Analytical approximations for the current sheet adiabatic invariant

Current sheets are observed in a range of collisionless space plasma, including the Earth's magnetotail. The current layers can be supported in part by the current carried by Speiser-type orbits. Outside the layer, a Speiser particle follows a regular cyclotron motion, while closer to the layer it abruptly changes to a meandering motion across the current sheet. The action integral over the periodic orbit motion perpendicular to the current sheet, Jz∝∮vzdz, is known as the current sheet adiabatic invariant. The quantity has proven useful for describing solutions to the Vlasov equation that governs the distribution functions of the plasma species. Considering a general current sheet geometry, we here derive analytical approximations for Jz. The utility of the approximations is emphasized through an application to the electron current layer that forms during symmetric anti-parallel magnetic reconnection.