# Resonant Fast-Alfvén Wave Coupling in a 3D Coronal Arcade

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model and Method

#### 2.1. Equilibrium and Coordinates

#### 2.2. Linearised Equations

**u**and

**b**, respectively. V is the background Alfvén speed, $\nu $ is a linear drag term and $\eta $ the resistivity. These equations have been made dimensionless by normalising using the equilibrium magnetic field strength, ${B}_{0}$, and density, ${\rho}_{0}$, at appropriate reference locations, as well as distance, ${L}_{0}$, of the loop apex from the origin. These quantities may be used to obtain the normalising speed, ${V}_{0}={B}_{0}/\sqrt{{\mu}_{0}{\rho}_{0}}$, and time, ${T}_{0}={L}_{0}/{V}_{0}$.

#### 2.3. Boundary Conditions

#### 2.4. Normal Modes

#### 2.5. Numerical Details

## 3. Results

#### 3.1. Location of Resonant Alfvén Waves

#### 3.2. Polarisation of Resonant Alfvén Waves

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**A sketch of the model field geometry. (

**a**) A 3D (three-dimensional) view of the arcade based on a line dipole aligned with the z axis and sitting below the photosphere. The blue lines represent the portion of the magnetic field lines above the photosphere. (

**b**) A cross-sectional view of the simulation domain and its boundaries in terms of field-aligned coordinates, $\alpha $ and $\gamma $ (1). (

**c**) The variation of Alfvén speed (V) in the vertical $(x,z)$ plane (also the $\gamma =0$ plane) that runs the length of the arcade. The vertical white lines indicate the z positions used to study the Alfvén wave fields displayed in Figure 2.

**Figure 2.**Magnitude of the normal mode’s field aligned vorticity, $|{\omega}_{\Vert}|$ (

**a**,

**c**,

**e**), and current, $|{j}_{\Vert}|$ (

**b**,

**d**,

**f**) at $z=0.5$ (

**a**,

**b**), 0.24 (

**c**,

**d**), and 0.0 (

**e**,

**f**).

**Figure 3.**(

**a**) Time-averaged energy density in the plane $\beta =z=0.5$. The intersection with three surfaces of the field-aligned coordinate (${\gamma}_{1}=0$, $\gamma ={\gamma}_{2}$ and $\gamma ={\gamma}_{3}$) are also shown. (

**b**) The location of where the energy density maximizes ${\alpha}_{W}(\beta ,\gamma )$ for the three surfaces $\gamma ={\gamma}_{1}$ (black symbols), ${\gamma}_{2}$ (red line) and ${\gamma}_{3}$ (blue dashed line). (

**c**) A close-up view of the energy density on the field lines carrying a 3D Alfvén resonance (identified by the red rectangle in (

**b**)) plotted in the $\gamma ={\gamma}_{1}=0$ plane (equivalent to the $y=0$ plane) with velocity vectors overplotted. See text for details.

**Figure 4.**(

**a**) The surface containing the resonant Alfvén waves (identified from Figure 3b as a maximum of the time-averaged energy density). (

**b**) A cut in the vertical $y=0$ plane of the magnitude of the field-aligned vorticity. Overplotted is the Resonance Map showing permissible Resonant Paths (white lines) and the boundaries of the Resonant Zone (red lines). See text for details.

**Figure 5.**(

**a**) The variation of natural Alfvén frequency, ${\omega}_{A}$, with polarisation angle, $\theta $, for six different field lines, labelled 1–6. The horizontal blue line denotes the driving frequency, ${\omega}_{d}$. (

**b**) The intersections of these field lines with the vertical $(x,z)$ plane that runs the length of the arcade are shown as blue dots. The blue arrows indicate the polarisation required for the ${\omega}_{A}$ to equal ${\omega}_{d}$, i.e., the resonance condition. The solid black line represents a permissible Resonant Path, and the red lines indicate the boundaries of the Resonant Zone.

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**MDPI and ACS Style**

Wright, A.; Elsden, T.
Resonant Fast-Alfvén Wave Coupling in a 3D Coronal Arcade. *Physics* **2023**, *5*, 310-321.
https://doi.org/10.3390/physics5010023

**AMA Style**

Wright A, Elsden T.
Resonant Fast-Alfvén Wave Coupling in a 3D Coronal Arcade. *Physics*. 2023; 5(1):310-321.
https://doi.org/10.3390/physics5010023

**Chicago/Turabian Style**

Wright, Andrew, and Thomas Elsden.
2023. "Resonant Fast-Alfvén Wave Coupling in a 3D Coronal Arcade" *Physics* 5, no. 1: 310-321.
https://doi.org/10.3390/physics5010023