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Description
Kinetic Alfvén waves (KAWs) have long wavelengths parallel to magnetic field lines and perpendicular wavelengths comparable to the ion Larmor radius. KAWs possess a parallel electric field component ($\delta E_{\parallel}$) that accelerates electrons along magnetic field lines [e.g., Hasegawa, 1976]. These waves are frequently observed in the terrestrial magnetosphere during substorms [e.g., Stasiewicz et al., 2000]. They can accelerate electrons to energies of several keV parallel to magnetic field lines [e.g., Wygant et al., 2002]. In the process of electron acceleration by KAWs, electrons satisfying the Landau resonance condition between their parallel velocity ($v_{\parallel}$) and the wave phase speed ($V_{\mathrm{ph} \parallel}$) can be trapped by the wave and transported to higher latitudes while being accelerated [e.g., Artemyev et al., 2015]. Additionally, it has been pointed out that trapped electrons with a first adiabatic invariant ($\mu$) that is not small are affected by the mirror force and cannot reach the ionosphere due to escaping the trap [Watt and Rankin, 2009]. The details of these escape processes from KAWs remain poorly understood.
In this study, we apply the theory of the second-order resonance of charged particles trapped by coherent electromagnetic waves [e.g., Omura et al., 2008] to the electron acceleration process by KAWs. By considering the simple harmonic motion of the wave phase as viewed from the electron ($\psi$) and the inhomogeneity factor ($S$) due to background magnetic field gradients, we can describe the motion of electrons trapped by KAWs in velocity phase space. As trapped electrons move parallel to a magnetic field line, the background magnetic field gradients and $S$, as viewed from the electron, change accordingly. The range of energies at which electrons can be trapped narrows with increasing $S$, facilitating the escape of electrons from the trapped region as they move to higher latitudes with larger magnetic field gradients.
We focus on the detrapped state, where electrons have escaped from the KAW’s trap. When considering an $L=9$ magnetic field line and assuming $\delta E_{\parallel}$ around $1 \, \mathrm{mV/m}$ at the magnetic equator, electrons detrapped between $10^{\circ}$ and $20^{\circ}$ magnetic latitude with initial energies under $2 \, \mathrm{keV}$ can be accelerated up to approximately $10.5 \, \mathrm{keV}$ at the ionosphere, particularly when $\psi$ is in the range $(-\pi,0)$, where the wave effectively accelerates the electrons. Our results also reveal that the energy spectrum of electrons precipitating into the ionosphere is influenced by the position where electrons are detrapped, with the potential for a broadband energy distribution resulting from the acceleration of a monochromatic KAW. Considering the maximum kinetic energy at the ionosphere and the conservation of $\mu$ during acceleration, it is found that electrons must have $\mu$ less than approximately $0.23 \, \mathrm{eV/nT}$ to reach the ionosphere.
