# Modeling the Formation and Evolution of Solar Wind Microstreams: From Coronal Plumes to Propagating Alfvénic Velocity Spikes

Bahaeddine Gannouni, Victor Réville, Alexis P. Rouillard

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Bahaeddine Gannouni, Victor Réville, Alexis P. Rouillard. Modeling the Formation and Evolution of Solar Wind Microstreams: From Coronal Plumes to Propagating Alfvénic Velocity Spikes. The Astrophysical Journal, 2023, 958, 10.3847/1538-4357/acfef3. insu-04473121

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# Modeling the Formation and Evolution of Solar Wind Microstreams: From Coronal Plumes to Propagating Alfvénic Velocity Spikes

Bahaeddine Gannouni, Victor Réville, Alexis P. Rouillard

► To cite this version:

Bahaeddine Gannouni, Victor Réville, Alexis P. Rouillard. Modeling the Formation and Evolution of Solar Wind Microstreams: From Coronal Plumes to Propagating Alfvénic Velocity Spikes. The Astrophysical Journal, 2023, 958, 10.3847/1538-4357/acfef3. insu-04473121

HAL Id: insu-04473121

https://insu.hal.science/insu-04473121v1

Submitted on 23 Feb 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

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Distributed under a Creative Commons Attribution 4.0 International License
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# Modeling the Formation and Evolution of Solar Wind Microstreams: From Coronal Plumes to Propagating Alfvénic Velocity Spikes

Bahaeddine Gannouni ID, Victor Réville ID, and Alexis P. Rouillard ID

IRAP, Université Toulouse III—Paul Sabatier, CNRS, CNES, Toulouse, France

Received 2023 July 5; revised 2023 September 18; accepted 2023 September 29; published 2023 November 16

## Abstract

We investigate the origin of mesoscale structures in the solar wind called microstreams, defined as enhancements in the solar wind speed and temperature that last several hours. They were first clearly detected in Helios and Ulysses solar wind data and are now omnipresent in the “young” solar wind measured by the Parker Solar Probe and Solar Orbiter. These recent data reveal that microstreams transport a profusion of Alfvénic perturbations in the form of velocity spikes and magnetic switchbacks. In this study, we use a very-high-resolution 2.5D MHD model of the corona and the solar wind to simulate the emergence of magnetic bipoles interacting with the preexisting ambient corona and the creation of jets that become microstreams propagating in the solar wind. Our high-resolution simulations reach sufficiently high Lundquist numbers that capture the tearing mode instability that develops in the reconnection region and produces plasmoids released with the jet into the solar wind. Our domain runs from the lower corona to $20 R_⊙$, which allows us to track the formation process of plasmoids and their evolution into Alfvénic velocity spikes. We obtain perturbed solar wind flows lasting several hours with velocity spikes occurring at characteristic periodicities of about 19 minutes. We retrieve several properties of the microstreams measured in the pristine solar wind by the Parker Solar Probe, namely an increase in wind velocity of about $100 \text{ km s}^{-1}$ during a stream’s passage together with superposed velocity spikes of also about $100 \text{ km s}^{-1}$ released into the solar wind.

*Unified Astronomy Thesaurus concepts: Solar coronal plumes (2039); Solar wind (1534); Solar magnetic reconnection (1504); Magnetohydrodynamics (1964)*

*Supporting material: animation*

## 1. Introduction

White-light images of total solar eclipses and coronagraphs reveal fine ray-like structures emanating from polar coronal holes. These “plumes” extend outward from the base of the corona and are observed in white-light and extreme-ultraviolet (EUV) images. They are most commonly found in polar coronal holes, but can also be observed in equatorial coronal holes (Woo 1996). Plasma and magnetic field data obtained by the two Helios solar probes showed that the scales of these rays are preserved in the evolving interplanetary high-speed solar wind measured close to the Sun. Further studies based on in situ measurements made in the polar solar wind by the Ulysses mission confirmed this result and identified “micro-streams” in the form of velocity fluctuations of $40 \text{ km s}^{-1}$, higher kinetic temperatures, and slightly higher proton fluxes (Neugebauer et al. 1995). Raouafi et al. (2008) showed that X-ray jets are precursors of polar plumes and in some cases cause brightenings of plumes. Microstreams could therefore be the interplanetary manifestation of X-ray jets released during the formation of a plume inside a coronal hole (Neugebauer 2012). The aim of the present study is to investigate, through high-resolution magnetohydrodynamic (MHD) simulations, the mechanisms driving the formation and evolution of plumes and microstreams and their dynamic properties discussed in the next paragraphs.

Plumes are typically hazy and are routinely detected in the EUV at wavelengths of 171 and 193 Å (Raouafi & Stenborg 2014). It has been debated as to whether coronal plumes or interplume regions may be the source regions of the fast solar wind (Wilhelm et al. 2011; Poletto 2015). Plumes appear to form after magnetic bipoles erupt in the open magnetic field of coronal holes. According to DeForest et al. (2001), plumes extend away from photospheric flux concentrations and can last from hours to several weeks, reaching lengths of about $30 R_⊙$. Plumelets, which are small features within plumes, often exhibit intensity fluctuations on shorter timescales than the overall plume (DeForest et al. 1997; DeForest 2007; Uritsky et al. 2021). Data from STEREO/EUVI images show that these fluctuations, known as propagating disturbances, can have periods ranging from 5 to 30 minutes (McIntosh et al. 2010; Tian et al. 2011). The formation of a plume is typically preceded by recurrent jets that emerge from random flux emergence and cancellation, and the plume itself goes through phases of brightening and decay, during which subplumes may be visible (Raouafi & Stenborg 2014). The emergence of magnetic flux in the dominant polarity of coronal holes plays an essential role in the heating and outflow of plasma, and EUV brightening (Panesar et al. 2018, 2019). This is likely due to an interchange reconnection process, which takes place when emerging loop systems encounter an open background magnetic field (Vlahos 2002). It is also an efficient means of releasing plasma that is otherwise confined to closed field regions into the heliosphere and perhaps contributes to the mass flow of fast and slow solar winds emerging from coronal holes (Wang et al. 1996).

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Original content from this work may be used under the terms of the [Creative Commons Attribution 4.0 licence](https://creativecommons.org/licenses/by/4.0/). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
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# Modeling the Formation and Evolution of Solar Wind Microstreams: From Coronal Plumes to Propagating Alfvénic Velocity Spikes

Bahaeddine Gannouni ID, Victor Réville ID, and Alexis P. Rouillard ID

IRAP, Université Toulouse III—Paul Sabatier, CNRS, CNES, Toulouse, France

Received 2023 July 5; revised 2023 September 18; accepted 2023 September 29; published 2023 November 16

## Abstract

We investigate the origin of mesoscale structures in the solar wind called microstreams, defined as enhancements in the solar wind speed and temperature that last several hours. They were first clearly detected in Helios and Ulysses solar wind data and are now omnipresent in the “young” solar wind measured by the Parker Solar Probe and Solar Orbiter. These recent data reveal that microstreams transport a profusion of Alfvénic perturbations in the form of velocity spikes and magnetic switchbacks. In this study, we use a very-high-resolution 2.5D MHD model of the corona and the solar wind to simulate the emergence of magnetic bipoles interacting with the preexisting ambient corona and the creation of jets that become microstreams propagating in the solar wind. Our high-resolution simulations reach sufficiently high Lundquist numbers that capture the tearing mode instability that develops in the reconnection region and produces plasmoids released with the jet into the solar wind. Our domain runs from the lower corona to $20 R_⊙$, which allows us to track the formation process of plasmoids and their evolution into Alfvénic velocity spikes. We obtain perturbed solar wind flows lasting several hours with velocity spikes occurring at characteristic periodicities of about 19 minutes. We retrieve several properties of the microstreams measured in the pristine solar wind by the Parker Solar Probe, namely an increase in wind velocity of about $100 \text{ km s}^{-1}$ during a stream’s passage together with superposed velocity spikes of also about $100 \text{ km s}^{-1}$ released into the solar wind.

Unified Astronomy Thesaurus concepts: Solar coronal plumes (2039); Solar wind (1534); Solar magnetic reconnection (1504); Magnetohydrodynamics (1964)

Supporting material: animation

## 1. Introduction

White-light images of total solar eclipses and coronagraphs reveal fine ray-like structures emanating from polar coronal holes. These “plumes” extend outward from the base of the corona and are observed in white-light and extreme-ultraviolet (EUV) images. They are most commonly found in polar coronal holes, but can also be observed in equatorial coronal holes (Woo 1996). Plasma and magnetic field data obtained by the two Helios solar probes showed that the scales of these rays are preserved in the evolving interplanetary high-speed solar wind measured close to the Sun. Further studies based on in situ measurements made in the polar solar wind by the Ulysses mission confirmed this result and identified “micro-streams” in the form of velocity fluctuations of $40 \text{ km s}^{-1}$, higher kinetic temperatures, and slightly higher proton fluxes (Neugebauer et al. 1995). Raouafi et al. (2008) showed that X-ray jets are precursors of polar plumes and in some cases cause brightenings of plumes. Microstreams could therefore be the interplanetary manifestation of X-ray jets released during the formation of a plume inside a coronal hole (Neugebauer 2012). The aim of the present study is to investigate, through high-resolution magnetohydrodynamic (MHD) simulations, the mechanisms driving the formation and evolution of plumes and microstreams and their dynamic properties discussed in the next paragraphs.

Plumes are typically hazy and are routinely detected in the EUV at wavelengths of 171 and 193 Å (Raouafi & Stenborg 2014). It has been debated as to whether coronal plumes or interplume regions may be the source regions of the fast solar wind (Wilhelm et al. 2011; Poletto 2015). Plumes appear to form after magnetic bipoles erupt in the open magnetic field of coronal holes. According to DeForest et al. (2001), plumes extend away from photospheric flux concentrations and can last from hours to several weeks, reaching lengths of about $30 R_⊙$. Plumelets, which are small features within plumes, often exhibit intensity fluctuations on shorter timescales than the overall plume (DeForest et al. 1997; DeForest 2007; Uritsky et al. 2021). Data from STEREO/EUVI images show that these fluctuations, known as propagating disturbances, can have periods ranging from 5 to 30 minutes (McIntosh et al. 2010; Tian et al. 2011). The formation of a plume is typically preceded by recurrent jets that emerge from random flux emergence and cancellation, and the plume itself goes through phases of brightening and decay, during which subplumes may be visible (Raouafi & Stenborg 2014). The emergence of magnetic flux in the dominant polarity of coronal holes plays an essential role in the heating and outflow of plasma, and EUV brightening (Panesar et al. 2018, 2019). This is likely due to an interchange reconnection process, which takes place when emerging loop systems encounter an open background magnetic field (Vlahos 2002). It is also an efficient means of releasing plasma that is otherwise confined to closed field regions into the heliosphere and perhaps contributes to the mass flow of fast and slow solar winds emerging from coronal holes (Wang et al. 1996).

<figure><img src="image_3.png" /></figure>

Original content from this work may be used under the terms of the [Creative Commons Attribution 4.0 licence](https://creativecommons.org/licenses/by/4.0/). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Interchange reconnection has also been one of the suggested mechanisms for the formation of magnetic switchbacks and velocity spikes measured ubiquitously by the Parker Solar Probe (PSP) in the nascent solar wind (Bale et al. 2019; Kasper et al. 2019). Switchbacks are characterized by large-amplitude Alfvénic fluctuations that propagate away from the Sun, with an extensive range of magnetic deflection angles from a few degrees to a full inversion (Fargette et al. 2022). The origins of these features are still under debate, particularly whether they are generated locally in the solar wind or in the lower corona. For instance, the work of Schwadron & McComas (2021), Squire et al. (2020), and Shoda et al. (2021) suggests that switchbacks may be generated locally in the solar wind through processes involving velocity shears or turbulent flows. On the other hand, other studies, such as the work of Fisk & Kasper (2020) and Drake et al. (2021), propose that switchbacks are formed through interchange reconnection in the lower corona (Yamauchi et al. 2004; Fisk 2005; Fisk & Kasper 2020).

Switchbacks and velocity spikes come in bursts or patches whose spatial and timescales are comparable to those of microstreams (Bale et al. 2021; Fargette et al. 2021). These patches of disturbances are particularly intense in streamer flows, but are also very clear in solar wind flows originating from deep inside coronal holes (Rouillard et al. 2020; Fargette et al. 2021). Statistical analysis of these patches of switchbacks/velocity spikes (Fargette et al. 2021) as well as the analysis of solar wind composition (Bale et al. 2021) point toward an origin of these patches in sudden energy releases at the boundaries of supergranules. Since photospheric transport processes force an accumulation of magnetic elements (loops and open fields) near the boundaries of granules and supergranules, interchange magnetic reconnection could occur frequently in these regions. Moreover, in the study of Shi et al. (2022), the analysis of PSP corotation periods revealed temporal signatures in addition to spatial structure associated with switchback patches. Disentangling spatial and temporal scales in the data and identifying the corresponding processes at the solar surface will be key to evaluating the idea that switchbacks have indeed a solar origin.

