partially ionized plasmas the Ohmic diffusivity coefficient is modified to include the presence of collisions with neutrals; see the expression for η in Eq. 43.

4.2 Hall Effect

The Hall effect is a non-ideal effect present in classical ion–electron plasmas. Its presence does not require the action of particles other than ions and electrons. However, other particles (e.g., neutrals) may assist the action of the Hall effect and increase its magnitude.

The importance of the Hall effect is frequently measured in terms of the so-called Hall parameter, βH, which is the ratio between the electron/ion cyclotron frequency and the collisional frequency with other heavy particles.

βH,e=ωceνei+νen;βH,i=ωciνii+νin.(53)

Since ωce=eB/me, and ωci=eB/mi depend on the magnetic field, the Hall parameter in a plasma can, in principle, take any value. The Hall effect is produced when the electron cyclotron frequency is high, that is, when the electrons can freely gyrate around the magnetic field lines following curved trajectories without much interruption by collisions. For ions, the cyclotron frequency is three orders of magnitude lower owing to the large ion mass. It may therefore happen that, even for similar collisional frequencies, the electrons stay magnetized (i.e., attached to the magnetic field), but that the ions do not. The misalignment between the ion and electron trajectories produces the change in the direction of the current, J, leading to variations in the electric field. In the case of the solar atmosphere, this situation happens throughout the photosphere and lower chromosphere, depending on the magnetic field strength, as discussed by Pandey, Vranjes, and Krishan (2008).

The Hall effect in astrophysical plasmas is thought to operate on short spatial/temporal scales close to the collisional scales. In order to evaluate its significance for a given physical process, one usually compares the typical frequency of this process with the ion-cyclotron frequency in terms of the parameter ϵ,

ϵ=ω/ωci.(54)

In a fully ionized plasma the values of ϵ are typically extremely small for the frequencies of interest. For example, in the case of the Sun, typical waves have frequencies in the range of mHz, while ωi varies in the range of kHz–MHz. Nevertheless, in partially ionized plasmas the ϵ parameter must be modified to take into account the ionization fraction, ξi=ρi/ρ, leading to the expression (Cally & Khomenko, 2015):

ϵ=ω/ωciξi.(55)

This means that, for low ionization fractions, as in the solar atmosphere, ξi104, the Hall parameter may become important at significantly lower frequencies.

The Hall effect leads to the appearance and modification of a number of instabilities and to wave dispersion. It also affects reconnection processes. The additional current produced by the Hall effect may lead to the intensification of solar magnetic flux tubes (see Khodachenko & Zaitsev, 2002).

4.3 Battery Effect

The magnetic induction equation with only the battery term can be expressed as:

Bt=×(u×B+peene).(56)

The battery term is independent of the magnetic field and acts as a source term in the induction equation. A non-zero magnetic field can be produced from this effect even if no magnetic field exists initially in the medium. This effect was first investigated by Biermann in 1950. The battery effect was proposed to enable seeds for galactic dynamos to be provided (Kulsrud & Zweibel, 2008) and was recently investigated for the case of the solar local dynamo (Khomenko, Vitas, Collados, & de Vicente, 2017).

The above induction equation (56) can be rewritten in the same form as the equation for the evolution of the vorticity, ω=×u (see Kulsrud & Zweibel, 2008):

Bt=×(u×B)+p×ρρ2μe,ωt=×(u×ω)p×ρρ2.(57)

Comparison between the induction and vorticity equations provides a way of evaluating the strength of the initial magnetic field produced by the battery effect in unit time from the simple relation:

e|B|μ|×u|.(58)

The field generated by the battery effect can be further amplified by the flow via the dynamo effect. The battery effect is present in ion–electron plasmas and is affected by the presence of other particles (such as neutrals) only indirectly, just as for a Hall effect.

Yet another source term in the induction equation (52) is χ(ucun), which is potentially able to generate a seed magnetic field from the relative charge–neutral motion.

4.4 Ambipolar Diffusion

The ambipolar term (frequently called the “Cowling” term in ionospheric physics) is present in the single-fluid induction equation (51), with a corresponding counterpart in the energy equation. In general plasma physics, in a plasma without neutral particles, the term “ambipolar diffusion” refers to the diffusion of positive and negative particles at the same rate owing to their collisional interaction, which maintains the charge neutrality at scales larger than the Debye length. Unlike in astrophysics and ionospheric physics, “ambipolar diffusion” refers to the decoupling of neutral and charged plasma components. Similar to the Ohmic term, it provides additional diffusion of the magnetic field through the plasma. However, unlike Ohmic diffusion, the nature of ambipolar diffusion is related to the presence of neutral atoms. The charged particles are frozen into the magnetic field, and drag it along with themselves owing to collisions with neutrals, thereby causing diffusion of the magnetic field throughout the neutral gas.

Taking into account Ohm’s law (Eq. 45), the equation for internal energy evolution written for the plasma as a single fluid takes the form,

DeDt+γeu=ηJ2+ηAJ2.(59)

It can be seen that the right-hand side of this equation contains Joule heating, which includes two parts, Ohmic ( ηJ2) and ambipolar ( ηAJ2), where J=b^×J×b^ is the component of the current perpendicular to the magnetic field ( b^ is a unit vector in the direction of magnetic field). Ohmic term was discussed above, Section 4.1. The ambipolar term acts similarly to the Ohmic one and allows us to convert magnetic energy into heat by changing the magnetic configuration. There are, however, two important differences:

1.

