  Modeling the interface region between solar photosphere and corona is
challenging, because the relative importance of magnetic and plasma forces
change by several orders of magnitude. While the solar corona can be modeled by
the force-free assumption, we need to take care about plasma forces (pressure
gradient and gravity) in photosphere and chromosphere, here within the
magneto-hydro-static (MHS) model. We solve the MHS equations with the help of
an optimization principle and use vector magnetogram as boundary condition.
Positive pressure and density are ensured by replacing them with two new basic
variables. The Lorentz force during optimization is used to update the plasma
pressure on the bottom boundary, which makes the new extrapolation works even
without pressure measurement on the photosphere. Our code is tested by using a
linear MHS model as reference. From the detailed analyses, we find that the
newly developed MHS extrapolation recovers the reference model at high
accuracy. The MHS extrapolation is, however, numerically more expensive than
the nonlinear force-free field (NLFFF) extrapolation and consequently one
should limit their application to regions where plasma forces become important,
e.g. in a layer of about 2 Mm above the photosphere.