Several studies have recently appeared in the literature that make a tentative association between microstreams, plumes, and individual switchbacks with jetlets (Kumar et al. 2023a; Hou et al. 2023; Raouafi et al. 2023). In this paper, we investigate the idea that interchange reconnection can arise from the emergence of magnetic flux that contributes to the formation of coronal plumes. In particular, we wish to study the effect of the rate and amplitude of flux emergence on the formation of coronal plumes, interchange reconnection, and the structure of the resulting jets. An association between plasmons formed during reconnection and switchbacks was proposed by Drake et al. (2021) using kinetic simulations (Bale et al. 2023). Recent advanced MHD simulations have also looked into the evolution of reconnection outflows produced in 2.5D compressible Alfvén waves (Zhao et al. 2018; He et al. 2021) and in 3D torsional Alfvén waves (Pariat et al. 2009; Wyper et al. 2022) escaping the solar corona. In the present paper, we examine whether the magnetic islands produced through the tearing-mode instability in the reconnection layer could be the source of individual velocity spikes and switchbacks. In order to capture the development of the tearing-mode instability in the solar corona at the adequate time and spatial scales we limit our study to 2.5D MHD but extend the domain

out to $20 R_⊙$. This allows us to simulate the lifetime of an entire microstream and the associated release of multiple microjets to reproduce the form of microstreams measured in situ by PSP.

The paper is organized as follows. In Section 2, we describe the numerical model. In Section 3, we show the results for the main simulation setup, along with some discussion of the results. In Section 4, we conclude this study and comment on possible future work.

## 2. Simulation Details

### 2.1. Model Description

We investigate the dynamics of magnetic reconnection in a solar corona undergoing flux emergence. Flux emergence is a process by which new magnetic field lines emerge from the solar surface and enter the corona, leading to the formation of current sheets (CSs; see Shibata et al. 1989, 1992b). One type of reconnection process that has received significant attention is the tearing instability, which occurs when small perturbations in a CS grow and lead to the breakup of the sheet into magnetic islands, or “plasmons.” The tearing mode allows reaching fast reconnection, provided that the Lundquist number $S = Lν_A/η$ is high enough ($L$ is the length of CS, $ν_A$ is the Alfvén velocity, and $η$ is the magnetic diffusivity), typically $10^4$ in 2D configurations (Biskamp 1986; Loureiro et al. 2007; Bhattacharjee et al. 2009). Particularly interesting dynamics occur when a CS system is thinning. Linear theory shows that the tearing mode is then triggered as soon as the sheet aspect ratio reaches $\sim S^{-1/3}$ (Pucci & Velli 2014; Réville et al. 2020). In our case, as the bipolar flux emerges, a CS is formed at the contact of the opposite polarities and interchange reconnection occurs. The current layer is then expected to develop magnetic islands or plasmons.

To study this process, we solve the 2.5D compressible resistive MHD equations, using the PLUTO code (Mignone et al. 2007), a finite-volume shock-capturing code. We employ a second-order Runge-Kutta method to calculate the time step and a fourth-order spatial scheme provided by a parabolic reconstruction. We use a Harten-Lax-van Leer discontinuities (HLLD) solver (Miyoshi & Kusano 2005). The solenoidal constraint on the magnetic field is ensured through the constrained transport method (Balsara & Spicer 1999).

The equations can be written as follows:

$$ \frac{\partial}{\partial t} \rho + \nabla \cdot (\rho \mathbf{v}) = 0, \qquad (1) $$

$$ \frac{\partial}{\partial t} \rho \mathbf{v} + \nabla \cdot (\rho \mathbf{v} \mathbf{v} - \mathbf{B} \mathbf{B} + \mathbf{I} \mathbf{p}) = -\rho \nabla \Phi, \qquad (2) $$

$$ \frac{\partial}{\partial t} (E + \rho \Phi) + \nabla \cdot [(E + p + \rho \Phi)\mathbf{v} - \mathbf{B}(\mathbf{v} \cdot \mathbf{B}) + (\eta \cdot \mathbf{J}) \times \mathbf{B}] = Q_h - Q_c - Q_r, \qquad (3) $$

$$ \frac{\partial}{\partial t} \mathbf{B} + \nabla \cdot (\mathbf{v} \mathbf{B} - \mathbf{B} \mathbf{v}) - \eta \nabla^2 \mathbf{B} = 0, \qquad (4) $$

where $E \equiv \rho e + \rho v^2/2 + B^2/2$ is the background flow energy, $\mathbf{B}$ is the magnetic field, $\rho$ is the mass density, $\mathbf{v}$ is the velocity field, $p = p_{th} + B^2/2$ is the total pressure (thermal and magnetic), $\mathbf{I}$ is the identity matrix, $\mathbf{J} = \nabla \times \mathbf{B}$ is the electric current, $e = \frac{p_{th}}{(\gamma-1)\rho}$ is the specific internal energy density, and $\eta$ is the magnetic diffusivity. Finally, $\gamma = 5/3$ is the ratio of specific heats, and the terms $Q_*$ represent the terms of
Interchange reconnection has also been one of the suggested mechanisms for the formation of magnetic switchbacks and velocity spikes measured ubiquitously by the Parker Solar Probe (PSP) in the nascent solar wind (Bale et al. 2019; Kasper et al. 2019). Switchbacks are characterized by large-amplitude Alfvénic fluctuations that propagate away from the Sun, with an extensive range of magnetic deflection angles from a few degrees to a full inversion (Fargette et al. 2022). The origins of these features are still under debate, particularly whether they are generated locally in the solar wind or in the lower corona. For instance, the work of Schwadron & McComas (2021), Squire et al. (2020), and Shoda et al. (2021) suggests that switchbacks may be generated locally in the solar wind through processes involving velocity shears or turbulent flows. On the other hand, other studies, such as the work of Fisk & Kasper (2020) and Drake et al. (2021), propose that switchbacks are formed through interchange reconnection in the lower corona (Yamauchi et al. 2004; Fisk 2005; Fisk & Kasper 2020).

Switchbacks and velocity spikes come in bursts or patches whose spatial and timescales are comparable to those of microstreams (Bale et al. 2021; Fargette et al. 2021). These patches of disturbances are particularly intense in streamer flows, but are also very clear in solar wind flows originating from deep inside coronal holes (Rouillard et al. 2020; Fargette et al. 2021). Statistical analysis of these patches of switchbacks/velocity spikes (Fargette et al. 2021) as well as the analysis of solar wind composition (Bale et al. 2021) point toward an origin of these patches in sudden energy releases at the boundaries of supergranules. Since photospheric transport processes force an accumulation of magnetic elements (loops and open fields) near the boundaries of granules and supergranules, interchange magnetic reconnection could occur frequently in these regions. Moreover, in the study of Shi et al. (2022), the analysis of PSP corotation periods revealed temporal signatures in addition to spatial structure associated with switchback patches. Disentangling spatial and temporal scales in the data and identifying the corresponding processes at the solar surface will be key to evaluating the idea that switchbacks have indeed a solar origin.

Several studies have recently appeared in the literature that make a tentative association between microstreams, plumes, and individual switchbacks with jetlets (Kumar et al. 2023a; Hou et al. 2023; Raouafi et al. 2023). In this paper, we investigate the idea that interchange reconnection can arise from the emergence of magnetic flux that contributes to the formation of coronal plumes. In particular, we wish to study the effect of the rate and amplitude of flux emergence on the formation of coronal plumes, interchange reconnection, and the structure of the resulting jets. An association between plasmoids formed during reconnection and switchbacks was proposed by Drake et al. (2021) using kinetic simulations (Bale et al. 2023). Recent advanced MHD simulations have also looked into the evolution of reconnection outflows produced in 2.5D compressible Alfvén waves (Zhao et al. 2018; He et al. 2021) and in 3D torsional Alfvén waves (Pariat et al. 2009; Wyper et al. 2022) escaping the solar corona. In the present paper, we examine whether the magnetic islands produced through the tearing-mode instability in the reconnection layer could be the source of individual velocity spikes and switchbacks. In order to capture the development of the tearing-mode instability in the solar corona at the adequate time and spatial scales we limit our study to 2.5D MHD but extend the domain

out to $20 R_⊙$. This allows us to simulate the lifetime of an entire microstream and the associated release of multiple microjets to reproduce the form of microstreams measured in situ by PSP.

The paper is organized as follows. In Section 2, we describe the numerical model. In Section 3, we show the results for the main simulation setup, along with some discussion of the results. In Section 4, we conclude this study and comment on possible future work.

## 2. Simulation Details

### 2.1. Model Description

We investigate the dynamics of magnetic reconnection in a solar corona undergoing flux emergence. Flux emergence is a process by which new magnetic field lines emerge from the solar surface and enter the corona, leading to the formation of current sheets (CSs; see Shibata et al. 1989, 1992b). One type of reconnection process that has received significant attention is the tearing instability, which occurs when small perturbations in a CS grow and lead to the breakup of the sheet into magnetic islands, or “plasmoids.” The tearing mode allows reaching fast reconnection, provided that the Lundquist number $S = Lν_A/η$ is high enough ($L$ is the length of CS, $ν_A$ is the Alfvén velocity, and $η$ is the magnetic diffusivity), typically $10^4$ in 2D configurations (Biskamp 1986; Loureiro et al. 2007; Bhattacharjee et al. 2009). Particularly interesting dynamics occur when a CS system is thinning. Linear theory shows that the tearing mode is then triggered as soon as the sheet aspect ratio reaches $\sim S^{-1/3}$ (Pucci & Velli 2014; Réville et al. 2020). In our case, as the bipolar flux emerges, a CS is formed at the contact of the opposite polarities and interchange reconnection occurs. The current layer is then expected to develop magnetic islands or plasmoids.

To study this process, we solve the 2.5D compressible resistive MHD equations, using the PLUTO code (Mignone et al. 2007), a finite-volume shock-capturing code. We employ a second-order Runge-Kutta method to calculate the time step and a fourth-order spatial scheme provided by a parabolic reconstruction. We use a Harten-Lax-van Leer discontinuities (HLLD) solver (Miyoshi & Kusano 2005). The solenoidal constraint on the magnetic field is ensured through the constrained transport method (Balsara & Spicer 1999).

The equations can be written as follows:

$$ \frac{\partial}{\partial t} \rho + \nabla \cdot (\rho \mathbf{v}) = 0, \qquad (1) $$

$$ \frac{\partial}{\partial t} \rho \mathbf{v} + \nabla \cdot (\rho \mathbf{v} \mathbf{v} - \mathbf{B} \mathbf{B} + \mathbf{I} \mathbf{p}) = - \rho \nabla \Phi, \qquad (2) $$

$$ \frac{\partial}{\partial t} (E + \rho \Phi) + \nabla \cdot [(E + p + \rho \Phi)\mathbf{v} - \mathbf{B}(\mathbf{v} \cdot \mathbf{B}) + (\eta \cdot \mathbf{J}) \times \mathbf{B}] = Q_h - Q_c - Q_r, \qquad (3) $$

$$ \frac{\partial}{\partial t} \mathbf{B} + \nabla \cdot (\mathbf{v} \mathbf{B} - \mathbf{B} \mathbf{v}) - \eta \nabla^2 \mathbf{B} = 0, \qquad (4) $$

where $E \equiv \rho e + \rho v^2/2 + B^2/2$ is the background flow energy, $\mathbf{B}$ is the magnetic field, $\rho$ is the mass density, $\mathbf{v}$ is the velocity field, $p = p_{th} + B^2/2$ is the total pressure (thermal and magnetic), $\mathbf{I}$ is the identity matrix, $\mathbf{J} = \nabla \times \mathbf{B}$ is the electric current, $e = \frac{p_{th}}{(\gamma-1)\rho}$ is the specific internal energy density, and $\eta$ is the magnetic diffusivity. Finally, $\gamma = 5/3$ is the ratio of specific heats, and the terms $Q_*$ represent the terms of
volumetric energy gain and loss, heating, thermal conduction, and radiative losses. The system is solved in spherical coordinates $(r, \theta)$, and the gravity potential:

$$ \Phi = -\frac{GM_{\odot}}{r}. \qquad (5) $$

The heating term is defined as

$$ Q_h = F_h/H \left(\frac{R_{\odot}}{r}\right)^2 \exp\left(-\frac{r-R_{\odot}}{H}\right). \qquad (6) $$

The energy flux from the photosphere, denoted as $F_h$, has a value of $1.5 \times 10^5$ erg cm$^{-2}$ s$^{-1}$ (Withbroe 1988) and $H \sim 1 R_{\odot}$.

The thermal conduction is written as

$$ Q_c = \nabla \cdot (\alpha \mathbf{q}_s + (1-\alpha)\mathbf{q}_p), \qquad (7) $$

where $q_s = -\kappa_0 T^{5/2} \nabla T$ is the Spitzer-Härm thermal conduction with $\kappa_0 = 9 \times 10^{-7}$ cgs, and $\mathbf{q}_p = 3/2 p_{\text{th}} \mathbf{v}_e$ is the electron collisionless heat flux described in Hollweg (1986). The coefficient $\alpha = 1/(1 + (r - R_{\odot})^4/(r_{\text{coll}} - R_{\odot})^4)$ creates a smooth transition between the two regimes at a characteristic height of $r_{\text{coll}} = 5 R_{\odot}$. We have used an optically thin radiation cooling prescription:

$$ Q_r = n^2 \Lambda(T), \qquad (8) $$

with $n$ the electron density and $T$ the electron temperature. $\Lambda(T)$ follows the prescription of Athay (1986).

