[Paper Review] Quasineutral limit of the Vlasov-Poisson system with massless electrons

Daniel Han-Kwan|arXiv (Cornell University)|Apr 6, 2010

Navier-Stokes equation solutions29 references17 citations

TL;DR

This paper rigorously derives fluid limits for the Vlasov-Poisson system with massless electrons in the quasineutral regime using the relative entropy method. It shows that cold ions converge to the isothermal Euler or inviscid shallow water equations as the Debye length tends to zero, and further establishes the combined quasineutral and strong magnetic field limit, yielding a reduced kinetic model with enhanced scaling in the electron Larmor radius and Debye length.

ABSTRACT

In this paper, we study the quasineutral limit (in other words the limit when the Debye length tends to zero) of Vlasov-Poisson like equations describing the behaviour of ions in a plasma. We consider massless electrons, with a charge density following a Maxwell-Boltzmann law. For cold ions, using the relative entropy method, we derive the classical Isothermal Euler or the (inviscid) Shallow Water systems from fluid mechanics. In a second time, we study the combined quasineutral and strong magnetic field regime for such plasmas.

Motivation & Objective

• To mathematically justify the quasineutral limit of the Vlasov-Poisson system for plasmas with massless electrons and Maxwell-Boltzmann distributed electrons.
• To derive the classical isothermal Euler or inviscid shallow water equations as fluid limits for cold ions in the quasineutral regime.
• To analyze the combined quasineutral and strong magnetic field regime, incorporating the cyclotron frequency and electron Larmor radius scaling.
• To establish convergence results using the relative entropy method under appropriate scaling assumptions.
• To provide a rigorous mathematical framework for modeling ion dynamics in tokamak plasmas under extreme physical limits.

Proposed method

• Uses the relative entropy method to analyze the quasineutral limit of the Vlasov-Poisson system with massless electrons.
• Applies a dimensionless scaling where the Debye length is proportional to a small parameter $\epsilon$, leading to $\lambda_D^2 / L^2 = \epsilon$.
• Imposes the Maxwell-Boltzmann law for electron density: $n_e \propto e^{eV/k_B T_e}$, ensuring quasineutrality in the limit.
• Considers the limit $\epsilon \to 0$ in the scaled Vlasov-Poisson system, leading to a reduced system with $\epsilon \Delta V_\epsilon = \int f_\epsilon dv - d e^{V_\epsilon}$.
• Introduces a strong magnetic field scaling via $\Omega \tau = 1/\epsilon$, with $\Omega$ the cyclotron frequency, to model intense magnetic confinement.
• Derives the combined limit by setting $\lambda_D^2 / L^2 = \epsilon^{2\alpha}$ and $r_L / L = \epsilon$, with $\alpha > 1$ for physical relevance.

Experimental results

Research questions

• RQ1 How does the Vlasov-Poisson system for ions converge to the isothermal Euler or inviscid shallow water equations in the quasineutral limit with massless electrons?
• RQ2 What is the role of the relative entropy method in justifying fluid limits for kinetic equations with singular scaling?
• RQ3 How does the inclusion of a strong magnetic field affect the quasineutral limit of the Vlasov-Poisson system?
• RQ4 What scaling relations govern the simultaneous vanishing of the Debye length and electron Larmor radius in the plasma model?
• RQ5 What are the resulting fluid or kinetic equations in the combined quasineutral and strong magnetic field regime?

Key findings

• The quasineutral limit of the Vlasov-Poisson system with massless electrons and Maxwell-Boltzmann distributed electrons leads to the isothermal Euler equations for cold ions.
• For cold ions, the limit system is the inviscid shallow water equations, derived rigorously via the relative entropy method.
• The convergence is established in the limit $\epsilon \to 0$ under the scaling $\lambda_D^2 / L^2 = \epsilon$, where $\epsilon$ is the small parameter representing the Debye length scale.
• In the combined quasineutral and strong magnetic field regime, the limit system features a scaled magnetic term $v \wedge b / \epsilon$ in the Lorentz force, reflecting fast gyro-motion.
• The electron density is given by $n_e = d e^{V_\epsilon}$, with $d$ a normalization constant, ensuring quasineutrality in the limit.
• The analysis confirms that the physical regime $\lambda_D \ll r_L$ corresponds to $\alpha > 1$, which is consistent with typical tokamak conditions.