Magnetohydrodynamic Equilibrium Code Research Articles

Constructing stellarators with quasisymmetry to high order

A method is given to rapidly compute quasisymmetric stellarator magnetic fields for plasma confinement, without the need to call a three-dimensional magnetohydrodynamic equilibrium code inside an optimization iteration. The method is based on direct solution of the equations of magnetohydrodynamic equilibrium and quasisymmetry using Garren & Boozer’s expansion about the magnetic axis (Phys Fluids B, vol. 3, 1991, p. 2805), and it is several orders of magnitude faster than the conventional optimization approach. The work here extends the method of Landremanet al. (J. Plasma Phys., vol. 85, 2019, 905850103), which was limited to flux surfaces with elliptical cross-section, to higher order in the aspect-ratio expansion. As a result, configurations can be generated with strong shaping that achieve quasisymmetry to high accuracy. Using this construction, we give the first numerical demonstrations of Garren and Boozer’s ideal scaling of quasisymmetry breaking with the cube of the inverse aspect ratio. We also demonstrate a strongly non-axisymmetric configuration (vacuum rotational transform$\unicode[STIX]{x1D704}>0.4$) in which symmetry-breaking mode amplitudes throughout a finite volume are${<}2\times 10^{-7}$, the smallest ever reported. To generate boundary shapes of finite-minor-radius configurations, a careful analysis is given of the effect of substituting a finite minor radius into the near-axis expansion. The approach here can provide analytic insight into the space of possible quasisymmetric stellarator configurations, and it can be used to generate good initial conditions for conventional stellarator optimization.

Comparative collisionless alpha particle confinement in stellarator reactors with the XGC gyrokinetic code

Whole volume simulations of stellarators are necessary to address a number of important physics and engineering issues, including turbulent transport prediction and optimising divertor operation. In this work, advances in the development of a whole volume stellarator capability for the global gyrokinetic particle-in-cell code XGC are detailed. A 3D interpolation of the equilibrium magnetic field to the last close flux surface, calculated using the VMEC ideal magnetohydrodynamic equilibrium code, has been implemented. Orbit tracing with the XGC code is verified for the Wendelstein 7-X stellarator by comparison with the BEAMS3D and EUTERPE codes. The XGC stellarator version has then been used to consider collisionless alpha particle confinement in representative quasi-isodynamic and quasi-axisymmetric designs. It has been shown that, at least without further optimisation for energetic particle confinement, the Wendelstein 7-X-like quasi-isodynamic reactor design exhibits better alpha particle confinement.

Investigation of island formation due to RMPs in DIII-D plasmas with the SIESTA resistive MHD equilibrium code

The SIESTA magnetohydrodynamic (MHD) equilibrium code has been used to compute a sequence of ideally stable equilibria resulting from numerical variation of the helical resonant magnetic perturbation (RMP) applied to an axisymmetric DIII-D plasma equilibrium. Increasing the perturbation strength at the dominant $m=2$, $n=-1$ resonant surface leads to lower MHD energies and increases in the equilibrium island widths at the $m=2$ (and sidebands) surfaces, in agreement with theoretical expectations. Island overlap at large perturbation strengths leads to stochastic magnetic fields which correlate well with the experimentally inferred field structure. The magnitude and spatial phase (around the dominant rational surfaces) of the resonant (shielding) component of the parallel current are shown to change qualitatively with the magnetic island topology.

Inclusion of Pressure and Flow in the KITES MHD Equilibrium Code

One of the simplest self-consistent models of a plasma is single-fluid magnetohydrodynamic (MHD) equilibrium with no bulk fluid flow under axisymmetry. However, both fluid flow and non-axisymmetric effects can significantly impact plasma equilibrium and confinement properties: in particular, fluid flow can produce profile pedestals, and non-axisymmetric effects can produce islands and stochastic regions. There exist a number of computational codes which are capable of calculating equilibria with arbitrary flow or with non-axisymmetric effects. Previously, a concept for a code to calculate MHD equilibria with flow in non-axisymmetric systems was presented, called the KITES (Kyoto ITerative Equilibrium Solver) code [D. Raburn and A. Fukuyama, Plasma Fusion Res. 7, 240318 (2012)]. Since then, many of the computational modules for the KITES code have been completed, and the work-in-progress KITES code has been used to calculate non-axisymmetric force-free equilibria [D. Raburn and A. Fukuyama, Proceedings of the 9th EPS Conference on Plasma Physics, Stockholm, Sweden (2012)]. Additional computational modules are required to allow the KITES code to calculate equilibria with pressure and flow. Here, the authors report on the approaches used in developing these modules and provide a sample calculation with pressure.

