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Electric turbulence in a plasma subject to a strong magnetic field

G. Loeper, A. Vasseur

TL;DR

This work analyzes how a stochastic electric field in a strongly magnetized plasma induces a Landau-type energy transfer, deriving a diffusion limit for the gyroaveraged electron distribution. By leveraging gyroaveraging and a Vasicek/Poupaud–Vasseur style diffusion limit, the authors obtain a Spherical Harmonics Expansion (SHE)–type model for the energy variable with an explicitly computable diffusion coefficient a(e) that depends on the turbulent correlation A. The main result proves convergence to the SHE diffusion equation ∂t ρ - ∂e(a(e) ∂e ρ)=0, under precise mixing hypotheses, and provides explicit expressions and conditions for abnormal diffusion in energy. The analysis advances the understanding of energy transfer in magnetized plasmas under stochastic forcing and offers a rigorous route to quantify diffusion in energy space from turbulent electric fields.

Abstract

We consider in this paper a plasma subject to a strong deterministic magnetic field and we investigate the effect on this plasma of a stochastic electric field. We show that the limit behavior, which corresponds to the transfer of energy from the electric wave to the particles (Landau phenomena), is described by a Spherical Harmonics Expansion (SHE) model.

Electric turbulence in a plasma subject to a strong magnetic field

TL;DR

This work analyzes how a stochastic electric field in a strongly magnetized plasma induces a Landau-type energy transfer, deriving a diffusion limit for the gyroaveraged electron distribution. By leveraging gyroaveraging and a Vasicek/Poupaud–Vasseur style diffusion limit, the authors obtain a Spherical Harmonics Expansion (SHE)–type model for the energy variable with an explicitly computable diffusion coefficient a(e) that depends on the turbulent correlation A. The main result proves convergence to the SHE diffusion equation ∂t ρ - ∂e(a(e) ∂e ρ)=0, under precise mixing hypotheses, and provides explicit expressions and conditions for abnormal diffusion in energy. The analysis advances the understanding of energy transfer in magnetized plasmas under stochastic forcing and offers a rigorous route to quantify diffusion in energy space from turbulent electric fields.

Abstract

We consider in this paper a plasma subject to a strong deterministic magnetic field and we investigate the effect on this plasma of a stochastic electric field. We show that the limit behavior, which corresponds to the transfer of energy from the electric wave to the particles (Landau phenomena), is described by a Spherical Harmonics Expansion (SHE) model.

Paper Structure

This paper contains 8 sections, 9 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Explicit computation of the diffusion parameter
  4. Proof of the result
  5. Properties of the operators
  6. Computation of ${\mathbf E}(J\theta_t^\epsilon f_\epsilon)$
  7. Convergence to the SHE model
  8. Explicit computation of the diffusion coefficient

Key Result

Theorem 2.1

Let V^\epsilon be a stochastic potential satisfying assumptions (H) and independent of the initial data f^0_\epsilon\in L^2(\mathbb R^4). Let a(e) be the function defined by: This function is non negative. Assume that there is a constant C_0 such that \|f^0_\epsilon\|_{L^2(\mathbb R^{4})}\leq C_0 and: Let \rho_\epsilon be the gyro-average of f_\epsilon defined by : for every v such that |v|^2/2

Theorems & Definitions (9)

  • Theorem 2.1
  • Lemma 4.1
  • Proposition 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Proposition 4.5
  • Lemma 4.6
  • Lemma 4.7
  • Proposition 4.8