Transport of charged particles under fast oscillating magnetic fields (Q2902726)

In this paper the author addresses a problem of plasma confinement within the context of the kinetic description of plasma. The starting point of the analysis is the Vlasov-Maxwell system. A standard configuration for obtaining confinement is to apply a strong magnetic field. The limit of this configuration as the magnetic field strength goes to infinity is the guiding-center approximation. Since the energy is quadratic with respect to the magnetic field a problem with confinement models that rely on very large magnetic fields is that they require very large energy. As an alternative to this, the author proposes a confinement model that relies on rapidly oscillating magnetic fields. The present paper shows that plasma confinement can be obtained through the use off magnetic fields of the form NEWLINE\[NEWLINE B^{\epsilon}(t,x) = \theta(t/\epsilon)B(x)b(x),\;\; 0 < \epsilon \ll 1, \tag{1}NEWLINE\]NEWLINE in which \(B(x)\) is a scalar positive function, \(b(x)\) is a field of unitary vectors, and \(\theta = \theta(s)\) is a \(T\) periodic function of class \(C^1\). The magnetic energy is of the order \(|B|^2\), and it can be much lower for fields of the form (1) than for the guiding-center approximation. As a simplified version of (1) the author first considers the model with the magnetic field of the form \(B^{\epsilon}(t) = \theta(t/\epsilon)(0,0,B)\) with \(B\) a positive constant. Letting \(\omega_c = eB/m\) where \(e\) is particle charge, \(m\) is particle mass, and \(^{\perp}p = (p_2,-p_1,0)\) the Vlasov equation to be studied has the form NEWLINE\[NEWLINE \partial_t f^{\epsilon} + \frac{P}{m} \cdot \nabla_x f^{\epsilon} + \Big (eE(t,x) + \frac{m\omega_c} {2\epsilon} \theta'(t/\epsilon)^{\perp}x + \omega_c \theta(t/\epsilon)^{\perp}p \Big ) \cdot \nabla_p f^{\epsilon} = 0,\tag{2}NEWLINE\]NEWLINE NEWLINE\[NEWLINE \mathrm{div}_xE(t) = - \Delta_x \phi(t) = \frac{e}{\varepsilon_0} \Big \{ \int_{R^3} f^{\epsilon}(t,x,p) - n_0(x)dp \Big \},\; t \in \mathbb R_+,\; x \in \mathbb R^3. NEWLINE\]NEWLINE The analysis of (2) involves two time scales, \(t\), and \(s=t/\epsilon\). A first theorem that is proved gives the result that as \(\epsilon \rightarrow 0\) the weak solution of (2) two-scale converges to a solution of NEWLINE\[NEWLINE\begin{multlined} \partial_t f^0 + \Big (\frac{P}{m} - \frac{\omega_c}{2}(\theta(s)- \langle \theta \rangle) ^{\perp}x \Big ) \cdot \nabla_x f^0\\ + \Big(eE(t,x) + \frac{\omega_c}{2}(\theta(s)- \langle \theta \rangle) ^{\perp}p + \frac{m\omega_c^2} {4} (\langle \theta^2 \rangle- \theta^2(s))^{\perp \perp}x \Big ) \cdot \nabla_p f^0 = 0. \end{multlined}\tag{3}NEWLINE\]NEWLINE A second theorem then provides the result that for appropriate initial conditions the solution to (3) has the confinement property \(\mathrm{supp} f^0(t,s,.,.) \subset \{(x,p) : |^{\perp}x| \leq R \}\). A result on strong convergence is also given which shows that \(f^{\epsilon}(t,\cdot,\cdot)\) converges to \(f^0(t,t/\epsilon,\cdot, \cdot)\) in the \(L^2\) sense as \(\epsilon\) approaches zero. In the last section of the paper the analysis is extended to the more general fields of the form (1). The conclusion drawn from the paper is that magnetic fields of the form (1) can have good containment properties.