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Charged particle motion in a strong magnetic field: Applications to plasma confinement

Ugo Boscain, Wadim Gerner

TL;DR

This work rigorously justifies the zero-order guiding-center limit for charged-particle motion in strong magnetic fields by proving convergence of the gyro-dynamics as the gyro-frequency $\omega\to\infty$, deriving a limit motion $\dot{x}=h(t)b(x)$ with conserved magnetic moment $\mu$, and establishing a displacement formula for the pressure along particle trajectories with explicit $1/\omega$ corrections and a double-exponential error bound. It also identifies the leading drift time scale $t\sim\sqrt{\ln(\ln\omega)}$ for the validity of the first-order correction and analyzes resonant surfaces in quasi-symmetric equilibria, showing linear drift on resonant surfaces and $\mathcal{O}(1/\omega)$ drift off-resonance. These results connect rigorous PDE/dynamical-systems analysis with plasma-physics confinement theory and provide design insights for stellarators by highlighting risks on resonant surfaces and the role of magnetic-field regularity and growth at infinity.

Abstract

We derive the zero order approximation of a charged particle under the influence of a strong magnetic field in a mathematically rigorous manner and clarify in which sense this approximation is valid. We use this to further rigorously derive a displacement formula for the pressure of plasma equilibria and compare our findings to results in the physics literature. The main novelty of our results is a qualitative estimate of the confinement time for optimised plasma equilibria with respect to the gyro frequency. These results are of interest in the context of plasma fusion confinement.

Charged particle motion in a strong magnetic field: Applications to plasma confinement

TL;DR

This work rigorously justifies the zero-order guiding-center limit for charged-particle motion in strong magnetic fields by proving convergence of the gyro-dynamics as the gyro-frequency , deriving a limit motion with conserved magnetic moment , and establishing a displacement formula for the pressure along particle trajectories with explicit corrections and a double-exponential error bound. It also identifies the leading drift time scale for the validity of the first-order correction and analyzes resonant surfaces in quasi-symmetric equilibria, showing linear drift on resonant surfaces and drift off-resonance. These results connect rigorous PDE/dynamical-systems analysis with plasma-physics confinement theory and provide design insights for stellarators by highlighting risks on resonant surfaces and the role of magnetic-field regularity and growth at infinity.

Abstract

We derive the zero order approximation of a charged particle under the influence of a strong magnetic field in a mathematically rigorous manner and clarify in which sense this approximation is valid. We use this to further rigorously derive a displacement formula for the pressure of plasma equilibria and compare our findings to results in the physics literature. The main novelty of our results is a qualitative estimate of the confinement time for optimised plasma equilibria with respect to the gyro frequency. These results are of interest in the context of plasma fusion confinement.

Paper Structure

This paper contains 4 sections, 7 theorems, 113 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Proof of \ref{['IT1']} & \ref{['IC2']}
  3. Proof of \ref{['IT3']}
  4. Proof of \ref{['IT6']}

Key Result

Theorem 1.2

Let $B\in C^2(\mathbb{R}^3,\mathbb{R}^3)$ be a no-where vanishing, div-free vector field defined on $\mathbb{R}^3$ and fix $x_0,v_0\in \mathbb{R}^3$. Let $x_{\omega}$ be the (unique) solution to (IE1) for given $\omega >0$. Then there exists some $x\in C^{3}_{\operatorname{loc}}(\mathbb{R},\mathbb{R The limit trajectory $x(t)$ satisfies $\dot{x}(t)=h(t)b(x(t))$, where $b(y):=\frac{B(y)}{|B(y)|}$ a

Figures (1)

  • Figure 1: Motion of a charged particles in a uniform magnetic field. We plotted the trajectories $x_{\omega}$ for two different values of $\omega$. As $\omega$ increases the radius of the helix decreases and the oscillation becomes faster.

Theorems & Definitions (18)

  • Definition 1.1: Minimal growth at infinity
  • Theorem 1.2: First Main Theorem: Zero order approximation
  • Remark 1.3
  • Corollary 1.4: Magnetic moment preservation
  • Theorem 1.5: Second Main Theorem: Displacement formula
  • Remark 1.6
  • Definition 1.7: Quasi-symmetric plasma equilibrium
  • Theorem 1.8: Third Main Theorem: Resonant surfaces
  • Lemma 2.1: Existence of global particle path
  • proof : Proof of \ref{['L1']}
  • ...and 8 more