Calculation of three-dimensional MHD equilibria with islands and stochastic regions

Abstract

A three-dimensional MHD equilibrium code is described that does not assume the existence of good flux surfaces. Given an initial guess for the magnetic field, the code proceeds by calculating the pressure-driven current and then by updating the field using Ampere's law. The numerical algorithm to solve the magnetic differential equation for the pressure-driven current is described, and demonstrated for model fields having islands and stochastic regions. The numerical algorithm which solves Ampere's law In three dimensions is also described. Finally, the convergence of the code is illustrated for a particular steilarator equilibrium with no large islands."Presented as an invited talk at the 8th Europhysics Conference on Computational Physics, Elbsee, Fed. Rep. of Germany, May 13-16, 1986. To appear in a special issue of Computer Physics Communications. In this paper we describe a three-dimensional MHD equilibrium code that does not assume the existence of good flux surfaces. The code uses an algorithm for solving the equilibrium equations that is different from the energy-minimizing algorithms which have been developed extensively and successfully for 3D equilibrium codes that assume good surfaces [1][2][3]. We describe the algorithm, our reasons for choosing It, and the numerical methods we have developed to implement it. We also describe the numerical issues raised by the presence of Islands and stochastic regions, and how we have dealt with those issues. In particular, we describe how the code solves magnetic differential equations, using as examples model fields with islands and stochastic regions. Finally, we describe the convergence of the code for a particular stellarator equilibrium with no large inlands.We specialize In this paper to the case of zero net current within each flux surface, as is appropriate for applications to stellarators. Our code has been written to deal with the more general case of arbitrary net currents, except for some small modifications that need to be made to deal with the currents in the islands and stochastic regions.The applications we have in mind for the code are the study of the breaking of flux surfaces in stellarators, and the study of The magnetic field becomes increasingly distorted, it is not known under what conditions the field produced by the pressure-driven currents will cause the formation of large islands and stochastic regions. In particular, it is important to know whether there is an equilibrium beta limit due to such an effect, and what that beta limit is. The convention In evaluating stellarator designs has been to estimate the equilibrium beta limit to be the value of beta at which the magnetic axis shifts halfway to the wall. An analytical calculation of equilibrium island formation in one proposed helical axis stellarator found a much lower beta limit [4].Nominally axisymmetric devices, such as tokamaks, are in practice generally nonaxisymmetrlc because of the presence of tearing instabilities, magnetic ripple from discrete coils, and field error...

Recent advances in the design of quasiaxisymmetric stellarator plasma configurations

“…7 The VMEC code used to calculate three-dimensional equilibria for our stability and transport assessments uses a representation of the magnetic field that assumes nested flux surfaces. The PIES code is a three-dimensional equilibrium code that uses a general representation for the field, and is therefore capable of calculating islands and stochastic field line trajectories.…”

“…[3][4][5][6] The work described in this paper has advanced the study in two major respects: 1) two strategies for further configuration improvement have been explored, leading to an examination of two types of quasi-axisymmetric stellarator configurations whose physics properties we have not previously studied; 2) the requirement of good equilibrium flux surfaces has now been added as a design objective, and the flux surfaces have been evaluated with the PIES three-dimensional equilibrium code. 7 Throughout this paper, recently generated quasi-axisymmetric (QA) configurations will be compared with an earlier reference QA configuration denoted Configuration C82. 5 The plasma boundary shape of Configuration C82 is shown in Fig.…”

“…The equilibrium flux surface calculations described in this paper have been have been done using the PIES code. 7 For this purpose, a num-ber of modifications have been made to the code to improve its speed, allowing it to be used to routinely evaluate candidate configurations. These modifications are described in an appendix to this paper.…”

Magnetic island effects on axisymmetric equilibria

“…Numerical techniques exist for solving the 3D equilibrium problem. 2 However, for a number of reasons, it would be helpful to have a 2D equation with respect to only the 2D axisymmetric coordinates. Then, the plasma MHD equilibrium with a magnetic island can be effectively treated, with a 3D island structure grafted onto the 2D equilibrium in the vicinity of the rational surface.…”

Magnetic-Surface Quality in Nonaxisymmetric Plasma Equilibria

“…The breakup of magnetic surfaces in a given equilibrium can be studied using codes such as pies [3] and hint [4]. Approximate non-axisymmetric equilibria can be calculated with far less computational effort using the vmec code [5], which exploits the assumption of nested magnetic surfaces.…”