In the PLUTO code, all quantities are expressed in dimensionless units derived from physical quantities divided by normalization units, appropriate for solar wind conditions. We consider unit length $L_0 = 1 R_{\odot} \sim 6.9570 \times 10^5$ km and unit density $\rho_0 = 1.67 \times 10^{-12}$ kg m$^{-3}$. The velocities are normalized to the Keplerian velocity $v_0 \sim 437$ km s$^{-1}$. The characteristic magnetic field and unit time are $B_0 = \sqrt{4\pi\rho_0 v_0^2} \sim 2$ G and $t_0 = L_0/U_0 = 1593$ s, respectively, and we set the magnetic diffusivity $\eta$ to $10^{12}$ cm$^2$ s$^{-1}$. The simulation is integrated on a nonuniform grid with a strong refinement of $\Delta r = 10^{-4}$ in the 0.1 $R_{\odot}$ region in code units and a coarser grid extending up to 20 $R_{\odot}$. The range for $\theta$ is $[\pi/2 - 0.145, \pi/2 + 0.305]$ with a stretched grid cell size of $\Delta\theta = 2 \times 10^{-4}$, and the total grid size is $1536 \times 1536$. We consider reflective boundaries for the velocity components across $\theta_{\text{min}}$ and $\theta_{\text{max}}$. At $r = 20 R_{\odot} > r_{A_f}$ the fast magneto-sonic point, we use an outflow boundary condition.

We initialize the atmosphere with a supersonic Parker wind and a purely radial field of 2 G at the surface. We assume coronal conditions and do not include a chromosphere or transition region. We let the system reach a steady state before starting the flux emergence. For the inner boundary condition at $r = 1 R_{\odot}$, we fix the temperature and density at the initial value $n = 2 \times 10^8$ cm$^{-3}$ and $T = 1$ MK. The magnetic field is set to its background or emerging (see next) values, while the velocity field is computed to set the parallel electric field to zero. These boundary conditions are very close to the ones described in Parenti et al. (2022).

We then apply a time-dependent boundary condition for the magnetic field component ($B_r, B_{\theta}$) to control the emergence rate of the two polarities. A bipole of 20 G amplitude and $\sim 10^\circ$ of latitudinal extent is projected on the spherical harmonics base (with $l_{\text{max}} = 65$), and each coefficient is added to the background field at the boundary and increased linearly during

the emergence phase. The spherical harmonics decomposition ensures that $\nabla \cdot B = 0$ at all times. We focus on four setups with emergence rates of 3.61 G hr$^{-1}$, 4.52 G hr$^{-1}$, 6 G hr$^{-1}$, and 9 G hr$^{-1}$, which are described in Figure 1. The duration of emergence for each setup is 5.53, 4.4, 3.3, and 2.2 hr, respectively.

In 2.5D, the Lundquist number $S = L v_A / \eta$ should be higher than a critical value of approximately $10^4$ to trigger the tearing instability (Biskamp 1986; Loureiro et al. 2007; Bhattacharjee et al. 2009). Hence, we set an explicit value of $\eta$ to be comfortably above this threshold (note that this depends on the length of the CS, and will vary with time as the flux emergence proceeds). But we must also be careful that the numerical resistivity remains smaller than or equal to the value of $\eta$ chosen. Based on our experience of the onset of the tearing mode with the PLUTO code (Réville et al. 2020), we find that a value of $\eta = 10^{12}$ cm$^2$ s$^{-1}$ and the above-described grid resolution satisfy these requirements with $S_{\eta} \sim S_{\text{num}} \sim 10^4$. We have also performed some simulations with $\eta = 10^{11}$ cm$^2$ s$^{-1}$ and $\eta = 10^{13}$ cm$^2$ s$^{-1}$ and did not notice significant changes during the bursty reconnection phase (see Figure 9).

## 3. Results

### 3.1. Study of the CS Formation and Aspect Ratio

Figure 2 shows the time evolution of the simulation for a rate of emergence of 3.61 G hr$^{-1}$. We chose three characteristic phases of the emergence and relaxation phases, showing the logarithm of the out-of-the-plane current density and the radial velocity (top and bottom panels). In the early stages of emergence, the CS is created immediately and the tearing instability is triggered. The CS is disrupted several times but continues to lengthen as the emergence continues and reaches its plateau. Close to the peak of the emergence (middle panel), reconnection occurs and plasmoids are ejected on both sides of the CS. Finally, after the emergence has stopped, the CS slowly decays and shortens, thus stopping the reconnection process. This stage can be seen in the right panel of Figure 2.

The CS’s length ($L$) is automatically measured by calculating $L = |\mathbf{B}|/|\nabla \times \mathbf{B}|$ then fixing a threshold to locate the CS (see, e.g., Nóbrega-Siverio & Moreno-Insertis 2022). Figure 3 shows the evolution of the CS length obtained by this method.

Pucci & Velli (2014) introduced the concept of “ideal” tearing, in which the growth rate of the instability is independent of the Lundquist number. Assume that the ratio of the thickness of the CS $a$ to its length scale $L$ scales as $S^{-\alpha}$, where $S$ is the Lundquist number and $\alpha$ is the power-law index. There is a critical value of $\alpha$ at which the growth rate is constant, $\gamma t_A \sim S^{-1/2}$, $\alpha = \text{cnst}$, which is equal to $1/3$. If $\alpha$ is greater than $1/3$, the growth rate tends to diverge with increasing Lundquist number, while if $\alpha$ is less than $1/3$, the growth rate tends to zero. We thus expect the reconnection to occur precisely at $\alpha = 1/3$, when the CS forms from large aspect ratios.

In the first panel of Figure 3, we show the estimated value of $\alpha$ as a function of time. We notice that during the flux emergence phase, $\alpha$ increases and reaches values very close to $1/3$, when reconnection begins. This suggests that the tearing mode is effectively ideal and that the reconnection rate obtained in the simulations should be close to realistic values. Interestingly, as shown in Figure 3, the Alfvén time $t_A = L/v_A$ varies much less than the CS length and remains close to 3
volumetric energy gain and loss, heating, thermal conduction, and radiative losses. The system is solved in spherical coordinates $(r, \theta)$, and the gravity potential:

$$ \Phi = -\frac{GM_{\odot}}{r} \qquad (5) $$

The heating term is defined as

$$ Q_h = F_h/H \left(\frac{R_{\odot}}{r}\right)^2 \exp\left(-\frac{r-R_{\odot}}{H}\right). \qquad (6) $$

The energy flux from the photosphere, denoted as $F_h$, has a value of $1.5 \times 10^5$ erg cm$^{-2}$ s$^{-1}$ (Withbroe 1988) and $H \sim 1 R_{\odot}$.

The thermal conduction is written as

$$ Q_c = \nabla \cdot (\alpha q_s + (1 - \alpha) q_p), \qquad (7) $$

where $q_s = -\kappa_0 T^{5/2} \nabla T$ is the Spitzer-Härm thermal conduction with $\kappa_0 = 9 \times 10^{-7}$ cgs, and $q_p = 3/2 p_{th} v_e$ is the electron collisionless heat flux described in Hollweg (1986). The coefficient $\alpha = 1/(1 + (r - R_{\odot})^4/(r_{coll} - R_{\odot})^4)$ creates a smooth transition between the two regimes at a characteristic height of $r_{coll} = 5 R_{\odot}$. We have used an optically thin radiation cooling prescription:

$$ Q_r = n^2 \Lambda(T), \qquad (8) $$

with $n$ the electron density and $T$ the electron temperature. $\Lambda(T)$ follows the prescription of Athay (1986).

In the PLUTO code, all quantities are expressed in dimensionless units derived from physical quantities divided by normalization units, appropriate for solar wind conditions. We consider unit length $L_0 = 1 R_{\odot} \sim 6.9570 \times 10^5$ km and unit density $\rho_0 = 1.67 \times 10^{-12}$ kg m$^{-3}$. The velocities are normalized to the Keplerian velocity $v_0 \sim 437$ km s$^{-1}$. The characteristic magnetic field and unit time are $B_0 = \sqrt{4\pi\rho_0 v_0^2} \sim 2$ G and $t_0 = L_0/U_0 = 1593$ s, respectively, and we set the magnetic diffusivity $\eta$ to $10^{12}$ cm$^2$ s$^{-1}$. The simulation is integrated on a nonuniform grid with a strong refinement of $\Delta r = 10^{-4}$ in the 0.1 $R_{\odot}$ region in code units and a coarser grid extending up to 20 $R_{\odot}$. The range for $\theta$ is $[\pi/2 - 0.145, \pi/2 + 0.305]$ with a stretched grid cell size of $\Delta\theta = 2 \times 10^{-4}$, and the total grid size is $1536 \times 1536$. We consider reflective boundaries for the velocity components across $\theta_{min}$ and $\theta_{max}$. At $r = 20 R_{\odot} > r_{A,f}$ the fast magneto-sonic point, we use an outflow boundary condition.

We initialize the atmosphere with a supersonic Parker wind and a purely radial field of 2 G at the surface. We assume coronal conditions and do not include a chromosphere or transition region. We let the system reach a steady state before starting the flux emergence. For the inner boundary condition at $r = 1 R_{\odot}$, we fix the temperature and density at the initial value $n = 2 \times 10^8$ cm$^{-3}$ and $T = 1$ MK. The magnetic field is set to its background or emerging (see next) values, while the velocity field is computed to set the parallel electric field to zero. These boundary conditions are very close to the ones described in Parenti et al. (2022).

We then apply a time-dependent boundary condition for the magnetic field component ($B_r, B_{\theta}$) to control the emergence rate of the two polarities. A bipole of 20 G amplitude and $\sim 10^\circ$ of latitudinal extent is projected on the spherical harmonics base (with $l_{max} = 65$), and each coefficient is added to the background field at the boundary and increased linearly during

the emergence phase. The spherical harmonics decomposition ensures that $\nabla \cdot B = 0$ at all times. We focus on four setups with emergence rates of 3.61 G hr$^{-1}$, 4.52 G hr$^{-1}$, 6 G hr$^{-1}$, and 9 G hr$^{-1}$, which are described in Figure 1. The duration of emergence for each setup is 5.53, 4.4, 3.3, and 2.2 hr, respectively.

In 2.5D, the Lundquist number $S = L v_A / \eta$ should be higher than a critical value of approximately $10^4$ to trigger the tearing instability (Biskamp 1986; Loureiro et al. 2007; Bhattacharjee et al. 2009). Hence, we set an explicit value of $\eta$ to be comfortably above this threshold (note that this depends on the length of the CS, and will vary with time as the flux emergence proceeds). But we must also be careful that the numerical resistivity remains smaller than or equal to the value of $\eta$ chosen. Based on our experience of the onset of the tearing mode with the PLUTO code (Réville et al. 2020), we find that a value of $\eta = 10^{12}$ cm$^2$ s$^{-1}$ and the above-described grid resolution satisfy these requirements with $S_{\eta} \sim S_{num} \sim 10^4$. We have also performed some simulations with $\eta = 10^{11}$ cm$^2$ s$^{-1}$ and $\eta = 10^{13}$ cm$^2$ s$^{-1}$ and did not notice significant changes during the bursty reconnection phase (see Figure 9).

# 3. Results

## 3.1. Study of the CS Formation and Aspect Ratio

Figure 2 shows the time evolution of the simulation for a rate of emergence of 3.61 G hr$^{-1}$. We chose three characteristic phases of the emergence and relaxation phases, showing the logarithm of the out-of-the-plane current density and the radial velocity (top and bottom panels). In the early stages of emergence, the CS is created immediately and the tearing instability is triggered. The CS is disrupted several times but continues to lengthen as the emergence continues and reaches its plateau. Close to the peak of the emergence (middle panel), reconnection occurs and plasmoids are ejected on both sides of the CS. Finally, after the emergence has stopped, the CS slowly decays and shortens, thus stopping the reconnection process. This stage can be seen in the right panel of Figure 2.

The CS’s length ($L$) is automatically measured by calculating $L = |\mathbf{B}|/|\nabla \times \mathbf{B}|$ then fixing a threshold to locate the CS (see, e.g., Nóbrega-Siverio & Moreno-Inserti 2022). Figure 3 shows the evolution of the CS length obtained by this method.

Pucci & Velli (2014) introduced the concept of “ideal” tearing, in which the growth rate of the instability is independent of the Lundquist number. Assume that the ratio of the thickness of the CS $a$ to its length scale $L$ scales as $S^{-\alpha}$, where $S$ is the Lundquist number and $\alpha$ is the power-law index. There is a critical value of $\alpha$ at which the growth rate is constant, $\gamma t_A \sim S^{-1/2}$, cnst, which is equal to $1/3$. If $\alpha$ is greater than $1/3$, the growth rate tends to diverge with increasing Lundquist number, while if $\alpha$ is less than $1/3$, the growth rate tends to zero. We thus expect the reconnection to occur precisely at $\alpha = 1/3$, when the CS forms from large aspect ratios.

