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Control of a Uniformly Magnetized Plasma with External Electric Fields

Peiyi Chen, Rogerio Jorge, Qin Li, Yukun Yue

Abstract

Stabilizing plasma dynamics through externally applied electric and magnetic fields is a fundamental control problem. We study this question for a plasma evolving under a uniform external magnetic field. Although the governing dynamics are nonlinear, a linear analysis based on the Laplace-Fourier transform yields actionable insight. In particular, by controlling the location of the roots of the dispersion relation, we propose a general control strategy that restores stability, with the free-streaming solution recovered as a special case. Numerical experiments for Gaussian equilibria and for the Dory-Guest-Harris instability show that the proposed control suppresses the unstable modes and stabilizes the dynamics, in agreement with our theoretical predictions.

Control of a Uniformly Magnetized Plasma with External Electric Fields

Abstract

Stabilizing plasma dynamics through externally applied electric and magnetic fields is a fundamental control problem. We study this question for a plasma evolving under a uniform external magnetic field. Although the governing dynamics are nonlinear, a linear analysis based on the Laplace-Fourier transform yields actionable insight. In particular, by controlling the location of the roots of the dispersion relation, we propose a general control strategy that restores stability, with the free-streaming solution recovered as a special case. Numerical experiments for Gaussian equilibria and for the Dory-Guest-Harris instability show that the proposed control suppresses the unstable modes and stabilizes the dynamics, in agreement with our theoretical predictions.

Paper Structure

This paper contains 17 sections, 4 theorems, 66 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Linear analysis
  3. Stability condition.
  4. Two case studies
  5. Gaussian equilibrium.
  6. Dory-Guest-Harris instability
  7. Control with an external electric field
  8. Control Strategy 1: set 0 in (3.6).
  9. Control Strategy 2: set c to be constant in (3.6).
  10. Numerical experiments
  11. Dory-Guest-Harris instability.
  12. Initial perturbation as a sine wave in space
  13. Initial perturbation as a Gaussian in space.
  14. Gaussian equilibrium
  15. Conclusion
  16. ...and 2 more sections

Key Result

Proposition 2.1

Vlasov-Poisson system with external magnetic field eqn: magnet VP satisfies the following properties:

Figures (17)

  • Figure 1: Isotropic Gaussian $\mu$ and $|P(\mathbf{k},\lambda)|$. The minimum is taken at $(\mathrm{Re}(\lambda) = 0.001, \mathrm{Im}(\lambda) = 1.164)$ with value $\min |P(0.001, 1.164)|=2\times 10^{-3}.$
  • Figure 2: Equilibrium distribution \ref{['eq:DGH6_equilibrium']} and $|P(\mathbf{k},\lambda)|$\ref{['eq:penrose_Pk_lam']}. The minimum is taken at $(\lambda_R=0.0059, \lambda_I=1.0207)$ with minimal value $1.5 \times 10^{-5}.$
  • Figure 3: The left panel shows the evolution of the electric energy $\mathcal{E}(t)$, and the right panel shows the first coordinate of $|\mathbf{E}_{(1,0)^\top}(t)|$. Both plots are presented on the semi-log scale.
  • Figure 4: Distribution $f(x_1, x_2=0, v_1, v_2=0)$ snapshot at different time steps, without control. The distribution is asymmetric in the spatial domain, and the effect is rooted in the asymmetry in the perturbation in the initial data.
  • Figure 5: Distribution $f(x_1, x_2=0, v_1, v_2)$ at five $x_1$ locations, without control.
  • ...and 12 more figures

Theorems & Definitions (11)

  • Proposition 2.1
  • Proposition 2.2: Penrose condition
  • Remark 2.1
  • proof : Proof of Proposition \ref{['prop:penrose_cond']}
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Remark 3.1
  • Theorem 3.1
  • proof
  • ...and 1 more