Our aim is to model the 3D magnetic field structure of the upper solar atmosphere,
including regions of non-negligible plasma beta. We use high-resolution photospheric
magnetic field measurements from SUNRISE/IMaX as boundary condition for a magneto-static
magnetic field model. The high resolution of IMaX allows us to resolve the interface
region between photosphere and corona, but modelling this region is challenging for
the following reasons. While the coronal magnetic field is thought to be force-free
(the Lorentz-force vanishes), this is not the case in the mixed plasma \(\beta\) environment
in the photosphere and lower chromosphere. In our model, pressure gradients and gravity
forces are taken self-consistently into account and compensate the non-vanishing Lorentz-force.
Above a certain height (about 2 Mm) the non-magnetic forces become very weak and consequently
the magnetic field becomes almost force-free. Here we apply a linear approach, where
the electric current density consists of a superposition of a field-line parallel
current and a current perpendicular to the Sun's gravity field. We illustrate the
prospects and limitations of this approach and give an outlook for an extension towards
a non-linear model.