Analysis of anomalous transport based on radial fractional diffusion equation

The anomalous transport in magnetically confined plasmas is investigated by the radial fractional transport equations. It is shown that for fractional transport models, hollow density profiles are formed and uphill transports can be observed regardless of whether the fractional diffusion coefficients (FDCs) are radially dependent or not. When a radially dependent FDC Dα(r)<1 is imposed, compared with the case under Dα(r)=1.0, it is observed that the position of the peak of the density profile is closer to the core. Besides, it is found that when FDCs at the positions of source injections increase, the peak values of density profiles decrease. The non-local effect becomes significant as the order of fractional derivative α→1 and causes the uphill transport. However, as α→2, the fractional diffusion model returns to the standard model governed by Fick’s law.

Uniform lifetime for classical solutions to the Hot, Magnetized, Relativistic Vlasov Maxwell system

<p style='text-indent:20px;'>This article is devoted to the kinetic description in phase space of magnetically confined plasmas. It addresses the problem of stability near equilibria of the Relativistic Vlasov Maxwell system. We work under the Glassey-Strauss compactly supported momentum assumption on the density function <inline-formula><tex-math id="M1">\begin{document}$ f(t,\cdot) $\end{document}</tex-math></inline-formula>. Magnetically confined plasmas are characterized by the presence of a strong <i>external</i> magnetic field <inline-formula><tex-math id="M2">\begin{document}$ x \mapsto \epsilon^{-1} \mathbf{B}_e(x) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> is a small parameter related to the inverse gyrofrequency of electrons. In comparison, the self consistent <i>internal</i> electromagnetic fields <inline-formula><tex-math id="M4">\begin{document}$ (E,B) $\end{document}</tex-math></inline-formula> are supposed to be small. In the non-magnetized setting, local <inline-formula><tex-math id="M5">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula>-solutions do exist but do not exclude the possibility of blow up in finite time for large data. Consequently, in the strongly magnetized case, since <inline-formula><tex-math id="M6">\begin{document}$ \epsilon^{-1} $\end{document}</tex-math></inline-formula> is large, standard results predict that the lifetime <inline-formula><tex-math id="M7">\begin{document}$ T_\epsilon $\end{document}</tex-math></inline-formula> of solutions may shrink to zero when <inline-formula><tex-math id="M8">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> goes to <inline-formula><tex-math id="M9">\begin{document}$ 0 $\end{document}</tex-math></inline-formula>. In this article, through field straightening, and a time averaging procedure we show a uniform lower bound (<inline-formula><tex-math id="M10">\begin{document}$ 0&lt;T&lt;T_\epsilon $\end{document}</tex-math></inline-formula>) on the lifetime of solutions and uniform Sup-Norm estimates. Furthermore, a bootstrap argument shows <inline-formula><tex-math id="M11">\begin{document}$ f $\end{document}</tex-math></inline-formula> remains at a distance <inline-formula><tex-math id="M12">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> from the linearized system, while the internal fields can differ by order 1 for well prepared initial data.</p>

Available energy of magnetically confined plasmas

The concept of the available energy of a collisionless plasma is discussed in the context of magnetic confinement. The available energy quantifies how much of the plasma energy can be converted into fluctuations (including nonlinear ones) and is thus a measure of plasma stability, which can be used to derive linear and nonlinear stability criteria without solving an eigenvalue problem. In a magnetically confined plasma, the available energy is determined by the density and temperature profiles as well as the magnetic geometry. It also depends on what constraints limit the possible forms of plasma motion, such as the conservation of adiabatic invariants and the requirement that the transport be ambipolar. A general method based on Lagrange multipliers is devised to incorporate such constraints in the calculation of the available energy, and several particular cases are discussed for which it can be calculated explicitly. In particular, it is shown that it is impossible to confine a plasma in a Maxwellian ground state relative to perturbations with frequencies exceeding the ion bounce frequency.