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R C Wolf - One of the best experts on this subject based on the ideXlab platform.
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serpent neutronics model of wendelstein 7 x for 14 1 mev neutrons
Symposium On Fusion Technology, 2021Co-Authors: C Biedermann, S Akaslompolo, J P Koschinsky, Joona Kontula, S Bozhenkov, T Kurkisuonio, Jaakko Leppanen, A Snicker, R C WolfAbstract:Abstract In this work, a Serpent 2 neutronics model of the Wendelstein 7-X (W7-X) stellarator is prepared, and an response function for the Scintillating-Fibre neutron detector (SciFi) is calculated using the model. The neutronics model includes the simplified geometry for the key components of the stellarator itself as well as the torus hall. The objective of the model is to assess the 14.1 MeV neutron flux from deuteron-triton fusions in W7-X, where the neutrons are modelled only until they have slowed down to 1 MeV energy. The key messages of this article are: demonstration of unstructured mesh geometry usage for Stellarators, W7-X in particular; technical documentation of the model and first insights in fast neutron behaviour in W7-X, especially related to the SciFi: the model indicates that the superconducting coils are the strongest scatterers and block neutrons from large parts of the plasma. The back-scattering from e.g. massive steel support structures is found to be small. The SciFi will detect neutrons from an extended plasma volume in contrast to having an effective line-of-sight.
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magnetic configuration effects on the wendelstein 7 x stellarator
Nature Physics, 2018Co-Authors: A Dinklage, C D Beidler, P Helander, G Fuchert, H Maasberg, K Rahbarnia, Sunn T Pedersen, Y Turkin, R C WolfAbstract:The two leading concepts for confining high-temperature fusion plasmas are the tokamak and the stellarator. Tokamaks are rotationally symmetric and use a large plasma current to achieve confinement, whereas Stellarators are non-axisymmetric and employ three-dimensionally shaped magnetic field coils to twist the field and confine the plasma. As a result, the magnetic field of a stellarator needs to be carefully designed to minimize the collisional transport arising from poorly confined particle orbits, which would otherwise cause excessive power losses at high plasma temperatures. In addition, this type of transport leads to the appearance of a net toroidal plasma current, the so-called bootstrap current. Here, we analyse results from the first experimental campaign of the Wendelstein 7-X stellarator, showing that its magnetic-field design allows good control of bootstrap currents and collisional transport. The energy confinement time is among the best ever achieved in Stellarators, both in absolute figures (τE > 100 ms) and relative to the stellarator confinement scaling. The bootstrap current responds as predicted to changes in the magnetic mirror ratio. These initial experiments confirm several theoretically predicted properties of Wendelstein 7-X plasmas, and already indicate consistency with optimization measures. Results from the first experimental campaign of the Wendelstein 7-X stellarator demonstrate that its magnetic-field design grants good control of parasitic plasma currents, leading to long energy confinement times.
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magnetic configuration effects on the wendelstein 7 x stellarator
Nature Physics, 2018Co-Authors: A Dinklage, C D Beidler, P Helander, G Fuchert, H Maasberg, K Rahbarnia, Sunn T Pedersen, Y Turkin, R C WolfAbstract:The two leading concepts for confining high-temperature fusion plasmas are the tokamak and the stellarator. Tokamaks are rotationally symmetric and use a large plasma current to achieve confinement, whereas Stellarators are non-axisymmetric and employ three-dimensionally shaped magnetic field coils to twist the field and confine the plasma. As a result, the magnetic field of a stellarator needs to be carefully designed to minimize the collisional transport arising from poorly confined particle orbits, which would otherwise cause excessive power losses at high plasma temperatures. In addition, this type of transport leads to the appearance of a net toroidal plasma current, the so-called bootstrap current. Here, we analyse results from the first experimental campaign of the Wendelstein 7-X stellarator, showing that its magnetic-field design allows good control of bootstrap currents and collisional transport. The energy confinement time is among the best ever achieved in Stellarators, both in absolute figures (τE > 100 ms) and relative to the stellarator confinement scaling. The bootstrap current responds as predicted to changes in the magnetic mirror ratio. These initial experiments confirm several theoretically predicted properties of Wendelstein 7-X plasmas, and already indicate consistency with optimization measures. Results from the first experimental campaign of the Wendelstein 7-X stellarator demonstrate that its magnetic-field design grants good control of parasitic plasma currents, leading to long energy confinement times.
