# Structure of Current Sheets Formed in 2D Magnetic Configurations with X-Type Null Lines in the Presence of the Hall Currents and Inverse Currents

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## Abstract

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## 1. Introduction

## 2. Magnetic Field with the X-Line and Experimental CS-3D Device

## 3. Formation of a Current Sheet

## 4. Characteristics of 2D Current Sheets Formed in a Magnetic Field with an X-Line

## 5. Hall Currents and Out-of-Plane Magnetic Fields in the Current Sheet

## 6. Plasma Acceleration in Current Sheets and the Generation of Inverse Currents

## 7. Appearance of the Out-of-Plane Magnetic Field ${\mathit{B}}_{\mathit{z}}$ in the Opposite Direction

## 8. Braking Ampère’s Forces Responsible for Terminating High-Speed Plasma Flows

## 9. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**A 2D magnetic field with the X-type null line on the z-axis. The color map shows the magnetic field magnitude, and the arrows indicate the directions of the magnetic field on the (x,y) plane.

**Figure 2.**Experimental setup CS-3D (Prokhorov General Physics Institute, Russian Academy of Sciences). (

**a**) Cross-section and (

**b**) side view: 1—the system of direct conductors with currents for excitation of the 2D magnetic field with the null line at the z-axis; the field lines are shown in the panel (

**a**) by the dashed lines with the arrows; 2—the cylindrical vacuum chamber; its axis is aligned with the null line; 3—the system of coils of the Θ discharge used to produce the initial plasma; 4—the current sheet formed after excitation of the plasma current ${J}_{Z}$; 5—grid electrodes; 6—quartz windows; 7—coils for exciting the magnetic field. ${B}_{z}$, AA’, BB’, and CC’ are the lines of displacement of the magnetic probes.

**Figure 3.**(

**a**) Scheme of the magneto-sonic wave motion in the initial 2D magnetic field $\overrightarrow{B}=\{hy;hx;0\}$ with the null line on the z-axis; the front of the wave is shown by a dotted circle; red arrows show the directions of the plasma flows behind the wave front. (

**b**) Profiles of magnetic field perturbations ${\delta B}_{x}$ along the BB’ line at successive instants of time. ${B}_{x}^{0}$ is the x-component of the initial magnetic field.

**Figure 4.**Profiles of currents and magnetic fields across the sheet (

**a**) and parallel to the sheet surface (

**b**).

**Figure 5.**A 2D color map of the electron density during the formation of the current sheet (in units of ${10}^{15}{\mathrm{cm}}^{-3}$).

**Figure 6.**(

**a**,

**b**) The distributions of the out-of-plane magnetic field components ${B}_{z}(y)$ and ${B}_{z}(x)$ and the Hall current densities ${j}_{x}(y)$ and ${j}_{y}(x)$ along the BB’ and AA’ directions, respectively. (

**c**) The scheme of the structure of the out-of-plane quadrupole magnetic field ${B}_{z}$.

**Figure 7.**Profiles of the linear current ${I}_{Z}\left(x\right)=\int {j}_{z}(x,y)dy$ near the current sheet neutral plane for three moments of time: 1.5 μs (red curve), 2.4 μs (green curve), and 4.5 μs (blue curve).

**Figure 8.**Distributions of the out-of-plane magnetic field ${B}_{z}(x)$ and the linear current ${I}_{Z}\left(x\right)$ for two moments of time: 1.5 μs (

**a**) and 2.4 μs (

**b**).

**Figure 9.**Distributions of the current sheet parameters along the sheet width: ${B}_{y}\left(x\right)$ is the normal magnetic field component, ${I}_{Z}\left(x\right)=\int {j}_{z}\left(x,y\right)dy$ is the linear current within the sheet thickness at $\left|y\right|\le 0.8\mathrm{c}\mathrm{m}$, ${F}_{x}\left(x\right)=\int {f}_{x}\left(x,y\right)dy$ is the Ampère force within the same region.

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**MDPI and ACS Style**

Frank, A.G.; Savinov, S.A.
Structure of Current Sheets Formed in 2D Magnetic Configurations with X-Type Null Lines in the Presence of the Hall Currents and Inverse Currents. *Symmetry* **2024**, *16*, 103.
https://doi.org/10.3390/sym16010103

**AMA Style**

Frank AG, Savinov SA.
Structure of Current Sheets Formed in 2D Magnetic Configurations with X-Type Null Lines in the Presence of the Hall Currents and Inverse Currents. *Symmetry*. 2024; 16(1):103.
https://doi.org/10.3390/sym16010103

**Chicago/Turabian Style**

Frank, Anna G., and Sergey A. Savinov.
2024. "Structure of Current Sheets Formed in 2D Magnetic Configurations with X-Type Null Lines in the Presence of the Hall Currents and Inverse Currents" *Symmetry* 16, no. 1: 103.
https://doi.org/10.3390/sym16010103