Equilibrium and stability of mirror-confined nonneutral E layers

Abstract

The steady‐state Vlasov‐Maxwell equations are used to study the equilibrium properties of a nonneutral $E$ layer confined both axially and radially, by a static external mirror field, mirror field, $B_0^{\mathrm{ext}}\mathrm{(x)}$ . Equilibrium properties are calculated for the electron distribution function $f_e^0{\mathrm{\hspace{0.17em}(H\hspace{0.17em},\hspace{0.17em}P\hspace{0.17em}}}_{\theta })$ in which all electrons have the same total energy $\mathrm{(H)}$ and the same canonical angular momentum ${\mathrm{(P\hspace{0.17em}}}_{\theta })$ , i.e., $f_e^0{\mathrm{\hspace{0.17em}(H\hspace{0.17em},\hspace{0.17em}P\hspace{0.17em}}}_{\theta }{\mathrm{)\hspace{0.17em}=\hspace{0.17em}N}}_0{\mathrm{\delta (H\hspace{0.17em}-\hspace{0.17em}H}}_0{\mathrm{)\delta (P}}_{\theta }{\mathrm{\hspace{0.17em}-\hspace{0.17em}P}}_0)$ , where ${\mathrm{N\hspace{0.17em}}}_0$ , ${\mathrm{H\hspace{0.17em}}}_0$ , and ${\mathrm{P\hspace{0.17em}}}_0$ are positive constants. For a low‐density $E$ layer, the electrostatic potential energy of an electron, ${\mathrm{-\hspace{0.17em}e\hspace{0.17em}\phi }}_0\mathrm{(r\hspace{0.17em},\hspace{0.17em}z\hspace{0.17em})}$ , is small in comparison with ${\mathrm{H\hspace{0.17em}}}_0$ . Neglecting terms of order ${\mathrm{|e\hspace{0.17em}\phi }}_0{\mathrm{(r\hspace{0.17em},\hspace{0.17em}z)/\hspace{0.17em}H\hspace{0.17em}}}_0|$ , a closed zero‐order expression for the $\mathrm{r\hspace{0.17em}‐\hspace{0.17em}z}$ boundary of the $E$ layer is obtained. Iterating, the equilibrium electrostatic potential ${\phi }_0\mathrm{(r\hspace{0.17em},\hspace{0.17em}z)}$ is then computed to lowest order, together with ${\mathrm{O\hspace{0.17em}[e\hspace{0.17em}\phi }}_0{\mathrm{(r\hspace{0.17em},\hspace{0.17em}z)\hspace{0.17em}/\hspace{0.17em}H}}_0]$ corrections to the boundary of the $E$ layer. Computer simulation experiments are used to investigate electrostatic stability properties for the case of azimuthally symmetric perturbations $\text{(∂/∂θ\hspace{0.17em}=\hspace{0.17em}0)}$ . The code allows only $r$ and $z$ spatial variations but self‐consistently follows the three velocity components of 6264 macroparticles. When the electrons are initially loaded close to the equilibrium state described by $f_e^0{\mathrm{(H\hspace{0.17em},\hspace{0.17em}P}}_{\theta }{\mathrm{)\hspace{0.17em}=\hspace{0.17em}N\hspace{0.17em}}}_0{\mathrm{\delta (H\hspace{0.17em}-\hspace{0.17em}H\hspace{0.17em}}}_0{\mathrm{)\delta (P\hspace{0.17em}}}_{\theta }{\mathrm{\hspace{0.17em}-\hspace{0.17em}P\hspace{0.17em}}}_0)$ , the $E$ layer is observed to achieve a stable quasisteady configuration in about 15 ${\omega }_{\mathrm{p\hspace{0.17em}e}}^{\text{−1}}$ with negligible particle loss out the mirrors.