Particle Acceleration and Plasma Heating in the Chromosphere

We propose a new mechanism of electron acceleration and plasma heating in the solar chromosphere, based on the magnetic Rayleigh–Taylor instability. The instability develops at the chromospheric footpoints of a flare loop and deforms the local magnetic field. As a result, the electric current in the loop varies, and a resulting inductive electric field appears. A pulse of the induced electric field, together with the pulse of the electric current, propagates along the loop with the Alfvén velocity and begins to accelerate electrons up to an energy of about 1 MeV. Accelerated particles are thermalized in the dense layers of the chromosphere with the plasma density \(n \approx10^{14}\,\mbox{--}\,10^{15}~\mbox{cm}^{-3}\), heating them to a temperature of about several million degrees. Joule dissipation of the electric current pulse heats the chromosphere at heights that correspond to densities \(n \le10^{11}\,\mbox{--}\,10^{13}~\mbox{cm}^{-3}\). Observations with the New Solar Telescope at Big Bear Solar Observatory indicate that chromospheric footpoints of coronal loops might be heated to coronal temperatures and that hot plasma might be injected upwards, which brightens ultra-fine loops from the photosphere to the base of the corona. Thereby, recent observations of the Sun and the model we propose stimulate a déjà vu – they are reminiscent of the concept of the chromospheric flare.

This work is supported partially by RFBR grants No 14-02-00133 and 15-02-08028, by the Programs of Presidium of RAS No 9 and 41, grant of Leading Scientific School NSh 1041.2014.2, as well as by grant of Ministry of Education and Sciences 14.Z50.31.0007. The authors thank D. Melrose and one of the Guest Editors for valuable remarks.

Authors and Affiliations

1. Institute of Applied Physics, Ulianova str. 46, Nizhny Novgorod, 603950, Russia

   V. V. Zaitsev

2. Pulkovo Observatory, Pulkovo chaussee 65, Saint Petersburg, 196140, Russia

   A. V. Stepanov

Corresponding author

Correspondence to A. V. Stepanov.

Solar and Stellar Flares: Observations, Simulations, and Synergies

Guest Editors: Lyndsay Fletcher and Petr Heinzel

Appendix: Generation of Accelerating Electric Field

We consider a cylindrical magnetic flux tube of radius \(a\) with the longitudinal magnetic field \(B_{\mathrm{z}}\), uniform over the flux tube cross section, and with the electric current density \(j_{\mathrm{z}}(z,t)\), homogeneous over the tube cross section. To analyze the magnetic and electric fields arising in the course of the propagation of the current pulse, we start from the equations for the plasma

Equations (20) and (22) for the fields, and Ohm’s law

In a cylindrical coordinate system, we assume axial symmetry, i.e. \(\frac{\partial}{\partial \varphi } = 0\). We also assume that the plasma is incompressible, i.e. \(\rho= \mathit{const}\). Then from Equation (32) we obtain

We assume that in the first approximation for the amplitude of the electric current (\(B_{\varphi }\)), the radial velocity component \(V_{\mathrm{r}} = 0\) (provided this approximation is valid). In this case, it follows from Equation (35) that \(V_{\mathrm{z}} = \mathit{const} = 0\), as the velocity is zero at infinity. Then, from Equations (33) and (20) it follows

From Equations (37) and (38), we obtain

i.e., the pulse of the electric current propagates along the magnetic flux tube with the Alfvén velocity \(V_{\mathrm{A}} = B_{\mathrm {z}0}/\sqrt{4\pi\rho}\). If the amplitude of the electric current is sufficiently small, then in the classical Alfvén wave the pressure gradients by the coordinates \(r\), \(z\) are balanced by the corresponding magnetic field gradients, and it follows from Equations (39) and (40) that the approximation of \(V_{\mathrm{r}} = 0, V_{\mathrm{z}} =0\) is valid. From Equations (20), (22), and (37), we obtain

i.e., in this case, there is no longitudinal electric field (Pikel’ner 1964). However, if an electric current is sufficiently large, so that the condition \(p \ll B_{\varphi }^{2}/8\pi\) is fulfilled, the velocity components \(V_{\varphi }\), \(V_{\mathrm{z}}\) are nonzero, and for them the following approximate equations can be written

From the conditions \(\partial/\partial \varphi = 0\) and \(E_{\varphi } = 0\), it follows that \(B_{\mathrm{r}} = 0\), \(B_{\mathrm{z}} = B_{\mathrm{z}0}\),