Ambipolar diffusion only acts on perpendicular currents. This means that ambipolar diffusion cannot by itself produce the reconnection of magnetic field lines, but it does allow the magnetic field lines to be brought closer together to form sharp (or singular) structures in the magnetic field (Brandenburg & Zweibel, 1994; Bulanov & Sakai, 1998). The final state of magnetic field relaxation under ambipolar diffusion is a force-free structure with only parallel currents.

2.

The timescales imposed by ηA and by η,

tAL2/ηA;tL2/η

can be significantly different; see Figure 1.

Ambipolar diffusion has been proposed as a possible mechanism for the escape of magnetic flux from dense molecular clouds by Mestel and Spitzer (1956) and is believed to affect the dynamics of self-gravitating gases (Shu, 1983). Ambipolar diffusion is considered to be the dominant non-ideal plasma mechanism due to neutrals acting in the solar chromosphere.

5. Current Ideas: Interstellar Medium

The interstellar medium, that is, the space between the stars in our and other galaxies, contains relatively dense and cold molecular clouds with much less dense plasma in between. It is believed that most star formation happens in dense molecular clouds. The ionization fraction in the interstellar medium is below 107. Apart from molecules, plasma in the interstellar medium contains charged dust grains. The collisional coupling between all these components is weak; therefore, plasma in the interstellar medium is susceptible to multi-fluid effects. Historically, multi-fluid effects related to the very weak degree of ionization in molecular clouds and the interstellar medium were studied much earlier than other fields of astrophysics (Mestel & Spitzer, 1956). Ambipolar diffusion is a central mechanism discussed in the context of molecular cloud dynamics, the mechanisms of stellar formation, and the interpretation of observed properties of interstellar shocks (Draine & McKee, 1993; Shu, 1992), but other mechanisms, such as Ohmic diffusion and the Hall effect, have also entered into consideration. A good recent review of the evolution of the field of star formation in molecular clouds influenced by multi-fluid effects is provided by Ballester et al. (2018), and is closely followed in this article.

5.1 Gravitational Collapse and Star Formation in Molecular Clouds

Molecular clouds are observed to contain high-speed non-thermal motions. The characteristics of these motions indicate that no global gravitational collapse is occurring in the clouds, but they do contain small-scale turbulence (Zuckerman & Evans, 1974) or Alfvénic incompressible waves. Therefore, a mechanism is needed to prevent global gravitational collapse of a cloud in order to maintain a relatively stationary situation and allow for star formation. The collapse leading to star formation occurs in the dense cores of molecular clouds, and its origin is still one of the key issues to be solved. Several scenarios have been proposed for support against global gravitational collapse, such as magnetic support, turbulent support, or hierarchical collapse models. In all cases, the support prevents global collapse but enables the hierarchical collapse on smaller scales. Ambipolar diffusion was proposed as a mechanism to enable star formation in the magnetic support model.

The magnetic support model was sustained by the first measurements of strong magnetic fields in molecular clouds (recent measurements have provided much lower field strengths; see Crutcher, 2012). This model requires equipartition between magnetic and gravitational energies. When the magnetic energy slightly dominates, the clouds are labeled as magnetically subcritical and therefore globally stable. Nevertheless, in order to produce star formation some loss of the support is needed. It was proposed that ambipolar diffusion is responsible for the loss of support. If neutrals are present, they would drag the magnetic field frozen into the plasma with them due to charge-neutral collisions (a mechanism of ambipolar diffusion explained in Section 4.4). This would cause diffusion and the loss of some of magnetic field, with subsequent loss of support and dynamical collapse. Discussion of this process can be found in Shu, Adams, and Lizano (1987). While attractive, the mechanism of stability loss due to ambipolar diffusion is nowadays considered rather inefficient because of several issues. One of these is related to the timescale imposed by ambipolar diffusion in molecular clouds, Eq. 60.

This characteristic timescale was found to be an order of magnitude longer than the free-fall time. Therefore, only some minor part of the densest cores would be able to collapse through this mechanism to permit the star formation, and the overall process would be inefficient.

5.2 Magnetic Flux Removal from Forming Protostellar Disks

As a result of gravitational collapse, the material in a protostellar cloud starts to rotate at an increasing rate. The rotation would generate strong centrifugal forces and the cloud would break into small-mass fragments, unable to collapse, so the star formation would stop. This problem is known as the angular momentum problem (Hoyle, 1945). To avoid it, the excess angular momentum must be transported outside the collapsing cloud.

The magnetic field provides a natural mechanism for angular momentum removal. A frozen-in magnetic field would be dragged with the plasma and rotate with the same angular velocity. This would accelerate the material outwards along the magnetic field lines, transferring angular momentum outside the protostar. Such a mechanism is known as magnetic braking (Mouschovias, 1977). The strength of the magnetic field measured in molecular clouds is enough to provide the work of this mechanism.

The magnetic braking mechanism was shown to be too strong under ideal MHD conditions since it removed too much angular momentum. As a result, a disc formed in ideal MHD does not follow the Keplerian regime and is not rotationally supported. Ohmic diffusion, ambipolar diffusion, and Hall effect were all proposed as possible solutions to this problem. While ambipolar diffusion only redistributes the magnetic flux, Ohmic diffusion allows the removal of part of it, thereby allowing it to reach the desired regime (see the review by Li et al., 2014), and therefore may be a better candidate to facilitate the mechanism of magnetic braking.

