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The three-dimensional finite Larmor radius approximation. (English) Zbl 1191.35267

This paper is devoted to the behavior of a plasma which is submitted to a large external magnetic field. It is known that such a field induces fast small oscillations for the particles and consequently introduces a new small time scale which is very restrictive and inconvenient from the numerical point of view. The mathematical model used here is the following Vlasov-Poisson system: \[ \partial_tf+v\cdot \nabla_xV+(E+v\wedge B)\cdot \nabla_vf=0, \]
\[ E=-\nabla_xV, \;\;-\triangle_xV=\int fdv, \;\;f_{t=0}=f_0, \] where \(E\) and \(B\) are the electric and magnetic fields, respectively, \(f(t,x,v)\) is the density of ions, with \(t\in \mathbb{R}^+\), \(x,v\in \mathbb{R}^{d}\) or \(\mathbb{R}^{d}/\mathbb{Z}^{d}\) (\(d=2\) or \(3\)), the differential form \(f(t,x,v)dxdv\) gives the number of ions in the infinitesimal volume \([x,x+dx]\times [v,v+dv]\) at time \(t\). Here the electrons are for the moment neglected. It turns out that the particles move on a helix whose axis is the direction of the magnetic field. The rotation period (around the axis) is the inverse of the cyclotron frequency \(\Omega = |q||B|m^{-1}\), where \(q\) is the charge. The Larmor radius is defined by \(r_L=|v_{\bot }|\Omega^{-1}\). Taking into account both the quasineutrality and local thermodynamic equilibrium of the electrons the author considers the finite Larmor radius regime for the plasma. If assume that \(|B|\sim 1/\varepsilon \) with \(\varepsilon \to 0\) then \(\Omega \sim 1/\varepsilon \), \(r_{L} \sim 1/\varepsilon \). The classical guiding center approximation corresponds to the scaling for the Vlasov-Poisson system including \(\varepsilon \). The asymptotic gyrokinetic limit of the rescaled and modified Vlasov-Poisson system in three-dimensional setting is discussed.

MSC:

35Q83 Vlasov equations
82D10 Statistical mechanics of plasmas
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