CAN PLASMA EXPANSION EXPLAIN THE OBSERVED ACCELERATION OF Ne⁷⁺ IONS IN A CORONAL MAGNETIC FUNNEL?

Tu et al. (2005) reported that the solar wind originates in coronal funnels, in which ions are accelerated between heights from 5 Mm to 20 Mm above the solar photosphere. There are several modeling efforts attempting to explain the origin and acceleration of the solar wind. We refer to a recent paper by Cranmer et al. (2010) for a discussion on the wave turbulence-driven and magnetic loop reconnection driven models. There is another type of model dealing with the collisionless exospheric plasma expansion from the hot coronal plasma at large heights, and not from the chromosphere (Pierrard et al. 2004; Marsch 2006). Boswell et al. (2006) suggested that the accelerations of the ions in the coronal funnels could be caused by current- free double layers (CFDLs), as seen in laboratory experiments in Helicon Plasma Device (HPD; Charles et al. 2007). The CFDLs form in HPD due to plasma expansion in abruptly diverging magnetic fields. Since the scale length of the diverging magnetic field (B), L_B, in HPD is microscopic and comparable to the ion Larmor radius ρ_i (∼10 cm), the HPD mechanism could not be operative in the funnels, in which ρ_i is a few millimeters and L_B is millions of meters (Singh 2011).

The gravitational potential energy of an Ne⁷⁺ ion at the height h = 20 Mm is W_g = M_Ne g_sh = 1.754×10⁻¹⁶ J, where M_Ne = 3.2×10⁻²⁶ kg, and g_s = 274 m s⁻². Overcoming the gravity requires an accelerating electrical potential φ_g = 1.754×10⁻¹⁷/(7 ×1.6 ×10⁻¹⁹) = 157 V. Thus a mechanism for accelerating the Ne⁷⁺ ions detected in Tu et al. (2005) must not only impart a kinetic energy (10 eV) corresponding to the measured drift V_Ne ≈ 10 km s⁻¹, but it must overcome the gravitational pull also, requiring total electrical potential drop φ_tot = 157 + 10/7 ≈ 158.4 V.

The large-scale plasma expansion (LASPE) involving H⁺ as major ions can generate such a potential drop provided the bottom of the funnels in the chromosphe is populated by hot electrons. Funnels are fed with plasma when the magnetic field lines of the magnetic loops reconnect with the open field lines of the funnel (Tu et al. 2005). Such magnetic loops might already contain the hot electrons before the onset of reconnection (Fisk 2003; Gloeckler et al. 2003). Furthermore, kinetic simulations show that the acceleration/heating of electrons naturally occurs in the reconnection process (Singh et al. 2010; Drake et al. 2003; Hoshino et al. 2001), developing an energetic tail in the electron velocity distribution function (EVDF). This generates two electron populations, a cold Maxwellian bulk and a hot tail. When plasmas having such two populations of electrons expand, the ambipolar potential is localized in a DL as established theoretically (Bezzerides et al. 1978) and in lab experiments (Hairapetian & Stenzel 1988, 1991). We discuss here the LASPE in a coronal funnels having the scenarios of both bulk and tail heated electrons. We first highlight the features of LASPE in a diverging flux tube using prior studies on the acceleration of both light and heavy ions in the terrestrial polar wind (Barakat & Schunk 1983, 1984) followed by a discussion on coronal funnels.

During the 1980s it emerged that LASPE with a hot/warm electron population could be an effective mechanism for the ion acceleration in the terrestrial polar wind (Singh & Schunk 1982; Barakat & Schunk 1983, 1984). Barakat & Schunk (1983) modeled the outflow of H⁺ and O⁺ ions from an ionospheric boundary at a geocentric distance r = r_o = 10.8 Mm = 1.7R_e (R_e as Earth's radius) to r = 10R_e = 63 Mm by increasing the temperature of the bulk ionospheric electrons. They demonstrated that when the nominal ionospheric electron temperature T_e = 0.3 eV is enhanced to T_e ∼ 1 eV, heavy oxygen ions overcome Earth's gravity and they outflow with a large flux. The Vlasov equations for the ion velocity distribution functions (IVDFs) under the influence of the ambipolar electric fields, mirror force on the ions in the diverging B(r), and gravitational force were solved subject to the prescribed boundary condition on the IVDFs at r = r_o. The magnetic field varied with the geocentric distance r as B(r) = B₀ (r_o/r)³, with B_o being the magnetic field at r = r_o. The ambipolar electric field E was calculated assuming that the plasma is quasi-neutral and that the electrons obey the Boltzmann law, n = n_o exp(eϕ/KT_e) with a constant temperature T_e. Accordingly, the electric field is given by

$$\begin{equation} E = - (KT_{\rm e} /e)\;\partial ({\rm In}(n/n_{\rm o}))/\partial r, \end{equation} \tag{ 1 }$$

where n_o is the plasma density at the ionospheric boundary r = r_o, where the electrostatic potential φ is zero and K is the Boltzmann constant.

