Nonlinear Evolution of a 3D Inertial Alfvén Wave and Its Implication in Particle Acceleration

Abstract

A simulation based on a pseudo-spectral method has been performed in order to study particle acceleration. A model for the acceleration of charged particles by field localization is developed for the low-$\upbeta$ plasma. For this purpose, a fractional diffusion approach has been employed. The nonlinear interaction between a 3D inertial Alfvén wave and a slow magnetosonic wave has been examined, and the dynamical equations of these two waves in the presence of ponderomotive nonlinearity have been solved numerically. The nonlinear evolution of the inertial Alfvén wave in the presence of slow magnetosonic wave undergoes a filamentation instability and results in field intensity localization. The results obtained show the localization and power spectrum of inertial Alfvén wave due to nonlinear coupling. The scaling obtained after the first break point of the magnetic power spectrum has been used to calculate the formation of the thermal tail of energetic particles in the solar corona.

Appendix: Expressions for Ponderomotive Force

The ponderomotive force components for electrons are given by

$$\begin{aligned} F_{\mathrm{e}x} &= \biggl( \frac{e^{2}\lambda_{\mathrm{e}}^{2}k_{0 \bot}^{2}}{4m_{\mathrm{e}}c^{2}} \biggr)\frac{\partial \vert A_{z} \vert ^{2}}{\partial x}, \end{aligned}$$

(8)

$$\begin{aligned} F_{\mathrm{e}y} &= \biggl( \frac{c^{2}\alpha^{2}m_{\mathrm{e}}k_{0x}k_{0y}}{4B_{0}^{2}} \biggr)\frac{\partial \vert A_{z} \vert ^{2}}{\partial x}, \end{aligned}$$

(9)

and

$$ F_{\mathrm{e}z} = - \biggl( \frac{e^{2}\lambda_{\mathrm{e}}^{4}k_{0 \bot}^{4}}{4m_{\mathrm{e}}c^{2}} \biggr)\frac{\partial \vert A_{z} \vert ^{2}}{\partial z}. $$

(10)

Similarly the ponderomotive force components for ions are given by

$$\begin{aligned} F_{\mathrm{i}x} &= \biggl( - \frac{m_{\mathrm{i}}P^{2}c^{2}k_{0y}^{2}\alpha^{2}}{4B_{0}^{2}} \biggr)\frac{\partial \vert A_{z} \vert ^{2}}{\partial x} + \biggl( \frac{eQck_{0x}k_{0z}\alpha^{2}}{4\omega_{0}B_{0}} \biggr)\frac{\partial \vert A_{z} \vert ^{2}}{\partial z}, \end{aligned}$$

(11)

$$\begin{aligned} F_{\mathrm{i}y} &= \biggl( \frac{m_{\mathrm{i}}P^{2}c^{2}k_{0x}k_{0y}\alpha^{2}}{4B_{0}^{2}} \biggr)\frac{\partial \vert A_{z} \vert ^{2}}{\partial x} + \biggl( \frac{eQck_{0y}k_{0z}\alpha^{2}}{4\omega_{0}B_{0}} \biggr)\frac{\partial \vert A_{z} \vert ^{2}}{\partial z}, \end{aligned}$$

(12)

and

$$ F_{\mathrm{i}z} = \biggl( \frac{eQck_{0x}k_{0z}\alpha^{2}}{4\omega_{0}B_{0}} \biggr)\frac{\partial \vert A_{z} \vert ^{2}}{\partial x} - \biggl( \frac{e^{2}\alpha^{2}k_{0z}^{2}}{4\omega_{0}^{2}m_{\mathrm{i}}} + \frac{e^{2}}{4m_{\mathrm{i}}c^{2}} - \frac{e^{2}\alpha k_{0z}}{2\omega_{0}cm_{\mathrm{i}}} \biggr) \frac{\partial \vert A_{z} \vert ^{2}}{\partial z}. $$

(13)

Here

$$P = \frac{\omega_{\mathrm{ci}}^{2}}{\omega_{\mathrm{ci}}^{2} - \omega_{0}^{2}},\qquad Q = \frac{\omega_{\mathrm{ci}}\omega_{0}}{\omega_{\mathrm{ci}}^{2} - \omega_{0}^{2}},\quad \mbox{and}\quad \alpha = \frac{\omega_{0}}{ck_{0z}} \bigl( 1 + \lambda_{\mathrm{e}}^{2}k_{0 \bot}^{2} \bigr). $$