## The power spectrum of solar convection flows from high-resolution observations and 3D simulations

^{1}
Instituto de Astrofísica de Canarias,
via Lactea, s/n, 38205
La Laguna, Tenerife,
Spain

e-mail:
fmi@iac.es

^{2}
Dept. of Astrophysics, Universidad de La Laguna,
38200,
La Laguna, Tenerife,
Spain

Received:
31
July
2013

Accepted:
16
January
2014

*Context. *Understanding solar surface magnetoconvection requires the
study of the Fourier spectra of the velocity fields. Nowadays, observations are available
that resolve very small spatial scales, well into the subgranular range, almost reaching
the scales routinely resolved in numerical magnetoconvection simulations. Comparison of
numerical and observational data at present can provide an assessment of the validity of
the observational proxies.

*Aims. *Our aims are: (1) to obtain Fourier spectra for the photospheric
velocity fields using the spectropolarimetric observations with the highest spatial
resolution so far (~120 km),
thus reaching for the first time spatial scales well into the subgranular range; (2) to
calculate corresponding Fourier spectra from realistic 3D numerical simulations of
magnetoconvection and carry out a proper comparison with their observational counterparts
considering the residual instrumental degradation in the observational data; and (3) to
test the observational proxies on the basis of the numerical data alone, by comparing the
actual velocity field in the simulations with synthetic observations obtained from the
numerical boxes.

*Methods. *(a) For the observations, data from the SUNRISE/IMaX
spectropolarimeter are used. (b) For the simulations, we use four series of runs obtained
with the STAGGER code for different average signed vertical magnetic field values
(0, 50, 100, and 200 G). Spectral line profiles are
synthesized from the numerical boxes for the same line observed by IMaX (Fe I 5250.2 Å)
and degraded to match the performance of the IMaX instrument. Proxies for the velocity
field are obtained via Dopplergrams (vertical component) and local correlation tracking
(LCT, for the horizontal component). Fourier power spectra are calculated and a comparison
between the synthetic and observational data sets carried out. (c) For the internal
comparison of the numerical data, velocity values on constant optical depth surfaces are
used instead of on horizontal planes.

*Results. *A very good match between observational and simulated Fourier
power spectra is obtained for the vertical velocity data for scales between
200 km and 6 Mm. Instead, a clear vertical shift is
obtained when the synthetic observations are not degraded to emulate the degradation in
the IMaX data. The match for the horizontal velocity data is much less impressive because
of the inaccuracies of the LCT procedure. Concerning the internal comparison of the direct
velocity values of the numerical boxes with those from the synthetic observations, a high
correlation (0.96) is obtained
for the vertical component when using the velocity values on the log *τ*_{500} = −1
surface in the box. The corresponding Fourier spectra are near each other. A lower maximum
correlation (0.5) is reached
(at log *τ*_{500} = 0) for the horizontal
velocities as a result of the coarseness of the LCT procedure. Correspondingly, the
Fourier spectra for the LCT-determined velocities is well below that from the actual
velocity components.

*Conclusions. *As measured by the Fourier spectra, realistic numerical
simulations of surface magnetoconvection provide a very good match to the observational
proxies for the photospheric velocity fields at least on scales from several Mm down to
around 200 km. Taking into
account the spatial and spectral instrumental blurring is essential for the comparison
between simulations and observations. Dopplergrams are an excellent proxy for the vertical
velocities on constant-*τ* isosurfaces, while LCT is a much less reliable
method of determining the horizontal velocities.

Key words: Sun: photosphere / Sun: granulation / convection / magnetohydrodynamics (MHD) / turbulence

*© ESO, 2014*