Supergranules in the quiet Sun are outlined by a web-like structure of enhanced magnetic field strength, the so-called magnetic network. We aim to map the magnetic network field around the average supergranule near disk center. We use observations of the line-of-sight component of the magnetic field from the Helioseismic and Magnetic Imager (HMI) onboard the Solar Dynamics Observatory (SDO). The average supergranule is constructed by coaligning and averaging over 3000 individual supergranules. We determine the positions of the supergranules with an image segmentation algorithm that we apply to maps of the horizontal flow divergence measured using time-distance helioseismology. In the center of the average supergranule, the magnetic (intranetwork) field is weaker by about 2.2 Gauss than the background value (3.5 Gauss), whereas it is enhanced in the surrounding ring of horizontal inflows (by about 0.6 Gauss on average). We find that this network field is significantly stronger west (prograde) of the average supergranule than in the east (by about 0.3 Gauss). With time-distance helioseismology, we find a similar anisotropy. The observed anisotropy of the magnetic field adds to the mysterious dynamical properties of solar supergranulation.
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