Euler potentials'discontinuity in the presence of field line-aligned currents

Since their use in the study of charged particle motion in the 1960s,Euler potentials(α ,β)have been widely employed as magnetic field coordinates in both space plasma and fusion plasma studies.People related them to magnetic vector potential A via the relation A=α▽β subject to gauge condition A·B=0(B is the magnetic induction).For a given magnetic field,the Euler potentials are often constructed with the relation B·△ S=△α△βon a surface that crosses the field lines,where △ S is the area-element surrounding by two line-elements corresponding to the changes in α and β,then mapping the values of α and βalong field lines into space.In this short paper,we show that in the presence of field line-aligned currents,the mapping does not work and the orthogonality gauge condition is not satisfied.

Since their use in the study of charged particle motion in the 1960s, Euler potentials （α,β） have been widely employed as mag- netic field coordinates in both space plasma and fusion plasma studies. People related them to magnetic vector potential A via the relation A = α▽β, subject to gauge condition A · B = 0 （B is the magnetic induction）. For a given magnetic field, the Euler poten- tials are often constructed with the relation B·△S = AaAflon a surface that crosses the field lines, where AS is the area-element surrounding by two line-elements corresponding to the changes in α and β, then mapping the values of ct and β along field lines into space. In this short paper, we show that in the presence of field line-aligned currents, the mapping does not work and the or- thogonality gauge condition is not satisfied.