Figures 1(a) and (b) are reproduced from Barakat & Schunk (1983). The parameters used here are T_e = 3000 K = 0.28 eV (dashed curves) and T_e = 10,000 K = 0.94 eV (solid curves); densities n_o(H⁺) = 100 cm^{− 3}, n_o(O⁺) = 50 cm^{− 3}, T_o(H⁺) = T_o(O⁺) = 0.28 eV, and velocities U_o(H⁺) = 11 km s⁻¹, and U_o(O⁺) = 0. The electron density at the base is n_o = n_o(H⁺) + n_o(O⁺). The subscript "o" refers to the quantities at the bottom of the flux tube at r = r_o as specified boundary conditions. Figure 1(a) on the left shows the altitude profile of the ion densities n(H⁺) and n(O⁺) while Figure 1(b) on the right shows their drift velocities U(H⁺) and U(O⁺) for T_e = 0.28 eV and T_e = 0.94 eV. The effect of the increase in T_e on the outflow of the light H⁺ ions is not so dramatic, but it is dramatic for the outflow of the heavy O⁺ ions. For the H⁺ ions outflow velocity is slightly enhanced while the density is slightly reduced with the increase in T_e as expected for the gravitationally unbound light H⁺ ions. The dramatic effects on the heavy O⁺ ions are as follows. The outflow velocity increased from 700 m s⁻¹ for T_e = 0.28 eV to 4 km s⁻¹ for T_e = 0.94 eV at the top boundary. The density of O⁺ ions at altitudes r > 8 R_e reduces to n(O⁺) < 10⁻⁵ cm⁻³ for T_e = 0.28 eV, but with the increased T_e = 0.94 eV, n(O⁺) is increased by several orders of magnitude all over the modeled flux tube reaching n(O+) ∼ 0. 02 cm⁻³ at the top (r = 10R_e). The increased T_e enhances the ambipolar electric fields (Equation (1)), which accelerate the heavy ions upward against the gravity.

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In another study Barakat & Schunk (1984) modeled the effects of a minor hot electron population on the outflow of ions, instead of the bulk heated electrons as discussed above. In this model, the densities of the cold (n_c) and hot (n_h) electrons are given by

$$\begin{equation} n_{\rm c} = n_{{\rm co}} \;\exp (e\varphi /KT_{{\rm ec}});\quad n_{\rm h} = n_{{\rm ho}} \exp (e\varphi /KT_{{\rm eh}}), \end{equation} \tag{ 2 }$$

the total electron density n_e = n_c + n_h = n_i, the total ion density. It is assumed that φ = 0, when n_c = n_co, n_h = n_ho and n_o = n_co+ n_ho at the bottom boundary at r = r_o. T_ec and T_eh are the constant cold and hot electron temperatures, respectively.