In the first panel of Figure 3, we show the estimated value of $\alpha$ as a function of time. We notice that during the flux emergence phase, $\alpha$ increases and reaches values very close to $1/3$, when reconnection begins. This suggests that the tearing mode is effectively ideal and that the reconnection rate obtained in the simulations should be close to realistic values. Interestingly, as shown in Figure 3, the Alfvén time $t_A = L/v_A$ varies much less than the CS length and remains close to 3
<figure><img src="image_4.png" /><figcaption>Figure 1. Figure illustrating various simulated setups of flux emergence evolution over time.</figcaption></figure>

minutes throughout the emergence phase. This is due to the linear dependence of $L$, which follows the linear increase of $v_A$ during the emergence ($v_A$ is computed in the bipole, away from the CS, as in usual tearing-mode analyses). The value of $t_A$ is also constant for different emergence rates, which will have consequences on the measured periodicity of the reconnection jets (see Section 3.2).

Once the flux emergence phase is complete, we observe a decrease in both the CS length and the value of $\alpha$. This indicates that the magnetic field is settling and converging to the X-point. The decrease in $\alpha$ indicates that the CS becomes thicker and more diffuse, and the magnetic field lines are less tightly packed. Once the CS starts thickening, the tearing reconnection essentially stops.

As the emergence phase ceases, a quasi-steady reconnecting phase begins, during which the CS starts to diffuse. To quantify the behavior of the CS, we calculated $L$ over time for all our simulations, as illustrated in Figure 4. There is a clear linear relationship between $L$ and the rate of flux emergence, suggesting that the rate of emergence is a key factor in the evolution of $L$. Furthermore, we observed that the decay of the CS followed a linear trend with the same slope for the same magnetic diffusivity. As expected for quasi-steady reconnection, the decay rate depends on the chosen $\eta$ value: a higher (lower) $\eta$ leads to a slower (faster) decay of $L$ (Pontin & Priest 2022).

## 3.2. Formation and Propagation of Jets and Reconnection Plasmoids

In all the setups, we observed recurrent jets and velocity spikes. Plasmoids repeatedly form and are ejected from the CS, triggering sequences of perturbations of the plasma feeding the solar wind above the cusp of the forming pseudostreamer. In Figure 5, we present the maximum value of the radial velocity for each setup of the flux emergence rate. It shows that the flux emergence rate has a direct impact on the speed of the triggered jet, which varies between 50 and 200 km s⁻¹. The higher the emergence rate the higher the amplitude of the jet. Furthermore, the reconnection process heats the plasma and creates density structures that can be related to EUV observations.

### 3.2.1. Synthetic EUV Emission

A 2.5D MHD simulation provides electron densities and temperatures in $(r, \theta, \phi)$ inside a 2D plane. We can construct a 3D

cube necessary to compute EUV images by multiplying the 2D plane over a depth $L_{los}$ along the line of sight assumed perpendicular to the 2D simulation plane. The emission ($dI_j$) from the plasma in each cell ($j$) of the cube can then be calculated. For the Solar Dynamics Observatory (SDO) EUV bands, the dominant process assumed for the observed emission is excitation by electron-ion collisions followed by spontaneous emission. The expression for the emission is given by

$$dI_j = A_f \cdot G(T_j, n_j) \cdot n_j^2 \cdot dV. \quad (9)$$

Here, $A_f$ represents the spectral response function of the instrument being simulated and $G(T_j, n_j)$ encapsulates the atomic physics involved in the spectral line formation, which is dependent on the local electron density ($n$) and temperature ($T$). The values of $A_f$ are provided by the SDO/Atmospheric Imaging Assembly (AIA) instrument team and distributed in the SolarSoft library, while $G(T_j, n_j)$ is the contribution function calculated using the CHIANTI atomic physics database version 7.1.3 and ChiantiPy interface (Dere et al. 1997; Dere 2013) by assuming ionization equilibrium and coronal composition. The values of $n_j$ and $T_j$ are derived from the output of the MHD model.

The simulated AIA images are generated in units of DN s⁻¹, which is the calibrated data unit. Using this method, the synthetic images should capture the emission properties of the corona and allow comparison with the observational data obtained by SDO/AIA. Figure 6 shows the EUV emissions observed at 193 and 171 Å. We can see clearly that the plasmoid is discernible at 193 Å. For this reason, the 193 Å band was chosen due to its response function peaking at a temperature of approximately 1.5 MK reached by the plasmoids.

Figure 7 shows the integrated intensity of the emission at 193 Å for, respectively, several values of magnetic diffusivity and different flux emergence rates where we have captured the plasmoids more clearly so that we can focus on the emission triggered by magnetic reconnection.

We observe that the rate of flux emergence influences the amplitude of brightening at EUV 193 Å. A fast magnetic flux emergence rate of 9 G hr⁻¹ results in an emission amplitude of 3.10⁴ DN s⁻¹, while a slow flux emergence rate of 3.61 G hr⁻¹ results in an EUV brightness that is 10 times smaller.
<figure><img src="image_4.png" /><figcaption>Figure 1. Figure illustrating various simulated setups of flux emergence evolution over time.</figcaption></figure>

minutes throughout the emergence phase. This is due to the linear dependence of $L$, which follows the linear increase of $v_A$ during the emergence ($v_A$ is computed in the bipole, away from the CS, as in usual tearing-mode analyses). The value of $t_A$ is also constant for different emergence rates, which will have consequences on the measured periodicity of the reconnection jets (see Section 3.2).

Once the flux emergence phase is complete, we observe a decrease in both the CS length and the value of $\alpha$. This indicates that the magnetic field is settling and converging to the X-point. The decrease in $\alpha$ indicates that the CS becomes thicker and more diffuse, and the magnetic field lines are less tightly packed. Once the CS starts thickening, the tearing reconnection essentially stops.

As the emergence phase ceases, a quasi-steady reconnecting phase begins, during which the CS starts to diffuse. To quantify the behavior of the CS, we calculated $L$ over time for all our simulations, as illustrated in Figure 4. There is a clear linear relationship between $L$ and the rate of flux emergence, suggesting that the rate of emergence is a key factor in the evolution of $L$. Furthermore, we observed that the decay of the CS followed a linear trend with the same slope for the same magnetic diffusivity. As expected for quasi-steady reconnection, the decay rate depends on the chosen $\eta$ value: a higher (lower) $\eta$ leads to a slower (faster) decay of $L$ (Pontin & Priest 2022).

## 3.2. Formation and Propagation of Jets and Reconnection Plasmoids

In all the setups, we observed recurrent jets and velocity spikes. Plasmoids repeatedly form and are ejected from the CS, triggering sequences of perturbations of the plasma feeding the solar wind above the cusp of the forming pseudostreamer. In Figure 5, we present the maximum value of the radial velocity for each setup of the flux emergence rate. It shows that the flux emergence rate has a direct impact on the speed of the triggered jet, which varies between 50 and 200 km s$^{-1}$. The higher the emergence rate the higher the amplitude of the jet. Furthermore, the reconnection process heats the plasma and creates density structures that can be related to EUV observations.

### 3.2.1. Synthetic EUV Emission

A 2.5D MHD simulation provides electron densities and temperatures in $(r, \theta, \phi)$ inside a 2D plane. We can construct a 3D

cube necessary to compute EUV images by multiplying the 2D plane over a depth $L_{los}$ along the line of sight assumed perpendicular to the 2D simulation plane. The emission ($dI_j$) from the plasma in each cell ($j$) of the cube can then be calculated. For the Solar Dynamics Observatory (SDO) EUV bands, the dominant process assumed for the observed emission is excitation by electron-ion collisions followed by spontaneous emission. The expression for the emission is given by

$$dI_j = A_f \cdot G(T_j, n_j) \cdot n_j^2 \cdot dV. \qquad (9)$$

Here, $A_f$ represents the spectral response function of the instrument being simulated and $G(T_j, n_j)$ encapsulates the atomic physics involved in the spectral line formation, which is dependent on the local electron density ($n$) and temperature ($T$). The values of $A_f$ are provided by the SDO/Atmospheric Imaging Assembly (AIA) instrument team and distributed in the SolarSoft library, while $G(T_j, n_j)$ is the contribution function calculated using the CHIANTI atomic physics database version 7.1.3 and ChiantiPy interface (Dere et al. 1997; Dere 2013) by assuming ionization equilibrium and coronal composition. The values of $n_j$ and $T_j$ are derived from the output of the MHD model.

The simulated AIA images are generated in units of DN s$^{-1}$, which is the calibrated data unit. Using this method, the synthetic images should capture the emission properties of the corona and allow comparison with the observational data obtained by SDO/AIA. Figure 6 shows the EUV emissions observed at 193 and 171 Å. We can see clearly that the plasmoid is discernible at 193 Å. For this reason, the 193 Å band was chosen due to its response function peaking at a temperature of approximately 1.5 MK reached by the plasmoids.

Figure 7 shows the integrated intensity of the emission at 193 Å for, respectively, several values of magnetic diffusivity and different flux emergence rates where we have captured the plasmoids more clearly so that we can focus on the emission triggered by magnetic reconnection.

We observe that the rate of flux emergence influences the amplitude of brightening at EUV 193 Å. A fast magnetic flux emergence rate of 9 G hr$^{-1}$ results in an emission amplitude of $3.10^4$ DN s$^{-1}$, while a slow flux emergence rate of 3.61 G hr$^{-1}$ results in an EUV brightness that is 10 times smaller.
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Development and decay of the CS during flux emergence. The plot from left to right shows the logarithm of the out-of-plane current (top panels) and the radial velocity (bottom panels) at three different times. The first phase is a lengthening and thinning CS. The middle panel shows a well-developed CS undergoing reconnection through tearing. Finally, after the emergence, the CS starts decaying and reconnection stops. Velocity jets can be seen inward to the bipole footpoints and outward in the plume's fan.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 3.** Evolution of the power-law index $\alpha$, $L$, and the Alfvén time $t_A$ during the flux emergence and relaxation phase with a rate of 4.52 G hr$^{-1}$. The black vertical line indicates the time when the emergence has finished. The green horizontal line represents the critical $\alpha$ triggering the Sweet-Parker reconnection ($1/2$), and the red horizontal line denotes the critical $\alpha$ for the establishment of ideal tearing ($1/3$). The third panel shows the evolution of $t_A = L/v_A$. One can see that it remains close to 3 minutes during the whole emergence phase, while the CS length doubles in size and follows linearly the evolution of the Alfvén speed increase in the bipole.</figcaption></figure>

### 3.2.2. Periodicity of Jetlet-associated Brightenings versus Periodicity of Outflows

Our simulations reveal that the emerging bipole exhibits rapid brightening in the EUV 193 Å channel as it interacts with the overlying magnetic field. This brightening is mainly driven by magnetic energy being converted into heat in the CS layer (Ohmic heating), as well as increased densities in the sequence of outflowing jets via magnetic reconnection. We have checked whether the brightening could arise from adiabatic compression rather than reconnection associated heating. Within the CS, the component $-p\nabla \cdot v$ predominantly cools the local plasma, implying that the major contribution to the temperature rise is indeed due to the Ohmic heating term $\eta J^2$ during magnetic reconnection. EUV brightenings and jets should consequently be considered as the macroscopic signatures of local and bursty energy releases that develop along the reconnecting layer at the surface of the emerging bipole.

Investigating the quasiperiodicity of these energy releases is important because it provides insights into the relationship between the energy releases and the rate of magnetic flux emergence, which drives the underlying footpoint exchange mechanism studied here. We performed a wavelet analysis using the wavelet software package of Torrence & Compo (1998) to study the periodicity of the oscillating emission intensity and the
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Development and decay of the CS during flux emergence. The plot from left to right shows the logarithm of the out-of-plane current (top panels) and the radial velocity (bottom panels) at three different times. The first phase is a lengthening and thinning CS. The middle panel shows a well-developed CS undergoing reconnection through tearing. Finally, after the emergence, the CS starts decaying and reconnection stops. Velocity jets can be seen inward to the bipole footpoints and outward in the plume's fan.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 3.** Evolution of the power-law index $\alpha$, L, and the Alfvén time $t_A$ during the flux emergence and relaxation phase with a rate of 4.52 G hr$^{-1}$. The black vertical line indicates the time when the emergence has finished. The green horizontal line represents the critical $\alpha$ triggering the Sweet-Parker reconnection ($1/2$), and the red horizontal line denotes the critical $\alpha$ for the establishment of ideal tearing ($1/3$). The third panel shows the evolution of $t_A = L/v_A$. One can see that it remains close to 3 minutes during the whole emergence phase, while the CS length doubles in size and follows linearly the evolution of the Alfvén speed increase in the bipole.</figcaption></figure>

### 3.2.2. Periodicity of Jetlet-associated Brightenings versus Periodicity of Outflows

Our simulations reveal that the emerging bipole exhibits rapid brightening in the EUV 193 Å channel as it interacts with the overlying magnetic field. This brightening is mainly driven by magnetic energy being converted into heat in the CS layer (Ohmic heating), as well as increased densities in the sequence of outflowing jets via magnetic reconnection. We have checked whether the brightening could arise from adiabatic compression rather than reconnection associated heating. Within the CS, the component $-p\nabla \cdot v$ predominantly cools the local plasma, implying that the major contribution to the temperature rise is indeed due to the Ohmic heating term $\eta J^2$ during magnetic reconnection. EUV brightenings and jets should consequently be considered as the macroscopic signatures of local and bursty energy releases that develop along the reconnecting layer at the surface of the emerging bipole.

