Photospheric flux density of magnetic helicity

Several recent studies have developed the measurement of magnetic helicity flux from the time evolution of photospheric magnetograms. The total flux is computed by summing the flux density over the analyzed region. All previous analyses used the density G(A) (=-2(A center dot u)B(n)) which involves the vector potential A of the magnetic field. In all the studied active regions, the density G(A) has strong polarities of both signs with comparable magnitude. Unfortunately, the density G(A) can exhibit spurious signals which do not provide a true helicity flux density. The main objective of this study is to resolve the above problem by defining the flux of magnetic helicity per unit surface. In a first step, we define a new density, G(theta), which reduces the fake polarities by more than an order of magnitude in most cases (using the same photospheric data as G(A)). In a second step, we show that the coronal linkage needs to be provided in order to define the true helicity flux density. It represents how all the elementary flux tubes move relatively to a given elementary flux tube, and the helicity flux density is defined per elementary flux tube. From this we define a helicity flux per unit surface, G(Phi). We show that it is a field-weighted average of G(theta) at both photospheric feet of coronal connections. We compare these three densities (G(A), G(theta), G(Phi)) using theoretical examples representing the main cases found in magnetograms (moving magnetic polarities, separating polarities, one polarity rotating around another one and emergence of a twisted flux tube). We conclude that G(theta) is a much better proxy of the magnetic helicity flux density than G(A) because most fake polarities are removed. Indeed G(theta) gives results close to G(Phi) and should be used to monitor the photospheric injection of helicity (when coronal linkages are not well known). These results are applicable to the results of any method determining the photospheric velocities. They can provide separately the flux density coming from shearing and advection motions if plasma motions are known.