Appendix A: Multicomponent MHD equations

  
A.1 Single fluid description

Let us consider a fluid composed of $\alpha$ species of numerical density $n_\alpha$, mass $m_\alpha$, charge $q_\alpha$ and velocity $\vec v_\alpha$. All species are assumed to be coupled enough so that they have the same temperature T. To get a single fluid description, we then define

$$\rho \Sigma_\alpha n_\alpha m_\alpha$$ =

$$\rho \vec v \Sigma_\alpha m_\alpha n_\alpha \vec v_\alpha$$ =

$$\Sigma_\alpha n_\alpha k_{{\rm B}} T$$ P = (A.1)

$$\vec J \Sigma_\alpha n_\alpha q_\alpha \vec v_\alpha$$ =

as being the flow density $\rho$, velocity $\vec{v}$, pressure P and current density $\vec J$. We consider now a fluid composed of three species, namely electrons (e), ions (i) and neutrals (n). The equations of motion for each species are

   

$$\rho_{\rm n} \frac{D\vec{v}\rm _n}{Dt} -\vec{\nabla}P_{\rm n} - \rho_{\rm n} \vec{\nabla} \Phi_{\rm G} + \vec{F}_{\rm en} + \vec{F}_{\rm in}$$ = (A.2)

$$\rho_{\rm i} \frac{D\vec{v}\rm _i}{Dt} -\vec{\nabla}P_{\rm i} - \rho_{\rm i} \vec{\nabla} \Phi_{\rm G} +... ...\rm i} n_{\rm i} \left(\vec{E} +\frac{\vec{v}_{\rm i}}{c} \times \vec{B}\right)$$ = (A.3)

$$\rho_{\rm e} \frac{D\vec{v}_{\rm e}}{Dt} -\vec{\nabla}P_{\rm e} - \rho_{\rm e} \vec{\nabla} \Phi_{\rm G} +... ...\rm e}\,n_{\rm e} \left(\vec{E} +\frac{\vec{v}_{\rm e}}{c}\times \vec{B}\right)$$ = (A.4)

where $\Phi_{\rm G}$ is the gravitational potential and the collisional force of particles $\alpha$ on particles $\beta$ is given by $\vec{F}_{\alpha \beta} = m_{\alpha\beta} n_{\alpha} \nu_{\alpha\beta} (\vec{v}_\alpha - \vec{v}_\beta)$, $m_{\alpha\beta}= m_{\alpha} m_{\beta}/(m_{\alpha}+m_{\beta})$ being the reduced mass, $\nu_{\alpha\beta} = n_\beta \langle\sigma_{\alpha\beta} v\rangle$ the collisional frequency and $\langle\sigma_{\alpha\beta} v\rangle$ the averaged momentum transfer rate coefficient.

A single fluid dynamical description of several species is relevant whenever they are efficiently collisionally coupled, namely if they fulfill $\Vert\vec{v}_{\alpha=\rm e,n,i}- \vec{v} \Vert \ll \Vert \vec{v}\Vert$. Under this assumption and using Newton's principle ( $\Sigma_{\alpha, \beta} \vec F_{\alpha \beta}= \vec 0$), we get the usual MHD momentum conservation equation for one fluid

 

$$\rho \frac{D\vec{v}}{Dt} = -\vec{\nabla}P - \rho \vec{\nabla} \Phi_{\rm G} + \frac{1}{c}\vec{J} \times \vec{B}$$ (A.5)

by adding all equations for each specie. The Lorentz force acting on the mean flow is

 

$$\frac{1}{c}\vec{J} \times \vec{B} = (1 + X) (\vec F_{\rm in} + \vec F_{\rm en}) \ ,$$ (A.6)

where $X = \rho\rm _i/\rho\rm _n$. Even if the bulk of the flow is neutral, collisions with charged particles give rise to magnetic effects. In turn, the magnetic field is coupled to the flow by the currents generated there. This feedback is provided by the induction equation, which requires the knowledge of the local electric field $\vec E$. Its expression is obtained from the electrons momentum equation

$$\vec{E} + \frac{\vec{v}_{\rm e}}{c} \times \vec{B} \frac{1}{en_{\rm e}}\big(m_{\rm ie} n_{\rm i}\nu_{\rm ie} \vec{v}... ...nu_{\rm ne} \vec{v}_{\rm ne} \big) - \frac{\vec{\nabla} P_{\rm e}}{e n_{\rm e}}$$ = (A.7)

where $\vec{v}_{\alpha \beta} \equiv \vec v_\alpha - \vec v_\beta$ is the drift velocity between the two species. Due to their negligible contribution to the mass of the bulk flow, all terms involving the electrons inertia have been neglected (electrons quite instantly adjust themselves to the other forces).

