Inference of horizontal velocity fields from the induction equation in the solar atmosphere

I. Analytical and numerical solutions in 2D

¹ Institut für Sonnenphysik, Georges-Köhler-Allee 401a, D-79110 Freiburg, Germany
² Departamento de Astrofísica, Universidad de La Laguna, E-38205 La Laguna, Tenerife, Spain
³ Instituto de Astrofísica de Canarias, Avd. Vía Láctea s/n, E-38205 La Laguna, Tenerife, Spain

^★ Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 12 December 2025
Accepted: 17 February 2026

Abstract

Context. Spectroscopic and spectropolarimetric observations, which rely on the Doppler effect, only provide access to the line-of-sight component of the solar plasma velocity (i.e., v_z). However, many dynamic processes in the solar atmosphere involve strong horizontal motions (i.e., in the plane perpendicular to the line of sight: v_x, v_y). Existing methods for estimating horizontal velocities are generally insensitive to variations in height (i.e., the z-coordinate), providing them only on a single plane perpendicular to the line of sight: v_x(x, y), v_y(x, y).

Aims. Motivated by the fact that modern analysis techniques (i.e., Stokes inversion) allow us to retrieve the height dependence of v_z 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, develop, 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.

Methods. The components of the two-dimensional magnetic induction equation are discretized via finite differences, leading to an overdetermined system whose solution provides v_y(y, z). The method assumes that B, its time variation B˙, as well as v_z are known. This is currently possible through modern Stokes inversion techniques applied to spatially and temporally resolved spectropolarimetric observations.

Results. Using analytically prescribed values and two-dimensional magnetohydrodynamic simulations of the solar surface, we demonstrate that, in these idealized cases, the horizontal velocity component in a two-dimensional domain, v_y(y, z), can be successfully recovered with a mean error of about 1%. We observe that in the regions where either the modulus of the velocity or its horizontal components are close to zero, its retrieval worsens in comparison to the rest of the domain.

Conclusions. 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.