An adjoint method for determining the sensitivity of island size to magnetic field variations

Abstract

An adjoint method to calculate the gradient of island width in stellarators is presented and applied to a set of magnetic field configurations. The underlying method for calculation of the island width is that of Cary & Hanson (Phys. Fluids B, vol. 3, issue 4, 1991, pp. 1006–1014) (with a minor modification), and requires that the residue of the island centre be small. Therefore, the gradient of the residue is calculated in addition. Both the island width and the gradient calculations are verified using an analytical magnetic field configuration introduced by Reiman & Greenside (Comput. Phys. Commun., vol. 43, issue 1, 1986, pp. 157–167). The method is also applied to the calculation of the shape gradient of the width of a magnetic island in a National Compact Stellarator Experiment (NCSX) vacuum configuration with respect to positions on a coil. A gradient-based optimization is applied to a magnetic field configuration studied by Hanson & Cary (Phys. Fluids, vol. 27, issue 4, 1984, pp. 767–769) to minimize stochasticity by adding perturbations to a pair of helical coils. Although only vacuum magnetic fields and an analytical magnetic field model are considered in this work, the adjoint calculation of the island width gradient could also be applied to a magnetohydrodynamic (MHD) equilibrium if the derivative of the magnetic field, with respect to the equilibrium parameters, is known. Using the island width gradient calculation presented here, more general gradient-based optimization methods can be applied to design stellarators with small magnetic islands. Moreover, the sensitivity of the island size may itself be optimized to ensure that coil tolerances, with respect to island size, are kept as high as possible.

Improving the stellarator through advances in plasma theory

“…Shape gradients on coils can provide tolerance information, as they represent displacements of the coils to which the plasma physics properties are sensitive [91], as shown in figure 8. Adjoint methods have recently been derived for many quantities of interest for stellarator design, including collisional transport, coil complexity, and the width of magnetic islands [58,[92][93][94][95][96]. Derivative-based optimization using these adjoint methods has been demonstrated in the past year for obtaining desirable MHD properties [94] and quasisymmetry [72] and to maximize the volume of good surfaces [96].…”

“…Adjoint methods have recently been derived for many quantities of interest for stellarator design, including collisional transport, coil complexity, and the width of magnetic islands [58,[92][93][94][95][96]. Derivative-based optimization using these adjoint methods has been demonstrated in the past year for obtaining desirable MHD properties [94] and quasisymmetry [72] and to maximize the volume of good surfaces [96]. Two examples are shown in figures 9 and 10.…”

Single-stage gradient-based stellarator coil design: stochastic optimization

“…Both challenges are related but not identical: a relatively simple coil system whose performance degrades strongly with manufacturing and alignment errors may not be as desirable as a more complex coil system with more robust performance. Work has therefore also been lately devoted to the design of efficient numerical methods for the evaluation of the sensitivity of error fields to coil perturbations [45,46], and the sensitivity of physical quantities to error fields [13,23]. These methods can be included in deterministic stellarator optimization codes, and serve to narrow the search to configurations with lower sensitivity or with sensitivity with respect to perturbations that are more easily controlled.…”