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

TL;DR

This work tackles recovering the solar atmosphere's horizontal velocity components from magnetic-field evolution by leveraging the full induction equation under ideal MHD. The authors discretize the 2D induction equation in the $(y,z)$ plane, forming an overdetermined linear system for $v_y$ using known ${f B}$, ${\dot{\bf B}}$, and $v_z$, and solve it with a least-squares approach, including careful boundary treatment. Through analytical test fields and CO5BOLD 2D MHD simulations, they demonstrate that $v_y(y,z)$ can be retrieved with a mean error around 1% and near-unity correlation to the true field, establishing a foundation for extending the method to 3D and to more realistic observational conditions. The work advances velocity inference beyond fixed-height, line-of-sight analyses and highlights practical considerations such as boundary handling and degeneracies, with significant implications for energy and helicity transport studies in the solar atmosphere.

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.

Figures (9)

• Figure 1: Structure of the matrix of the system $\widehat{{\bf A}}$ for a $N \times M = 10 \times 10$ domain, defined by $k$ rows and $r$ columns. In red we show the position of the $\widehat{A}_{kr}$ coefficients related to Eq. \ref{['eq:induction_eq_2D_yz_y']} and in blue those related to Eq. \ref{['eq:induction_eq_2D_yz_z']}.
• Figure 2: Example of a 2D grid for a $10 \times 10$ domain. The solution of the system is evaluated in the blue points. These points are called inner cells. The boundary condition must be specified in all red points. These are ghost or boundary cells. The white points in the corners are not used in the scheme considered here. In a different numerical scheme, they may play a role.
• Figure 3: 2D analytical magnetic field. The color map shows the magnitude of the magnetic field $\|{\bf B}\|$. The green arrows show its direction. The colorbar is saturated at $400$ G to show the lower values of the magnetic field. The maximum value of the magnetic field at the selected region is $\approx 415$ G.
• Figure 4: Velocity field maps in the plane $(y,z)$. Panel (a) shows the velocity field defined analytically: ${\rm\bf {v}} = (v_y,v_z)$ (see Eqs. \ref{['eq:yz_vy_ex']} and \ref{['eq:yz_vz_ex']}), whereas panel (b) provides the inferred velocity field: $\widetilde{{\rm\bf {v}}} = (\widetilde{v_y},v_z)$. The modulus of the vectors is encoded in the color of the arrows as indicated in the colorbar.
• Figure 5: Errors of the inference of $\widetilde{v_{y}}$ in the analytical 2D case. The left panel shows the relative error $\varepsilon_{\widetilde{v_{y}}}$ of the inferred velocity. The right panel shows the angle between the analytical velocity (${\bf v}$) and the inferred one (${\widetilde{\bf v}}$). See text for more details. Both plots also indicate the mean and the median of these errors.