5.3 Width of Filamentary Structures in Molecular Clouds

Non-ideal MHD was invoked to explain the width of filaments in molecular clouds observed by the Herschel telescope (André et al., 2010). The filamentary structures were observed to have nearly the same width, independently of their central density. It was suggested that this width is a universal property of the filaments and is set by some fundamental physical process.

Interstellar material is accreted across the filaments, producing flows across their axes, and also flows along them towards the dense cores. Hennebelle and André (2013) proposed that these flows drive turbulence in the central part of the filaments, and that this turbulence is in virial equilibrium, that is, the turbulent pressure is equal to the hydrodynamical pressure of the accretion flow. The stationary state is determined by the rate at which the turbulence dissipates. The model by Hennebelle and André (2013) predicts that the width of the filaments is almost independent of the central density and has values similar to those observed if dissipation of the turbulence is due to ion–neutral friction, unlike turbulent dissipation.

Figure 2. Classification of interstellar shocks from Draine and McKee (1993).

5.4 Interstellar Shock Waves

The interstellar medium is constantly disrupted by violent events such as supernova explosions, fast-moving clumps of interstellar gas, and stellar winds. These high-speed flows propagate into the interstellar medium plasma in the form of shock waves. The structure of interstellar shocks is strongly influenced by multi-fluid effects. Theoretical models of the formation of shocks in the interstellar medium and molecular clouds were developed in the 1980s and can be consulted in the original works (Chernoff, 1987; Draine, 1986; Draine & McKee, 1993; Hollenbach & McKee, 1989; Roberge & Draine, 1990; Wardle, 1990).

The peculiarity of interstellar shocks consists in the fact that, despite the very low ionization fractions, the magnetic field is relatively strong and affects the dynamics. The ion-cyclotron frequency is significantly higher than the ion–neutral collisional frequency, that is, the magnetic field is effectively frozen into the charged fluid. Ion-neutral collisions are typically too weak to effectively couple ion and neutral fluids and these components have to be considered separately using a two-fluid approximation.

The structure of the discontinuity at the shock front and emission from the shocked hot gas can be set by a number of mechanisms such as viscosity or radiation, the latter being the main mechanism of shocks cooling in the interstellar medium (Draine & McKee, 1993). Depending on the shock amplitude, different spectral lines act as cooling agents. Ion–neutral frictional heating (Section 3.2) plays the major role in heating associated with shocks in the partially ionized plasma of the interstellar medium.

Because of the low ionization fraction, the Alfvén speeds associated with the ionized part and the whole fluid are significantly different. The total Alfvén speed of the plasma in the interstellar medium can reach about 2 km/second. In contrast, the Alfvén speed, which takes into account only ionized fluid, is about 100 km/s. Fast waves can propagate forming shocks non-parallel to the magnetic field with speeds close to the ion Alfvén speed. Ion–neutral collisions would tend to dampen these waves. However, because of their high propagation speeds, ions “communicate” information ahead of the wave front that propagates with the Alfvén speed of the whole fluid. Therefore, no shock can be formed in the ionized fluid.

Neutrals and ions are accelerated through different forces and may have significantly different streaming velocities at the shock fronts. Ion–neutral collisions heat and accelerate (or decelerate) both ion and neutral fluids. The resulting shock structure was classified to fall into the main three categories as sketched in Figure 2, adopted from Draine and McKee (1993). The particular shape depends on the efficiency of the cooling mechanisms at the shock front and on the strength of the shock itself. For example, if the cooling is very efficient (or the shock is weak), the perturbation in a neutral fluid remains subsonic and no discontinuity is formed. In this case the shock is of type “C” and all the variables are continuous at the shock front, with ions lagging behind the neutrals. If the ion–neutral frictional heating is strong enough to raise the neutral gas temperature and sound speed, the shock can be supersonic far upstream (where the neutral fluid is still cold), but subsonic downstream (where the neutral fluid is already heated). The neutral fluid makes a sharp supersonic–subsonic transition, and a shock of type “J” is formed (middle panel of Figure 2). In particular cases, the supersonic–subsonic transition is still smooth, leading to a type “C*” shock (bottom panel).

Under certain circumstances the fronts of interstellar shocks may become unstable for a small perturbation. Several instabilities of different types were studied for interstellar shocks, such as thermal instability (McCray, Stein, & Kafatos, 1975), the Wardle instability (Wardle, 1990), and blast wave instability (Ryu & Vishniac, 1987).

6. Current Ideas: The Ionosphere

Ionospheric partially ionized plasma is one of the most accessible for study and direct measurement by satellites. The ionosphere is an upper layer of the Earth’s atmosphere (and other planets generally), where neutral gas becomes ionized. For the Earth’s atmosphere, the ionosphere occupies a region between 90 and 1,000 km from the surface. The ionosphere borders the magnetosphere above it and mesosphere below. Typical multi-fluid processes in the ionosphere are related to the degree of ion–neutral coupling, the creation and presence of global currents, and the presence of multiple plasma drifts due to forces created by internal and external currents, and neutral winds. Further information on the conditions of planetary ionospheres can be found in the monographs by Bauer (1973), Cravens (1997), Schunk & Nagy (2004), and Kelley (2009), and in the recent reviews by Pfaff (2012), Leake et al. (2014), and Ballester et al. (2018).

