A&A 439, 1191-1203 (2005)

DOI: 10.1051/0004-6361:20052663

## Photospheric flux density of magnetic helicity

**E. Pariat**

^{1}, P. Démoulin^{1}and M. A. Berger^{2}^{1}Observatoire de Paris, LESIA, UMR 8109 (CNRS), 92195 Meudon, France

e-mail: [etienne.pariat;pascal.demoulin]@obspm.fr

^{2}Department of Mathematics, University College London, UK

e-mail: m.berger@ucl.ac.uk

(Received 10 January 2005 / Accepted 12 May 2005 )

** Abstract **

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} (=
) which involves the vector
potential 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,
, 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,
. We show that it is a field-weighted average of
at both photospheric feet of coronal connections. We compare these three densities (*G*_{A},
, ) 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
is a much better proxy of the magnetic helicity
flux density than *G*_{A} because most fake polarities are removed. Indeed
gives results close to 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.

**Key words:**Sun: magnetic fields

**--**Sun: photosphere

**--**Sun: corona

**©**

*ESO 2005*