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Magnetic configuration effects on the Wendelstein 7-X stellarator
Nature Publishing Group, 2018Co-Authors: Dinklage A., R C Wolf, Maaßberg H., Beidler C D, Helander P, Fuchert G, Rahbarnia K, Ts Sunn-pedersen, Turkin Y, Andreeva TAbstract:\u3cp\u3e The two leading concepts for confining high-temperature fusion plasmas are the tokamak and the stellarator. Tokamaks are rotationally symmetric and use a large plasma current to achieve confinement, whereas Stellarators are non-axisymmetric and employ three-dimensionally shaped magnetic field coils to twist the field and confine the plasma. As a result, the magnetic field of a stellarator needs to be carefully designed to minimize the collisional transport arising from poorly confined particle orbits, which would otherwise cause excessive power losses at high plasma temperatures. In addition, this type of transport leads to the appearance of a net toroidal plasma current, the so-called bootstrap current. Here, we analyse results from the first experimental campaign of the Wendelstein 7-X stellarator, showing that its magnetic-field design allows good control of bootstrap currents and collisional transport. The energy confinement time is among the best ever achieved in Stellarators, both in absolute figures (τ \u3csub\u3eE\u3c/sub\u3e > 100 ms) and relative to the stellarator confinement scaling. The bootstrap current responds as predicted to changes in the magnetic mirror ratio. These initial experiments confirm several theoretically predicted properties of Wendelstein 7-X plasmas, and already indicate consistency with optimization measures. \u3c/p\u3
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density related operational limit in the limiter phase of wendelstein 7 x
European Conference on Circuit Theory and Design, 2017Co-Authors: G Fuchert, Sunn T Pedersen, R C Wolf, D Zhang, R Burhenn, S Bozhenkov, M Jakubowski, H Niemann, E Pasch, G A WurdenAbstract:The stellarator Wendelstein 7-X (W7-X) has been designed to show that optimized Stellarators can achieve and sustain fusion relevant plasma conditions. One of the main optimization criteria was the reduction of the neoclassical transport, which is considered as one of the most critical issues of the stellarator concept. While the demonstration of the neoclassical optimization will definitely be one of the most important aspects of W7-X, several additional topics are equally important due to the boundary condition that fusion relevant conditions should be demonstrated and sustained. Hence, low neoclassical transport needs to be achieved in a scenario which is compatible with divertor operation, high densities, and an acceptable impurity concentration without further accumulation during stable operation. So far, these issues have mostly not been tackled experimentally in the first experimental campaign. W7-X had its first plasma in December 2015 and the first operational campaign (OP1.1) was mainly intended for testing and commissioning purposes. In OP1.1, W7-X was operated in a limiter configuration without divertor and at low densities. Nevertheless, valuable insights could be gained which help to prepare and understand the first experiments with a test divertor planned to start in the second half of 2017. As will be shown in this contribution, one of the interesting observations in OP1.1 was the presence of an operational limit, where above a critical density the power balance seems to be dominated by radiative losses. Such a behavior is well-known from other stellarator experiments and while it can be expected that this critical density is particularly low for OP1.1 (high impurity concentration connected to the limiter operation and conservative wall conditioning), this observation also shows that impurity related radiation losses will be an important issue to keep track of as the density is progressively increased in the next experimental campaigns of W7-X.
P Helander - One of the best experts on this subject based on the ideXlab platform.
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magnetic configuration effects on the wendelstein 7 x stellarator
Nature Physics, 2018Co-Authors: A Dinklage, C D Beidler, P Helander, G Fuchert, H Maasberg, K Rahbarnia, Sunn T Pedersen, Y Turkin, R C WolfAbstract:The two leading concepts for confining high-temperature fusion plasmas are the tokamak and the stellarator. Tokamaks are rotationally symmetric and use a large plasma current to achieve confinement, whereas Stellarators are non-axisymmetric and employ three-dimensionally shaped magnetic field coils to twist the field and confine the plasma. As a result, the magnetic field of a stellarator needs to be carefully designed to minimize the collisional transport arising from poorly confined particle orbits, which would otherwise cause excessive power losses at high plasma temperatures. In addition, this type of transport leads to the appearance of a net toroidal plasma current, the so-called bootstrap current. Here, we analyse results from the first experimental campaign of the Wendelstein 7-X stellarator, showing that its magnetic-field design allows good control of bootstrap currents and collisional transport. The energy confinement time is among the best ever achieved in Stellarators, both in absolute figures (τE > 100 ms) and relative to the stellarator confinement scaling. The bootstrap current responds as predicted to changes in the magnetic mirror ratio. These initial experiments confirm several theoretically predicted properties of Wendelstein 7-X plasmas, and already indicate consistency with optimization measures. Results from the first experimental campaign of the Wendelstein 7-X stellarator demonstrate that its magnetic-field design grants good control of parasitic plasma currents, leading to long energy confinement times.