Conditions in planetary ionospheres vary strongly with height, longitude and latitude, and day and night, as a consequence of varying chemical composition, the ionizing properties of radiation, and magnetic field. The global magnetic field of the Earth is imposed on the ionospheric plasma. The degree of ionization of the Earth’s ionosphere is a sensitive function of photoionization by solar EUV and X-ray radiation; it is affected by the presence of aurorae and locally by meteoric impact. The chemical composition of the ionosphere (usually divided into F, E, and D layers) changes with height. In the lowest, D, layer the dominant neutral is molecular nitrogen ( N2), while its dominant ion is nitric oxide ( NO+). In the E layer, in addition to nitrogen there are also molecular oxygen ions O2+. In the highest, F, layer, atomic oxygen is dominant in both neutral and ionized states (Leake et al., 2014).

Ionospheric plasma has, on average, clearly distinct neutral, ion, and electron temperatures above a few hundred kilometers. The ionosphere is therefore usually described as a three-fluid system. The change with height of the relation between the cyclotron frequencies and collisional frequencies in the ionosphere is similar to that of the solar atmosphere. The lower ionosphere is collisionally dominated, with both ion–neutral and electron–neutral collisional frequencies being higher than the ion-cyclotron frequency (Leake et al., 2014; Song, Gombosi, & Ridley, 2001). The electron-cyclotron frequency dominates all the rest, so that electrons are magnetized throughout the region. In the upper layers, collisions are less important and all the particles become magnetized.

6.1 Ionospheric Dynamo Effect

One of the most prominent manifestations of ion–neutral interactions in the ionosphere is the ionospheric dynamo effect. The mechanism of dynamo circulation is the following. When solar radiation heats the Earth’s atmosphere, it creates a system of neutral winds, expanding from the sub-solar point. The motion of the neutral gas drags ions via ion–neutral collisions and produces their motion across the north–south-directed global magnetic field of the Earth. While ions are dragged across the field, electrons are still magnetized and remain attached to the field. This creates a small imbalance in the drift speeds of the electrons and ions, and a system of horizontal global-scale electric currents develops. The currents generate an eastward horizontal polarization electric field to fulfill the divergence-free condition. This system of currents, forming current loops at mid-latitudes, is called the solar quiet dynamo.

The eastward-directed electric field created as explained previously produces in its turn a vertical E×B drift. Only electrons are susceptible to this vertical drift, since ions are stopped by collisions with neutrals. Charge separation between ions and electrons (the Hall effect) establishes a vertically polarized electric field Ep. This polarization field creates a horizontal Ep×B drift, which reinforces the small existing dynamo current created by neutral winds. Overall, this effect forms the strongest current system, known as the equatorial electroject.

Figure 3. Values of the parallel, Pedersen, and Hall conductivities in the Earth’s ionosphere from Leake et al. (2014).

6.2 Conductivity Tensor

Plasma in the ionosphere is permeated by the global magnetic field of the Earth. This causes the particle distribution functions, fa, in the presence of a magnetic field to be highly anisotropic. Taking into account the anisotropy of the medium, it is possible to express the generalized Ohm’s law (Eq. 48) in the form of the diffusivity tensor (here, the contribution from the battery term has been neglected):

E*=[E+un×B]=η^J=ηJ||+ηpJ+ηHJ×b^,(61)

where ηp=η+ηA is the Pedersen diffusivity, defined as the sum of the Ohmic and ambipolar (Cowling) diffusivities, and J is the perpendicular current, defined in Section 4.4. Expressions for the η coefficients are defined in Section 3.5.

This relation can be inverted and written in terms of the conductivity tensor, σ^,

J=σ^E*=σ||E||*+σpE*σHE*×b^,(62)

where the tensor components are given by the equation

σ||=1η;σp=ηpηp2+ηH2;σH=ηHηp2+ηH2.(63)

The conductivity tensor has the following form:

σ^=[σpσH0σHσp000σ||].(64)

Therefore, the ionospheric conductivity is anisotropic and couples neutral and ionized species via σ coefficients, organized by the large-scale magnetic field. Figure 3 illustrates typical values of σ coefficients in the Earth’s ionosphere. Parallel (Ohmic) conductivity dominates in the higher layers, and Hall conductivity peaks at intermediate heights, below the peak in the Pedersen conductivity. The peak in Hall conductivity is responsible for the ionospheric dynamo effect mentioned previously. The height distribution of conductivities in ionosphere is similar to that of the solar atmosphere (Leake et al., 2014).

6.3 Joule Heating

As discussed in Section 4.4, the relative motion of ions and neutrals produces Joule heating (see Eq. 59). Satellites located in the ionosphere measure Joule heating in this region at different heights. Since neutral winds constantly create drifts in the ionosphere, the associated Joule heating is a natural response and is an important agent of the ionospheric energy balance.

Apart from drifts, Joule heating can be produced by energetic particles in aurorae or by magnetic storms. The presence of energetic particles entering the ionosphere at high latitudes is responsible for the increase in local plasma conductivity (via an increase in the number density of electrons) and heating. The incoming beam of energetic electrons interacts with the upper atmosphere, which suffers collisional ionization, thus creating hot plasma and heating the surrounding neutral gas.

