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 α-distribution on one polarity and the initial potential magnetic field to calculate the electric current density with Equation (12View Equation) and to update the new magnetic field B from the Biot–Savart equation (11View Equation). This scheme is repeated iteratively until a stationary state is reached, where the magnetic field does not change anymore. Amari et al. (1997Jump To The Next Citation Point, 1999) implemented the Grad–Rubin method on a finite difference grid and decomposes Equations (2View Equation) – (4View Equation) into a hyperbolic part for evolving α along the magnetic field lines and an elliptic one to update the magnetic field from Ampere’s law:
B (k) ⋅ ∇ α(k) = 0, (54 ) (k) α |S± = α0±. (55 )
This evolves α from one polarity on the boundary along the magnetic field lines into the volume above. The value of α 0± is given either in the positive or negative polarity:
∇ × B (k+1) = α (k)B (k), (56 ) ∇ ⋅ B (k+1) = 0, (57 ) B (k+1)| ± = B , (58 ) z (k+S1) z0 lim |B | = 0. (59 ) |r|→ ∞
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 α, ensure the existence and unique NLFFF solutions at least for small values of α 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 α 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 (11View Equation) – (12View Equation). The horizontal component of the measured magnetic field is then used to compute the distribution of α on the boundary using Equation (14View Equation). While α is computed this way on the entire lower boundary, the Grad–Rubin method requires only the prescription of α 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. (2009Jump To The Next Citation Point). 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 α-distribution on both polarities with Equation (37View Equation).

As a further step to derive one unique solution the Grad–Rubin approach has been extended by Wheatland and Régnier (2009Jump To The Next Citation Point) and Amari and Aly (2010Jump To The Next Citation Point) by using these two different solutions (from different polarities) to correct the α-distribution on the boundaries and to find finally one consistent solution by an outer iterative loop, which changes the α-distribution on the boundary. An advantage in this approach is that one can specify where the α-distribution, as computed by Equation (14View Equation), 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 α on both polarities, allows also to specify where the initial distribution of α is trustworthy.

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