### 6.3 MHD relaxation method

MHD relaxation method means that a reduced set of time-dependent MHD equations is used to
compute stationary equilibria:
Here, is a fictitious viscosity, the fluid velocity, and the electric field. For general MHD
equilibria the approach was proposed by Chodura and Schlüter (1981). Applications to force-free coronal
magnetic fields can be found in MikiÄ‡ and McClymont (1994), Roumeliotis (1996), and McClymont et al.
(1997). In principle, any time-dependent MHD code can be used for this aim. The first NLFFF
implementation of this methods used the code developed by MikiÄ‡ et al. (1988). MHD relaxation
means that an initial non-equilibrium state is relaxed towards a stationary state, here NLFFF.
The initial non-equilibrium state is often a potential field in the 3D-box, where the bottom
boundary field has been replaced by the measurements. This leads to large deviations from the
equilibrium close to this boundary. As a consequence one finds a finite plasma flow velocity in
Equation (60) because all non-magnetic forces accumulate in the velocity field. This velocity
field is reduced during the relaxation process and the force-free field equations are obviously
fulfilled when the left-hand side of Equation (60) vanishes. The viscosity is usually chosen as
with constant. By combining Equations (60), (61), (62), and (64) one gets a relaxation process for
the magnetic field
For details regarding a currently-used implementation of this approach see Valori et al. (2005).