# 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

## References

- Southwood, D.J. Some features of field line resonances in the magnetosphere. Planet. Space Sci.
**1974**, 22, 483–491. [Google Scholar] [CrossRef] - Goossens, M.; Erdélyi, R.; Ruderman, M.S. Resonant MHD waves in the solar atmosphere. Space Sci. Rev.
**2011**, 158, 289–338. [Google Scholar] [CrossRef] - Van Doorsselaere, T.; Srivastava, A.K.; Antolin, P.; Magyar, N.; Vasheghani Farahani, S.; Tian, H.; Kolotkov, D.; Ofman, L.; Guo, M.; Arregui, I.; et al. Coronal heating by MHD waves. Space Sci. Rev.
**2020**, 216, 140. [Google Scholar] [CrossRef] - Allan, W.; White, S.P.; Poulter, E.M. Impulse-excited hydromagnetic cavity and field-line resonances in the magnetosphere. Planet. Space Sci.
**1986**, 34, 371–385. [Google Scholar] [CrossRef] - Poedts, S.; Goossens, M.; Kerner, W. Numerical simulation of coronal heating by resonant absorption of Alfvén waves. Sol. Phys.
**1989**, 123, 83–115. [Google Scholar] [CrossRef] - Goossens, M.; Ruderman, M.S.; Hollweg, J.V. Dissipative MHD solutions for resonant Alfvén waves in 1-dimensional magnetic flux tubes. Sol. Phys.
**1995**, 157, 75–102. [Google Scholar] [CrossRef] - Wright, A.N.; Allan, W. Structure, phase motion, and heating within Alfvén resonances. J. Geophys. Res. Space Phys.
**1996**, 101, 17399–17408. [Google Scholar] [CrossRef] - Wright, A.N.; Thompson, M.J. Analytical treatment of Alfvén resonances and singularities in nonuniform magnetoplasmas. Phys. Plasmas
**1994**, 1, 691–705. [Google Scholar] [CrossRef] [Green Version] - Tirry, W.J.; Goossens, M. Dissipative MHD solutions for resonant Alfvén waves in two-dimensional poloidal magnetoplasmas. J. Geophys. Res. Space Phys.
**1995**, 100, 23687–23694. [Google Scholar] [CrossRef] - Terradas, J.; Soler, R.; Luna, M.; Oliver, R.; Ballester, J.L.; Wright, A.N. Solar prominences embedded in flux ropes: Morphological features and dynamics from 3D MHD simulations. Astrophys. J.
**2016**, 820, 125. [Google Scholar] [CrossRef] [Green Version] - Degeling, A.W.; Rankin, R.; Kabin, K.; Rae, I.J.; Fenrich, F.R. Modeling ULF waves in a compressed dipole magnetic field. J. Geophys. Res. Space Phys.
**2010**, 115, A10212. [Google Scholar] [CrossRef] [Green Version] - Wright, A.N.; Elsden, T. The theoretical foundation of 3D Alfvén resonances: Normal modes. Astrophys. J.
**2016**, 833, 230. [Google Scholar] [CrossRef] - Degeling, A.W.; Rae, I.J.; Watt, C.E.J.; Shi, Q.Q.; Rankin, R.; Zong, Q.G. Control of ULF wave accessibility to the inner magnetosphere by the convection of plasma density. J. Geophys. Res. Space Phys.
**2018**, 123, 1086–1099. [Google Scholar] [CrossRef] [Green Version] - Elsden, T.; Wright, A.; Degeling, A. A review of the theory of 3-D Alfvén (field line) resonances. Front. Astron. Space Sci.
**2022**, 9, 917817. [Google Scholar] [CrossRef] - Wright, A.; Degeling, A.W.; Elsden, T. Resonance maps for 3D Alfvén waves in a compressed dipole field. J. Geophys. Res. Space Phys.
**2022**, 127, e2022JA030294. [Google Scholar] [CrossRef] - Wright, A.N.; Elsden, T. Simulations of MHD wave propagation and coupling in a 3-D magnetosphere. J. Geophys. Res. Space Phys.
**2020**, 125, e27589. [Google Scholar] [CrossRef] - Halberstadt, G.; Goedbloed, J.P. The continuous Alfvén spectrum of line-tied coronal loops. Astron. Astrophys.
**1993**, 280, 647–660. Available online: https://ui.adsabs.harvard.edu/abs/1993A%26A...280..647H (accessed on 27 February 2023). - Prokopyszyn, A.P.K.; Wright, A.N.; Hood, A.W. Line-tied boundary conditions can cause resonant absorption models to generate unphysically large boundary layers. Astrophys. J.
**2021**, 914, 15. [Google Scholar] [CrossRef] - Elsden, T.; Wright, A.N. The theoretical foundation of 3-D Alfvén resonances: Time-dependent solutions. J. Geophys. Res. Space Phys.
**2017**, 122, 3247–3261. [Google Scholar] [CrossRef] [Green Version] - Elsden, T.; Wright, A.N. The Broadband excitation of 3-D Alfvén resonances in a MHD waveguide. J. Geophys. Res. Space Phys.
**2018**, 123, 530–547. [Google Scholar] [CrossRef] [Green Version] - Elsden, T.; Wright, A.N. The effect of fast normal mode structure and magnetopause forcing on FLRs in a 3-D waveguide. J. Geophys. Res. Space Phys.
**2019**, 124, 178–196. [Google Scholar] [CrossRef] [Green Version] - Dungey, J.W. Electrodynamics of the Outer Atmosphere: Report to National Science Foundation on Work Carried on under Grant NSF-G450; Pennsylvania State University, Ionosphere Research Laboratory: State College, PA, USA, 1954. [Google Scholar]
- Elsden, T. Numerical Modelling of Ultra Low Frequency Waves in Earth’s Magnetosphere. Ph.D. Thesis, University of St Andrews, St. Andrews, UK, 2016. [Google Scholar] [CrossRef] [Green Version]
- Singer, H.J.; Southwood, D.J.; Walker, R.J.; Kivelson, M.G. Alfvén wave resonances in a realistic magnetospheric magnetic field geometry. J. Geophys. Res. Space Phys.
**1981**, 86, 4589–4596. [Google Scholar] [CrossRef] [Green Version] - Leonovich, A.S.; Mazur, V.A. A theory of transverse small-scale standing Alfvén waves in an axially symmetric magnetosphere. Planet. Space Sci.
**1993**, 41, 697–717. [Google Scholar] [CrossRef] - Klimushkin, D.Y.; Leonovich, A.S.; Mazur, V.A. On the propagation of transversally small-scale standing Alfvén waves in a three-dimensionally inhomogeneous magnetosphere. J. Geophys. Res. Space Phys.
**1995**, 100, 9527–9534. [Google Scholar] [CrossRef] - Elsden, T.; Wright, A.N. Evolution of high-m poloidal Alfvén waves in a dipole magnetic field. J. Geophys. Res. Space Phys.
**2020**, 125, e28187. [Google Scholar] [CrossRef]

**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