Force balance in current sheets in collisionless plasma

Abstract

In this paper, we derive a divergent form of the force balance equation for collisionless plasma in the quasineutrality approximation, in which the electric field and current density are excluded. For a stationary spatially one-dimensional current sheet with a constant normal component of the magnetic field and magnetized electrons, the general form of the force balance equation has been obtained for the first time in the form of a conservation law. An equation in this form is necessary for the correct formulation of boundary conditions when modeling asymmetric current sheets, as well as for the control of the stationarity of the numerical solution obtained in the model. Furthermore, the fulfillment of this equation is considered for two types of stationary configurations of a thin current sheet, which are obtained using a numerical model. The derived equation makes it possible to develop models of asymmetric current sheets, in particular current sheets on the magnetopause flanks in the magnetotail.

Fast Tearing Mode Driven by Demagnetized Electrons

“…The dynamics of the magnetized electron population is described in the guiding-center drift approximation (Haggerty et al, 2015;Malova et al, 2013;Zelenyi et al, 2004aZelenyi et al, , 2011. The velocity distribution function of their guiding centers was chosen in the Maxwell-Boltzmann form (Mingalev et al, 2021;Whipple et al, 1991) (see Text S1 in Supporting Information S1). Increasing a pressure anisotropy coefficient of magnetized electrons Λ Me = μ 0 (p ‖ p ⊥ )/B 2 leads to higher intensity and thinning of the TCS (Tsareva et al, 2023;Zelenyi et al, 2004a).…”

Nonlinear Equilibrium Structure of Super Thin Current Sheets: Influence of Quasi‐Adiabatic Electron Population

“…Since the magnetic moment $${\mu }_{e}\left({v}_{\perp }^{2},z\right)={m}_{e}{v}_{\perp }^{2}/2B(z)$$ and the total energy H e ( v 2 , z ) = m e v 2 /2 − eϕ ( z ) are the independent integrals of the electron motion, the general solution of the stationary Vlasov equation in the guiding center approximation has the form of an arbitrary function of these two integrals $${f}^{Me}\left({v}_{\Vert }^{2},{v}_{\perp }^{2},z\right)={\Phi }\left({\mu }_{e}\left({v}_{\perp }^{2},z\right),{H}_{e}\left({v}^{2},z\right)\right)$$. In the 1D anisotropic model (Mingalev et al., 2021; Whipple et al., 1991), the electron distribution function was chosen in the bi‐Maxwellian form: $$${f}^{Me}(\mathbf{v},z)=C\frac{{n}_{0}}{\sqrt{{\pi }^{3}}\,{v}_{Te\perp }^{2}{v}_{Te\Vert }}\mathrm{exp}\left(\frac{e\phi (z)}{{k}_{B}{T}_{e0}}\right)\times \mathrm{exp}\left(-\frac{{v}_{\Vert }^{2}}{{v}_{Te\Vert }^{2}}-\frac{{v}_{\perp }^{2}}{{v}_{Te\perp }^{2}}\right),$$$ where C −1 = ∫d 3 v f Me ( v , ∞ ) is the normalization constant; …”

“…According to this theory, corresponding drift electron currents are E × B , gradient and curvature ones. And the current density is described by the expression (Krall & Trivelpiece, 1973; Leontovich, 1965; Mingalev et al., 2021) $$${\mathbf{j}}_{\perp }^{Me}=-e{n}^{Me}\frac{\mathbf{E}\times \mathbf{B}}{{B}^{2}}+\frac{\mathbf{B}}{{B}^{2}}\times \nabla {p}_{\perp }^{Me}+\left({p}_{\Vert }^{Me}-{p}_{\perp }^{Me}\right)\frac{\mathbf{B}}{{B}^{2}}\times \left(\frac{\mathbf{B}}{B}\cdot \nabla \right)\frac{\mathbf{B}}{B},$$$ $$B=\sqrt{{B}_{x}^{2}+{B}_{n}^{2}}$$ is the total magnetic field, $$\mathbf{E}=-\vec{\nabla }\phi $$ is the ambipolar electric field arising from the requirement of charge quasi‐neutrality and determined from the condition of the field‐aligned force balance of electrons, which is obtained by neglecting the electron inertia for the fast parallel motion in the momentum flux equation $${m}_{e}{n}^{Me}\mathrm{d}{v}_{\Vert }^{Me}/\mathrm{d}t=-e{n}^{Me}{E}_{\Vert }-\left(\mathbf{B}\cdot \text{div}{\hat{\mathbf{P}}}_{Me}\right)/B$$, where the gyrotropic pressure tensor …”

“…According to this theory, corresponding drift electron currents are E × B, gradient and curvature ones. And the current density is described by the expression (Krall & Trivelpiece, 1973;Leontovich, 1965;Mingalev et al, 2021)…”

“…. In the 1D anisotropic model (Mingalev et al, 2021;Whipple et al, 1991), the electron distribution function was chosen in the bi-Maxwellian form:…”