Context :

Aims : We study the properties of internal gravity waves ( IGWs ) detected in synthetic observations that are obtained from realistic numerical simulation of the solar atmosphere .

Methods : We used four different simulations of the solar magneto-convection performed using the CO ^ { 5 } BOLD code .
A magnetic-field-free model and three magnetic models were simulated .
The latter three models start with an initial vertical , homogeneous field of 10 , 50 , and 100 G magnetic flux density , representing different regions of the quiet solar surface .
We used the NICOLE code to compute synthetic spectral maps from all the simulated models for the two magnetically insensitive neutral iron lines Fe I \lambda \lambda 5434 Å and 5576 Å .
We carried out Fourier analyses of the intensity and Doppler velocities to derive the power , phase , and coherence in the k _ { h } - \omega diagnostic diagram to study the properties of internal gravity waves .

Results : We find the signatures of the internal gravity waves in the synthetic spectra to be consistent with observations of the real Sun .
The effect of magnetic field on the wave spectra is not as clearly discernible in synthetic observations as in the case of numerical simulations .
The phase differences obtained using the spectral lines are significantly different from the phase differences in the simulation .
The phase coherency between two atmospheric layers in the gravity wave regime is height dependent and is seen to decrease with the travel distance between the observed layers .
In the studied models , the lower atmosphere shows a phase coherency above the significance level for a height separation of \sim 400 km , while in the chromospheric layers it reduces to \sim 100–200 km depending on the average magnetic flux density .

Conclusions : We conclude that the energy flux of IGWs determined from the phase difference analysis may be overestimated by an order of magnitude .
Spectral lines that are weak and less temperature sensitive may be better suited to detecting internal waves and accurately determining their energy flux in the solar atmosphere .