  This paper is the third in a series of papers working towards the
construction of a realistic, evolving, non-linear force-free coronal field
model for the solar magnetic carpet. Here, we present preliminary results of 3D
time-dependent simulations of the small-scale coronal field of the magnetic
carpet. Four simulations are considered, each with the same evolving
photospheric boundary condition: a 48 hr time series of synthetic magnetograms
produced from the model of Meyer, Mackay, van Ballegooijen and Parnell, 2011,
Solar Phys., 272, 29. Three simulations include a uniform, overlying coronal
magnetic field of differing strength, the fourth simulation includes no
overlying field. The build-up, storage and dissipation of magnetic energy
within the simulations is studied. In particular, we study their dependence
upon the evolution of the photospheric magnetic field and the strength of the
overlying coronal field. We also consider where energy is stored and dissipated
within the coronal field. The free magnetic energy built up is found to be more
than sufficient to power small-scale, transient phenomena such as nanoflares
and X-ray bright points, with the bulk of the free energy found to be stored
low down, between 0.5-0.8 Mm. The energy dissipated is presently found to be
too small to account for the heating of the entire quiet Sun corona. However,
the form and location of energy dissipation regions are in qualitative
agreement with what is observed on small scales on the Sun. Future MHD
modelling using the same synthetic magnetograms may lead to a higher energy
release.