Control of magnetic islands in the STOR-M tokamak using resonant helical fields

The resonant interaction between magnetohydrodynamic (MHD) instability modes and the externally applied helical magnetic field is demonstrated in the Saskatchewan Torus-Modified (STOR-M) tokamak. The study is conducted both numerically and experimentally using a 2D MHD equilibrium code in the former and an (l = 2, n = 1) helical coil carrying a short current pulse in the latter. It is shown numerically that the resonant helical current can efficiently suppress the magnetic islands resonating on the (m = 2, n = 1) magnetic surface when the value of the safety factor at the plasma edge is relatively low (⩽4). It is also found numerically that (2, 1) islands are induced when the applied helical current exceeds a threshold. The experimental study is performed in STOR-M during low-q ohmic discharges with high MHD activities. The amplitude and frequency of (2, 1) Mirnov fluctuations are significantly reduced after the activation of the resonant field. Lesser suppression in sideband islands is also observed. Moreover, a phase of improved plasma confinement, characterized by a reduction in Hα emission level, a reduction in loop voltage and an increase in the soft x-ray emission, is induced after application of the resonant field.

Predictive modelling for EAST divertor operation

The predictive modelling study of the divertor operation in EAST tokamak [B. Wan et al., Nucl. Fusion 49, 104011 (2009)] with double null (DN) configuration is carried out by using the two-dimensional edge plasma code B2.5-SOLPS5.0 [D. P. Coster, X. Bonnin et al., J. Nucl. Mater. 337–339, 366 (2005)]. The modelling study includes the particle and power balance in the scrape-off-layer (SOL), the operation parameters of plasma density, temperature and plasma heat fluxes at the separatrix, the target plates and the wall, and the effect of the gas puffing, drifts, and vertical target plate on the divertor operation. The fluid model for the edge plasma is applied using the real magnetohydrodynamic (MHD) equilibrium from the MHD equilibrium code EFIT [L. L. Lao et al., Nucl. Fusion 25, 1611 (1985)] and the real divertor geometry in the device. Before EAST tokamak starts its experimental programme of divertor operation, the modelling plays an important role in the design of its experimental programme and the optimization of the divertor operation parameters. Based on the modelling results, EAST divertor can operate over a large wide of plasma parameters with different regimes. For a heating power of 8 MW and an edge density at core-SOL interface Nedge = 0.8 × 10191/m3 and Nedge = 1.3 × 10191/m3, the EAST divertor begins access to the high recycling operation regime at the outer and inner target plates, respectively, where the plasma temperature and the heat fluxes at the target plates decrease. The gas puffing can increase the plasma density at the separatrix and trigger the transition from the high recycling operation into detachment at the target plates. When E × B and B × ▿B drifts are taken into account, the asymmetry of plasma parameters and heat fluxes between up-down SOLs can be found. The vertical target plate in EAST divertor can reduce the peak values of heat fluxes at the target plate and enables detachment at lower plasma density. The divertor with the vertical target plate can be considered as a possible option for EAST divertor upgrade in the future.

Helical ITER hybrid scenario equilibria

Numerical computations of ITER equilibria in the hybrid scenario using a three-dimensional (3D) magnetohydrodynamic equilibrium code with nested magnetic flux surfaces demonstrate the formation of internal 3D helical cores similar to saturated ideal internal kink modes under reversed magnetic shear conditions when the minimum value of the inverse rotational transform is in the neighbourhood of unity.

Observation of Equilibria with a Double Magnetic Axis in LHD

Finite-beta equilibria with a double magnetic axis have been realized in the Large Helical Device (LHD). Since the rotational transform is weak in the central region of the LHD, the effect of an externally applied quadrupole field is more pronounced in the central region than in the periphery. Consequently, the magnetic axis splits due to a moderate elongation. In the case of vertical elongation, the figure-eight structure of the magnetic surfaces has been observed in a soft X-ray image. Degradation of confinement due to the appearance of the separatrix is suggested in the case of horizontal elongation, which is closely related to the equilibrium beta limit. The three-dimensional magnetohydrodynamic equilibrium code HINT, which does not assume the existence of nested flux surfaces, provides physical pictures consistent with the experimental observations.

