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The origin and evolution of the magnetic helicity in the solar corona are notwell understood. For instance, the magnetic helicity of an active region isoften about $10^{42}$ Mx$^2$ ($10^{26}$ Wb$^{2}$), but the observed processeswhereby it is thought to be injected into the corona do not yet provide anaccurate estimate of the resulting magnetic helicity budget or time evolution.The variation in magnetic helicity is important for understanding the physicsof flares, coronal mass ejections, and their associated magnetic clouds. Toshed light on this topic, we investigate here the changes in magnetic helicitydue to electric currents in the corona for a single twisted flux tube that maymodel characteristic coronal structures such as active region filaments,sigmoids, or coronal loops. For a bipolar photospheric magnetic field andseveral distributions of current, we extrapolated the coronal field as anonlinear force-free field. We then computed the relative magnetic helicity, aswell as the self and mutual helicities. Starting from a magnetic configurationwith a moderate amount of current, the amount of magnetic helicity can increaseby 2 orders of magnitude when the maximum current strength is increased by afactor of 2. The high sensitivity of magnetic helicity to the current densitycan partially explain discrepancies between measured values on the photosphere,in the corona, and in magnetic clouds. Our conclusion is that the magnetichelicity strongly depends on both the strength of the current density and alsoon its distribution. Only improved measurements of current density at thephotospheric level will advance our knowledge of the magnetic helicity contentin the solar atmosphere.

Relationship between photospheric currents and coronal magnetic helicity for force-free bipolar fields