Doppler-shift properties of the solar transition region (TR) and low corona are investigated in relation to the underlying chromospheric supergranular network, with particular regard to the role of the magnetic field. EUV line properties were obtained from a large raster scan of the solar chromosphere, transition region, and corona acquired by the SUMER spectrometer on board SOHO. The observed region includes an equatorial coronal hole, as well as surrounding quiet-Sun areas. The chromospheric supergranular network pattern is enhanced using a filter defined previously by Aiouaz and coworkers applied to a Lyman continuum image. The filtered continuum image and Dopplergrams are used to produce dispersion plots. We find correlations between the chromospheric network, and the N IV (765.15 Å), O IV (790.19 Å), S V (786.50 Å), O V (760.45 Å) Doppler shifts in quiet Sun and coronal hole. It is established that the maximum inflow (redshift) at transition region temperatures appears statistically toward the center of the network lanes in the quiet-Sun areas and toward the boundary of the network lanes in the coronal hole. Furthermore, while the strong redshifts in the TR lines complement spatially blueshifts from the low corona (Ne VIII 770.41 Å) in a puzzle-like pattern, bidirectional flows (i.e., cospatial strong red- and blueshifts in the TR/Corona) appear predominantly in the coronal hole. The bidirectional flows congregate almost systematically at boundaries of magnetic field concentrations seen in SOHO MDI magnetograms. These results support a coherent interpretation of almost 30 years of unexplained statistical Doppler-shift variations in the transition region and corona for quiet Sun and coronal hole. The proposed scenario involves reconnections between the strong network magnetic field and continuously advected weak field from the supergranular cell interior
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