7. Current Ideas: Solar Physics

Interest in the study of multi-fluid effects in solar physics has recently increased, and the subject is actively developing. Unlike the other fields discussed here, the justification for studying multi-fluid effects in solar plasma stems from theoretical considerations. Only a few frequently uncertain and indirect observations of multi-fluid effects exist so far.

The solar atmosphere consists of four distinct layers: the photosphere, the chromosphere, the transition region, and the corona. Owing to the rather cool temperatures of the photosphere and chromosphere of the Sun, the degree of ionization there is very small, reaching values as low as 104 at the temperature minimum, and remaining well below unity at greater heights (see, e.g., Vernazza, Avrett, & Loeser, 1981). The plasma becomes completely ionized in the corona. Clouds of chromospheric material can frequently be found suspended in the solar corona thanks to the action of magnetic forces, in structures known as solar prominences. The matter in prominences is almost half neutral.

In the photosphere the collisions are strong enough to exceed the ion-cyclotron frequency for a magnetic field strength typical of solar structures. Nevertheless, the electrons stay magnetized. This leads to conditions similar to the ionospheric dynamo (except for the presence of the global magnetic field and neutral winds), with ions being disrupted by collisions with neutrals, a Hall current being created in this way. In the middle and upper chromosphere, the collisional frequency is not strong enough and ions also become magnetized. Since ions and neutrals move under the action of different forces, this creates favorable conditions for ion–neutral drifts, and the ambipolar diffusion mechanism sets in. The first studies of the influence of ambipolar diffusion on waves can be found in, for example, Osterbrock (1961) or Piddington (1956).

Depending on the degree of collisional coupling, it may be sufficient to apply a single-fluid theory, together with the generalized Ohm’s law, or it may be necessary to treat the neutral and charged fluids separately, or even to treat the different neutral and charged components with different masses in a separate way. A number of works have recently used a multi-fluid formalism for the description of solar plasma (Khomenko et al., 2014a; Leake et al., 2014; Meier & Shumlak, 2012). Reviews of the subject can be found in Martínez-Sykora, De Pontieu, Hansteen, and Carlsson (2015), Khomenko (2017), and Ballester et al. (2018).

7.1 Generation, Propagation, and Damping of Magnetohydrodynamic Waves

The effects of ion–neutral collisions on the propagation, excitation, and damping of different types of magnetohydrodynamic waves in the solar plasma have been extensively studied theoretically (see the review in Ballester et al. (2018)). The vast majority of these studies are not directly relevant to addressing the observed properties of solar waves, since ion–neutral effects become important at high frequencies (of the order of kHz to MHz), while observed solar oscillations have frequencies in the range of MHz. However, the solar oscillation spectrum is broad and, despite not having been measured yet, the presence of high-frequency waves is justified on theoretical grounds (Musielak, Rosner, Stein, & Ulmschneider, 1994). Therefore, ion–neutral effects on waves must be taken into account when considering the energy balance of the solar atmosphere. In short, neutrals were observed to produce several effects on waves: (a) damping when the wave frequencies are comparable to the collisional frequencies; (b) the presence of cut-off frequencies defining the regions of wave numbers where the wave propagation is suppressed; and (c) the appearance of new wave modes.

Ion–neutral collisions are found to be an important mechanism for wave damping in the photosphere, chromosphere, and prominence plasma (Khodachenko et al., 2004). This mechanism is efficient in dissipating incompressible Alfvén waves, which are significantly more difficult to dissipate than compressional acoustic waves (Osterbrock, 1961; Piddington, 1956). Collisional friction was proposed as one of the reasons for the damping of coronal loop oscillations and prominence oscillations.

Multiple new wave modes appear under a multi-fluid treatment. Some of these modes are unable to propagate in the atmosphere, depending on their frequency, that is, cut-off frequencies (or cut-off wave numbers) appear, when waves are modeled using a single-fluid frame (Section 2.3). The physical reason for these cut-offs can be explained by taking as an example Alfvén waves, whose restoring force is magnetic tension. As discussed in Soler, Carbonell, Ballester, and Terradas (2013), for a range of intermediate wave numbers, the disturbance in the magnetic field may decay owing to ion–neutral friction before the ion–neutral coupling has had time to transfer the restoring properties of magnetic tension to the neutral fluid. In this situation, neutral–ion collisions are efficient enough to dissipate perturbations in the magnetic field, but they are not efficient enough to transfer significant inertia to neutrals before the magnetic field perturbations have decayed. Hence, oscillations of the magnetic field become suppressed. The cut-offs due to ion–neutral interactions disappear in the two-fluid treatment (Zaqarashvili et al., 2011) since ion–neutral drift velocity can be fully considered.

