### 6.2 Grad–Rubin method

The Grad–Rubin method has been originally proposed (but not numerically implemented) by Grad and
Rubin (1958) for the application to fusion plasma. The first numerical application to coronal magnetic fields
was carried out by Sakurai (1981). The original Grad–Rubin approach uses the -distribution on one
polarity and the initial potential magnetic field to calculate the electric current density with Equation (12)
and to update the new magnetic field from the Biot–Savart equation (11). This scheme
is repeated iteratively until a stationary state is reached, where the magnetic field does not
change anymore. Amari et al. (1997, 1999) implemented the Grad–Rubin method on a finite
difference grid and decomposes Equations (2) – (4) into a hyperbolic part for evolving along
the magnetic field lines and an elliptic one to update the magnetic field from Ampere’s law:
This evolves from one polarity on the boundary along the magnetic field lines into the volume above.
The value of is given either in the positive or negative polarity:
An advantage from a mathematical point of view is that the Grad–Rubin approach solves the nonlinear
force-free equations as a well-posed boundary value problem. As shown by Bineau (1972) the
Grad–Rubin-type boundary conditions, the vertical magnetic field and for one polarity the distribution of
, ensure the existence and unique NLFFF solutions at least for small values of and weak
nonlinearities. See Amari et al. (1997, 2006) for more details on the mathematical aspect of this approach.
The largest-allowed current and the corresponding maximum values of for which one can expect
convergence of the Grad–Rubin approach have been studied in Inhester and Wiegelmann (2006).
Starting from an initial potential field the NLFFF equations are solved iteratively in the form of
Equations (11) – (12). The horizontal component of the measured magnetic field is then used to compute
the distribution of on the boundary using Equation (14). While is computed this way on the
entire lower boundary, the Grad–Rubin method requires only the prescription of for one
polarity. For measured data which contain noise, measurement errors, finite forces, and other
inconsistencies the two solutions can be different: However, see for example the extrapolations from
Hinode data carried out in DeRosa et al. (2009). While both solutions are based on well-posed
mathematical problems, they are not necessary consistent with the observations on the entire
lower boundary. One can check the consistency of the -distribution on both polarities with
Equation (37).
As a further step to derive one unique solution the Grad–Rubin approach has been extended by
Wheatland and Régnier (2009) and Amari and Aly (2010) by using these two different solutions (from
different polarities) to correct the -distribution on the boundaries and to find finally one consistent
solution by an outer iterative loop, which changes the -distribution on the boundary. An advantage in
this approach is that one can specify where the -distribution, as computed by Equation (14), is
trustworthy (usually in strong field regions with a low measurement error in the transverse field) and
where not (in weak field regions). This outer iterative loop, which aims at finding a consistent
distribution of on both polarities, allows also to specify where the initial distribution of is
trustworthy.