Figure 2 shows the essential features of the effects of the minor hot electron population on the outflow of the ionospheric ions. The parameters in the model are the same as for Figure 1 except that the minor hot electron density n_ho/n_o = 0.01 and the temperature ratio η = T_eh/T_ec = 100, giving T_eh = 28 eV for T_ec = 0.28 eV. The top and bottom panels in the left column of Figure 2 show the altitude profiles of density and flow velocity of the H⁺ ions, respectively. Like wise the right-hand column shows the profiles for the heavy oxygen ions. Each panel in Figure 2 has two sets of curves; the broken line curves show the expansion with no hot electrons, n_ho = 0, n_co = n_o, and T_ec = 0.28 eV. The solid line curves show the case for n_ho = 0.01n_o, T_eh = 100 T_ec, n_ec = 0.99n_o and T_ec = 0.28 eV. The noteworthy feature of the solid lines curves are the discontinuities in the altitude profiles at a critical altitude r_c. The critical altitude approximately corresponds to where the total ion density in the expansion has decreased to the level of the hot electrons, that is n(r = r_c) ≈ n_h (r = r_c), as expected theoretically (Bezzerides et al. 1978). At the critical altitude the density and flow velocity profiles change discontinuously; the density sharply decreases and ions are suddenly accelerated. The electric potential also changes discontinuously, but not shown in Figure 2. The discontinuous changes in the profiles are attributed to the formation of a DL, which traps the cold electrons and accelerates the ions. In Barakat & Schunk (1984) the spatial resolution was insufficient to resolve the sharp transitions in the profiles and hence the profiles show the discontinuities. Formation of such DLs in expansion of laser plasmas with two-temperature electron populations is well known in plasma physics since the early theoretical work by Bezzerides et al. (1978), which was later verified in laboratory experiments (Hairapetian & Stenzel 1988, 1991) and also by kinetic simulations resolving the DL profile (Singh et al. 2005). The DL forms above a theoretically determined critical temperature of the hot electrons, T_eh > (5 + (24)^1/2) T_ec = 9.9T_ec (Bezzerides et al. 1978). The potential drop across the DL scales as φ_DL ∼ T_eh/e and, therefore, the ion outflow velocities above the discontinuity approximately scale as U ∼ T_eh^1/2.

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We suggest that the mechanism for the acceleration of the detected Ne⁷⁺ as a tracer for the solar wind acceleration in the coronal funnels (Tu et al. 2005) is a problem similar to the extraction of heavy ions from the terrestrial ionosphere, albeit with much stronger solar gravity. We present here simple analytical calculations to illustrate that the enhanced electron temperature at the bottom of the funnels, footed in the chromospheric magnetic networks, can be instrumental in the origin of the solar wind. The electrons could be either already heated in the reconnecting magnetic loops or they could be heated during the reconnection between the magnetic fields of the funnels and the loops in the chromospheric magnetic networks. The reconnection supplies the plasma from the magnetic loops into the funnel (Peter 2007; Tu et al. 2005; McKenzie et al. 1998).

First, we consider the scenario that the plasma with bulk heated electrons is injected at the bottom of the funnel below a height h_B ∼ 5 Mm (Tu et al. 2005). As this plasma expands upward, the accelerated heavy ions with large charge to mass ratio (q/M) are detected at a height h_T ∼ 20 Mm. The acceleration of the heavy minor ions occurs in the ambipolar electric field set up by the expansion of the major H⁺ ions and the hot electrons. We estimate that the electric potential drop associated with the ambipolar electric field in the expansion is

$$\begin{equation} \varphi _0 \sim (T_{\rm e} /e)\;{\rm In}(n_{\rm T} /n_{\rm B}) = (T_{\rm e} /e)\;{\rm In}(B_{\rm T} /B_{\rm B}), \end{equation} \tag{ 3 }$$

where n_B(B_B) and n_T(B_T) are the plasma densities (B) at the bottom and top of the funnel, respectively. In writing the latter equality in Equation (3), we assume that n varies in proportion to B. We find that this density variation for the H⁺ ions is approximately obeyed in Figure 1(a) for the polar wind. The acceleration of the ions by the ambipolar electric field is countered by the solar gravity. Thus, the flow velocities V_j of the j type of ions with mass M_j and charge q_j at a height h above the photosphere is given by

$$\begin{equation} V_{\rm j} = 2^{1/2} [(T_{\rm e} /M_{\rm j})\;(q_{\rm j} /e)\;{\rm In(}B_{\rm B} /B_{\rm T}) - g_{\rm s} h]^{1/2}. \end{equation} \tag{ 4 }$$

From Tu et al. (2005), we find B_B/B_T ≈ 12. We have g_s = 274 ms⁻², and for Ne⁷⁺ we have M_Ne = 20 M_H, q_Ne/e = 7. Using (4) we find T_e = 63.6 eV, required for producing V_Ne = 10 km s⁻¹ at h = 20 Mm. With this T_e, Equation (4) yields the flow velocity for the major H⁺ ions V_H+ = 123 km s⁻¹.

For T_e = 63.6 eV, Equation (3) yields a potential drop φ₀ = 158 V over the height from h = 0 to 20 Mm. This potential drop could impart Ne⁷⁺ a kinetic energy of 1106 eV. But overcoming the gravity for Ne⁷⁺ requires 1096 eV and only 10 eV goes in imparting the ions a velocity of 10 km s⁻¹ as observed in Tu et al. (2005).

