Minimum energy state and minimum angle rotation of the magnetic field in a current sheet with sheared magnetic field

Abstract

In order to answer why and how “the minimum angle rotation of the magnetic field” is realized in a current sheet with a sheared magnetic field, Taylor's helicity constraint, which is valid for low‐β plasmas, is applied to a one‐dimensional planar current sheet with a sheared magnetic field. A single constant helicity is defined for the total rectangular volume surrounding the current sheet and is shown to be gauge‐invariant. The minimization of the magnetic energy with the constraint of the constant total helicity shows that the field is described by the constant α force‐free equation and that the current sheet is a special class of tangential discontinuities with a constant field strength, or a “perpendicular rotational discontinuity.” The total rotational angle of the magnetic field across the current sheet is proportional to the ratio of the total magnetic energy/helicity in the force‐free state. It is proposed that among an infinite number of force‐free states the current sheet relaxes into a unique force‐free state with the absolute minimum ratio of energy/helicity and thus into the absolute minimum energy state for a given constant helicity. Therefore, in the relaxed state the total rotational angle of the magnetic field across the current sheet is minimum and less than 180°. Although the present study of the relaxed state is applicable only to a tangential discontinuity, a qualitative resemblance of the model prediction with observations in situ and simulations of quasi‐perpendicular rotational discontinuities suggests that the observed minimum angle rotation of the magnetic field in a current sheet with a sheared magnetic field is an emergence of plasma relaxation or self‐organization in space plasmas.