All drift velocities can be easily obtained. The electron-ion drift velocity is directly provided by $\vec{v}_{\rm ie} = \vec J/e n_{\rm e}$. Using Eq. (A.6) and noting that $\rho_{\rm e} \nu_{\rm en} \ll \rho_{\rm i} \nu_{\rm in}$ we get the ion-neutral drift velocity

$$\vec{v}_{\rm in}=\frac{\frac{1}{c}\vec{J}\times\vec{B}}{(1+X) m_{... ..._{\rm en}}{m_{\rm in} n_{\rm i} \nu_{\rm in}} \frac{\vec{J}}{ e n_{\rm e}}\cdot$$ (A.8)

On the same line of thought, the electrons velocity is $\vec{v}_{\rm e} = \vec{v} -(\vec{v}-\vec{v}_{\rm e})$ where

$$\vec{v} - \vec{v}_{\rm e} \frac{\vec{v}_{\rm n} - \vec{v}_{\rm e}}{1+X} + \frac{X}{1+X}(\vec{v}_{\rm i} - \vec{v}_{\rm e})$$ = (A.9)

$$\simeq \frac{\vec J}{e n_{\rm e}} - \frac{1}{(1+X)^2} \frac{\frac{1}{c} \vec{J}\times\vec{B}}{m_{\rm in} n_{\rm i}\nu_{\rm in}}\cdot$$ (A.10)

Gathering these expressions for all drift velocities, we obtain the generalized Ohm's law

 

$$\vec{E} + \frac{\vec{v} }{c} \times \vec{B} \eta \vec{J} + \frac{\frac{1}{c}\vec{J}\times\vec{B}}{e n_{\rm e}} - \frac{\vec{\nabla} P_{\rm e}}{en_{\rm e}} - \frac{1}{(1+X)^2} \frac{\frac{1}{c^2}(\vec{J}\times\vec{B})\times\vec{B}}{m_{\rm in} n_{\rm i} \nu_{\rm in}}$$ (A.11)

where $\eta=(m_{\rm ne}n_{\rm n} \nu_{\rm ne} + m_{\rm ie} n_{\rm i}\nu_{\rm ie})/(en_{\rm e})^2$is the electrical resistivity due to collisions. The corresponding MHD heating rate writes

$$\Gamma_{\rm MHD}= \vec{J}\cdot\vec{E'} = \vec{J} \cdot \left (\vec{E} + \frac{\vec{v}}{c} \times \vec{B} \right )$$ (A.12)

where $\vec{E'}$ is the electrical field in the comoving frame. This expression leads to Eq. (17).

The generalization of this derivation for a mixture of several chemical elements has been done in a quite straightforward way. The bulk flow density becomes $\rho= \overline{\rho_{\rm i}} + \overline{\rho_{\rm n}} + \rho_{\rm e}$, where the overline stands for a sum over all elements (ions and neutrals), with $X=\overline{\rho_{\rm i}}/\overline{\rho_{\rm n}}$. The neutrals and ions velocities are means over all elements, $\langle \vec{v}_{\rm n,i}\rangle \equiv \sum_{\rm n,i} \rho_{\rm n,i} \vec{v}_{\rm n,i}/\overline{\rho_{\rm n,i}}$. The conductivity and collision terms are also sums over all elements, namely $\overline{\eta}=(\overline{m_{\rm ne}n_{\rm n}\nu_{\rm ne}} + \overline{m_{\rm ie} n_{\rm i}\nu_{\rm ie}})/(en_{\rm e})^2$ and $\overline{m_{\rm in} n_{\rm i} \nu_{\rm in}}$, and are computed using the expressions for the collision frequencies.

