On acoustic and gravity waves in the solar photosphere and their energy transport

Aims. We study acoustic and atmospheric gravity waves in the quiet Sun to estimate their energy transport to the chromosphere. Methods. A two-dimensional time sequence from quiet Sun disc centre was analysed with simultaneous spectroscopic observations in Fe i 5576 angstrom and Fe i 5434 angstrom (both with Lande factor g = 0). We calculated response functions of the velocities for the line minimum shifts and atmospheric transmissions of waves for the two lines. For this, NLTE line formation in granular and intergranular model atmospheres from numerical simulations were performed. For the interpretation of the observed waves and for the estimates of energy fluxes, we assumed adiabatic propagation of plane waves in an isothermal model atmosphere. Fourier analyses of intensity and velocity fluctuations were carried out. They yield power, phase, and coherence as functions of frequency nu (from temporal Fourier transforms) and in the k(h-nu) plane (from three-dimensional transforms). The power spectra, together with the mass densities at velocity formation heights, give then the energy fluxes. Results. The rms velocities found here in the acoustic and gravity wave domains are lower by a factor similar to 1.5 as in earlier work. We therefore admit a factor of 2 for an upward correction of the estimated fluxes. For acoustic waves we find: 1) upward propagating waves are present on the Sun with frequencies up to 14-15 mHz (periods U approximate to 70 s); 2) the approximation of plane adiabatic waves in an isothermal atmosphere appears adequate for estimating the energy fluxes; 3) the acoustic energy fluxes are in the same range as found in our earlier work from ground-based, two-dimensional spectroscopy, 1500-3100 Wm(-2) at an atmospheric height of similar to 380 km and 1300-2700 Wm(-2) at 570 km. The energy flux carried by gravity waves is difficult to determine. We find: 1) phase and coherence spectra between continuum and velocity fluctuations show that convective overshoot and gravity waves are superimposed. We account for the convective flows using these coherence spectra. 2) At low frequencies, the vertical wavelength Lambda(z) can be short (<< 300 km), yielding large corrections for atmospheric transmissions (factors >100). We thus exclude from the flux estimates waves with |k(z)| > 20 Mm(-1) and with vertical group velocities nu(gr,z) < 0.3 kms(-1). They are likely to be strongly reduced in amplitude by radiative damping. 3) With these caveats, the energy fluxes carried by gravity waves are found in the range of 4000-8200 Wm(-2) at 380 km and 700-1400 Wm(-2) at 570 km. Gravity waves thus also contribute to the energy transport into the chromosphere.