The linear theory considers only wave damping, but not dissipation. The wave energy equation in the linear theory is adiabatic, since the Joule dissipation term and the frictional heating term are quadratic in perturbations and are necessarily neglected (Ballester et al., 2018). In order to fully take into account the impact produced by the dissipation of waves due to ion–neutral collisions, non-linear equations must be solved. A number of studies have considered this issue and concluded that the dissipation of perpendicular currents produced by waves allows the efficient conversion of the magnetic energy of waves into thermal energy. It may produce significant heating of the magnetized chromosphere above the magnetic elements (Goodman, 2000; Song & Vasyliūnas, 2011) (see Section 7.5). Alfvén waves are considered in solar physics as very good candidates for transporting energy to the corona and contributing to maintaining its 106 degree temperature. In a recent study, Soler, Terradas, Oliver, and Ballester (2019) modeled torsional Alfvén waves propagating in a magnetic flux tube from the photosphere to the corona, ejecting a broadband wave spectrum, and considering Ohmic diffusion and ion–neutral collisions as dissipation mechanisms in a partially ionized hydrogen–helium plasma. An illustration of the magnetic field lines and velocity vectors in this model is shown in Figure 4. Despite the fact that only about 1% of the initial energy flux was able to reach coronal heights in this model, Soler et al. (2019) concluded that this amount may be sufficient to partly compensate for the total coronal energy loss. Fully non-linear two-fluid simulations of magneto–acoustic wave propagation in the stratified solar atmosphere have been reported by Maneva, Alvarez Laguna, Lani, and Poedts (2017), showing a complicated dynamics, and both heating and cooling events in the chromosphere.

The efficiency of the generation of Alfvén waves by convection in the photosphere was called into question by Vranjes, Poedts, Pandey, and Pontieu (2008), who suggested that, thanks to the very low degree of ionization, the amplitudes of ion perturbations may be greatly reduced. This conclusion was questioned by Tsap, Stepanov, and Kopylova (2011) and Soler et al. (2013), who argued that the motions of neutrals was not taken into account. An alternative way of generating Alfén waves was put forward by Cally and Khomenko (2015), who demonstrated that the Hall effect acting in the partially ionized photosphere is able to produce Alfvén waves through mode transformation from fast magneto–acoustic waves.

Figure 4. Visualization of the magnetic field and velocity perturbations due to torsional Alfvén waves in a solar flux tube extending from the photosphere to the corona. The red and blue lines follow selected magnetic field lines. The arrows correspond to the velocity of ions at three horizontal planes located 1, 2.5, and 4 Mm above the photosphere, with the color coding indicating the speed values in the color bar on the left. Note that the vertical and horizontal axes have different scales. Results from Soler et al. (2019).

Figure 5. Rayleigh–Taylor instability at the border between a partially ionized solar prominence and the fully ionized corona. The images show the total plasma density at different stages of the instability. Results from Khomenko et al. (2014b).

7.2 Instabilities

As for waves, studies of instabilities in partially ionized solar plasma are essentially theoretical. Ion–neutral interactions lead to a modification of the onset criteria and growth rates of classical instabilities such as the Rayleigh–Taylor (RTI) and Kelvin–Helmholtz (KHI) instabilities, and also to the appearance of new ones typical of a partially ionized medium.

Contact instabilities, such as magnetic RTI and KHI, are thought to arise at the interfaces between coronal and prominence material, or at the borders of magnetic flux tubes. Linear analyses and numerical simulations show that the presence of neutrals in a partially ionized plasma removes the critical wavelength imposed by the magnetic field, thus making the perturbations unstable over the whole wavelength range (Díaz, Khomenko, & Collados, 2014; Khomenko, Díaz, de Vicente, Collados, & Luna, 2014b; Soler, Díaz, Ballester, & Goossens, 2012). Nevertheless, the growth rate is typically very small. The instability threshold of the compressional KHI is very much sensitive to the value of the flow and is lower than in the fully ionized case owing to ion–neutral coupling, reaching sub-Alfvénic values (Soler et al., 2012).

An example of fully non-linear simulations of the later phase of the RTI at an interface between a partially ionized solar prominence and the fully ionized corona is shown in Figure 5, taken from Khomenko et al. (2014b). These simulations included ambipolar diffusion as the main non-ideal mechanism. They largely confirm the linear theory regarding the removal of the cut-off frequency of the instability and show that, thanks to the presence of neutrals, the prominence-corona interface is always unstable.

The two-stream Farley–Buneman instability (FBI) was suggested to arise under chromospheric conditions due to the drift motions of charged particles when electrons are strongly magnetized, but ions are unmagnetized owing to collisions with neutrals. The triggering of FBI can be produced by waves or flows of quasi-neutral gas from the photosphere, creating cross-field motion in a partially ionized plasma (Fontenla, 2005; Madsen, Dimant, Oppenheim, & Fontenla, 2014). Conditions in the chromosphere meet the instability criteria if the electron drift trigger velocity is slightly below the sound speed (Madsen et al., 2014). It has been suggested that this instability may lead to chromospheric heating; however, the currents necessary to provide an electron drift velocity of 2–4 km s–1 are two orders of magnitude greater that actually measured (Socas-Navarro, 2007).

7.3 Flux Emergence

Another phenomenon affected by ion–neutral interactions is magnetic flux emergence (Arber, Haynes, & Leake, 2007; Leake & Arber, 2006). Magnetic flux emergence on large scales is thought to be a central mechanism allowing for the formation of solar active regions and sunspots. A magnetic flux rope formed deep in the solar convection zone rises from the solar interior to the surface thanks to its buoyancy. In the upper part of the convection zone the flux rope may be destroyed by strong convective motions preventing efficient emergence. The rising flux would encounter a layer of almost neutral gas in the photosphere. It then becomes affected by ambipolar diffusion removing perpendicular currents and modifying the structure of the emerged field. By performing 2D and 3D numerical simulations, Leake and Arber (2006) and Arber et al. (2007) have shown that the amount of emergent flux can be greatly increased by the presence of this diffusive layer of partially ionized plasma. In the chromosphere, the order of magnitude stronger dissipation of currents perpendicular to the magnetic field, compared with that of longitudinal currents, has been found to facilitate the creation of force-free field structures (Arber, Botha, & Brady, 2009).

