Do Current and Magnetic Helicities Have the Same Sign?

Current helicity, Hc, and magnetic helicity, Hm, are two main quantities used to characterize magnetic fields. For example, such quantities have been widely used to characterize solar active regions and their ejecta (magnetic clouds). It is commonly assumed that Hc and Hm have the same sign, but this has not been rigorously addressed beyond the simple case of linear force-free fields. We aim to answer whether HmHc ≥ 0 in general, and whether it is true over some useful set of magnetic fields. This question is addressed analytically and with numerical examples. The main focus is on cylindrically symmetric straight flux tubes, referred to as flux ropes (FRs), using the relative magnetic helicity with respect to a straight (untwisted) reference field. Counterexamples with HmHc < 0 have been found for cylindrically symmetric FRs with finite plasma pressure, and for force-free cylindrically symmetric FRs in which the poloidal field component changes direction. Our main result is a proof that HmHc ≥ 0 is true for force-free cylindrically symmetric FRs where the toroidal field and poloidal field components are each of a single sign, and the poloidal component does not exceed the toroidal component. We conclude that the conjecture that current and magnetic helicities have the same sign is not true in general, but it is true for a set of FRs of importance to coronal and heliospheric physics.