3 Comments on the electron heat flux

Figure: Electron heat flux profile (solid line) and thermal Knudsen number $K_T\equiv \overline\lambda_{\rm ep} \vert \partial \ln T_{\rm e\parallel}(r)/\partial r\vert$ (dashed line) for the most strongly collisional case $N=6400$ normalized to $q_0\equiv n_0 m_{\rm e} v_{\rm e0}^3$, where $v_{\rm e0}$ is the electron thermal velocity at the base of the system at $r=r_0$. The triangle on the heat flux axis indicates the heat flux expected near the base of the system using the Spitzer-Härm formula (Spitzer & Härm, 1953). The dot-dash line shows the $-T_{\rm e\parallel}^{5/2} \partial T_{\rm e\parallel}/\partial r$ law normalized to the measured flux at $r=r_0$.

Fig. 6 shows the heat flux and the thermal Knudsen number $K_T$ measured in the $N=6400$ run. One observes that while it remains approximately constant in the supersonic region above the $r\approx10 r_0$ level, $K_T$ grows steeply in the subsonic region, where it increases from $10^{-3}$ to $10^{-2}$. Since $K_T \lesssim 2~10^{-2}$ in whole simulation domain, it is not particularly surprising that the heat flux closely follows a $T_{\rm e\parallel}^{5/2} \partial T_{\rm e\parallel}/\partial r$ dependence (dot-dash curve) as in the case of the classical Spitzer-Härm heat conduction formula (Spitzer & Härm, 1953). This conclusion is misleading, since, despite the smallness of the Knudsen number, the heat flux is strongly non classical. As already pointed out by several authors in the past the classical heat conduction formulation breaks down either because the heat flux intensity exceeds a value of the order $10^{-2}q_0$ (Gray & Kilkenny, 1980), or because the Knudsen number is larger than $10^{-3}$ (Shoub, 1983), or because the flow velocity is a significant fraction of the sound speed (Hollweg, 1974; Alexander, 1993), or even because the electric field in the system is of the order of the Dreicer field $\mathcal{E}_{D}\equiv kT_{\rm e} /(e\overline\lambda_{\rm ep})$ (Scudder, 1996b). Indeed, in the $N=6400$ simulation the electric field at the sonic point is $\mathcal{E}\approx \mathcal{E}_{D}$, reaching an intensity $\mathcal{E} \approx 8 \mathcal{E}_{D}$ in the $N=1600$ case. Concerning the heat flux intensity, Fig.6 shows that it is small enough for the the low heat flux intensity condition established by Gray & Kilkenny (1980) to be satisfied. One can therefore conclude that the heat flux intensity is low enough for the plasma to be capable to support a Spitzer-Härm flux. Let us now address the problem of the heat flux in a flowing, and weakly collisional plasma. As discussed by Hollweg (1974) and Alexander (1993) a non negligible fraction of the of the energy is carried by a collisionless term of the form $q_{\rm NC}=(3/2)\alpha nv k_{\rm B}T_{\rm e}$ where $\alpha$ is a positive numerical factor of order unity (note that the electron temperature has been supposed to be isotropic by these authors). Given that collisions are still relatively important in our simulations we make the ansatz that the observed electron heat flux is made of the sum of a classical (collisional) term (e.g., Braginskii, 1965) $q_{\rm SH}$ and a collisionless term $q_{\rm NC}$

$$q_{\rm e} = q_{\rm SH} + q_{\rm NC}$$

$$= -3.16~{ {n k_{\rm B}^2 T_{\rm e}}\over { m_{\rm e}{\overline\nu_{... ...{\partial T_{\rm e}}\over{\partial r}} +{3\over 2}~\alpha~nv k_{\rm B}T_{\rm e}$$ (14)

where $\alpha$ is a positive constant of order unity whose numerical value depends on the assumptions of the specific heat flux model (Hollweg, 1974; Alexander, 1993). As a guide, Hollweg's estimate of $\alpha$ for the solar wind are in the range 2.0 to 7.0 (Hollweg, 1974). Eq. (14) shows that the two heat conduction terms have an extremely different dependence on the macroscopic moments of the plasma. The Spitzer-Härm heat conduction does only depend on the temperature, and its radial variation, while the collisionless term $q_{\rm NC}$ depends on both the electron number flux and the temperature (but not on the temperature gradient). This situation is reminiscent of the heat conduction in a plasma confined to the space between two parallel plates separated by a distance $L$ at temperatures $T_0$ and $T_{\rm L}$, respectively (Landi & Pantellini, 2001). If the plasma is dominated by collisions the heat conduction between the two plates just equal to $q_{\rm SH}$. However, if the plasma is sufficiently diluted for a non negligible number of electrons to be able to proceed from one plate to the other without colliding with other particles in the system, the heat flux is best described by the collisionless approximation $q_{\rm NC} \propto n(T_0 T_{\rm L}^{1/2}-T_{\rm L}T_0^{1/2})$ which (unlike $q_{\rm SH}$) is a function of the number density $n$.

