Parallel Electric Field and Particle Acceleration in Oblique Magnetosonic Shock Waves

The electric field parallel to the magnetic field, $E_\parallel$, in nonlinear magnetosonic waves is studied with theory and particle simulations, and its results are applied to the investigation of the effect of $E_\parallel$ on particle acceleration in shock waves. In the ideal MHD, $E_\parallel$ is zero, and it was generally thought that $E_\parallel$ was quite weak. Our studies, however, show that it can be strong in nonlinear magnetosonic waves. In a shock wave with its amplitude $\epsilon \sim O(1)$, the magnitude of the integral of $E_\parallel$ along the magnetic field, $F=-\int E_\parallel ds$, is given as $eF \sim \epsilon (m_i v_A^2 + \Gamma _e T_e)$, where $\Gamma _e$ is the specific heat ratio. Furthermore, particle motions in three acceleration mechanisms are calculated with two different test particle methods: In the first method, the total electric field {\boldmath $E$} is used in the equation of motion, while in the second one, $E_\parallel$ is omitted. Comparison of these calculations confirms that $E_\parallel$ is unimportant in the incessant acceleration of relativistic ions. However, $E_\parallel$ is essential for the acceleration of trapped electrons and for the acceleration of positrons along the magnetic field.