  In the quiet solar photosphere, the mixed polarity fields form a magnetic
carpet, which continuously evolves due to dynamical interaction between the
convective motions and magnetic field. This interplay is a viable source to
heat the solar atmosphere. In this work, we used the line-of-sight (LOS)
magnetograms obtained from the Helioseismic and Magnetic Imager (HMI) on the
\textit{Solar Dynamics Observatory} (\textit{SDO}), and the Imaging
Magnetograph eXperiment (IMaX) instrument on the \textit{Sunrise} balloon-borne
observatory, as time dependent lower boundary conditions, to study the
evolution of the coronal magnetic field. We use a magneto-frictional relaxation
method, including hyperdiffusion, to produce time series of three-dimensional
(3D) nonlinear force-free fields from a sequence of photospheric LOS
magnetograms. Vertical flows are added up to a height of 0.7 Mm in the modeling
to simulate the non-force-freeness at the photosphere-chromosphere layers.
Among the derived quantities, we study the spatial and temporal variations of
the energy dissipation rate, and energy flux. Our results show that the energy
deposited in the solar atmosphere is concentrated within 2 Mm of the
photosphere and there is not sufficient energy flux at the base of the corona
to cover radiative and conductive losses. Possible reasons and implications are
discussed. Better observational constraints of the magnetic field in the
chromosphere are crucial to understand the role of the magnetic carpet in
coronal heating.