Figure 7: Electron heat flux calculated using the collisionless approximation $q_{\rm NC}=(3/2)\alpha nv T_{\rm e\parallel}$ (top panel), the classical Spitzer-Härm approximation $q_{\rm SH}= -{\rm constant}\times T_{\rm e\parallel}^{5/2}{{\partial T_{\rm e\parallel}}/{\partial r}}$ (middle panel). The lower panel shows the heat flux profiles effectively measured in the simulations. Fluxes are in arbitrary units, but the same normalization has been used for all simulations.

Fig. 7 compares the Spitzer-Härm estimate and the collisionless estimates of the electron heat flux with the observed heat flux for the four simulations. All profiles in the figure have been obtained using $T_{\rm e\parallel}$ in place of the temperature $T_{\rm e}$ which appears in Eq. (14). Even though Hollweg's collisionless approximation is not expected to provide an accurate approximation of the heat flux in the simulated systems, it appears that the measured heat flux varies significantly from one simulation to the other, in good qualitative agreement with the non collisional flux $q_{\rm NC}$ obtained using Alexander's model (Alexander, 1993) to compute $\alpha$ in Eq. (14) after replacing $T_{\rm e}$ by $T_{\rm e\parallel}$. The Spitzer-Härm prediction of an equal heat flux intensity for all four simulations (based on the fact that the radial profiles of $T_{\rm e\parallel}$ are very similar cf. Fig. 4) is completely at odds with the measured intensities. But why is this so, despite the smallness of the Knudsen number? The answer is hidden in Eq. (14). Indeed, from Eq. (14), after replacing $T_{\rm e}$ by $T_{\rm e\parallel}$, it follows that the ratio of the two contributions to the total heat flux is given by

>From Eq. (15) it follows that the condition for the heat flux in the system to be dominated by the classical term $q_{\rm SH}$ one must have $K_T \gg \alpha v/v_{\rm e\parallel}$. For example, at the sonic point one has $v/v_{\rm e\parallel}=(m_{\rm e}/m_{\rm p})^{1/2}=1/20$ and $K_T\approx 10^{-2}$ from where one can estimate $q_{\rm NC}/q_{\rm SH}\approx 5\alpha$, which is substantially larger that unity for any reasonable value of $\alpha$. Thus, for the heat flux to be of the Spitzer-Härm type in the vicinity the sonic point requires the thermal Knudsen number to be larger than $(m_{\rm e}/m_{\rm p})^{1/2}$. The simulations suggest that this is not easily achievable because the driving of the wind to supersonic velocities does precisely requires the plasma to be sufficiently collisional at the sonic point. As already stated, the way around this restriction may be the presence of an additional scattering mechanism (e.g. waves) in the plasma. However, in that case the Spitzer-Härm formulation of the heat transport would not be the relevant one anyway. As already pointed out in the introduction recent multi-moment simulations of the solar wind yield a close to classical electron heat flux (e.g. Li, 1999; Lie-Svendsen et al., 2001; Olsen & Leer, 1999). The discrepancy may be due to the fact that physical conditions of the wind we simulate are quite different from those used in these multi-moment simulations or, eventually, to the fact that the heat flow equations in the multi-moment models are affected by the closure scheme. The simplified version of the Coulomb collision operator used in our model or even the one-dimensionality of the model may also contribute to the observed discrepancy. The reason for the radial dependence of the heat flux measured in the simulation (cf Fig. 6) to be roughly of the Spitzer-Härm type stems from the fact that the radial dependences of both terms in Eq. (14) are quite similar for the given temperature profiles. Indeed, if we replace $T_{\rm e}$ by $T_{\rm e\parallel}$ in Eq. (14) and use the fair approximation $T_{\rm e}\propto r^{-0.4}$ (from Fig. 4) it follows that both $q_{\rm SH}$ and $q_{\rm NC}$ vary approximately as $r^{-2.4}$.

Figure 8: Dependence of the Mach number and the proton potential energy on the proton to electron mass ratio for the simulation with $N=6400$. Even though the qualitative behavior is similar it appears that a high mass ratio is in favor of a stronger acceleration of the wind. The definitions for the Mach number and the normalizing energy $\Psi _0$ are the same as in Fig. 1.