Figure 6. Reconnection experiment in a partially ionized plasma, including the effects of radiation. The figure shows snapshots of current and magnetic field lines at the same simulation time, but at different ionization levels (marked by the symbol Ψ0, indicated in the figure). Results from Alvarez Laguna et al. (2017).

7.4 Magnetic Reconnection

The influence of partial ionization of the solar plasma on reconnection rates has been the subject of a number of theoretical studies. While ambipolar diffusion affects only perpendicular currents, and therefore cannot directly produce reconnection, it can modify the configuration of the magnetic field lines, making the conditions for reconnection more favorable. Initial studies have shown that the reconnection rates do indeed strongly depend on the collisional coupling between ionized and neutral species (Zweibel, 1989). Owing to the action of ambipolar diffusion, oppositely orientated magnetic field lines can be brought sufficiently close to facilitate reconnection (Brandenburg & Zweibel, 1994, 1995). The current sheet can become thin with regard to the scale of the neutral–ion mean free path (Murphy & Lukin, 2015). A two-fluid numerical model of reconnection used by Sakai, Tsuchimoto, and Sokolov (2006) demonstrated that reconnection leads to proton heating and jet-like phenomena with different temperatures of neutral and ionized species. More advanced numerical modeling of reconnection in a Harris current including ion–neutral scattering collisions, ionization, recombination, optically thin radiative loss, collisional heating, and thermal conduction (Leake, Lukin, Linton, & Meier, 2012) has shown a more complex picture, where the neutral and ionized fluid becomes uncoupled upstream from the reconnection site. It undergoes ion recombination, which, combined with Alfvénic outflows, leads to fast reconnection rates independent of the Lundquist number. The effects of radiation on reconnection in partially ionized plasmas have been thoroughly studied in Alvarez Laguna, Lani, Mansour, Deconinck, and Poedts (2017). It was confirmed that reconnection occurs more quickly in weakly ionized plasmas, owing to the effects of ambipolar diffusion and fast recombination. Radiation cooling was shown to have a strong effect because of its influence on the concentration of ions inside the current sheet. Altogether, this results in a thinning of the current sheet, accelerating the appearance of the tearing mode instability and enhanced reconnection (see Figure 6).

Asymmetric reconnection can also occur in the chromosphere when the emergent magnetic flux interacts with an already existing one. This situation was studied by Murphy and Lukin (2015), who showed that the ion and neutral outflows are strongly coupled, but that the inflows are asymmetrically decoupled. Yet another kind of reconnection induced by slow mode shock waves was modeled by Hillier, Takasao, and Nakamura (2016), who showed that the system undergoes several stages from weak to strong coupling, finally reaching a quasi-steady state. It is characterized initially by an over-pressurized neutral region that explosively expands outwards from the reconnection site, and frictional heating is produced across the shock front owing to the strong drift velocity. The shock transition was found to be continuous in physical variables from a subsonic velocity upstream to a sharp jump followed by a relaxation downstream (from C-shock to J-shock; see Section 5.4). Altogether, these models suggest that such phenomena could be responsible for the foot-point heating of coronal loops, explosive events, chromospheric, transition-region, and sunspot penumbrae jets, and for producing spicules.

7.5 Chromospheric Heating

From the point of view of energy balance, the additional dissipation of perpendicular currents due to ion–neutral interaction (ambipolar diffusion) can lead to Joule heating several orders of magnitude higher than that in a fully ionized plasma (Arber et al., 2007; Khodachenko et al., 2004). It has been pointed out that current dissipation, enhanced by the presence of neutrals in a plasma not entirely coupled by collisions, can play an important role in the energy balance of the chromosphere and above (De Pontieu & Haerendel, 1998; Judge, 2008; Krasnoselskikh, Vekstein, Hudson, Bale, & Abbett, 2010). Numerical modeling has confirmed that the amount of heating is indeed sufficiently high and the timescales associated with it sufficiently short for it to compensate easily for the radiative energy losses of the magnetized chromosphere and explain the chromospheric temperature increase (Khomenko & Collados, 2012; Martínez-Sykora, De Pontieu, & Hansteen, 2012). A necessary condition for the heating is the existence of a non-force-free magnetic field, producing perpendicular currents. Such currents can exist in the form of relatively stationary currents in magnetic structures, but can also be dynamically created by waves and flows. An illustration of this effect is shown in Figure 7, taken from Khomenko, Vitas, Collados, and de Vicente (2018). It shows a snapshot of realistic simulations of solar dynamo quiet Sun magnetic fields that take into account ambipolar diffusion as the main non-ideal effect. The magnetic fields in these simulations are structured on small scales, resulting in currents that dissipate owing to the ambipolar mechanism, which produces heating of the plasma. The locations of such heating events form filamentary structures following expanding inclined magnetic fields in the chromosphere.

