# Viscous Effects on Nonlinear Double Tearing Mode and Plasmoid Formation in Adjacent Harris Sheets

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## Abstract

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## 1. Introduction

## 2. Model Equations

## 3. Linear DTM Evolution

## 4. Nonlinear DTM and Plasmoids

#### 4.1. Nonlinear DTM Evolution

#### 4.2. Plasmoid Formation

#### 4.3. Flow Pattern and Vortices

## 5. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Double Harris sheet equilibrium magnetic field, (

**b**) current density and (

**c**) pressure profiles at different distances between the rational surfaces.

**Figure 2.**Linear mode structures of (

**a**) magnetic field component ${B}_{x}$, (

**b**) magnetic field component ${B}_{z}$, (

**c**) velocity component ${v}_{x}$, (

**d**) velocity component ${v}_{z}$ and (

**e**) pressure P at $d=1$.

**Figure 3.**(

**a**) Linear growth rates as functions of the resistivity for different distances between the rational surfaces using $\nu \to 0$. (

**b**) $\alpha $ (from ${\gamma}_{lin}$ = ${\eta}^{\alpha}$ ) is function of the distances between the rational surfaces.

**Figure 4.**(

**a**) Linear growth rates as functions of the viscosity at different distances between the rational surfaces for fixed value of $\eta =0.00028$. (

**b**) Linear growth rates as functions of the distances between the rational surfaces for different Prandtl numbers.

**Figure 5.**Kinetic energy evolution for different numerical resolutions at $d=2\left({x}_{0}\right)=1.1$.

**Figure 6.**Kinetic energy evolution for (

**a**) ${P}_{r}=0.33,\text{}0.5$, (

**b**) ${P}_{r}=1,\text{}5$ and (

**c**) ${P}_{r}=10,100$ at $d=1.1$.

**Figure 7.**Two-dimensional contours of current density distribution and 2D magnetic field lines with P

_{r}= 0.33 and $d=1.1$. (

**a**) t = 0; (

**b**) t = 25; (

**c**) t = 85; (

**d**) t = 90; (

**e**) t = 95; (

**f**) t = 100.

**Figure 8.**Two-dimensional contours of current density distribution and 2D magnetic field lines with P

_{r}= 1 and $d=1.1$. (

**a**) t = 85; (

**b**) t = 95; (

**c**) t = 100; (

**d**) t = 115.

**Figure 9.**Two-dimensional contours of current density distribution and 2D magnetic field lines with P

_{r}= 10 and $d=1.1$. (

**a**) t = 116; (

**b**) t = 120; (

**c**) t = 124; (

**d**) t = 132.

**Figure 10.**Two-dimensional contours of current density distribution and 2D magnetic field lines with P

_{r}= 100 and $d=1.1$. (

**a**) t = 160; (

**b**) t = 185; (

**c**) t = 250; (

**d**) t = 270.

**Figure 11.**(

**a**) Monster plasmoid evolution; (

**b**) monster plasmoid width as a function of ${P}_{r}$ with ${P}_{r}=0.33,1,5$ and 10.

**Figure 12.**Two-dimensional contours of the current density distribution and flow field stream lines (

**a**–

**d**), 2D contours of the flow field in the z-direction and 2D magnetic field lines (

**e**–

**h**), P

_{r}= 0.33 with $d=1.1$.

**Figure 13.**Two-dimensional contours of the current density distribution and flow field stream lines (

**a**–

**d**), 2D contours of the flow field in the z-direction and 2D magnetic field lines (

**e**–

**h**), P

_{r}= 1 with $d=1.1$.

**Figure 14.**Two-dimensional contours of the current density distribution and flow field stream lines (

**a**–

**d**), 2D contours of the flow field in the z-direction and 2D magnetic field lines (

**e**–

**h**), P

_{r}= 10 with $d=1.1$.

**Figure 15.**Two-dimensional contours of the current density distribution and flow field stream lines (

**a**–

**d**), 2D contours of the flow field in the z-direction and 2D magnetic field lines (

**e**–

**h**), P

_{r}=100 with $d=1.1$.

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**MDPI and ACS Style**

Ahmad, N.; Zhu, P.; Shen, C.; Ali, A.; Zeng, S.
Viscous Effects on Nonlinear Double Tearing Mode and Plasmoid Formation in Adjacent Harris Sheets. *Magnetochemistry* **2023**, *9*, 205.
https://doi.org/10.3390/magnetochemistry9090205

**AMA Style**

Ahmad N, Zhu P, Shen C, Ali A, Zeng S.
Viscous Effects on Nonlinear Double Tearing Mode and Plasmoid Formation in Adjacent Harris Sheets. *Magnetochemistry*. 2023; 9(9):205.
https://doi.org/10.3390/magnetochemistry9090205

**Chicago/Turabian Style**

Ahmad, Nisar, Ping Zhu, Chao Shen, Ahmad Ali, and Shiyong Zeng.
2023. "Viscous Effects on Nonlinear Double Tearing Mode and Plasmoid Formation in Adjacent Harris Sheets" *Magnetochemistry* 9, no. 9: 205.
https://doi.org/10.3390/magnetochemistry9090205