Investigating the quasiperiodicity of these energy releases is important because it provides insights into the relationship between the energy releases and the rate of magnetic flux emergence, which drives the underlying footpoint exchange mechanism studied here. We performed a wavelet analysis using the wavelet software package of Torrence & Compo (1998) to study the periodicity of the oscillating emission intensity and the
<figure><img src="image_7.png" /><figcaption>**Figure 4.** In the upper panel, we plot the evolution of CS length (L) for all setups with $\eta = 10^{12} \text{ cm}^2 \text{ s}^{-1}$ and the second panel shows the linear fit with the corresponding slope “s.”</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 5.** The maximum values of the radial velocity simulated in the plume above the CS at a distance of approximately 1.5 $R_\odot$ for the different setups of flux emergence. Magnetic diffusivity was fixed in all simulations at $\eta = 10^{12} \text{ cm}^2 \text{ s}^{-1}$.</figcaption></figure>

radial velocity $V_r$ of the jets for the different flux emergence rates. Figure 8 shows the wavelet spectral analysis of the radial velocity $V_r$ for a flux emergence with a rate of 4.52 G hr$^{-1}$. The signal is first detrended by subtracting a linear polynomial fit from the original data. The resulting detrended signal is then normalized by dividing it by its standard deviation to ensure that it has a zero mean and unit variance.

Figure 9 shows the periodicity of radial velocity spikes and the 193 Å EUV emission for several flux emergence rates and for several magnetic diffusivity values. First, we notice that the radial velocity oscillation is mostly unchanged with the flux emergence rate. Second, we observe a good correspondence between the periodicity of the jet outflows and the EUV brightenings. This is expected because, in the simulations, the two processes are directly related to the tearing-induced reconnection phase. It has

been, however, very surprising that the periodicity does not change significantly with the control parameters of the simulation.

This relatively constant periodicity across the various simulation parameters can be interpreted as follows: the Alfvén time $t_A$ away from the CS is approximately 3 minutes, and as we are in the ideal tearing reconnection regime $\gamma t_A$, the normalized growth rate, is close to unity. Hence, once the CS has reached the ideal tearing ratio, the periodicity of jets is directly related to the time for the CS to be disrupted, i.e., a few times $t_A$, which leads to the observed 19 minutes. Moreover, as shown in Figure 3, the Alfvén time stays roughly constant throughout the emergence phase, as the CS lengthens and adapts to the increasing magnetic field strength in the bipole. Indeed, in our setup, it is strictly equivalent to an increase in either the emergence speed or the bipole amplitude, and thus $t_A$ remains close to this 3 minute value in all the setups. Note that as $S \sim 10^4$, the $\gamma t_A \sim$ constant may not be fully reached yet.

Finally, plasmoids propagate within the CS either sunward or antisunward, while only antisunward plasmoids will launch jets propagating in the solar wind. As a result, the periodicity of the velocity spikes may be higher than the full CS disruption and reformation cycle, as the jets correspond only to the outward-moving plasmoids. This nonetheless suggests that the periodicity of EUV brightenings and velocity jets may follow a universal rule and depend only on the local value of $t_A$.

These results are also consistent with previous observations reported by Uritsky et al. (2021) and Kumar et al. (2023b), who also found that the oscillations in the plume exhibit a range of periods similar to those of the jet (10–20 minutes).

### 3.3. Alfvén Waves versus Magneto-acoustic Waves: Which Wave Mode Dominates the Coronal Jets?

There has been a debate in the community about whether propagation disturbances (PDs) in plumes observed in the solar corona are plasma outflows or slow-mode waves (Poletto 2015;
<figure><img src="image_7.png" /><figcaption>**Figure 4.** In the upper panel, we plot the evolution of CS length (L) for all setups with $\eta = 10^{12} \text{ cm}^2 \text{ s}^{-1}$ and the second panel shows the linear fit with the corresponding slope “s.”</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 5.** The maximum values of the radial velocity simulated in the plume above the CS at a distance of approximately 1.5 $R_\odot$ for the different setups of flux emergence. Magnetic diffusivity was fixed in all simulations at $\eta = 10^{12} \text{ cm}^2 \text{ s}^{-1}$.</figcaption></figure>

radial velocity $V_r$ of the jets for the different flux emergence rates. Figure 8 shows the wavelet spectral analysis of the radial velocity $V_r$ for a flux emergence with a rate of 4.52 G hr$^{-1}$. The signal is first detrended by subtracting a linear polynomial fit from the original data. The resulting detrended signal is then normalized by dividing it by its standard deviation to ensure that it has a zero mean and unit variance.

Figure 9 shows the periodicity of radial velocity spikes and the 193 Å EUV emission for several flux emergence rates and for several magnetic diffusivity values. First, we notice that the radial velocity oscillation is mostly unchanged with the flux emergence rate. Second, we observe a good correspondence between the periodicity of the jet outflows and the EUV brightenings. This is expected because, in the simulations, the two processes are directly related to the tearing-induced reconnection phase. It has

been, however, very surprising that the periodicity does not change significantly with the control parameters of the simulation.

This relatively constant periodicity across the various simulation parameters can be interpreted as follows: the Alfvén time $t_A$ away from the CS is approximately 3 minutes, and as we are in the ideal tearing reconnection regime $\gamma t_A$, the normalized growth rate, is close to unity. Hence, once the CS has reached the ideal tearing ratio, the periodicity of jets is directly related to the time for the CS to be disrupted, i.e., a few times $t_A$, which leads to the observed 19 minutes. Moreover, as shown in Figure 3, the Alfvén time stays roughly constant throughout the emergence phase, as the CS lengthens and adapts to the increasing magnetic field strength in the bipole. Indeed, in our setup, it is strictly equivalent to an increase in either the emergence speed or the bipole amplitude, and thus $t_A$ remains close to this 3 minute value in all the setups. Note that as $S \sim 10^4$, the $\gamma t_A \sim$ constant may not be fully reached yet.

Finally, plasmoids propagate within the CS either sunward or antisunward, while only antisunward plasmoids will launch jets propagating in the solar wind. As a result, the periodicity of the velocity spikes may be higher than the full CS disruption and reformation cycle, as the jets correspond only to the outward-moving plasmoids. This nonetheless suggests that the periodicity of EUV brightenings and velocity jets may follow a universal rule and depend only on the local value of $t_A$.

These results are also consistent with previous observations reported by Uritsky et al. (2021) and Kumar et al. (2023b), who also found that the oscillations in the plume exhibit a range of periods similar to those of the jet (10–20 minutes).

### 3.3. Alfvén Waves versus Magneto-acoustic Waves: Which Wave Mode Dominates the Coronal Jets?

There has been a debate in the community about whether propagation disturbances (PDs) in plumes observed in the solar corona are plasma outflows or slow-mode waves (Poletto 2015;
<figure><img src="image_9.png" /><figcaption>(a) 171 Å</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(b) 193 Å</figcaption></figure>

**Figure 6.** Synthetic EUV emission intensity maps produced from the simulation results for two wavelengths observed by the SDO. These emission intensity maps were produced from the simulation setup with a flux emergence rate of 4.52 G hr⁻¹.

<figure><img src="image_11.png" /></figure>

**Figure 7.** Synthetic emission intensities of the 193 Å EUV line for the four emergence rates simulated for a fixed $\eta = 10^{12}$ cm² s⁻¹.

<figure><img src="image_12.png" /></figure>

**Figure 8.** Wavelet analysis of the radial velocity for the setup of a flux emergence of 4.52 G hr⁻¹ and $\eta = 10^{12}$ cm² s⁻¹. (a) Radial velocity evolution over time at 1.5 $R_\odot$, the detrended radial velocity after subtracting it from the original gray (color). (b) The wavelet power spectrum of the detrended signal is displayed as well as (c) the global wavelet power spectrum.

Wang 2016). The work of Wang (2016) suggests that both of these interpretations may be correct, as reconnection at the footpoints of the plumelet drives flows that can be observed as “jetlets” and generate Alfvénic fronts, but the dense material in the jets travels more slowly and an inhomogeneous wake of shear and compressible turbulence should be observed between the jet and the Alfvénic front.
<figure><img src="image_9.png" /><figcaption>(a) 171 Å</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(b) 193 Å</figcaption></figure>

**Figure 6.** Synthetic EUV emission intensity maps produced from the simulation results for two wavelengths observed by the SDO. These emission intensity maps were produced from the simulation setup with a flux emergence rate of 4.52 G hr⁻¹.

<figure><img src="image_11.png" /></figure>

**Figure 7.** Synthetic emission intensities of the 193 Å EUV line for the four emergence rates simulated for a fixed $\eta = 10^{12}$ cm² s⁻¹.

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

**Figure 8.** Wavelet analysis of the radial velocity for the setup of a flux emergence of 4.52 G hr⁻¹ and $\eta = 10^{12}$ cm² s⁻¹. (a) Radial velocity evolution over time at 1.5 $R_⊙$, the detrended radial velocity after subtracting it from the original gray (color). (b) The wavelet power spectrum of the detrended signal is displayed as well as (c) the global wavelet power spectrum.

Wang 2016). The work of Wang (2016) suggests that both of these interpretations may be correct, as reconnection at the footpoints of the plumelet drives flows that can be observed as “jetlets” and generate Alfvénic fronts, but the dense material in the jets travels more slowly and an inhomogeneous wake of shear and compressible turbulence should be observed between the jet and the Alfvénic front.
<figure><img src="image_13.png" /><figcaption>**Figure 9.** Figure depicting the periodicity of oscillation of radial velocity and EUV 193 Å for (a) different emergence rates (G hr⁻¹) but a fixed diffusivity of 10¹² cm² s⁻¹ and (b) different magnetic diffusivity values for the same emergence rate 4.52 G hr⁻¹.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure 10.** Density and velocity perturbations as a function of time for an emergence rate of 9 G hr⁻¹ at 1.15 R☉ in the jet’s stalk.</figcaption></figure>

In our simulations, the flux emergence increases *L*, and when the tearing instability is triggered, plasmons are then generated that move up and down the X-line. It is important to understand what type of wave this process generates, transverse Alfvén wave or magneto-acoustic waves? To check this, we start with the linearized equation of ideal MHD. Three well-known modes can be distinguished: fast and slow magneto-acoustic waves, as well as (transverse) shear Alfvén waves. For shear Alfvén wave, we have

$$ egin{cases} v_{\text{pA}} = \frac{\omega}{k} = v_A \cos \theta \approx v_A, \\ \frac{\delta v}{v_A} = \pm \frac{\delta B}{B_0}, \\ \delta \rho = 0, \\ \delta |B| = 0, \end{cases} \quad (10) $$

where $v_A$ is the Alfvén speed, $\theta$ is the angle between the wavevector $k$ and the magnetic field $B$ and is close to zero, $\delta v$, $\delta B$, and $\delta \rho$ are the perturbed plasma velocity, magnetic fields, and

<figure><img src="image_15.png" /></figure>

plasma density, respectively, and $B_0$ is the background magnetic magnitude. For slow and fast waves, the phase speed is

$$ \begin{aligned} v_{\text{ph}} \pm 2 &= \left(\frac{\omega}{k}\right)^2 \\ &= \frac{1}{2}(v_S^2 + v_A^2) \pm \frac{1}{2}[(v_S^2 + v_A^2)^2 - 4v_S^2 v_A^2 \cos^2 \theta]^{1/2}. \end{aligned} \quad (11) $$

To simplify our analysis, we focus on the case where $v_S \ll v_A$ which is justified when $r < R_A$ (below the Alfvén radius). We then have for fast magneto-acoustic waves (Hollweg 1975)

$$ \begin{aligned} \frac{\delta \rho}{\rho_0} &= \frac{\delta |B|}{B_0}, \\ v_{\text{ph}+2} &= v_A^2, \\ |\frac{\delta \rho}{\rho_0}| &< 0.1, \end{aligned} \quad (12) $$

and for slow magneto-acoustic waves

$$ \begin{aligned} |\frac{\delta \rho}{\rho_0}| &= |\frac{\delta v}{v_S}|, \\ v_{\text{ph}-2} &= v_S^2 \cos^2 \theta, \\ \left(|\frac{\delta \rho}{\rho_0}| > 0.1\right), \\ \delta |B| &= 0. \end{aligned} \quad (13) $$

Figure 10 presents the plasma velocity fluctuation and density fluctuation at 1.15 R☉. It is evident that $|\frac{\delta \rho}{\rho_0}| \sim |\frac{\delta v}{v_S}|$. Given the additional constraint that $|\frac{\delta \rho}{\rho_0}| > 0.1$ and that we have checked that $\text{Max}(|\delta|B||) \approx 10^{-5} \ll 1$, the characteristics exhibited by the jets align remarkably well with the behavior expected from slow magneto-acoustic waves.