The magnetized plasma motions in the solar photosphere are able to create an electromagnetic Poynting energy flux sufficient to heat the upper solar atmosphere (Osterbrock, 1961), and a significant part of this flux may be produced in the form of Alfvén waves. An analysis by Shelyag, Khomenko, de Vicente, and Przybylski (2016) and Khomenko et al. (2018) has shown that up to 80% of the Poynting flux associated with these waves can be dissipated and converted into heat owing to the effect of ambipolar diffusion providing the chromosphere with an order of magnitude greater amount of energy than from the dissipation of stationary currents (Khomenko & Collados, 2012). Nevertheless, Arber, Brady, and Shelyag (2016) argue that heating produced by acoustic shocks is more important than that by Alfvén wave dissipation through ion–neutral collisions.

Figure 7. Three-dimensional view of magnetic field lines (colored lines), temperature (grayscale image at the bottom), and heating events (gray contours in the upper part of the domain) in 3D realistic simulations of the solar dynamo, taking into account partial ionization. Results from Khomenko et al. (2018).

7.6 Observations of Non-MHD Effects

A few attempts have been made to directly detect multi-fluid effects in the solar plasma through measurements of differences in ion and neutral velocities. By measuring the Doppler shift in ionic Fe ii and neutral Fe i lines, simultaneously over the same volume of plasma, Khomenko, Collados, Shchukina, and Díaz (2015) found that the Evershed outflow of neutral iron in sunspot penumbrae has a few m s–1 higher average velocities than that of ionized iron over the entire photosphere. Similarly, Khomenko, Collados, and Díaz (2016) have shown that non-negligible differences in He i and Ca ii velocities exist in solar prominences when observed with sufficiently high cadence. While the measured neutral and ion velocities were the same in most spatial locations most of the time, they were observed to decouple in the presence of strong spatial and temporal gradients, such as wave fronts. However, Anan, Ichimoto, and Hillier (2017) have concluded that the observed small differences in velocities of Hi 397 nm, Hi 434 nm, Ca ii 397 nm, and Ca ii 854 were indications of the motion of different components in the prominence along the line of sight, rather than indications of decoupling. Direct detection of decoupling effects would require higher spatial and temporal resolution, as well as a careful selection of spectral lines for the analysis.

Nonetheless, there exist other indirect indications of decoupling, such as those observed by Gilbert, Kilper, and Alexander (2007). These authors have compared He i1083 nm and H α data in multiple solar prominences in different phases of their life cycle, and were able to detect the drainage effects across a prominence magnetic field with different timescales for helium and hydrogen atoms. Such drainage was predicted earlier by Gilbert, Hansteen, and Holzer (2002) from simplified semi-analytical calculations. Other indirect evidence of ion–neutral decoupling was reported by de la Cruz Rodríguez and Socas-Navarro (2011), who deduced misalignment in the visible direction of chromospheric fibrils and the measured magnetic field vector. They noted that most of the fibrils are aligned with the magnetic field, although there are a few noteworthy cases where significant misalignments occur, well beyond the observational uncertainty. Misalignment of chromospheric fibrils and the magnetic field was confirmed by Martínez-Sykora, De Pontieu, Carlsson, and Hansteen (2016), who used advanced radiative magnetohydrodynamic simulations, including ambipolar diffusion. These authors showed that the magnetic field is indeed often not well aligned with chromospheric features. The misalignment occurs where the ambipolar diffusion is large, that is, the ions and neutral populations decouple as the ion–neutral collision frequency drops, thereby allowing the field to slip through the neutral population. The conditions for misalignment also require currents perpendicular to the field to be strong, and thermodynamic timescales to be longer than, or similar to, those of ambipolar diffusion.

8. Concluding Remarks

This article presents a brief overview of a wide field of multi-fluid effects in solar and astrophysical plasmas. The equations governing multi-fluid MHD are considered here, together with the most frequently used simplified approaches, such as single- or two-fluid MHD. This is followed by a discussion of the closure of the equations, collisional terms, and other factors affecting astrophysical plasmas, such as interaction with radiation. The generalized Ohm’s law is presented and different forms of it are discussed. A number of non-ideal terms from the generalized Ohm’s law lead to new physical effects in plasmas.

Multi-fluid effects play an important role in astrophysical and solar physics plasmas. In some fields, such as interstellar medium studies or ionospheric studies, the need to go beyond a more standard MHD theory arises from observational demands or from direct measurements. In others, as in the case of solar physics, the arguments is a theoretical one.

In interstellar medium plasmas, multi-fluid effects are involved in the theories of star formation in molecular clouds. They can either contribute directly to the gravitational collapse, or indirectly by allowing the removal of an excess of magnetic flux from forming protostellar discs. The dissipation of turbulence due to ion–neutral friction was suggested as a possible explanation of the observed width of filaments in molecular clouds. Interstellar shocks, produced by supernova explosions and other violent events, are influenced by multi-fluid effects, since they shape the structure of the shock transition.

In the ionosphere, the plasma properties can be directly measured by a number of satellites, and are quite well known. The presence of neutral winds, together with weak collisional coupling, and a strong and global magnetic field, leads to the appearance of electric fields and currents known as the ionospheric dynamo. Ion–neutral interactions lead to plasma anisotropy along and across the magnetic field, and ion–neutral drifts produce Joule heating, which play an important role in the energization of ionospheric plasma.

In solar physics, the study of multi-fluid effects is an actively developing field. While no direct observational confirmation of ion–neutral decoupling has been confidently measured so far, a number of theoretical studies demonstrate the importance of ion–neutral interaction effects on waves and instabilities, flux emergence, reconnection, and energetic events. Ambipolar diffusion is an important effect to consider also in chromospheric heating models.

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