Antoine J. Cerfon

“One size fits all” analytic solutions to the Grad–Shafranov equation

An extended analytic solution to the Grad–Shafranov equation using Solov’ev profiles is presented. The solution describes standard tokamaks, spherical tokamaks, spheromaks, and field reversed configurations. It allows arbitrary aspect ratio, elongation, and triangularity as well as a plasma surface that can be smooth or possess a double or single null divertor X-point. The solution can also be used to evaluate the equilibrium beta limit in a tokamak and spherical tokamak in which a separatrix moves onto the inner surface of the plasma.

A fast, high-order solver for the Grad-Shafranov equation

Abstract We present a new fast solver to calculate fixed-boundary plasma equilibria in toroidally axisymmetric geometries. By combining conformal mapping with Fourier and integral equation methods on the unit disk, we show that high-order accuracy can be achieved for the solution of the equilibrium equation and its first and second derivatives. Smooth arbitrary plasma cross-sections as well as arbitrary pressure and poloidal current profiles are used as initial data for the solver. Equilibria with large Shafranov shifts can be computed without difficulty. Spectral convergence is demonstrated by comparing the numerical solution with a known exact analytic solution. A fusion-relevant example of an equilibrium with a pressure pedestal is also presented.

Tokamak plasma high field side response to an n?=?3 magnetic perturbation: a comparison of 3D equilibrium solutions from seven different codes

In comparing equilibrium solutions for a DIII-D shot that is amenable to analysis by both stellarator and tokamak three-dimensional (3D) equilibrium codes, a significant disagreement has been seen between solutions of the VMEC stellarator equilibrium code and solutions of tokamak perturbative 3D equilibrium codes. The source of that disagreement has been investigated, and that investigation has led to new insights into the domain of validity of the different equilibrium calculations, and to a finding that the manner in which localized screening currents at low order rational surfaces are handled can affect global properties of the equilibrium solution. The perturbative treatment has been found to break down at surprisingly small perturbation amplitudes due to overlap of the calculated perturbed flux surfaces, and that treatment is not valid in the pedestal region of the DIII-D shot studied. The perturbative treatment is valid, however, further into the interior of the plasma, and flux surface overlap does not account for the disagreement investigated here. Calculated equilibrium solutions for simple model cases and comparison of the 3D equilibrium solutions with those of other codes indicate that the disagreement arises from a difference in handling of localized currents at low order rational surfaces, with such currents being absent in VMEC and present in the perturbative codes. The significant differences in the global equilibrium solutions associated with the presence or absence of very localized screening currents at rational surfaces suggests that it may be possible to extract information about localized currents from appropriate measurements of global equilibrium plasma properties. That would require improved diagnostic capability on the high field side of the tokamak plasma, a region difficult to access with diagnostics.

Efficiency enhancement of A 1.5-MW, 110-GHz gyrotron with a single-stage depressed collector

We report new experimental results from a 1.5-MW, 110-GHz gyrotron with a single-stage depressed collector. The gyrotron was operated in the TE22,6 mode with 3-μs pulse duration. An internal mode converter, which consists of a launcher and four mirrors, has been installed and tested. A highly Gaussian-like output beam was observed. A single-stage depressed collector has been operated for the study of efficiency enhancement using the same cavity V-2005 as was used in a previous experiment in the axial configuration, in which the output microwave beam propagated through a circular waveguide that also served as a collector. Output power of 1.5 MW, corresponding to 50% efficiency, was measured at 97 kV of beam voltage and 42 A of beam current at 25 kV of collector depression voltage. The results are compared between the axial configuration and the internal mode converter configuration.

An adaptive fast multipole accelerated Poisson solver for complex geometries

Abstract We present a fast, direct and adaptive Poisson solver for complex two-dimensional geometries based on potential theory and fast multipole acceleration. More precisely, the solver relies on the standard decomposition of the solution as the sum of a volume integral to account for the source distribution and a layer potential to enforce the desired boundary condition. The volume integral is computed by applying the FMM on a square box that encloses the domain of interest. For the sake of efficiency and convergence acceleration, we first extend the source distribution (the right-hand side in the Poisson equation) to the enclosing box as a C 0 function using a fast, boundary integral-based method. We demonstrate on multiply connected domains with irregular boundaries that this continuous extension leads to high accuracy without excessive adaptive refinement near the boundary and, as a result, to an extremely efficient “black box” fast solver.

Magnetohydrodynamic stability comparison theorems revisited

Magnetohydrodynamic (MHD) stability comparison theorems are presented for several different plasma models, each one corresponding to a different level of collisionality: a collisional fluid model (ideal MHD), a collisionless kinetic model (kinetic MHD), and two intermediate collisionality hybrid models (Vlasov-fluid and kinetic MHD-fluid). Of particular interest is the re-examination of the often quoted statement that ideal MHD makes the most conservative predictions with respect to stability boundaries for ideal modes. Some of the models have already been investigated in the literature and we clarify and generalize these results. Other models are essentially new and for them we derive new comparison theorems. Three main conclusions can be drawn: (1) it is crucial to distinguish between ergodic and closed field line systems; (2) in the case of ergodic systems, ideal MHD does indeed make conservative predictions compared to the other models; (3) in closed line systems undergoing perturbations that maintain...

Accurate spectral numerical schemes for kinetic equations with energy diffusion

We examine the merits of using a family of polynomials that are orthogonal with respect to a non-classical weight function to discretize the speed variable in continuum kinetic calculations. We consider a model one-dimensional partial differential equation describing energy diffusion in velocity space due to Fokker-Planck collisions. This relatively simple case allows us to compare the results of the projected dynamics with an expensive but highly accurate spectral transform approach. It also allows us to integrate in time exactly, and to focus entirely on the effectiveness of the discretization of the speed variable. We show that for a fixed number of modes or grid points, the non-classical polynomials can be many orders of magnitude more accurate than classical Hermite polynomials or finite-difference solvers for kinetic equations in plasma physics. We provide a detailed analysis of the difference in behavior and accuracy of the two families of polynomials. For the non-classical polynomials, if the initial condition is not smooth at the origin when interpreted as a three-dimensional radial function, the exact solution leaves the polynomial subspace for a time, but returns (up to roundoff accuracy) to the same point evolved to by the projected dynamics in that time. By contrast, using classical polynomials, the exact solution differs significantly from the projected dynamics solution when it returns to the subspace. We also explore the connection between eigenfunctions of the projected evolution operator and (non-normalizable) eigenfunctions of the full evolution operator, as well as the effect of truncating the computational domain.

A Spectral Transform Method for Singular Sturm--Liouville Problems with Applications to Energy Diffusion in Plasma Physics

We develop a spectrally accurate numerical method to compute solutions of a model PDE used in plasma physics to describe diffusion in velocity space due to Fokker--Planck collisions. The solution is represented as a discrete and continuous superposition of normalizable and nonnormalizable eigenfunctions via the spectral transform associated with a singular Sturm--Liouville operator. We present a new algorithm for computing the spectral density function of the operator that uses Chebyshev polynomials to extrapolate the value of the Titchmarsh--Weyl

Exact axisymmetric Taylor states for shaped plasmas

m

Analytic fluid theory of beam spiraling in high-intensity cyclotrons

-function from the complex upper half-plane to the real axis. The eigenfunctions and density function are rescaled, and a new formula for the limiting value of the