We show in Figure 11 a 2D map of the density variations relative to the local density in the high corona and the nascent solar wind. These density variations suggest compression waves are triggered and propagate along the plume that has formed above the cusp of the emerging bipoles. These density
<figure><img src="image_15.png" /><figcaption>**Figure 9.** Figure depicting the periodicity of oscillation of radial velocity and EUV 193 Å for (a) different emergence rates (G hr⁻¹) but a fixed diffusivity of 10¹² cm² s⁻¹ and (b) different magnetic diffusivity values for the same emergence rate 4.52 G hr⁻¹.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Figure 10.** Density and velocity perturbations as a function of time for an emergence rate of 9 G hr⁻¹ at 1.15 R☉ in the jet’s stalk.</figcaption></figure>

In our simulations, the flux emergence increases *L*, and when the tearing instability is triggered, plasmons are then generated that move up and down the X-line. It is important to understand what type of wave this process generates, transverse Alfvén wave or magneto-acoustic waves? To check this, we start with the linearized equation of ideal MHD. Three well-known modes can be distinguished: fast and slow magneto-acoustic waves, as well as (transverse) shear Alfvén waves. For shear Alfvén wave, we have

$$ egin{cases} v_{\text{pA}} = \frac{\omega}{k} = v_A \cos \theta \approx v_A, \\ \frac{\delta v}{v_A} = \pm \frac{\delta B}{B_0}, \\ \delta \rho = 0, \\ \delta |B| = 0, \end{cases} \qquad (10) $$

where $v_A$ is the Alfvén speed, $\theta$ is the angle between the wavevector $k$ and the magnetic field $B$ and is close to zero, $\delta v$, $\delta B$, and $\delta \rho$ are the perturbed plasma velocity, magnetic fields, and

<figure><img src="image_17.png" /></figure>

plasma density, respectively, and $B_0$ is the background magnetic magnitude. For slow and fast waves, the phase speed is

$$ \begin{aligned} v_{\text{ph}} \pm^2 &= \left(\frac{\omega}{k}\right)^2 \\ &= \frac{1}{2}(v_S^2 + v_A^2) \pm \frac{1}{2}[(v_S^2 + v_A^2)^2 - 4v_S^2 v_A^2 \cos^2 \theta]^{1/2}. \end{aligned} \qquad (11) $$

To simplify our analysis, we focus on the case where $v_S \ll v_A$ which is justified when $r < R_A$ (below the Alfvén radius). We then have for fast magneto-acoustic waves (Hollweg 1975)

$$ \begin{aligned} \frac{\delta \rho}{\rho_0} &= \frac{\delta |B|}{B_0}, \\ v_{\text{ph}+2} &= v_A^2, \\ |\frac{\delta \rho}{\rho_0}| &< 0.1, \end{aligned} \qquad (12) $$

and for slow magneto-acoustic waves

$$ \begin{aligned} |\frac{\delta \rho}{\rho_0}| &= |\frac{\delta v}{v_S}|, \\ v_{\text{ph}-2} &= v_S^2 \cos^2 \theta, \\ \left(\frac{\delta \rho}{\rho_0}\right) &> 0.1, \\ \delta |B| &\approx 0. \end{aligned} \qquad (13) $$

Figure 10 presents the plasma velocity fluctuation and density fluctuation at 1.15 R☉. It is evident that $|\frac{\delta \rho}{\rho_0}| \sim |\frac{\delta v}{v_S}|$. Given the additional constraint that $|\frac{\delta \rho}{\rho_0}| > 0.1$ and that we have checked that $\text{Max}(|\delta|B||) \approx 10^{-5} \ll 1$, the characteristics exhibited by the jets align remarkably well with the behavior expected from slow magneto-acoustic waves.

We show in Figure 11 a 2D map of the density variations relative to the local density in the high corona and the nascent solar wind. These density variations suggest compression waves are triggered and propagate along the plume that has formed above the cusp of the emerging bipoles. These density
<figure><img src="image_16.png" /><figcaption>**Figure 11.** A 2D map of density fluctuations ($\delta\rho$) relative to the local density $\rho$. $\delta\rho$ is computed by subtracting two consecutive frames separated by a 16 minute time lapse.</figcaption></figure>

fluctuations originate in the plasmons ejected outwards during the magnetic reconnection process. As the size of magnetic islands becomes larger, the medium surrounding the plasmons is compressed by the outflowing structure, leading to changes in the density.

We now focus on the Alfvénic character of the generated waves. We define the cross helicity, $H_c$, as

$$H_c = \frac{2 \cdot (\delta v_A \cdot \delta v)}{\delta v_A^2 + \delta v^2} \qquad (14)$$

$H_c$ quantifies the degree of correlation in the fluctuating velocity and magnetic field components. This provides information on the nature and direction of the propagating waves. In the case of pure shear Alfvén waves, we have $\delta v = \pm \delta v_A$, where $\delta v_A$ is the perturbed Alfvén velocity, and $H_c = \pm 1$. The propagation direction of Alfvén waves is indicated by the sign of the cross helicity. A “+” sign denotes propagation antiparallel to the local mean magnetic field $B_0$, while a “-” sign denotes propagation parallel to $B_0$. In the solar wind, particularly in the inner heliosphere, Alfvén waves predominantly propagate outward from the Sun. As our background field is positive, a negative cross helicity indicates fluctuations propagating outward into the heliosphere, while positive values denote inward-propagating fluctuations.

We use the maximum radial velocity to pinpoint the jet’s location. Subsequently, we calculate perturbations in magnetic field and plasma velocity, both in the radial and tangential directions. A linear function of time is employed to fit the background profile, which is then subtracted from the original data. The resulting perturbed signal is normalized by dividing it by its standard deviation, resulting in a zero mean and unit variance, denoted as $\delta Z$.

In Figure 12, we present a time series illustrating radial and tangential perturbations in Alfvén and plasma velocities at various radial positions for a simulation with an emergence rate of 9 G hr$^{-1}$. We selected four heliocentric radial distances [1.5, 3, 6, 12] in solar radii to monitor the jet’s evolution. The color map represents cross helicity, with blue and red colors indicating negative cross helicity during the passage of the wave packet, signifying outward-moving Alfvén waves.

The tangential perturbation exhibits a decreasing trend with solar radius, transitioning from 0.6 km s$^{-1}$ at 1.5 $R_\odot$ to 0.08 km s$^{-1}$ at 12 $R_\odot$. A similar trend is observed in the radial direction, with velocity decreasing from 2 to 0.6 km s$^{-1}$.

This behavior can be attributed to the diffusive nature of slow magneto-acoustic waves and associated damping, as well as the damping due to decreasing resolution at higher altitudes.

As we ascend in altitude, the jet displays reduced compressibility. Significant disparities exist between the tangential and radial perturbations. Tangential perturbations are characterized by a more pronounced anticorrelation as well as higher frequencies. Notably, when examining various altitudes of 1.5 $R_\odot$, 3 $R_\odot$, 6 $R_\odot$, and 12 $R_\odot$, we observe distinct patterns in the tangential perturbations.

At 1.5 $R_\odot$ we identify the presence of two distinct modes, with periodicities of 3 minutes and 17 minutes. The two modes survive up to 4 $R_\odot$ but disappear at 6 $R_\odot$. At the farthest radial distance of 12 $R_\odot$, we observe oscillations with a periodicity of 17.5 minutes, which are essentially the periodicity observed for the radial jets and the EUV emission. The frequency filtering at higher altitudes can also be attributed to the progressively coarser resolution as one moves farther away from the reconnection site.

Similar behavior is observed for different emergence rates, where faster emergence rates result in larger wave amplitudes.

### 3.4. Long-lasting Oscillation

Various studies have investigated the transverse motion of coronal jets (Shibata et al. 1992a; Canfield et al. 1996; Savcheva et al. 2007). In our simulations, we observe that the jet is displaced to higher latitudes with a velocity $v_\theta \sim 20$ km s$^{-1}$ comparable with those reported in these past studies. Figure 13 illustrates whip-like motion following the triggered reconnection, and the evolution of the CS leads to the propagation of velocity spikes characterized by developing fluctuations along the left side of the pseudostreamer stalk. Similar behavior can be observed in the density perturbation, as shown in Figure 11, where a wisp-like structure is omnipresent with velocity spikes that are initiated by magnetic reconnection within the CS and move away from the stalk as previously reported in observations by Shimojo et al. (2007). In particular, these whip-like oscillations consistently align on one side, contributing to the formation of the long-lasting transverse oscillation phenomenon seen in Figure 14.

The emergence rate has an effect on the jet expansion in the $\theta$ direction, where this lateral expansion was also reported in coronal observations by Moore et al. (2010) describing expanding jets as “curtain-like spires.” Our simulations reveal two driving mechanisms for these transverse motions: from the ideal tearing reconnection process and the expanding motion influenced by magnetic flux emergence.

## 4. Discussion

In this work, we study the effect of magnetic flux emergence in 2.5D resistive MHD simulations of the solar corona and wind. Our study shows that the emergence process leads naturally to interchange reconnection with the ambient coronal solution and creates jets, or velocity spikes, that then propagate into the solar wind. We observe two main phases of the reconnection process. First, shortly after the start of emergence, a CS is created, which then lengthens and thins until it reaches the aspect ratio $a/L \propto S^{-1/3}$. Fast reconnection then proceeds through the so-called “ideal” tearing instability, creating plasmons that are ejected either toward the Sun and inner boundary condition or toward the fan of the pseudostreamers
<figure><img src="image_18.png" /><figcaption>**Figure 11.** A 2D map of density fluctuations ($\delta\rho$) relative to the local density $\rho$. $\delta\rho$ is computed by subtracting two consecutive frames separated by a 16 minute time lapse.</figcaption></figure>

fluctuations originate in the plasmons ejected outwards during the magnetic reconnection process. As the size of magnetic islands becomes larger, the medium surrounding the plasmons is compressed by the outflowing structure, leading to changes in the density.

We now focus on the Alfvénic character of the generated waves. We define the cross helicity, $H_c$, as

$$H_c = \frac{2 \cdot (\delta v_A \cdot \delta v)}{\delta v_A^2 + \delta v^2} \quad (14)$$

$H_c$ quantifies the degree of correlation in the fluctuating velocity and magnetic field components. This provides information on the nature and direction of the propagating waves. In the case of pure shear Alfvén waves, we have $\delta v = \pm \delta v_A$, where $\delta v_A$ is the perturbed Alfvén velocity, and $H_c = \pm 1$. The propagation direction of Alfvén waves is indicated by the sign of the cross helicity. A “+” sign denotes propagation antiparallel to the local mean magnetic field $B_0$, while a “-” sign denotes propagation parallel to $B_0$. In the solar wind, particularly in the inner heliosphere, Alfvén waves predominantly propagate outward from the Sun. As our background field is positive, a negative cross helicity indicates fluctuations propagating outward into the heliosphere, while positive values denote inward-propagating fluctuations.

We use the maximum radial velocity to pinpoint the jet's location. Subsequently, we calculate perturbations in magnetic field and plasma velocity, both in the radial and tangential directions. A linear function of time is employed to fit the background profile, which is then subtracted from the original data. The resulting perturbed signal is normalized by dividing it by its standard deviation, resulting in a zero mean and unit variance, denoted as $\delta Z$.

In Figure 12, we present a time series illustrating radial and tangential perturbations in Alfvén and plasma velocities at various radial positions for a simulation with an emergence rate of 9 G hr$^{-1}$. We selected four heliocentric radial distances [1.5, 3, 6, 12] in solar radii to monitor the jet's evolution. The color map represents cross helicity, with blue and red colors indicating negative cross helicity during the passage of the wave packet, signifying outward-moving Alfvén waves.

The tangential perturbation exhibits a decreasing trend with solar radius, transitioning from 0.6 km s$^{-1}$ at 1.5 $R_\odot$ to 0.08 km s$^{-1}$ at 12 $R_\odot$. A similar trend is observed in the radial direction, with velocity decreasing from 2 to 0.6 km s$^{-1}$.

This behavior can be attributed to the diffusive nature of slow magneto-acoustic waves and associated damping, as well as the damping due to decreasing resolution at higher altitudes.

As we ascend in altitude, the jet displays reduced compressibility. Significant disparities exist between the tangential and radial perturbations. Tangential perturbations are characterized by a more pronounced anticorrelation as well as higher frequencies. Notably, when examining various altitudes of 1.5 $R_\odot$, 3 $R_\odot$, 6 $R_\odot$, and 12 $R_\odot$, we observe distinct patterns in the tangential perturbations.

At 1.5 $R_\odot$ we identify the presence of two distinct modes, with periodicities of 3 minutes and 17 minutes. The two modes survive up to 4 $R_\odot$ but disappear at 6 $R_\odot$. At the farthest radial distance of 12 $R_\odot$, we observe oscillations with a periodicity of 17.5 minutes, which are essentially the periodicity observed for the radial jets and the EUV emission. The frequency filtering at higher altitudes can also be attributed to the progressively coarser resolution as one moves farther away from the reconnection site.

Similar behavior is observed for different emergence rates, where faster emergence rates result in larger wave amplitudes.

### 3.4. Long-lasting Oscillation

Various studies have investigated the transverse motion of coronal jets (Shibata et al. 1992a; Canfield et al. 1996; Savcheva et al. 2007). In our simulations, we observe that the jet is displaced to higher latitudes with a velocity $v_\theta \sim 20$ km s$^{-1}$ comparable with those reported in these past studies. Figure 13 illustrates whip-like motion following the triggered reconnection, and the evolution of the CS leads to the propagation of velocity spikes characterized by developing fluctuations along the left side of the pseudostreamer stalk. Similar behavior can be observed in the density perturbation, as shown in Figure 11, where a wisp-like structure is omnipresent with velocity spikes that are initiated by magnetic reconnection within the CS and move away from the stalk as previously reported in observations by Shimojo et al. (2007). In particular, these whip-like oscillations consistently align on one side, contributing to the formation of the long-lasting transverse oscillation phenomenon seen in Figure 14.

