Observational constraints on well-posed reconstruction methods and the optimization-Grad-Rubin method

Abstract

Context. Grad-Rubin type methods are interesting candidates for reconstructing the force-free magnetic field of a solar coronal region. As input these methods, however, require the normal component B n of the field on the whole boundary of the numerical box and the force-free function α on the part of the boundary where B n > 0 (or B n < 0), while observations provide data only on its lower photospheric part. Moreover, they introduce an unpleasing asymmetry between the opposite polarity parts of the boundary, and certainly do not take full advantage of the available data on α. Aims. We address these issues resulting from observations. We present a possible way to supply the missing information about B n and α on the non-photospheric sides of the box, and to use more effectively the data provided by the measurements. Methods. We introduce the optimization-Grad-Rubin method (OGRM), which is in some sense midway between optimization methods and the standard Grad-Rubin methods. It is based on an iterative scheme in which the α used as a boundary condition is imposed to take identical values at both footpoints of any field line and to be as close as possible to the α provided by the measurements on the photosphere. The degree of "closeness" is measured by an "error functional" containing a weight function reflecting the confidence that can be placed on the observational data. Results. The new method is implemented in our code XTRAPOL, along with some technical improvements. It is thus tested for two specific choices of the weight function by reconstructing a force-free field from data obtained by perturbing in either a random or a non-random way boundary values provided by an exact solution.

Magnetic Field Extrapolations into the Corona: Success and Future Improvements

“…In the recent years, the optimization scheme has been developed to include more constraints and thus to obtain an equilibrium closer to a force-free field. -Grad-Rubin methods: the nonlinear force-free equation being a system of partial differential equations of mixed type, the method described by Grad and Rubin (1958) consists in separating the elliptic part (force-free equation) and the hyperbolic part (gradient of α) of the system, each system being then linear and easier to solve (Grad and Rubin, 1958;Sakurai, 1981;Amari et al, 1997;Amari, Boulmezaoud, and Mikic, 1999;Wheatland, 2004;Amari, Boulmezaoud, and Aly, 2006;Inhester and Wiegelmann, 2006;Wheatland, 2006;Wheatland and Régnier, 2009;Malanushenko, Longcope, and McKenzie, 2009;Amari and Aly, 2010;Malanushenko et al, 2012). The Grad-Rubin method requires as boundary conditions the vertical/radial component of the magnetic field on each boundary, as well as the distribution of α in a chosen polarity.…”

Guiding Nonlinear Force-Free Modeling Using Coronal Observations: First Results Using a Quasi-Grad-Rubin Scheme

“…Noise in α at the lower boundary and limits to the field of view prevent this condition from being satisfied, and the problem is in general ill-posed (Aly 1989). Techniques exist for "pre-processing" of the boundary data to attempt to mitigate this problem (e.g., Wiegelmann et al 2006Wiegelmann et al , 2008 and for formulating a well-posed problem given the uncertainties in the measurements (Amari & Aly 2010;Wheatland & Régnier 2009;Wheatland & Leka 2011).…”

Reconstruction of the solar coronal magnetic field in spherical geometry

“…We just recall here that the most frequently used ones are (i) the optimization method (Wiegelmann 2004), which uses all photospheric data and tries to minimize a cost function measuring the difference between the computed transverse field and the measured one; (ii) the vertical integration method, which also uses all photospheric data (Song et al 2006); (iii) the relaxation methods (Valori et al 2005), in which the coronal field is obtained as the result of an MHD evolution driven by boundary conditions depending on the observed field; and (iv) the Grad-Rubin methods (Sakurai 1981;Amari et al 1999Amari et al , 2006Wheatland 2007;Wheatland & Regnier 2009), which only use a part of the observational data. There is also a hybrid method called OGRM (Amari & Aly 2010) which combines a Grad-Rubin iterative scheme with an optimization scheme by selecting the most appropriate boundary values of the force-free function α at each step.…”