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

1. Problem Statement

Current spectroscopic and spectropolarimetric observations of the Sun rely on the Doppler effect, which only provides the line-of-sight (LOS) velocity component ($v_z$). However, many critical solar atmospheric phenomena (e.g., sub-granular flows, magnetic tornadoes, reconnection events) involve significant horizontal motions ($v_x, v_y$).

Existing methods for retrieving horizontal velocities, such as Local Correlation Tracking (LCT) and its variants (FLCT, deep learning), generally provide velocity maps on a single plane perpendicular to the LOS ($v_x(x,y), v_y(x,y)$). These methods lack height dependence (the $z$-coordinate), preventing a full 3D understanding of atmospheric dynamics. Furthermore, traditional "inductive techniques" often rely solely on the vertical component of the induction equation, which limits the retrieval of the full velocity vector at multiple heights.

The Goal: To develop and test a method capable of inferring the height-dependent horizontal velocity field ($v_h(x, y, z)$) by utilizing the full magnetic induction equation, assuming that the magnetic field vector ($\mathbf{B}$), its time derivative ($\dot{\mathbf{B}}$), and the LOS velocity ($v_z$) are known via modern Stokes inversion techniques.

2. Methodology

The authors propose a numerical method based on the ideal Magnetohydrodynamics (MHD) induction equation:
$$\frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{v} \times \mathbf{B})$$

2.1. Dimensional Reduction (2D Case)

To validate the approach, the study focuses on a 2D domain in the $(y, z)$ plane, where $z$ is the LOS and $y$ is the horizontal direction. The authors assume $v_x = B_x = 0$. Under these constraints, the induction equation yields two non-trivial components:

1. $\dot{B}_y = \frac{\partial}{\partial z} (v_y B_z - v_z B_y)$
2. $\dot{B}_z = \frac{\partial}{\partial y} (v_z B_y - v_y B_z)$

In this setup, $B_y, B_z, \dot{B}_y, \dot{B}_z$, and $v_z$ are treated as known inputs (derived from observations/inversions), leaving the horizontal velocity component $v_y(y, z)$ as the single unknown.

2.2. Discretization and Linear System

The method employs a finite difference scheme using a 5-point centered stencil to approximate spatial derivatives.

• This discretization transforms the partial differential equations into an overdetermined linear system of the form $\mathbf{A}\mathbf{x} = \mathbf{c}$.
• For a grid of $N \times M$ points, there are $N \times M$ unknowns ($v_y$ at each grid point) but $2 \times N \times M$ equations (one for $\dot{B}_y$ and one for $\dot{B}_z$ at each point).
• The matrix $\mathbf{A}$ is constructed using coefficients derived from the known magnetic field components and their derivatives.

2.3. Boundary Conditions

To handle the boundaries of the computational domain while maintaining the 5-point stencil, the authors introduce ghost cells. Two types of boundary conditions are discussed:

• Dirichlet: Requires known values of $v_y$ at the boundary (often unavailable in practice).
• Neumann (Symmetric): Assumes the derivative of $v_y$ is zero at the boundary (or symmetric behavior). The authors utilize vanishing Neumann boundary conditions for their tests, as they do not require prior knowledge of the velocity at the edges.

2.4. Solution

The overdetermined system is solved using a Least Squares approach:
$$\mathbf{x} = (\mathbf{A}^T \mathbf{A})^{-1} \mathbf{A}^T \mathbf{c}$$
The solution is computed using the LAPACK dgesv subroutine (LU decomposition).

3. Key Contributions

• Novel Formulation: Extends inductive velocity retrieval methods from a single 2D plane to a 2D vertical slice, enabling the recovery of the $z$-dependence of horizontal velocities.
• Overdetermined System: Demonstrates that utilizing both components of the induction equation in a 2D plane creates a solvable, overdetermined system for the horizontal velocity.
• Validation Framework: Establishes a rigorous testing protocol using both analytical functions and realistic MHD simulations to quantify retrieval accuracy.

4. Results

The method was tested in two scenarios:

4.1. Analytical Test

• Setup: Prescribed analytical functions for $\mathbf{B}$ and $\mathbf{v}$ on a $100 \times 100$ grid.
• Performance:
  • Mean Error: $\approx 0.82\%$ in the magnitude of $v_y$.
  • Median Error: $\approx 1.41\%$.
  • Directional Error: The angle between the true and retrieved velocity vectors had a mean of $\approx 0.5^\circ$.
  • Correlation: Pearson correlation coefficient ($r$) was $\approx 0.99985$.
• Limitations: Errors increased significantly in regions where $v_y \to 0$ or $\|\mathbf{v}\| \to 0$, as the relative error definition becomes unstable when the denominator approaches zero.

4.2. Numerical Simulation (CO5BOLD)

• Setup: Used data from a 2D magneto-hydrodynamic simulation of solar surface magneto-convection.
• Performance:
  • Mean Error: $\approx 1.06\%$.
  • Median Error: $\approx 0.11\%$.
  • Directional Error: Mean angle error $\approx 0.1^\circ$.
  • Correlation: $r \approx 0.99999$.
• Observation: The retrieved velocity field was perceptually identical to the original simulation data. Unlike the analytical case, errors did not strictly correlate with zero velocity magnitude but rather with regions where the horizontal component was small relative to the vertical component.

5. Significance and Future Work

• Foundation for 3D: This paper establishes the mathematical and numerical foundation for retrieving 3D horizontal velocity fields ($v_x, v_y$ as functions of $x, y, z$). The 3D extension would utilize all three components of the induction equation.
• Physical Insight: The ability to retrieve height-dependent flows is crucial for calculating magnetic energy flux and helicity injection into the corona, which drives solar flares and coronal mass ejections.
• Degeneracies: The authors address theoretical degeneracies (adding a gradient of a scalar to $\mathbf{v} \times \mathbf{B}$). They argue that in 3D, the known $v_z$ (from Doppler) breaks these degeneracies, allowing for a unique solution.
• Future Challenges: The authors plan to extend this to 3D (which increases computational cost significantly due to matrix size) and test the method under realistic observational constraints, including:
  • Noise and systematic errors in Stokes inversions.
  • Finite time sampling effects (non-zero $\Delta t$).
  • The impact of non-ideal MHD effects (magnetic diffusion).

Conclusion: The proposed method successfully retrieves the horizontal velocity field in a 2D vertical plane with high accuracy (mean error $\sim 1\%$) in idealized scenarios, proving the feasibility of inferring the vertical structure of solar atmospheric flows using the full induction equation.