Time-dependent scenario modeling of the Tokamak Physics Experiment using a theory-based transport model

Predictive transport and current drive simulations are presented for the time evolution of the temperature and density profiles in the Tokamak Physics Experiment (TPX). [W. M. Nevins et al., Proceedings of the International Conference on Plasma Physics and Controlled Nuclear Fusion Research, Würzburg, 1992 (International Atomic Energy Agency, Vienna, 1992), Vol. 3, p. 279]. A distinguishing feature of this study is that we use a theoretically derived transport model that has been empirically calibrated against Ohmic, low-confinement and high-confinement mode discharges from seven different tokamaks. Heating and deposition profiles predicted by the ACCOME current drive and magnetohydrodynamic equilibrium code [R. S. Devoto et al., Nucl. Fusion 32, 773 (1992)] are incorporated into the time-dependent BALDUR one-and-one-half dimensional transport code [C. E. Singer et al., Comput. Phys. Commun. 49, 275 (1988)]. We consider various scenarios of fast wave, lower hybrid, and neutral beam heating and current drive and evaluate the effects on the evolution of density and temperature profiles.

Experimental studies of linear high-beta heliac plasma configurations

The formation and quasi-equilibrium of a high-beta heliac plasma are investigated in the High-Beta Q Machine [Phys. Fluids 30, 2885 (1987)], a linear high-beta l=1 stellarator with an internal current carrying conductor (hardcore). The hardcore current rise time is varied from 9 μsec to smaller values comparable with that of the main compression field (450 nsec). Flux contours and plasma pressure calculated from internal magnetic-probe measurements are used to distinguish between nearly axisymmetric plasma confined near the hardcore and the heliac plasma confined near the magnetic axis. For the shorter hardcore rise times, the axisymmetric plasma pressure becomes small compared to that of the heliac. Further analysis of the data allows calculation of the currents flowing in the plasma, the rotational transform, and the magnetic well depth. Appreciable axial current is observed, consistent with induction by the changing magnetic fields during the heliac formation. The observed relationship between the axial current and rotational transform is confirmed by computer modeling using the hase magnetohydrodynamic equilibrium code [Nucl. Fusion 23, 1061 (1983)].

Monte carlo calculation of transport for three‐dimensional magnetohydrodynamic equilibria

AbstractA Monte Carlo algorithm has been implemented which couples the results of a three‐dimensional magnetohydrodynamic equilibrium code with particle transport calculations similar to those of Boozer and Kuo‐Petravic. The equilibrium results are in a form which lends itself to easy and accurate computation of the coefficients required to follow particle orbits in flux variables at finite pressure. In particular, the parallel current and the Clebsch potential for the current are obtained from Fourier series solutions of first‐order partial differential equations with constant coefficients.Confinement times are estimated from the exponential decay of expected values of appropriately chosen functionals of the particle distribution. The observation that these functionals satisfy boundary conditions helps us to compute confinement times over a wide range of collision frequencies, including cases where losses due to particle trapping are very high. Initial conditions are chosen to optimize the Monte Carlo calculation by using known information about the expected particle distribution.Results for actual and proposed stellarator experiment are given. Electron and ion confinement times are compared, since this is relevant to the issue of the effect of ambipolar electric fields on stellarator confinement. A spectral method is given to determine the electric field from charge neutrality, and numerical evidence is presented to suggest that anomalous electron transport may be due to small resonant terms in the electric potential. Also, comparisons are made with a simplified theory of particle transport already incorporated in the equilibrium code.

Multipinch – a reversed-field pinch with a magnetic well

The Multipinch is an axisymmetric, vertically elongated toroidal pinch with a reversed toroidal field and an internal figure-eight separatrix. Unlike the circular cross-section pinch, the Multipinch configuration has a magnetic well. A magnetohydrodynamic equilibrium code for a non-circular reversed-field pinch is described and equilibrium calculations for the experimental device are presented. Experimental results concerning plasma configuration and stability are given. It was found that the up-down symmetric configuration became asymmetric when the toroidal current was increased to a level at which the toroidal field at the conducting shell wall became zero. This was found to be consistent with the minimum-energy Taylor configuration for the vertically elongated Multipinch.

Experimental and computational study of high-beta tokamak equilibria

High-beta tokamak equilibria in Torus II are studied both experimentally and computationally. Electron temperature and density profile measurements from the multipoint Thomson scattering system and line-integrated density from a CO2 interferometer show that an equilibrium is obtained with a volume averaged beta of 10%. The peak electron temperature is 80 eV and the density is 1×1015 cm−3. A numerical computation of these equilibria is carried out using a 2-D, free-boundary, magnetohydrodynamic equilibrium code which is in good agreement with the experimental results.

Three-Dimensional Magnetohydrodynamic Equilibria of Toroidal Stellarators

Fully nonaxisymmetric magnetohydrodynamic equilibria have been calculated numerically for the Proto-Cleo stellarator utilizing a fully three-dimensional magnetohydrodynamic equilibrium code. It was found that recently observed anomalous circulating currents in stellarators may be a result of the equilibrium condition.