  
A.2 Momentum transfer rate coefficient

For ion-electron collisions we use the canonical from Schunk (1975), summed over all species:

$$\overline{m_{\rm ie} n_{\rm i} \nu_{\rm ie}}=m_{\rm e} n_{\rm e} ... ...B}} T_{\rm e}}{\pi m_{\rm e}}}\sum_i n_{\rm i} Z_{\rm i}^2 \ln{\Lambda_{\rm i}}$$ (A.13)

with the Coulomb factor $\Lambda_{\rm i}=(3/2 Z_{\rm i} e^3)\sqrt{(k_{{\rm B}}T_{\rm e})^3/ \pi n_{\rm e}}$.

For the collisions between electrons and neutrals we use the expression of Osterbrock (1961) for the collisional momentum transfer rate coefficient between a neutral and a charged particle, which corrects the classical one (e.g., Schunk 1975) for strong repulsive forces at close distances. Its expression is $\langle\sigma v\rangle_{\rm n,i-e} = 2.41 \pi e \sqrt{\alpha_{\rm n}/m_{\rm n,i-e}}$, where the polarizabilities $\alpha_{\rm n}$ used are also taken from Osterbrock. We thus obtain

$$\overline{m_{\rm en} n_{\rm n} \nu_{\rm ne}} = 2.41 \pi\, e \,n_{\rm e} \sum_{n={\rm H,He}} n_{\rm n} \sqrt{m_{\rm e} \alpha_{\rm n}}\,.$$ (A.14)

Finally, it is mainly the ion-neutral collision momentum transfer rate coefficient determines the ambipolar diffusion heating. It can be computed with the previous momentum transfer rate coefficient expression. However as noted by Draine (1980) the previous expression underestimates $\sigma$ at high velocities. Thus, as Draine, we take the "hard sphere'' value for the cross-section ( $\sigma_{\rm S} \simeq 10^{-15}$ cm²) whenever it is superior to the polarizability one. For intermediate to hight ionizations ( $f_{\mbox{\tiny H$^+$ }}\gtrsim10^{-4}$) the dominant ion-neutral collisions are those between H-H⁺. Charge exchange effects between these two species will amplify $\langle\sigma_{\mbox{\tiny H H$^+$ }}v\rangle$above the values expected by polarizability alone and thus it is computed separately (Eqs. (A.16) and (A.17)). We thus obtain for ion-neutral collisions

$$\overline{m_{\rm in} n_{\rm i} \nu_{\rm in}} = \frac{1}{2} m_... ..._{\rm S}\,\tilde{v}; \langle\sigma v\rangle_{{\rm He},\rm i})}$$ (A.15)

where $\tilde{v}=\sqrt{8k_{{\rm B}}T/\pi m_{\rm in} + v_{\rm in}^2}$. For $\tilde{v}<1000$ kms^-1.

The value of $\langle\sigma_{\mbox{\tiny H H$^+$ }}v\rangle$ which we used is given by Draine (1980),

 

$$\frac{\langle\sigma_{ \mbox{{\tiny H\,H$^+$ }}}v\rangle}{1 \,{\rm... ...10^{-9}\,\tilde{v}^{0.73}&\tilde{v}\geq 2 \:\mbox{km s}^{-1} \end{array}\right.$$ (A.16)

Safier (1993a) used the expression $\tilde{v}=v_{\rm in}$ which, as discussed in Sect. 4.8, results in a smaller momentum transfer rate coefficient. Geiss & Buergi (1986) computed another expression of the H-H⁺ momentum transfer rate coefficient, which provides

$$\langle\sigma_{ \mbox{{\tiny H\,H$^+$ }}}v\rangle\ = 1.12\ti... ...1+2\log{T_4})^2\big)\times10^{-9}\:\: {\rm cm}^3~{\rm s}^{-1}.$$ (A.17)

In Fig. 8 we compared both momentum transfer rate coefficients, they typically differ in 40%, which can be used as an estimate of their accuracy. It is thus the uncertainty in the H-H⁺momentum transfer rate coefficient that dominates the final intrinsic uncertainty of our calculations.