2 Wind acceleration

Let us now address the question of the wind acceleration mechanism. In order to do so we write the energy equation for the species $j$ for the case of a steady state and purely radial wind. Indeed, the second moment of Boltzmann's equation (e.g., Endeve & Leer, 2001) leads to the following expression

$$E_j={1\over 2} m_j v_j^2 + h_j(r)+\Psi_j + {{q_j}\over{n_j v_j}}$$ (6)

where $q_j$, $n_j$, $v_j$ are the heat flux, density and bulk velocity of the corresponding specie and where we have defined the enthalpy per particle of the species $j$

$$h_j(r)\equiv {3\over 2}k_{\rm B} T_{j\parallel}+k_{\rm B} T_{j\perp}.$$ (7)

We note in passing that the first moment of Boltzmann's equation leads to Jockers's Eq. (1.1) (Jockers, 1970)

$$m_j v_j {{\partial v_j}\over{\partial r}}= - {1\over n}{{\p... ...parallel}-T_{j\perp}) - {{\partial \Psi_j}\over{\partial r}}.$$ (8)

When applied to the electrons one may neglect the small terms proportional to the electron mass $m_{\rm e}$ in Eq. (8) which then reduces to the usual expression for the electric field $\mathcal{E}$

$$e\mathcal{E} = -\frac{1}{n_{\rm e}}\frac{\partial}{\partial ... ... - \frac{2}{r}k_{\rm B}(T_{{\rm e}\parallel}-T_{{\rm e}\perp})$$ (9)

But let us come back to Eq. (6). In all simulations $E_{\rm e}$ and $E_{\rm p}$ are separately approximately constant over the whole simulation domain (excepted for a small region near the $r=r_0$ boundary). This is shown in Fig. 2 for the $N=6400$ case.

Figure 2: Proton and electron energy profiles obtained by evaluating Eq. (6) for the $N=6400$ simulation. Note how both, $E_{\rm e}$ and $E_{\rm p}$ are separately constant over most of the simulation domain. The electrostatic profile $-e\phi$ is plotted as a reference.

At first, this seems to indicate that the net energy exchanges between protons and electrons are quantitatively small. This is not necessarily correct. Indeed, it appears that the system does merely organize itself in order to ensure a spatially constant proton to electron energy density ratio throughout the system. The ratio can be constant even if interspecies energy exchanges via collisions are strong. A simple example of such a system consists of a collisional proton-electron plasma under the effect a constant gravitational acceleration field $g$. In this case the temperatures of both, electrons and protons, must be equal, isotropic and spatially constant. Further, the heat flux must vanish and the charge neutralizing electric field is just the Pannekoek-Rosseland field $g(m_{\rm p}-m_{\rm e})/(2e)$ so that $E_{\rm p}=E_{\rm e} = (5/2)k_{\rm B} T + g(m_{\rm p}+m_{\rm e})z/2$, i.e. $E_{\rm p}/E_{\rm e}=1$, independently of the height $z$. On the other hand, as we shall see below, in the spherically symmetric case the heat flux (mainly conveyed by the electrons) is the dominant source of energy for the acceleration of the wind. In order to proof this affirmation it is useful to evaluate the mean energy per particle $\langle E\rangle$ as a function of $r$. Averaging the contribution of electrons and protons according to Eq. 6 leads to

$$\langle E\rangle \approx {1\over 2}\left[ {1\over 2} m_{\rm... ...{r_0}}\left(1- {{r_0}\over r}\right) + {{q}\over{nv}}\right]$$ (10)

where the enthalpy term $h(r)$ includes the temperature terms from all species (electrons and protons). In order to obtain Eq. (10) we made use of the fact that ${m_{\rm p}} \gg{m_{\rm e}}$ and that the proton and electron number fluxes are equal, i.e. $n_{\rm p} v_{\rm p}=n_{\rm e} v_{\rm e}\equiv nv$. As a reference, the number flux $nv$ has been found to be of the order $10^{-2} v_{\rm e 0} n_0$ in all runs. In particular for the $N=6400$ case we find $nv=1.2\, 10^{-2}\, n_0 v_{\rm e 0}$. The energy flux $F_E$ conveyed by the wind through the spherical shell located at a distance $r$ is the product of the particles mean energy at that distance (after deduction of their gravitational energy) times the number of particles crossing the shell per time unit. Since the total number density is equal to twice the number density of either species the particle flux is a constant given by $8\pi r^2 nv$ and the energy flux becomes

$$F_E(r) =8\pi r^2 nv\left[\langle E\rangle - {{GM m_{\rm p}}\over{2r_0}}\left(1- {{r_0}\over r}\right) \right]$$ (11)

Figure 3: Relative importance of each term in Eq. (10) for the most dense case $N=6400$ (top panel) and the most tenuous case $N=400$ (lower panel). The dense case supports a transonic wind (the vertical line in the top panel gives the position of the sonic point $r_*$) while the tenuous case does not. From the comparison of the two figures it appears that the wind acceleration is primarily driven by the heat flux term $q/(nv)$.