The emergence rate has an effect on the jet expansion in the $\theta$ direction, where this lateral expansion was also reported in coronal observations by Moore et al. (2010) describing expanding jets as “curtain-like spires.” Our simulations reveal two driving mechanisms for these transverse motions: from the ideal tearing reconnection process and the expanding motion influenced by magnetic flux emergence.

## 4. Discussion

In this work, we study the effect of magnetic flux emergence in 2.5D resistive MHD simulations of the solar corona and wind. Our study shows that the emergence process leads naturally to interchange reconnection with the ambient coronal solution and creates jets, or velocity spikes, that then propagate into the solar wind. We observe two main phases of the reconnection process. First, shortly after the start of emergence, a CS is created, which then lengthens and thins until it reaches the aspect ratio $a/L \propto S^{-1/3}$. Fast reconnection then proceeds through the so-called “ideal” tearing instability, creating plasmons that are ejected either toward the Sun and inner boundary condition or toward the fan of the pseudostreamers
<figure><img src="image_17.png" /><figcaption>**Figure 12.** Radial and tangential Alfvénic and velocity perturbations at $r/R_⊙$=[1.5, 3, 6, 12] for an emergence rate of 9 G hr$^{-1}$. Radial Alfvénic perturbations (represented by the blue line) and radial velocity perturbations (represented by the orange line) at various radial locations in the solar corona. The plot also displays the cross helicity (represented by the red and blue color map) at these locations.</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>**Figure 12.** Radial and tangential Alfvénic and velocity perturbations at $r/R_⊙$=[1.5, 3, 6, 12] for an emergence rate of 9 G hr$^{-1}$. Radial Alfvénic perturbations (represented by the blue line) and radial velocity perturbations (represented by the orange line) at various radial locations in the solar corona. The plot also displays the cross helicity (represented by the red and blue color map) at these locations.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>**Figure 13.** The figure illustrates a jet wisp triggered by magnetic reconnection, where white circles indicate the wisp signatures.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>**Figure 14.** The figure illustrates jet expansion during CS evolution.</figcaption></figure>

created in the corona. The plasmoid propagates at high speed along the CS and appears to hit the stalk of the pseudostreamer. The plasmoid is eroded and triggers slow magneto-acoustic waves with jets of amplitude up to 200 km s⁻¹. Following the completion of the flux emergence phase, we observe a decrease in both the CS length and α. This suggests that the magnetic field is settling into a more stable state. The CS then diffuses and no more bursty reconnection occurs. The recent study of Wyper et al. (2022), investigated interchange reconnection with 3D ideal MHD simulations of pseudostreamers interacting with the solar corona. Although they reach similar conclusions on the creation of Alfvénic structures in the fans of pseudostreamers, our work complements and differs in some important aspects. First, by precisely controlling the explicit resistivity of the model, we show that the tearing instability is in the ideal regime, which ensures that the reconnection properties should be relatively independent of the Lundquist number and close to the low coronal regime where S ~ 10¹⁴. Second, while Wyper et al. (2022) triggers reconnection by surface motions, emergence suffices in our case. Finally, by varying the emergence rate of the bipole, or equivalently the amplitude of the emerging flux, we have shown that the periodicity of the jets matches the periodicity of the EUV emission of the plasma and that it is roughly independent of the emergence rate. This is due to the fact the characteristic Alfvén time $t_A = L/v_A \sim 3$ minutes remains unchanged for higher emergence rates and magnetic field amplitudes, as the CS lengthens proportionally to the Alfvén speed. The time between each jet is thus the time for the CS to be disrupted, i.e., a few $t_A$, or 19 minutes. These findings are consistent with previous observations reported by Uritsky et al. (2021) and Kumar et al. (2023b), further

supporting the connection between plume oscillations and jet periodicity. Nevertheless, the rate of flux emergence has a significant impact on the observed emission amplitudes in the EUV range. Higher flux emergence rates correspond to larger emission amplitudes, as well as higher amplitudes of the velocity spikes.

Several recent studies based on observations of PSP suggest that switchbacks may be caused by jetlets originating from small bipoles located at the base of coronal plumes in coronal holes (Bale et al. 2021; Fargette et al. 2021; Kumar et al. 2022; Shi et al. 2022; Bale et al. 2023). Neugebauer et al. (1995) already suggested that jetlets may also generate microstreams, which are fluctuations in the solar wind speed and density observed in polar coronal holes. Although our current results do not directly demonstrate the generation of magnetic reversals through the emergence of bipoles, they do reveal the presence of Alfvénic perturbations that could potentially evolve into magnetic switchbacks. The study of Wyper et al. (2022) shows a somewhat different structure in 3D, with torsional Alfvén waves launched from the pseudostreamer fan. Yet, no full reversals (or switchbacks) seem able to survive outside of the closed magnetic structures in the simulations. Nonetheless, true switchbacks could be reformed later on as they propagate in the solar wind as nonlinear developments of seed Alfvén waves, as suggested by Squire et al. (2020) and Mallet et al. (2021). This emphasizes the need for further observations and MHD simulations to establish a definitive relationship between magnetic switchbacks and interchange reconnection in the chromosphere and transition region beneath plumes.

## Acknowledgments

The research of B.G., V.R., and A.P.R. was funded by the ERC SLOW_SOURCE (DLV-819189).

The authors are grateful to Kévin Dalmasse, Benoit Lavraud, Peter Wyper, Marco Velli, and Nour E. Raouafi for insightful discussions. The authors also thank A. Mignone and the PLUTO development team, on which the numerical work presented in this paper is based. The 2.5D MHD simulations were performed on the Toulouse CALMIP supercomputer and the Jean-Zay supercomputer (IDRIS), through the GENCI HPC allocation grant A0130410293.

This work benefited also from financial support from the Centre National des Études Spatiales (CNES).

## Appendix

Figure A1 depicts the initiation of the tearing instability for our simulation at an emergence rate of 6 G hr⁻¹. The still frame represents the time $t = 4.2$ hr, marking the onset of the tearing instability.

The accompanying animation, associated with Figure A1, presents the dynamic evolution of the following variables: radial velocity ($V_r$ in km s⁻¹), temperature (in MK), radial magnetic field ($B_r$ in G), and the logarithm of the out-of-plane current in response to the tearing mode and the multiple instances of instability onset.
<figure><img src="image_20.png" /><figcaption>**Figure 13.** The figure illustrates a jet wisp triggered by magnetic reconnection, where white circles indicate the wisp signatures.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>**Figure 14.** The figure illustrates jet expansion during CS evolution.</figcaption></figure>

created in the corona. The plasmoid propagates at high speed along the CS and appears to hit the stalk of the pseudostreamer. The plasmoid is eroded and triggers slow magneto-acoustic waves with jets of amplitude up to 200 km s⁻¹. Following the completion of the flux emergence phase, we observe a decrease in both the CS length and α. This suggests that the magnetic field is settling into a more stable state. The CS then diffuses and no more bursty reconnection occurs. The recent study of Wyper et al. (2022), investigated interchange reconnection with 3D ideal MHD simulations of pseudostreamers interacting with the solar corona. Although they reach similar conclusions on the creation of Alfvénic structures in the fans of pseudostreamers, our work complements and differs in some important aspects. First, by precisely controlling the explicit resistivity of the model, we show that the tearing instability is in the ideal regime, which ensures that the reconnection properties should be relatively independent of the Lundquist number and close to the low coronal regime where S ~ 10¹⁴. Second, while Wyper et al. (2022) triggers reconnection by surface motions, emergence suffices in our case. Finally, by varying the emergence rate of the bipole, or equivalently the amplitude of the emerging flux, we have shown that the periodicity of the jets matches the periodicity of the EUV emission of the plasma and that it is roughly independent of the emergence rate. This is due to the fact the characteristic Alfvén time $t_A = L/v_A \sim 3$ minutes remains unchanged for higher emergence rates and magnetic field amplitudes, as the CS lengthens proportionally to the Alfvén speed. The time between each jet is thus the time for the CS to be disrupted, i.e., a few $t_A$, or 19 minutes. These findings are consistent with previous observations reported by Uritsky et al. (2021) and Kumar et al. (2023b), further

supporting the connection between plume oscillations and jet periodicity. Nevertheless, the rate of flux emergence has a significant impact on the observed emission amplitudes in the EUV range. Higher flux emergence rates correspond to larger emission amplitudes, as well as higher amplitudes of the velocity spikes.

Several recent studies based on observations of PSP suggest that switchbacks may be caused by jetlets originating from small bipoles located at the base of coronal plumes in coronal holes (Bale et al. 2021; Fargette et al. 2021; Kumar et al. 2022; Shi et al. 2022; Bale et al. 2023). Neugebauer et al. (1995) already suggested that jetlets may also generate microstreams, which are fluctuations in the solar wind speed and density observed in polar coronal holes. Although our current results do not directly demonstrate the generation of magnetic reversals through the emergence of bipoles, they do reveal the presence of Alfvénic perturbations that could potentially evolve into magnetic switchbacks. The study of Wyper et al. (2022) shows a somewhat different structure in 3D, with torsional Alfvén waves launched from the pseudostreamer fan. Yet, no full reversals (or switchbacks) seem able to survive outside of the closed magnetic structures in the simulations. Nonetheless, true switchbacks could be reformed later on as they propagate in the solar wind as nonlinear developments of seed Alfvén waves, as suggested by Squire et al. (2020) and Mallet et al. (2021). This emphasizes the need for further observations and MHD simulations to establish a definitive relationship between magnetic switchbacks and interchange reconnection in the chromosphere and transition region beneath plumes.

## Acknowledgments

The research of B.G., V.R., and A.P.R. was funded by the ERC SLOW_SOURCE (DLV-819189).

The authors are grateful to Kévin Dalmasse, Benoit Lavraud, Peter Wyper, Marco Velli, and Nour E. Raouafi for insightful discussions. The authors also thank A. Mignone and the PLUTO development team, on which the numerical work presented in this paper is based. The 2.5D MHD simulations were performed on the Toulouse CALMIP supercomputer and the Jean-Zay supercomputer (IDRIS), through the GENCI HPC allocation grant A0130410293.

This work benefited also from financial support from the Centre National des Études Spatiales (CNES).

## Appendix

Figure A1 depicts the initiation of the tearing instability for our simulation at an emergence rate of 6 G hr⁻¹. The still frame represents the time $t = 4.2$ hr, marking the onset of the tearing instability.

The accompanying animation, associated with Figure A1, presents the dynamic evolution of the following variables: radial velocity ($V_r$ in km s⁻¹), temperature (in MK), radial magnetic field ($B_r$ in G), and the logarithm of the out-of-plane current in response to the tearing mode and the multiple instances of instability onset.
<figure><img src="image_20.png" /></figure>

**Figure A1.** Snapshot of the CS during flux emergence. The four panels display the following variables: radial velocity ($V_r$ in km s⁻¹), temperature (in MK), radial magnetic field ($B_r$ in G), and the logarithm of the out-of-plane current. This figure is available as an animation. The animation of the tearing instability for our simulation proceeds from *t* = 2.30 to 5.31 hr.

(An animation of this figure is available.)

