Inference of horizontal velocity fields from the induction equation in the solar atmosphere. I. Analytical and numerical solutions in 2D

Abstract

Spectroscopic and spectropolarimetric observations, which rely on the Doppler effect, only provide access to the line-of-sight component of the solar plasma velocity (vz). However, many dynamic processes in the solar atmosphere involve strong horizontal motions (in the plane perpendicular to the line-of-sight: vx, vy). Existing methods for estimating horizontal velocities are generally insensitive to variations in height (the z-coordinate), providing them only on a single plane perpendicular to the line-of-sight: vx(x,y), vy(x,y). Motivated by the fact that modern analysis techniques allow us to retrieve the height dependence of vz and B, our goal is to infer also this height dependence for the horizontal velocity field in the solar atmosphere. As a first step, we present, and test a method for the two-dimensional case on the (y,z) plane so as to show that the z dependence can be successfully retrieved. The components of the two-dimensional magnetic induction equation are discretized via finite differences, leading to an overdetermined system whose solution provides vy. The method assumes that B, its time variation, as well as vz are known. This is currently possible through modern Stokes inversion techniques applied to spatially and temporally resolved spectropolarimetric observations. Using analytically prescribed values and two-dimensional magneto-hydrodynamic simulations of the solar surface, we demonstrate that, in these idealized cases, the horizontal velocity component in a two-dimensional domain, can be successfully recovered with a mean error of about 1 %. The proposed method successfully retrieves the horizontal velocity field in the (y,z) plane, thereby establishing the foundation for future extensions to three-dimensional reconstructions of the horizontal velocity field.