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magnetic configuration effects on the wendelstein 7 x stellarator
Nature Physics, 2018Co-Authors: A Dinklage, C D Beidler, P Helander, G Fuchert, H Maasberg, K Rahbarnia, Sunn T Pedersen, Y Turkin, R C WolfAbstract:The two leading concepts for confining high-temperature fusion plasmas are the tokamak and the stellarator. Tokamaks are rotationally symmetric and use a large plasma current to achieve confinement, whereas Stellarators are non-axisymmetric and employ three-dimensionally shaped magnetic field coils to twist the field and confine the plasma. As a result, the magnetic field of a stellarator needs to be carefully designed to minimize the collisional transport arising from poorly confined particle orbits, which would otherwise cause excessive power losses at high plasma temperatures. In addition, this type of transport leads to the appearance of a net toroidal plasma current, the so-called bootstrap current. Here, we analyse results from the first experimental campaign of the Wendelstein 7-X stellarator, showing that its magnetic-field design allows good control of bootstrap currents and collisional transport. The energy confinement time is among the best ever achieved in Stellarators, both in absolute figures (τE > 100 ms) and relative to the stellarator confinement scaling. The bootstrap current responds as predicted to changes in the magnetic mirror ratio. These initial experiments confirm several theoretically predicted properties of Wendelstein 7-X plasmas, and already indicate consistency with optimization measures. Results from the first experimental campaign of the Wendelstein 7-X stellarator demonstrate that its magnetic-field design grants good control of parasitic plasma currents, leading to long energy confinement times.
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Magnetic configuration effects on the Wendelstein 7-X stellarator
'Springer Science and Business Media LLC', 2018Co-Authors: A Dinklage, P Helander, G Fuchert, K Rahbarnia, Y Turkin, D. C. Beidler, H. Maaßberg, T., Sunn Pedersen, C. R. Wolf, A. AlonsoAbstract:The two leading concepts for confining high-temperature fusion plasmas are the tokamak and the stellarator. Tokamaks are rotationally symmetric and use a large plasma current to achieve confinement, whereas Stellarators are non-axisymmetric and employ three-dimensionally shaped magnetic field coils to twist the field and confine the plasma. As a result, the magnetic field of a stellarator needs to be carefully designed to minimize the collisional transport arising from poorly confined particle orbits, which would otherwise cause excessive power losses at high plasma temperatures. In addition, this type of transport leads to the appearance of a net toroidal plasma current, the so-called bootstrap current. Here, we analyse results from the first experimental campaign of the Wendelstein 7-X stellarator, showing that its magnetic-field design allows good control of bootstrap currents and collisional transport. The energy confinement time is among the best ever achieved in Stellarators, both in absolute figures (τE> 100 ms) and relative to the stellarator confinement scaling. The bootstrap current responds as predicted to changes in the magnetic mirror ratio. These initial experiments confirm several theoretically predicted properties of Wendelstein 7-X plasmas, and already indicate consistency with optimization measures
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impurity transport in a mixed collisionality stellarator plasma
Physical Review Letters, 2017Co-Authors: P Helander, Sarah Newton, A Mollen, H M SmithAbstract:A potential threat to the performance of magnetically confined fusion plasmas is the problem of impurity accumulation, which causes the concentration of highly charged impurity ions to rise uncontrollably in the center of the plasma and spoil the energy confinement by excessive radiation. It has long been thought that the collisional transport of impurities in Stellarators always leads to such an accumulation (if the electric field points inwards, which is usually the case), whereas tokamaks, being axisymmetric, can benefit from "temperature screening," i.e., an outward flux of impurities driven by the temperature gradient. Here it is shown, using analytical techniques supported by results from a new numerical code, that such screening can arise in stellarator plasmas, too, and indeed does so in one of the most relevant operating regimes, where the impurities are highly collisional while the bulk plasma is at low collisionality.
