Abstract
We study the asymptotic behavior of solutions to the Vlasov equation in the presence of a strong external magnetic field. In particular we provide a mathematically rigorous derivation of the guiding-center approximation in the general three dimensional setting under the action of large inhomogeneous magnetic fields. First order corrections are computed and justified as well, including electric cross field, magnetic gradient and magnetic curvature drifts. We also treat long time behaviors on two specific examples, the two dimensional case in carte-sian coordinates and a poloidal axi-symmetric geometry, the former for expository purposes. Algebraic manipulations that underlie concrete computations make the most of the linearity of the stiffest part of the system of characteristics instead of relying on any particular variational structure.
Highlights
Since fusion configurations involve very hot plasmas, they typically require a careful design to maintain fast moving particles inside the core of the device on sufficiently long times
The unknowns are the number densities of particles, f ≡ f (t, x, v) depending on time t 0, position x ∈ Ω ⊂ R3 and velocity v ∈ R3. Such kinetic models provide an appropriate description of turbulent transport in a fairly general context, but in fusion configurations their numerical simulations require to solve a stiff six-dimensional problem, leading to a huge computational cost
The dynamical time scales we focus on are in any case much larger than the cyclotron period and we establish asymptotic descriptions in the limit ε → 0
Summary
Since fusion configurations involve very hot plasmas, they typically require a careful design to maintain fast moving particles inside the core of the device on sufficiently long times. Slow dynamics refer to dynamics where typical time derivatives are at most of order O(1) on short-time scales, and at most of order O(ε) on long-time scales so that on long time scales two kinds of fast dynamics may co-exist, principal ones at typical speed of order 1/ε and subprincipal ones at typical speed of order 1; see for instance [9] for a description of those various oscillations in a specific class of axi-symmetric geometries, without electric field and with a magnetic field nowhere toroidal and whose angle to the toroidal direction is independent of the poloidal angle. We benefit from the latter to prove for the first time a second-order description in full generality
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