The on-axis magnetic well and Mercier's criterion for arbitrary stellarator geometries

A simplified analytical form of the on-axis magnetic well and Mercier's criterion for interchange instabilities for arbitrary three-dimensional magnetic field geometries is derived. For this purpose, a near-axis expansion based on a direct coordinate approach is used by expressing the toroidal magnetic flux in terms of powers of the radial distance to the magnetic axis. For the first time, the magnetic well and Mercier's criterion are then written as a one-dimensional integral with respect to the axis arclength. When compared with the original work of Mercier, the derivation here is presented using modern notation and in a more streamlined manner that highlights essential steps. Finally, these expressions are verified numerically using several quasisymmetric and non-quasisymmetric stellarator configurations including Wendelstein 7-X.

Gradient-based optimization of 3D MHD equilibria

Using recently developed adjoint methods for computing the shape derivatives of functions that depend on magnetohydrodynamic (MHD) equilibria (Antonsen et al., J. Plasma Phys., vol. 85, issue 2, 2019; Paul et al., J. Plasma Phys., vol. 86, issue 1, 2020), we present the first example of analytic gradient-based optimization of fixed-boundary stellarator equilibria. We take advantage of gradient information to optimize figures of merit of relevance for stellarator design, including the rotational transform, magnetic well and quasi-symmetry near the axis. With the application of the adjoint method, we reduce the number of equilibrium evaluations by the dimension of the optimization space ( ${\sim }50\text {--}500$ ) in comparison with a finite-difference gradient-based method. We discuss regularization objectives of relevance for fixed-boundary optimization, including a novel method that prevents self-intersection of the plasma boundary. We present several optimized equilibria, including a vacuum field with very low magnetic shear throughout the volume.

Magnetic well and Mercier stability of stellarators near the magnetic axis

We have recently demonstrated that by expanding in small distance from the magnetic axis compared with the major radius, stellarator shapes with low neoclassical transport can be generated efficiently. To extend the utility of this new design approach, here we evaluate measures of magnetohydrodynamic interchange stability within the same expansion. In particular, we evaluate the magnetic well, Mercier's criterion, and resistive interchange stability near a magnetic axis of arbitrary shape. In contrast to previous work on interchange stability near the magnetic axis, which used an expansion of the flux coordinates, here we use the ‘inverse expansion’ in which the flux coordinates are the independent variables. Reduced expressions are presented for the magnetic well and stability criterion in the case of quasisymmetry. The analytic results are shown to agree with calculations from the VMEC equilibrium code. Finally, we show that near the axis, Glasser, Greene and Johnson's stability criterion for resistive modes approximately coincides with Mercier's ideal condition.

Progress on the small modular stellarator SCR-1: new diagnostics and heating scenarios

This work presents updates in the diagnostics systems, magnetohydrodynamics (MHD) calculations and simulations of microwave heating scenarios of the small modular Stellarator of Costa Rica 1 (SCR-1). Similarly, the design of a flexible bolometer and magnetic diagnostics (a set of Mirnov coils, Rogowski coils and two diamagnetic loops) are introduced. Furthermore, new MHD equilibrium calculations for the plasma of the SCR-1 device were performed using the VMEC code including the poloidal cross-section of the magnetic flux surfaces at different toroidal positions, profiles of the rotational transform, magnetic well, magnetic shear and total magnetic field norm. Charged particle orbits in vacuum magnetic field were computed by the magnetic field solver BS-SOLCTRA (Vargas et al. In 27th IAEA Fusion Energy Conference (FEC 2018), 2018. IAEA). A visualization framework was implemented using Paraview (Solano-Piedra et al. In 23rd IAEA Technical Meeting on the Research Using Small Fusion Devices (23rd TM RUSFD), 2017) and compared with magnetic mapping results (Coto-Vílchez et al. In 16th Latin American Workshop on Plasma Physics (LAWPP), 2017, pp. 43–46). Additionally, simulations of microwave heating scenarios were performed by the IPF-FDMC full-wave code. These simulations calculate the conversion of the ordinary waves to extraordinary waves and allow us to identify the location where the conversion takes place. Finally, the microwave heating scenarios for the $330^{\circ }$ toroidal position are presented. The microwave heating scenarios showed that the O–X–B mode conversion is around 12–14 %. It was possible to identify the spatial zone where the conversion takes place (upper hybrid frequency).

Adjoint approach to calculating shape gradients for three-dimensional magnetic confinement equilibria. Part 2. Applications

The shape gradient is a local sensitivity function defined on the surface of an object which provides the change in a characteristic quantity, or figure of merit, associated with a perturbation to the shape of the object. The shape gradient can be used for gradient-based optimization, sensitivity analysis and tolerance calculations. However, it is generally expensive to compute from finite-difference derivatives for shapes that are described by many parameters, as is the case for typical stellarator geometry. In an accompanying work (Antonsen, Paul & Landreman J. Plasma Phys., vol. 85 (2), 2019), generalized self-adjointness relations are obtained for magnetohydrodynamic (MHD) equilibria. These describe the relation between perturbed equilibria due to changes in the rotational transform or toroidal current profiles, displacements of the plasma boundary, modifications of currents in the vacuum region or the addition of bulk forces. These are applied to efficiently compute the shape gradient of functions of MHD equilibria with an adjoint approach. In this way, the shape derivative with respect to any perturbation applied to the plasma boundary or coil shapes can be computed with only one additional MHD equilibrium solution. We demonstrate that this approach is applicable for several figures of merit of interest for stellarator configuration optimization: the magnetic well, the magnetic ripple on axis, the departure from quasisymmetry, the effective ripple in the low-collisionality $1/\unicode[STIX]{x1D708}$ regime $(\unicode[STIX]{x1D716}_{\text{eff}}^{3/2})$ (Nemov et al. Phys. Plasmas, vol. 6 (12), 1999, pp. 4622–4632) and several finite-collisionality neoclassical quantities. Numerical verification of this method is demonstrated for the magnetic well figure of merit with the VMEC code (Hirshman & Whitson Phys. Fluids, vol. 26 (12), 1983, p. 3553) and for the magnetic ripple with modification of the ANIMEC code (Cooper et al. Comput. Phys. Commun., vol. 72 (1), 1992, pp. 1–13). Comparisons with the direct approach demonstrate that, in order to obtain agreement within several per cent, the adjoint approach provides a factor of $O(10^{3})$ in computational savings.

Effect of magnetic saline irrigation water with different gausses on the growth and yield of two tomato varieties grown in desert land

This experiments was conducted in plastic house of ahached guiding center/Karbala during the growing season of 2008-2009. Soil texture was sandy loam. The purpose was to study the effect of magnetic well water of different gausses (1000, 1250, 1750) G on the cultivars grown and yield components of two tomato Varieties (Shrouq and Locas). The RCBD with three replications was used. Results showed that the irrigated with magnetized water with 1750G was superior in number of fruit/ plant, fruit weigh, fruit diameter, number of shoots, Plant height, leaves area, leaves length, these characters were significantly increased for both cultivars. On the other hand there was significant impact of the treatment on the yield.