Heating of the chromospheric bulk electrons from less than 1 eV to 64 eV at the bottom of the funnels may not occur easily. But it is possible that magnetic reconnection might produce an energetic electron tail in the EVDF as seen in kinetic simulations (Hoshino et al. 2001), creating the two-temperature electron populations below h = 5 Mm. In this scenario, the plasma expansion could create abrupt transition in the ambipolar potential, across which the light and heavy ions are accelerated, like in Figure 2 for the polar wind. The potential drop across the DL is approximately φ_DL ∼ T_eh e⁻¹ (Bezzerides et al. 1978). Since the total ambipolar potential drop required to extract Ne⁷⁺ ions with velocity V_Ne = 10 km s⁻¹ at h = 20 Mm is φ = 158.4 V as calculated earlier, the required temperature for the electron tail is T_eh = 158.6 eV, which is larger than that for the bulk heating by a factor ln(B_B/B_T) = 2.48. Since the production of the observed neon ions need energetic electrons having T_e > 60 eV, the observation strongly suggests the presence of such a hot electron population. Thus, the required ratio T_eh/T_ec > 10 for DL formation can be easily met when the cold electron temperature in the chromosphere T_ec < 1 eV at the funnel bottom. The height of the DL formation strongly depends on the hot to cold electron density ratio; the smaller n_eh/n_o the farther the DL moves away from the funnel bottom (Hairapetian & Stenzel 1992). For the extraction of sufficient heavy ion flux, the DL must form closer to the bottom, requiring a significant fraction of electrons be heated by the reconnection.

Our main conclusions are summarized here. (1) We suggested that the LASPE with hot electrons at the bottom of the coronal funnels could facilitate the origination of the solar wind as reported in Tu et al. (2005). We discussed here the acceleration of Ne⁷⁺ ions to the observed flow velocity 10 km s⁻¹ at a height h = 20 Mm. (2) We discussed two scenarios for the LASPE involving (a) bulk heated electrons to temperatures T_e > 64 eV and (b) a small fraction of electrons forming energetic tail to the EVDF with T_eh > 158.4 eV. In the former case the φ(r) drops continuously along the length of the funnel while in the latter case with the two electron populations, a cold and a hot, φ (r) drops discontinuously forming a DL. The potential drop across the DL is KT_eh/e. (3) Most of the energy gained in the potential drops, in either case, is spent in encountering the solar gravity. (4) The mechanism for the ion acceleration involving CFDLs found in laboratory HPD with L_B ∼ ρ_i (Boswell et al. 2006) is not likely to be operative in the solar wind where L_B ≫ ρ_i (Singh 2011). (5) The electron temperatures found for the acceleration of Ne⁷⁺ ions (Tu et al. 2005) is consistent with the large electron energies (> 60 eV) required to produce the ions in the chromosphere.

Since the funnels are byproducts of the magnetic reconnection in the chromospheric magnetic networks (Tu et al. 2005), the heating/acceleration of electrons to 60–160 eV is not beyond expectations (Fisk 2003; Gloeckler et al. 2003; McKenzie et al. 1998). It is well known that the formation of current sheets and magnetic reconnection could significantly heat/accelerate the electrons, forming energetic tail to the EVDF (Hoshino et al. 2001; Drake et al. 2003; Singh et al. 2010).

The outflow of solar wind from the chromosphere has been studied extensively using sophisticated modeling (e.g., see Cranmer et al. 2010); in these models the outflow is powered by wave pressure or plasma heating by waves. In contrast, in our calculations the outflow is driven by hot electrons contained in the chromospheric loops (Gloeckler et al. 2003; Fisk 2003) or produced by the reconnection between the funnel and magnetic loops. Our model is based on simple physics (Barakat & Schunk 1983, 1984) and it seems to explain the observation of Tu et al. (2005) on the acceleration of Ne⁷⁺, provided that the enhanced electron temperature is T_e > 64 eV for the bulk heated electrons or T_e > 160 eV for the tail of the EVDF at the foot of the coronal funnel. Such electron temperatures are also needed for the production of the neon ions.

We presented here analytical estimates of the electron temperatures needed for the solar wind reported in Tu et al. (2005). A rigorous multi-fluid model of the LASPE including the effects of collisions, heat conduction, and radiative cooling is needed to rigorously develop the ideas presented in this paper.

This work was supported by the NSF grant ATM 0647157 and by the US Air Force Office for Scientific Research. Author acknowledges fruitful discussions with John Jasperse and Bamandas Basu.