Considering now

we obtain

Using Equation (34), we find

The sum of the last two terms in the right-hand side of Equation (48) for the uniform current distribution over the cross section of a flux tube is zero; therefore,

Assuming \(\frac{\partial}{\partial z} \approx- \frac{1}{V_{\mathrm{A}}}\frac{\partial}{\partial t}\), and using Equation (44), we obtain

The longitudinal electric field increases from the flux tube axis and reaches its maximum on the surface. It also increases with the increase in the electric current, and for \(B_{\varphi }^{\max} \approx\sqrt{4\pi \rho} V_{\mathrm{A}}\) reaches the values

which were used in Zaitsev, Stepanov, and Melnikov (2013).

As a result of the plasma effect, the field changes its sign relative to that in vacuum. At the leading edge of the current pulse, the derivative \(\partial B_{\varphi } /\partial t\) is positive and the electric field accelerates electrons upwards in the direction of the current pulse. In the tail of the pulse, \(B_{\varphi }\) decreases with time and the electric field accelerates electrons downwards to dense layers of the chromosphere. Given that \(B_{\varphi } = 2rI_{0}(z - V_{\mathrm{A}}t)/ca^{2}\), the formula (50) can be written as

The maximum field corresponds to \(r = a\):

The electric field averaged along the tube radius is

Equation (50) for the parallel component of the electric field can be written as

where \(k_{\parallel} \approx1/\Delta\xi\), \(k_{ \bot} \approx1/a\) are characteristic parallel and perpendicular wavenumbers of the electric current pulse, respectively, \(E_{\mathrm{r}} = - \frac{1}{c}(V_{\varphi } B_{\mathrm{z}0} - V_{\mathrm{z}}B_{\varphi } ) = \frac{V_{\mathrm{A}}}{c}B_{\varphi } (1 + \frac{1}{2}\frac{B_{\varphi }^{2}}{B_{\mathrm{z}0}^{2}}) \approx \frac{V_{\mathrm{A}}}{c}B_{\varphi }\) (if \(B_{\varphi }^{2} \ll B_{\mathrm {z}0}^{2}\)) is the perpendicular electric field in the current pulse. Because the current pulse has both longitudinal and transverse scales, i.e. there is an effective dispersion, this pulse corresponds to the nonlinear dispersive Alfvén wave. In the case of linear dispersive Alfvén waves (both inertial and kinetic) there is a parallel electric field as well. If the condition \(8\pi p/B^{2} \le m_{\mathrm{e}}/m_{\mathrm{i}}\) is fulfilled, the Alfvén waves are called inertial (Stasiewicz et al. 2000; Cramer 2001). In this case, the parallel electric field is determined by the formula (Stasiewicz et al. 2000)

where \(\lambda_{\mathrm{e}} = c/\omega_{\mathrm{pe}}\) is the electron skin depth. If \(m_{\mathrm{e}}/m_{\mathrm{i}} \le8\pi p/B^{2} \le1\) the Alfvén waves are called kinetic. In this case (Malovichko 2013),

where \(V_{\mathrm{Ti}}\), \(\omega_{\mathrm{Bi}}\) are ion thermal velocity and ion gyrofrequency, respectively. A parallel electric field (Equation (55)) in a nonlinear dispersive Alfvén wave can be higher by many orders of magnitude than the corresponding value in linear Alfvén waves. For example, for the case under consideration, \(\Delta\xi\approx a \approx10^{7}~\mbox{cm}\), we have \(k_{\parallel} \approx k_{ \bot} \approx10^{-7}~\mbox{cm}^{-1}\), and, as is easily seen from Equations (56) and (57), \(E_{\mathrm{z}} \approx2.5 \times10^{-19}E_{\mathrm{r}}\) in the case of inertial Alfvén waves, and \(E_{\mathrm{z}} \approx 10^{-14}E_{\mathrm{r}}\) in the case of kinetic Alfvén waves. For nonlinear dispersive Alfvén waves (Equation (55)), the parallel electric field can be on the order of the perpendicular one if the wave amplitude \(B_{\varphi } \sim B_{\mathrm{z}0}\). The approximation in which the electric field (Equation (55)) was obtained (\(m_{\mathrm{e}}/m_{\mathrm{i}} \le8\pi p/B^{2} \le1\)) formally corresponds to the nonlinear kinetic Alfvén wave. But as the appropriate solution is obtained within an MHD approach, it is more correctly called a nonlinear dispersive Alfvén wave.

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