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impurity transport and plasma flow in a mixed collisionality stellarator plasma
Proceedings of 26th IAEA Fusion Energy Conference Kyoto Japan, 2016Co-Authors: Sarah Newton, P Helander, A Mollen, H M Smith, Yu TurkinAbstract:Neoclassical accumulation of impurities in the core of hot stellarator plasmas is a known problem. The complexity of neoclassical transport in Stellarators means that few analytic studies are available to support numerical modelling efforts, and a robust understanding of the parameter dependence of the impurity flux is still lacking. Therefore we present an extension of the existing analytic treatment for highly collisional plasmas, into the experimentally relevant mixed collisionality regime -- where a dominant heavy, collisional, impurity is present in a collisionless bulk plasma, taken here to be in the 1/\nu regime. We find that temperature screening of the impurity flux by the bulk ion temperature gradient will arise. We also determine the bulk ion flow in the flux surface, and thus the effect of the impurity on the bulk ion contribution to the bootstrap current.
F Jenko - One of the best experts on this subject based on the ideXlab platform.
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turbulence suppression by energetic particle effects in modern optimized Stellarators
Physical Review Letters, 2020Co-Authors: Banon A Navarro, A Di Siena, F JenkoAbstract:Turbulent transport is known to limit the plasma confinement of present-day optimized Stellarators. To address this issue, a novel method to strongly suppress turbulence in such devices is proposed, namely the resonant wave-particle interaction of suprathermal particles---e.g., from ion-cyclotron-resonance-frequency heating---with turbulence-driving microinstabilities like ion-temperature-gradient modes. The effectiveness of this mechanism is demonstrated via large-scale gyrokinetic simulations, revealing an overall turbulence reduction by up to 65% in the case under consideration. Comparisons with a tokamak configuration highlight the critical role played by the magnetic geometry and the first steps into the optimization of fast particle effects in stellarator devices are discussed. These results hold the promise of new and still unexplored stellarator scenarios with reduced turbulent transport, essential for achieving burning plasmas in future devices.
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advances in stellarator gyrokinetics
Nuclear Fusion, 2015Co-Authors: P Helander, R Kleiber, J Riemann, F Jenko, G G Plunk, T Bird, J H E Proll, P. XanthopoulosAbstract:Recent progress in the gyrokinetic theory of stellarator microinstabilities and turbulence simulations is summarized. The simulations have been carried out using two different gyrokinetic codes, the global particle-in-cell code EUTERPE and the continuum code GENE, which operates in the geometry of a flux tube or a flux surface but is local in the radial direction. Ion-temperature-gradient (ITG) and trapped-electron modes are studied and compared with their counterparts in axisymmetric tokamak geometry. Several interesting differences emerge. Because of the more complicated structure of the magnetic field, the fluctuations are much less evenly distributed over each flux surface in Stellarators than in tokamaks. Instead of covering the entire outboard side of the torus, ITG turbulence is localized to narrow bands along the magnetic field in regions of unfavourable curvature, and the resulting transport depends on the normalized gyroradius ρ* even in radially local simulations. Trapped-electron modes can be significantly more stable than in typical tokamaks, because of the spatial separation of regions with trapped particles from those with bad magnetic curvature. Preliminary non-linear simulations in flux-tube geometry suggest differences in the turbulence levels in Wendelstein 7-X and a typical tokamak.
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nonlinear gyrokinetic simulations of ion temperature gradient turbulence for the optimized wendelstein 7 x stellarator
Physical Review Letters, 2007Co-Authors: P. Xanthopoulos, F Merz, T Gorler, F JenkoAbstract:Ion-temperature-gradient turbulence constitutes a possibly dominant transport mechanism for optimized Stellarators, in view of the effective suppression of neoclassical losses characterizing these devices. Nonlinear gyrokinetic simulation results for the Wendelstein 7-X stellarator [G. Grieger et al., in Proceedings of the IAEA Conference on Plasma Physics and Controlled Nuclear Fusion Research, 1990 (IAEA, Vienna, 1991) Vol. 3, p. 525]---assuming an adiabatic electron response---are presented. Several fundamental features are discussed, including the role of zonal flows for turbulence saturation, the resulting flux-gradient relationship, and the coexistence of ion-temperature-gradient modes with trapped ion modes in the saturated state.
