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Homotopy Formulas for the Magnetic Vector Potential and Magnetic Helicity  

Gary Webb   Submitted: 2010-02-05 13:45

Homotopy formulas are obtained for the magnetic vector potential A where B = curl A is the magnetic induction, which give alternative methods for calculating A for a given B, di fferent from the Biot-Savart formulas. The homotopy formulas are used to obtain A for multipole potential fi elds and other potential fi elds, 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 di fferential and integral forms of the helicity and relative helicity transport equations using both Eulerian and Lagrangian perspectives are discussed. The gauge free approach to magnetic helicity and magnetic helicity injection, based on a toroidal-poloidal decomposition of the fi eld, in which the fi eld 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 fi eld, in which there is either a at current sheet, or a warped current sheet in the helio-equatorial 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 confi gurations observed by the WIND spacecraft.

Authors: G. M. Webb, Q. Hu, B. Dasgupta and G. P. Zank

Publication Status: submitted to JGR; revised
Last Modified: 2010-05-20 08:51
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Homotopy Formulas for the Magnetic Vector Potential and Magnetic Helicity

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