## ORCID iDs

Bahaeddine Gannouni https://orcid.org/0000-0002-1711-1802

Victor Réville https://orcid.org/0000-0002-2916-3837

Alexis P. Rouillard https://orcid.org/0000-0003-4039-5767

## References

* Athay, R. G. 1986, ApJ, 308, 975

* Bale, S. D., Badman, S. T., Bonnell, J. W., et al. 2019, Natur, 576, 237

* Bale, S. D., Drake, J. F., McManus, M. D., et al. 2023, Natur, 618, 252

* Bale, S. D., Horbury, T. S., Velli, M., et al. 2021, ApJ, 923, 174

* Balsara, D. S., & Spicer, D. S. 1999, JCoPh, 149, 270

* Bhattacharjee, A., Huang, Y.-M., Yang, H., & Rogers, B. 2009, PhPl, 16, 112102

* Biskamp, D. 1986, PhFl, 29, 1520

* Canfield, R. C., Reardon, K. P., Leka, K. D., et al. 1996, ApJ, 464, 1016

* DeForest, C. E. 2007, ApJ, 661, 532

* DeForest, C. E., Hoeksema, J. T., Gurman, J. B., et al. 1997, SoPh, 175, 393

* DeForest, C. E., Lamy, P. L., & Llebaria, A. 2001, ApJ, 560, 490

* Dere, K., 2013 ChiantiPy: Python package for the CHIANTI atomic database, Astrophysics Source Code Library, ascl:1308.017

* Dere, K. P., Landi, E., Mason, H. E., Monsignori Fossi, B. C., & Young, P. R. 1997, A&AS, 125, 149

* Drake, J. F., Agapitov, O., Swisdak, M., et al. 2021, A&A, 650, A2

* Fargette, N., Lavraud, B., Rouillard, A. P., et al. 2021, ApJ, 919, 96

* Fargette, N., Lavraud, B., Rouillard, A. P., et al. 2022, A&A, 663, A109

* Fisk, L. A. 2005, ApJ, 626, 563

* Fisk, L. A., & Kasper, J. C. 2020, ApJL, 894, L4

* He, J., Zhu, X., Yang, L., et al. 2021, ApJL, 913, L14

* Hollweg, J. V. 1975, RvGSP, 13, 263

* Hollweg, J. V. 1986, JGR, 91, 4111

* Hou, C., Zhu, X., Zhuo, R., et al. 2023, ApJ, 950, 157

* Kasper, J. C., Bale, S. D., Belcher, J. W., et al. 2019, Natur, 576, 228

* Kumar, P., Karpen, J. T., Uritsky, V. M., et al. 2022, ApJ, 933, 21

* Kumar, P., Karpen, J. T., Uritsky, V. M., et al. 2023a, ApJL, 951, L15

* Kumar, P., Karpen, J. T., Uritsky, V. M., et al. 2023b, ApJL, 951, L15

* Loureiro, N. F., Schekochihin, A. A., & Cowley, S. C. 2007, PhPl, 14, 100703

* Mallet, A., Squire, J., Chandran, B. D. G., Bowen, T., & Bale, S. D. 2021, ApJ, 918, 62

* McIntosh, S. W., Innes, D. E., de Pontieu, B., & Leamon, R. J. 2010, A&A, 510, L2

* Mignone, A., Bodo, G., Massaglia, S., et al. 2007, ApJS, 170, 228

* Miyoshi, T., & Kusano, K. 2005, JCoPh, 208, 315

* Moore, R. L., Cirtain, J. W., Sterling, A. C., & Falconer, D. A. 2010, ApJ, 720, 757

* Neugebauer, M. 2012, ApJ, 750, 50

* Neugebauer, M., Goldstein, B. E., McComas, D. J., Suess, S. T., & Balogh, A. 1995, JGR, 100, 23389

* Nóbrega-Siverio, D., & Moreno-Insertis, F. 2022, ApJL, 935, L21

* Panesar, N. K., Sterling, A. C., Moore, R. L., et al. 2018, ApJL, 868, L27

* Panesar, N. K., Sterling, A. C., Moore, R. L., et al. 2019, ApJL, 887, L8

* Parenti, S., Réville, V., Brun, A. S., et al. 2022, ApJ, 929, 75

* Pariat, E., Antiochos, S. K., & DeVore, C. R. 2009, ApJ, 691, 61

* Poletto, G. 2015, LRSP, 12, 7

* Pontin, D. I., & Priest, E. R. 2022, LRSP, 19, 1

* Pucci, F., & Velli, M. 2014, ApJL, 780, L19

* Raouafi, N. E., Petrie, G. J. D., Norton, A. A., Henney, C. J., & Solanki, S. K. 2008, ApJL, 682, L137

* Raouafi, N. E., & Stenborg, G. 2014, ApJ, 787, 118

* Raouafi, N. E., Stenborg, G., Seaton, D. B., et al. 2023, ApJ, 945, 28

* Réville, V., Velli, M., Rouillard, A. P., et al. 2020, ApJL, 895, L20

* Rouillard, A. P., Kouloumvakos, A., Vourlidas, A., et al. 2020, ApJS, 246, 37

* Savcheva, A., Cirtain, J., Deluca, E. E., et al. 2007, PASJ, 59, S771

* Schwadron, N. A., & McComas, D. J. 2021, ApJ, 909, 95

* Shi, C., Panasenco, O., Velli, M., et al. 2022, ApJ, 934, 152

* Shibata, K., Ishido, Y., Acton, L. W., et al. 1992a, PASJ, 44, L173

* Shibata, K., Nozawa, S., & Matsumoto, R. 1992b, PASJ, 44, 265

* Shibata, K., Tajima, T., Steinolfson, R. S., & Matsumoto, R. 1989, ApJ, 345, 584

* Shimojo, M., Narukage, N., Kano, R., et al. 2007, PASJ, 59, S745

* Shoda, M., Chandran, B. D. G., & Cranmer, S. R. 2021, ApJ, 915, 52

* Squire, J., Chandran, B. D. G., & Meyrand, R. 2020, ApJ, 891, L2

* Tian, H., McIntosh, S. W., Habbal, S. R., & He, J. 2011, ApJ, 736, 130

* Torrence, C., & Compo, G. P. 1998, BAMS, 79, 61

* Uritsky, V. M., DeForest, C. E., Karpen, J. T., et al. 2021, ApJ, 907, 1

* Vlahos, L. 2002, in IAU Coll. 188, Proc. of the Magnetic Coupling of the Solar Atmosphere Euroconference, ed. H. Sawaya-Lacoste (Noordwijk: ESA Publications), 105

* Wang, T. J. 2016, in Low-Frequency Waves in Space Plasmas, ed. A. Keiling, D.-H. Lee, & V. Nakariakov, 216 (New York: Wiley), 395

* Wang, Y.-M., Hawley, S. H., & Sheeley, N. R. J. 1996, Sci, 271, 464

* Wilhelm, K., Abbo, L., Auchère, F., et al. 2011, A&ARv, 19, 35

* Withbroe, G. L. 1988, ApJ, 325, 442

* Woo, R. 1996, ApJL, 464, L95

* Wyper, P. F., DeVore, C. R., Antiochos, S. K., et al. 2022, ApJL, 941, L29

* Yamauchi, Y., Suess, S. T., Steinberg, J. T., & Sakurai, T. 2004, JGRA, 109, A03104

* Zhao, T.-L., Ni, L., Lin, J., & Ziegler, U. 2018, RAA, 18, 045
<figure><img src="image_22.png" /></figure>

**Figure A1.** Snapshot of the CS during flux emergence. The four panels display the following variables: radial velocity ($V_r$ in km s⁻¹), temperature (in MK), radial magnetic field ($B_r$ in G), and the logarithm of the out-of-plane current. This figure is available as an animation. The animation of the tearing instability for our simulation proceeds from *t* = 2.30 to 5.31 hr.

(An animation of this figure is available.)

## ORCID iDs

Bahaeddine Gannouni https://orcid.org/0000-0002-1711-1802

Victor Réville https://orcid.org/0000-0002-2916-3837

Alexis P. Rouillard https://orcid.org/0000-0003-4039-5767

## References

* Athay, R. G. 1986, ApJ, 308, 975

* Bale, S. D., Badman, S. T., Bonnell, J. W., et al. 2019, Natur, 576, 237

* Bale, S. D., Drake, J. F., McManus, M. D., et al. 2023, Natur, 618, 252

* Bale, S. D., Horbury, T. S., Velli, M., et al. 2021, ApJ, 923, 174

* Balsara, D. S., & Spicer, D. S. 1999, JCoPh, 149, 270

* Bhattacharjee, A., Huang, Y.-M., Yang, H., & Rogers, B. 2009, PhPl, 16, 112102

* Biskamp, D. 1986, PhFl, 29, 1520

* Canfield, R. C., Reardon, K. P., Leka, K. D., et al. 1996, ApJ, 464, 1016

* DeForest, C. E. 2007, ApJ, 661, 532

* DeForest, C. E., Hoeksema, J. T., Gurman, J. B., et al. 1997, SoPh, 175, 393

* DeForest, C. E., Lamy, P. L., & Llebaria, A. 2001, ApJ, 560, 490

* Dere, K., 2013 ChiantiPy: Python package for the CHIANTI atomic database, Astrophysics Source Code Library, ascl:1308.017

* Dere, K. P., Landi, E., Mason, H. E., Monsignori Fossi, B. C., & Young, P. R. 1997, A&AS, 125, 149

* Drake, J. F., Agapitov, O., Swisdak, M., et al. 2021, A&A, 650, A2

* Fargette, N., Lavraud, B., Rouillard, A. P., et al. 2021, ApJ, 919, 96

* Fargette, N., Lavraud, B., Rouillard, A. P., et al. 2022, A&A, 663, A109

* Fisk, L. A. 2005, ApJ, 626, 563

* Fisk, L. A., & Kasper, J. C. 2020, ApJL, 894, L4

* He, J., Zhu, X., Yang, L., et al. 2021, ApJL, 913, L14

* Hollweg, J. V. 1975, RvGSP, 13, 263

* Hollweg, J. V. 1986, JGR, 91, 4111

* Hou, C., Zhu, X., Zhuo, R., et al. 2023, ApJ, 950, 157

* Kasper, J. C., Bale, S. D., Belcher, J. W., et al. 2019, Natur, 576, 228

* Kumar, P., Karpen, J. T., Uritsky, V. M., et al. 2022, ApJ, 933, 21

* Kumar, P., Karpen, J. T., Uritsky, V. M., et al. 2023a, ApJL, 951, L15

* Kumar, P., Karpen, J. T., Uritsky, V. M., et al. 2023b, ApJL, 951, L15

* Loureiro, N. F., Schekochihin, A. A., & Cowley, S. C. 2007, PhPl, 14, 100703

* Mallet, A., Squire, J., Chandran, B. D. G., Bowen, T., & Bale, S. D. 2021, ApJ, 918, 62

* McIntosh, S. W., Innes, D. E., de Pontieu, B., & Leamon, R. J. 2010, A&A, 510, L2

* Mignone, A., Bodo, G., Massaglia, S., et al. 2007, ApJS, 170, 228

* Miyoshi, T., & Kusano, K. 2005, JCoPh, 208, 315

* Moore, R. L., Cirtain, J. W., Sterling, A. C., & Falconer, D. A. 2010, ApJ, 720, 757

* Neugebauer, M. 2012, ApJ, 750, 50

* Neugebauer, M., Goldstein, B. E., McComas, D. J., Suess, S. T., & Balogh, A. 1995, JGR, 100, 23389

* Nóbrega-Siverio, D., & Moreno-Insertis, F. 2022, ApJL, 935, L21

* Panesar, N. K., Sterling, A. C., Moore, R. L., et al. 2018, ApJL, 868, L27

* Panesar, N. K., Sterling, A. C., Moore, R. L., et al. 2019, ApJL, 887, L8

* Parenti, S., Réville, V., Brun, A. S., et al. 2022, ApJ, 929, 75

* Pariat, E., Antiochos, S. K., & DeVore, C. R. 2009, ApJ, 691, 61

* Poletto, G. 2015, LRSP, 12, 7

* Pontin, D. I., & Priest, E. R. 2022, LRSP, 19, 1

* Pucci, F., & Velli, M. 2014, ApJL, 780, L19

* Raouafi, N. E., Petrie, G. J. D., Norton, A. A., Henney, C. J., & Solanki, S. K. 2008, ApJL, 682, L137

* Raouafi, N. E., & Stenborg, G. 2014, ApJ, 787, 118

* Raouafi, N. E., Stenborg, G., Seaton, D. B., et al. 2023, ApJ, 945, 28

* Réville, V., Velli, M., Rouillard, A. P., et al. 2020, ApJL, 895, L20

* Rouillard, A. P., Kouloumvakos, A., Vourlidas, A., et al. 2020, ApJS, 246, 37

* Savcheva, A., Cirtain, J., Deluca, E. E., et al. 2007, PASJ, 59, S771

* Schwadron, N. A., & McComas, D. J. 2021, ApJ, 909, 95

* Shi, C., Panasenco, O., Velli, M., et al. 2022, ApJ, 934, 152

* Shibata, K., Ishido, Y., Acton, L. W., et al. 1992a, PASJ, 44, L173

* Shibata, K., Nozawa, S., & Matsumoto, R. 1992b, PASJ, 44, 265

* Shibata, K., Tajima, T., Steinolfson, R. S., & Matsumoto, R. 1989, ApJ, 345, 584

* Shimojo, M., Narukage, N., Kano, R., et al. 2007, PASJ, 59, S745

* Shoda, M., Chandran, B. D. G., & Cranmer, S. R. 2021, ApJ, 915, 52

* Squire, J., Chandran, B. D. G., & Meyrand, R. 2020, ApJ, 891, L2

* Tian, H., McIntosh, S. W., Habbal, S. R., & He, J. 2011, ApJ, 736, 130

* Torrence, C., & Compo, G. P. 1998, BAMS, 79, 61

* Uritsky, V. M., DeForest, C. E., Karpen, J. T., et al. 2021, ApJ, 907, 1

* Vlahos, L. 2002, in IAU Coll. 188, Proc. of the Magnetic Coupling of the Solar Atmosphere Euroconference, ed. H. Sawaya-Lacoste (Noordwijk: ESA Publications), 105

* Wang, T. J. 2016, in Low-Frequency Waves in Space Plasmas, ed. A. Keiling, D.-H. Lee, & V. Nakariakov, 216 (New York: Wiley), 395

* Wang, Y.-M., Hawley, S. H., & Sheeley, N. R. J. 1996, Sci, 271, 464

* Wilhelm, K., Abbo, L., Auchère, F., et al. 2011, A&ARv, 19, 35

* Withbroe, G. L. 1988, ApJ, 325, 442

* Woo, R. 1996, ApJL, 464, L95

* Wyper, P. F., DeVore, C. R., Antiochos, S. K., et al. 2022, ApJL, 941, L29

* Yamauchi, Y., Suess, S. T., Steinberg, J. T., & Sakurai, T. 2004, JGRA, 109, A03104

* Zhao, T.-L., Ni, L., Lin, J., & Ziegler, U. 2018, RAA, 18, 045