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Control strategies for magnetized plasma: a polar coordinates framework

Federica Ferrarese

TL;DR

The paper addresses controlling magnetized plasma described by the Vlasov-Poisson system using external magnetic fields within a 2D polar-coordinate framework suited for tokamak-like devices. It introduces two instantaneous feedback strategies that assign a piecewise-constant B_ext on a spatial cell grid or per particle with subsequent interpolation, all implemented through a PIC discretization. Numerical experiments on the Diocotron instability demonstrate effective reduction of boundary energy and stabilization of the plasma under both strategies, highlighting geometry-aware advantages and computational efficiency. The work provides a practical control framework that can be extended to more complex physics and higher dimensions, with potential applications in confinement devices and fusion-relevant plasma management.

Abstract

In this work, we provide an overview of various control strategies aimed at steering plasma toward desired configurations using an external magnetic field. From a modeling perspective, we focus on the Vlasov equation in a two-dimensional bounded domain, accounting for both a self-induced electric field and a strong external magnetic field. The results are presented in a polar coordinate framework, which is particularly well-suited for simulating toroidal devices such as Tokamaks and Stellarators. A key feature of the proposed control strategies is their feedback mechanism, which is based on an instantaneous prediction of the discretized system. Finally, different numerical experiments in the two-dimensional polar coordinate setting demonstrate the effectiveness of the approaches.

Control strategies for magnetized plasma: a polar coordinates framework

TL;DR

The paper addresses controlling magnetized plasma described by the Vlasov-Poisson system using external magnetic fields within a 2D polar-coordinate framework suited for tokamak-like devices. It introduces two instantaneous feedback strategies that assign a piecewise-constant B_ext on a spatial cell grid or per particle with subsequent interpolation, all implemented through a PIC discretization. Numerical experiments on the Diocotron instability demonstrate effective reduction of boundary energy and stabilization of the plasma under both strategies, highlighting geometry-aware advantages and computational efficiency. The work provides a practical control framework that can be extended to more complex physics and higher dimensions, with potential applications in confinement devices and fusion-relevant plasma management.

Abstract

In this work, we provide an overview of various control strategies aimed at steering plasma toward desired configurations using an external magnetic field. From a modeling perspective, we focus on the Vlasov equation in a two-dimensional bounded domain, accounting for both a self-induced electric field and a strong external magnetic field. The results are presented in a polar coordinate framework, which is particularly well-suited for simulating toroidal devices such as Tokamaks and Stellarators. A key feature of the proposed control strategies is their feedback mechanism, which is based on an instantaneous prediction of the discretized system. Finally, different numerical experiments in the two-dimensional polar coordinate setting demonstrate the effectiveness of the approaches.

Paper Structure

This paper contains 11 sections, 2 theorems, 45 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Problem setting and numerical methods
  3. PIC scheme in polar coordinates
  4. Control strategies
  5. Strategy one
  6. Strategy two
  7. Numerical experiments: Diocotron instability
  8. Validation test
  9. Test without control
  10. Test with control
  11. Conclusion

Key Result

Proposition 1

Assume the parameters to scale as then the feedback control at cell $C_k$ associated to eq:discr_J_strategy1 reads as follows where $\gamma>0$, and with $\mathbb{P}_{[-M,M]}(\cdot)$ denoting the projection over the interval $[-M,M]$. In the limit $h\to 0$ the control at the continuous level reads, with

Figures (8)

  • Figure 1: Diocotron instability. Initial density in polar coordinates (on the left) and in cartesian coordinates (on the right).
  • Figure 2: Diocotron instability. Error in time defined as in \ref{['eq:error_time']}.
  • Figure 3: Diocotron instability. Uncontrolled dynamics obtained by setting $B(t,{\bf r}) = 10$. Three snapshots of the dynamics taken at time $t=50$, $t=125$ and $t=200$. First row: polar coordinates. Second row: cartesian coordinates.
  • Figure 4: Diocotron instability: uncontrolled dynamics. Thermal energy at the boundaries computed as in \ref{['eq:energy']}.
  • Figure 5: Diocotron instability. Controlled dynamics obtained by setting $B(t,{\bf r})$ as in \ref{['eq:Bk_strategy1']} (strategy one). Three snapshots of the dynamics taken at time $t=50$, $t=125$ and $t=200$. First row: polar coordinates. Second row: cartesian coordinates.
  • ...and 3 more figures

Theorems & Definitions (3)

  • Proposition 1
  • Proposition 2
  • Remark 1