Fig. 3 illustrates the relative importance of each of the 4 terms in Eq. (10) as a function of the radial distance $r$ for the $N=6400$ (top) and the $N=400$ (bottom) run. Since the gravity term vanishes at $r=r_0$ and the bulk velocity is much smaller than the sound speed, only the enthalpy and the heat flux term contribute to the total energy there. The reason for the total energy being slightly larger in the low density case arises from the heat flux term $q/(nv)$ being stronger in that case. This is because at $r=r_0$ we do not have control over this term as we do for the enthalpy, which only depends on the temperature imposed at the boundary. Interestingly the run characterized by the largest heat flux term $q/(nv)$ (which does not necessarily mean that the heat flux $q$, or the specific heat flux $q/n$ are largest) is precisely the one where the wind remains subsonic. This is particularly surprising in the light of the fact that the velocity of the wind is clearly boosted by the heat flux term given that the enthalpy profile is seemingly identical for the two cases. However, a strong heat flux term near the bottom does not guarantee that the wind will be accelerated to supersonic velocities. A sufficient amount of collisions is needed to efficiently transform the energy transported outward by the electron heat flux into bulk plasma kinetic energy.

Figure 4: Radial and perpendicular temperature profiles for 4 different values of the $N$. Note the log-log axis of the top two panels.

Fig. 4 shows that the radial electron temperature profile $T_{\rm e\parallel}$ is essentially insensitive to the density while $T_{\rm e\perp}$ is not. Indeed, in the collisionless limit the parallel and perpendicular temperatures are independent of each other and one should observe $T_{\rm e\perp} \propto r^{-2}$ in order to satisfy to the angular momentum conservation law of individual particles (cf. Eq. (2)). As a consequence, in the rigorously collisionless case, $T_{\rm e\perp}$ should decrease by a factor $50^2$ from bottom to top of the simulation domain. Collisions do significantly contribute in limiting this bottom to top perpendicular electron temperature gradient which ranges from 10 to 30 depending on the value of $N$. On the other hand, for all four cases the parallel temperature only drops by a factor 3 from bottom to top, leading to strong temperature anisotropies $T_{\rm e\parallel}/T_{\rm e\perp}$ (bottom panel in Fig. 4). This is not particularly surprising as in the collisionless limit the parallel temperature of a plasma plunged in a potential field is constant as long as the velocity distribution function is close to Maxwellian.

We can now summarize the wind acceleration scenario from a kinetic point of view. The natural decrease of the temperature with distance (essentially due to the fact that in the collisionless limit $T_\perp \propto r^{-2}$) implies the existence of a radial heat flux predominantly conveyed by the electrons. The heat flux is transfered from the electrons to the protons which become accelerated in the outward direction. Since the momentum of the wind is mainly carried by the heavy protons (rather than by the light electrons) the plasma as a whole becomes accelerated in this way. This mechanism must be particularly efficient in the region located inside the spherical shell $r=r_\Psi$ (location of the maximum of $\Psi_{\rm p}$) where the protons have to climb uphill in order to escape from the potential energy well (cf. Fig. 1). As already stated above, collisions contribute in increasing the electric field strength. Since the electric field is directed outward, increasing the electric field favors the extraction of the protons from the gravitational well by reducing the height of the maximum in the proton potential $\Psi_{\rm p}$. The fluid estimate of the macroscopic electric field $\mathcal{E}$ for a spherically symmetric electro-proton plasma can be obtained by differentiating Eq. (6) for the electrons. Neglecting the small terms proportional to $m_{\rm e}$ and taking advantage of the fact that $E_e$ is approximately constant (in particular if one compares it to the $\phi(r)$ profile shown in Fig. 2) we then obtain

$$e\mathcal{E} \approx -k_{\rm B}{{\partial}\over{\partial r... ...) -{{\partial}\over{\partial r}}\left({{q}\over{nv}}\right).$$ (12)

This estimate is not as rigorous as the standard estimate based on Eq. (8) since it is based on the assumption of a constant $E_e(r)$ profile. The equation has the advantage of highlighting the role of the heat flux in the shaping of the electric field profile. For radially decreasing temperature profiles the first two terms on the right hand side of Eq. (12) are positive and favor the outward acceleration of the protons. They are reminiscent of the thermoelectric effect (e.g., Pantellini & Landi, 2001). Given that $q/nv$ has been seen to decrease with distance for the two extreme cases shown on Fig. 3 it follows that the third term on the right hand side of Eq. (12) is also positive for all simulation. However, the contribution of the latter to the acceleration is significantly stronger in the $N=6400$ case than in the $N=400$ case, where the radial dependence of $q/nv$ is seemingly weak.

