Exact solutions of the Vlasov-Maxwell system of equations and a two-scale structure of a magnetic dip

In the Earth's magnetosphere, NASA's Magnetospheric Multiscale (MMS) has been observing kinetic structures with current sheets, including magnetic dip structures. Nongyrotropic electrons are often detected, and a typical thickness is several electron skin depths. To explain those observations, we study the Vlasov-Maxwell system of equations considering nongyrotropy of particles. We find exact one-dimensional solutions for a magnetic dip, where the nongyrotropy produces electric currents perpendicular to the magnetic field. When we impose zero electric potential, the solutions become the same as obtained by Nicholson [1963], in which the electron nongyrotropy is negligibly small, and the thickness of the magnetic dip is several skin depths. Introducing a finite electric potential allows us to use a large electron nongyrotropy. When the electron nongyrotropy is large and ions are gyrotropic, the thickness of a magnetic dip becomes several electron skin depths. In contrast, when both ions and electrons show large nongyrotropy, the magnetic dip shows two scales: The inner region shows an electron-scale thin magnetic dip, which is surrounded by the outer region of an ion-scale thick magnetic dip. We compare these solutions with a magnetic dip structure observed by MMS.