Strong organized magnetic fields have been studied in the upper main sequence
chemically peculiar stars for more than half a century. However, only recently
have observational methods and numerical techniques become sufficiently mature
to allow us to record and interpret high-resolution four Stokes parameter
spectra, leading to the first assumption-free magnetic field models of these
stars. Here we present a detailed magnetic Doppler imaging analysis of the
spectropolarimetric observations of the prototypical magnetic Ap star alpha^2
CVn. The surface abundance distributions of Fe and Cr and a full vector map of
the stellar magnetic field are reconstructed in a self-consistent inversion
using our state-of-the-art magnetic Doppler imaging code Invers10. We succeeded
in reproducing most of the details of the available spectropolarimetric
observations of alpha^2 CVn with a magnetic map which combines a global
dipolar-like field topology with localized spots of higher field intensity. We
demonstrate that these small-scale magnetic structures are inevitably required
to fit the linear polarization spectra; however, their presence cannot be
inferred from the Stokes I and V observations alone. Our magnetic Doppler
imaging analysis of alpha^2 CVn and previous results for 53 Cam support the
view that the upper main sequence stars can harbour fairly complex surface
magnetic fields which resemble oblique dipoles only at the largest spatial
scales. Spectra in all four Stokes parameters are absolutely essential to
unveil and meaningfully characterize this field complexity in Ap stars. We
therefore suggest that understanding magnetism of stars in other parts of the
H-R diagram is similarly incomplete without investigation of their linear
polarization spectra.Comment: 16 pages, 12 figures; Accepted for publication by Astronomy &
Astrophysic