It is instructive to apply Eq. (12) to a a weakly collisional system. In such a case the parallel temperature $T_{e\parallel}$ is roughly constant and the perpendicular temperature $T_{e\perp}\propto r^{-2}$. This leads to the collisionless approximation

$$e\mathcal{E} \propto {{\rm constant}\over{r^3}} - {{\partial}\over{\partial r}}\left({{q}\over{nv}}\right)$$ (13)

where the constant is positive. From Eq. (13) it appears that when the heat flux term is constant, or only weakly spatially dependent, such that the first term on the right hand side of the equation dominates over the second term, the electric field decreases faster than the gravitational field (i.e. $\mathcal{E} \propto r^{-3}$). This is the reason for the proton potential energy $\Psi_{\rm p}$ to be a monotonically growing function of distance $r$ in the $N=400$ case shown on Fig.1. Increasing the number of particles in the system makes the plasma more collisional and forces the perpendicular temperature $T_{e\perp}$ to fall off more slowly than $r^{-2}$ (see Fig. 4) making it harder for the gravitational force acting on a proton to overcome the electric force. Eventually, if $T_{e\perp}$ decreases more slowly than $r^{-1}$, there must be a minimum distance beyond which the electric force on a proton overcomes the gravitational force and $\Psi_{\rm p}$ has a maximum. This is clearly the case for the $N=6400$ case shown in Fig. 4 where $T_{e\perp}\propto r^{-0.6}$. For the $N=400$ case the temperature profile is steeper, with an average radial dependence given by $T_{e\perp}\propto r^{-0.9}$. Thus, even though all profiles can be described by a power law which decreases more slowly than $r^{-1}$ (on average over the simulation domain) all profiles tend to steepen at large distances because of the plasma tendency to become less collisional. Eventually beyond some $N$-dependent threshold distance the $T_{e\perp}$ profiles becomes steeper than $r^{-1}$ so that the formation of a maximum of $\Psi_{\rm p}$ becomes impossible beyond this point. This is the case for the $N=400$ run. In the other three runs a maximum forms below the point where the $T_{e\perp}$ profile becomes steeper than $r^{-1}$. We can now reexamine Fig. 3 in the light of Eq. (12). Fig. 3 already told us that the enthalpy profile is not very sensitive on the plasma collisionality even though it contributes significantly in strengthening the electric field according to Eq. (12). The determinant contribution in accelerating the wind to supersonic velocities comes from the heat flux term $q/(nv)$ which has been seen to be much more sensitive on the plasma collisionality. As demonstrated by Eq. (12), the heat flux term contributes to the strengthening of the outward directed electric field, provided it decrease with distance. Fig. 3 shows that the heat flux term decreases for both simulations represented on the figure with the steepest profile being associated with the high density case which therefore produces the strongest electric field, according to the last term on the right-hand side of Eq. (12).

Figure 5: Parallel electron velocity distribution functions (solid lines) in arbitrary units at two different positions in the system, near the base and in the supersonic region. Shown are results of the $N=6400$ simulation. The dashed lines represent Maxwellian distributions for which the first three moments (density, mean velocity and temperature) are those of the measured distributions. Velocities are normalized to the electron thermal velocity $v_{\rm e0}=v_{\rm e}(r_0)$.

For the $N=6400$ simulation the parallel electron velocity distribution functions at the base of the system at $r=r_0$, and in the supersonic region at $r=45\,r_0$ are shown in Fig. 5. Since the collisional mean free path $\lambda$ is proportional to $v^3$ high energy electrons are nearly unaffected by collisions on their journey through the system. Thus, the velocity distribution of the high energy electrons flowing downward is the imprint of the upper boundary condition whereas the velocity distribution of the high energy electrons flowing upward is the imprint of the lower boundary at $r=r_0$. On the other hand, the low energy electrons, which populate the core of the velocity distribution function, are strongly affected by collisions. As a consequence, at low velocities, the electron velocity distributions are approximately isotropic Maxwellians which do not carry any heat flux. Instead, the heat flux is carried by the high energy electrons which are responsible for the asymmetry of the distribution function. As illustrated by the profiles on Fig. 5 the heat flux is due to a deficiency of downward flowing high energy electrons near the lower boundary and to an excess of upward flowing particles in the supersonic region.