  The average Doppler shift shown by spectral lines formed from the
chromosphere to the corona reveals important information on the mass and energy
balance of the solar atmosphere, providing an important observational
constraint to any models of the solar corona. Previous spectroscopic
observations of vacuum ultra-violet (VUV) lines have revealed a persistent
average wavelength shift of lines formed at temperatures up to 1 MK. At higher
temperatures, the behaviour is still essentially unknown. Here we analyse
combined SUMER/SoHO and EIS/Hinode observations of the quiet Sun around disk
centre to determine, for the first time, the average Doppler shift of several
spectral lines formed between 1 and 2 MK, where the largest part of the quiet
coronal emission is formed. The measurements are based on a novel technique
applied to EIS spectra to measure the difference in Doppler shift between lines
formed at different temperatures. Simultaneous wavelength-calibrated SUMER
spectra allow establishing the absolute value at the reference temperature of 1
MK. The average line shifts at 1 MK < T < 1.8 MK are modestly, but clearly
bluer than those observed at 1 MK. By accepting an average blue shift of about
(-1.8+/-0.6) km/s at 1 MK (as provided by SUMER measurements), this translates
into a maximum Doppler shift of (-4.4+/-2.2) km/s around 1.8 MK. The measured
value appears to decrease to about (-1.3+/-2.6) km/s at the Fe XV formation
temperature of 2.1 MK. The measured average Doppler shift between 0.01 and 2.1
MK, for which we provide a parametrisation, appears to be qualitatively and
roughly quantitatively consistent with what foreseen by 3-D coronal models
where heating is produced by dissipation of currents induced by photospheric
motions and by reconnection with emerging magnetic flux.
