Homotopy formulas for the magnetic vector potential and magnetic helicity: The Parker spiral interplanetary magnetic field and magnetic flux ropes

Homotopy formulas are obtained for the magnetic vector potential A, where B = del x A is the magnetic induction, which gives alternative methods for calculating A for a given B, different from the Biot-Savart formulas. The homotopy formulas are used to obtain A for multipole potential fields and other potential fields, which are useful in relative helicity calculations. However, the formulas do not work for monopole and split monopole magnetic fields. In the latter case, there is no global continuous solution for A, but a global discontinuous A can be constructed, which is continuous on two open sets that cover the sphere. The differential and integral forms of the helicity and relative helicity transport equations using both Eulerian and Lagrangian perspectives are discussed. The approach to magnetic helicity and magnetic helicity injection, based on a toroidal-poloidal decomposition of the field, in which the field is represented by Euler potentials is also used in the analysis. The relative magnetic helicity and helicity injection rate for the Parker spiral interplanetary magnetic field, in which there is either a flat current sheet or a warped current sheet in the helioequatorial plane, are discussed. One of the homotopy formulas is used to provide an efficient way to calculate the relative helicity of interplanetary magnetic flux rope configurations observed by the WIND spacecraft.