Tokamak plasma boundary reconstruction using toroidal harmonics and an optimal control method

This paper proposes a new fast and stable algorithm for the reconstruction of the plasma boundary from discrete magnetic measurements taken at several locations surrounding the vacuum vessel. The resolution of this inverse problem takes two steps. In the first one we transform the set of measurements into Cauchy conditions on a fixed contour $\Gamma\_O$ close to the measurement points. This is done by least square fitting a truncated series of toroidal harmonic functions to the measurements. The second step consists in solving a Cauchy problem for the elliptic equation satisfied by the flux in the vacuum and for the overdetermined boundary conditions on $\Gamma\_O$ previously obtained with the help of toroidal harmonics. It is reformulated as an optimal control problem on a fixed annular domain of external boundary $\Gamma\_O$ and fictitious inner boundary $\Gamma\_I$. A regularized Kohn-Vogelius cost function depending on the value of the flux on $\Gamma\_I$ and measuring the discrepency between the solution to the equation satisfied by the flux obtained using Dirichlet conditions on $\Gamma\_O$ and the one obtained using Neumann conditions is minimized. The method presented here has led to the development of a software, called VacTH-KV, which enables plasma boundary reconstruction in any Tokamak.Comment: Fusion Science and Technology, 201

A discontinuous Galerkin method for the Vlasov-Poisson system

A discontinuous Galerkin method for approximating the Vlasov-Poisson system of equations describing the time evolution of a collisionless plasma is proposed. The method is mass conservative and, in the case that piecewise constant functions are used as a basis, the method preserves the positivity of the electron distribution function and weakly enforces continuity of the electric field through mesh interfaces and boundary conditions. The performance of the method is investigated by computing several examples and error estimates associated system's approximation are stated. In particular, computed results are benchmarked against established theoretical results for linear advection and the phenomenon of linear Landau damping for both the Maxwell and Lorentz distributions. Moreover, two nonlinear problems are considered: nonlinear Landau damping and a version of the two-stream instability are computed. For the latter, fine scale details of the resulting long-time BGK-like state are presented. Conservation laws are examined and various comparisons to theory are made. The results obtained demonstrate that the discontinuous Galerkin method is a viable option for integrating the Vlasov-Poisson system.Comment: To appear in Journal for Computational Physics, 2011. 63 pages, 86 figure

A unified gas kinetic scheme for transport and collision effects in plasma

In this study, the Vlasov-Poisson equation with or without collision term for plasma is solved by the unified gas kinetic scheme (UGKS). The Vlasov equation is a differential equation describing time evolution of the distribution function of plasma consisting of charged particles with long-range interaction. The distribution function is discretized in discrete particle velocity space. After the Vlasov equation is integrated in finite volumes of physical space, the numerical flux across a cell interface and source term for particle acceleration are computed to update the distribution function at next time step. The flux is decided by Riemann problem and variation of distribution function in discrete particle velocity space is evaluated with central difference method. A electron-ion collision model is introduced in the Vlasov equation. This finite volume method for the UGKS couples the free transport and long-range interaction between particles. The electric field induced by charged particles is controlled by the Poisson's equation. In this paper, the Poisson's equation is solved using the Green's function for two dimensional plasma system subjected to the symmetry or periodic boundary conditions. Two numerical tests of the linear Landau damping and the Gaussian beam are carried out to validate the proposed method. The linear electron plasma wave damping is simulated based on electron-ion collision operator. Compared with previous methods, it is shown that the current method is able to obtain accurate results of the Vlasov-Poisson equation with a time step much larger than the particle collision time. Highly non-equilibrium and rarefied plasma flows, such as electron flows driven by electromagnetic field, can be simulated easily.Comment: 33 pages, 13 figure