6.3 MHD relaxation method

MHD relaxation method means that a reduced set of time-dependent MHD equations is used to compute stationary equilibria:
νv = (∇ × B ) × B, (60 ) e + v × B = 0, (61 ) ∂B--= − ∇ × e, (62 ) ∂t ∇ ⋅ B = 0. (63 )
Here, ν is a fictitious viscosity, v the fluid velocity, and e 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 Mikić 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 Mikić 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 The initial non-equilibrium state is often a potential field in the 3D-box, where the bottom boundary field has been replaced by the measurements.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 v in Equation (60View Equation) 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 (60View Equation) vanishes. The viscosity ν is usually chosen as
1 ν = --|B |2 (64 ) μ
with μ = constant. By combining Equations (60View Equation), (61View Equation), (62View Equation), and (64View Equation) one gets a relaxation process for the magnetic field
∂B-- ∂t = μFMHD, (65 ) ( ) FMHD = ∇ × [(∇-×--B)-×-B-] ×-B . (66 ) B2
For details regarding a currently-used implementation of this approach see Valori et al. (2005).
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