Sparse Reconstruction of Electric Fields from Radial Magnetic Data

Abstract

The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Preprint typeset using L A T E X style AASTeX6 v. ABSTRACTAccurate estimates of the horizontal electric field on the Sun's visible surface are important not only for estimating the Poynting flux of magnetic energy into the corona but also for driving time-dependent magnetohydrodynamic models of the corona. In this paper, a method is developed for estimating the horizontal field from a sequence of radial-component magnetic field maps. This problem of inverting Faraday's law has no unique solution. Unfortunately, the simplest solution (a divergence-free electric field) is not realistically localized in regions of non-zero magnetic field, as would be expected from Ohm's law. Our new method generates instead a localized solution, using a basis pursuit algorithm to find a sparse solution for the electric field. The method is shown to perform well on test cases where the input magnetic maps are flux balanced, in both Cartesian and spherical geometries. However, we show that if the input maps have a significant imbalance of flux -usually arising from data assimilation -then it is not possible to find a localized, realistic, electric field solution. This is the main obstacle to driving coronal models from time sequences of solar surface magnetic maps.

Optimization of Photospheric Electric Field Estimates for Accurate Retrieval of Total Magnetic Energy Injection

“…Kazachenko, Fisher, and Welsch (2014) argue that the error introduced by this is small and employ a central difference scheme also for the first order derivatives. However, we wish to reproduce ∂B z /∂t from Faraday's law exactly, and therefore employ the following consistent finite difference formulas for the spatial derivatives (used in a similar fashion also by Yeates, 2017):…”

The PDFI_SS Electric Field Inversion Software

“…Whichever polarity is the minority polarity then has the flux in each of its constituent pixels enhanced by a constant relative factor such that the total net radial flux is zero. This is similar to a technique proposed by Yeates (2017), except that in the latter case, both polarity regions are adjusted.…”

“…Weinzierl et al (2016a,b) presented solutions for the horizontal components of the electric field that combined a solution for the "inductive" contributions to the horizontal electric field components, determined from the time derivative of the radial magnetic field, with a non-inductive contribution that was determined from surface flux transport models. Yeates (2017) derived electric field solutions that combine solutions for the same inductive contribution as those above, but with the non-inductive contribution to the electric field determined from a "sparseness" constraint, to minimize unphysical artifacts of the horizontal electric field from the purely inductive solution. Lumme et al (2017); Price et al (2019) used solutions for all three components of the electric field using time derivatives for all three components of B, as described for the "PTD" solutions in KFW14, using a centered grid formalism.…”

A Comparison of Sparse and Non-sparse Techniques for Electric-Field Inversion from Normal-Component Magnetograms

“…Subsequently, the obtained E h does not strictly reproduce that expected from Ohm's law when considering the true physical system of a plasma velocity acting on a magnetic field. To counter this problem and include the missing inductive component when using only normal-component magnetograms, Yeates (2017) put forward a new method for deriving the horizontal electric field. The aim of this technique was to include a non-inductive component and minimise the number of locations where E h is non-zero, thus producing a "sparser" solution compared to standard Poisson-solver techniques.…”

“…This formulation was then solved using a least-squares technique using a basis pursuit algorithm that minimises the L 1 -norm. Yeates (2017) successfully tested this technique through a series of 2D idealised and data-based simulations of the normal magnetic field component in the photosphere, however, no 3D simulations of the corona were considered. Thus, the consequence of applying a sparse electric field determined through the L 1 -norm, in a 3D coronal magnetic field simulation is unclear.…”