Abstract
Methods for constructing synthetic multidimensional electron hole equilibria without using particle simulation are investigated. Previous approaches have various limitations and approximations that make them unsuitable within the context of expected velocity diffusion near the trapped-passing boundary. An adjustable model of the distribution function is introduced that avoids unphysical singularities there, and yet it is sufficiently tractable analytically to enable prescription of the potential spatial profiles. It is shown why simple models of the charge density as being a function only of potential cannot give solitary multidimensional electron holes, in contradiction of prior suppositions. Fully self-consistent axisymmetric electron holes in the drift-kinetic limit of electron motion (negligible gyro-radius) are constructed and their properties relevant to observational interpretation and finite-gyro-radius theory are discussed.
Highlights
Active space plasma regions are observed to contain long-lived solitary positive potential peaks whose spatial extent is a few Debye lengths;[1–13] they are mostly identified as electron holes, having an electron charge deficit on trapped orbits.[14]. It has been known for a long time that non-zero magnetic field is necessary for the sustainment of electron holes; if it is strong enough, the electron motion and trapping becomes one-dimensional
The present theory continues to calculate using only parallel particle dynamics, it qualitatively takes account of one recently discovered important effect of the transverse electric field in a multidimensional electron hole, namely, the resonant interaction of the trapped particle bounce motion with the gyro-motion. This interaction essentially always induces a region of stochastic orbits near zero parallel energy—the trapped-passing boundary of phase space—and the energy-depth of this stochastic layer increases with rL=L?
The anticipated result in the stochastic layer is a large effective phase-space diffusion rate which forces the distribution function to be approximately independent of energy in that region. It has been shown recently,[16] in contradiction of a long-standing suggestion, that regardless of the relative strength of the magnetic field, the screening of the trapped electron deficit charge is isotropic, having approximately Boltzmann dependence. This discredits one speculation concerning how the transverse scale of electron holes relates to magnetic field strength and reemphasizes the need to understand better multidimensional electron holes, especially since recent multisatellite measurements give unprecedented information about multidimensional holes in space plasmas.[17–19]
Summary
Active space plasma regions are observed to contain long-lived solitary positive potential peaks whose spatial extent is a few Debye lengths;[1–13] they are mostly identified as electron holes, having an electron charge deficit on trapped orbits.[14]. The present theory continues to calculate using only parallel particle dynamics, it qualitatively takes account of one recently discovered important effect of the transverse electric field in a multidimensional electron hole, namely, the resonant interaction of the trapped particle bounce motion with the gyro-motion This interaction essentially always induces a region of stochastic orbits near zero parallel energy—the trapped-passing boundary of phase space—and the energy-depth of this stochastic layer increases with rL=L?.15. The anticipated result in the stochastic layer is a large effective phase-space diffusion rate which forces the distribution function (phase-space density) to be approximately independent of energy in that region It has been shown recently,[16] in contradiction of a long-standing suggestion, that regardless of the relative strength of the magnetic field, the screening of the trapped electron deficit charge is isotropic, having approximately Boltzmann dependence. Ions are taken to be a uniform immobile neutralizing background
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