P. Xanthopoulos - One of the best experts on this subject based on the ideXlab platform.
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the parallel boundary condition for turbulence simulations in low magnetic shear devices
Plasma Physics and Controlled Fusion, 2018Co-Authors: Mike F Martin, P. Xanthopoulos, Matt Landreman, Noah Mandell, W DorlandAbstract:Flux tube simulations of plasma turbulence in Stellarators and tokamaks typically employ coordinates which are aligned with the magnetic field lines. Anisotropic turbulent fluctuations can be represented in such field-aligned coordinates very efficiently, but the resulting non-trivial boundary conditions involve all three spatial directions, and must be handled with care. The standard 'twist-and-shift' formulation of the boundary conditions (Beer et al 1995 Phys. Plasmas 2 2687) was derived assuming axisymmetry and is widely used because it is efficient, as long as the global magnetic shear is not too small. A generalization of this formulation is presented, appropriate for studies of non-axisymmetric, stellarator symmetric configurations, as well as for axisymmetric configurations with small global shear. The key idea is to replace the 'twist' of the standard approach (which accounts only for global shear) with the integrated local shear. This generalization allows one significantly more freedom when choosing the extent of the simulation domain in each direction, without losing the natural efficiency of field line-following coordinates. It also corrects some errors associated with naive application of axisymmetric boundary conditions to non-axisymmetric configurations. Simulations of stellarator turbulence that employ the generalized boundary conditions generally require much less resolution than simulations that use the conventional, axisymmetric boundary conditions. We also demonstrate the surprising result that (at least in some cases) an easily implemented but manifestly incorrect formulation of the boundary conditions does not change important predicted quantities, such as the turbulent heat flux.
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advances in stellarator gyrokinetics
Nuclear Fusion, 2015Co-Authors: P Helander, R Kleiber, J Riemann, F Jenko, G G Plunk, T Bird, J H E Proll, P. XanthopoulosAbstract:Recent progress in the gyrokinetic theory of stellarator microinstabilities and turbulence simulations is summarized. The simulations have been carried out using two different gyrokinetic codes, the global particle-in-cell code EUTERPE and the continuum code GENE, which operates in the geometry of a flux tube or a flux surface but is local in the radial direction. Ion-temperature-gradient (ITG) and trapped-electron modes are studied and compared with their counterparts in axisymmetric tokamak geometry. Several interesting differences emerge. Because of the more complicated structure of the magnetic field, the fluctuations are much less evenly distributed over each flux surface in Stellarators than in tokamaks. Instead of covering the entire outboard side of the torus, ITG turbulence is localized to narrow bands along the magnetic field in regions of unfavourable curvature, and the resulting transport depends on the normalized gyroradius ρ* even in radially local simulations. Trapped-electron modes can be significantly more stable than in typical tokamaks, because of the spatial separation of regions with trapped particles from those with bad magnetic curvature. Preliminary non-linear simulations in flux-tube geometry suggest differences in the turbulence levels in Wendelstein 7-X and a typical tokamak.
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collisionless microinstabilities in Stellarators iii the ion temperature gradient mode
Physics of Plasmas, 2014Co-Authors: G G Plunk, P. Xanthopoulos, P Helander, J W ConnorAbstract:We investigate the linear theory of the ion-temperature-gradient (ITG) mode, with the goal of developing a general understanding that may be applied to Stellarators. We highlight the Wendelstein 7X (W7-X) device. Simple fluid and kinetic models that follow closely from existing literature are reviewed and two new first-principle models are presented and compared with results from direct numerical simulation. One model investigates the effect of regions of strong localized shear, which are generic to stellarator equilibria. These “shear spikes” are found to have a potentially significant stabilizing affect on the mode; however, the effect is strongest at short wavelengths perpendicular to the magnetic field, and it is found to be significant only for the fastest growing modes in W7-X. A second model investigates the long-wavelength limit for the case of negligible global magnetic shear. The analytic calculation reveals that the effect of the curvature drive enters at second order in the drift frequency, co...
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nonlinear gyrokinetic simulations of ion temperature gradient turbulence for the optimized wendelstein 7 x stellarator
Physical Review Letters, 2007Co-Authors: P. Xanthopoulos, F Merz, T Gorler, F JenkoAbstract:Ion-temperature-gradient turbulence constitutes a possibly dominant transport mechanism for optimized Stellarators, in view of the effective suppression of neoclassical losses characterizing these devices. Nonlinear gyrokinetic simulation results for the Wendelstein 7-X stellarator [G. Grieger et al., in Proceedings of the IAEA Conference on Plasma Physics and Controlled Nuclear Fusion Research, 1990 (IAEA, Vienna, 1991) Vol. 3, p. 525]---assuming an adiabatic electron response---are presented. Several fundamental features are discussed, including the role of zonal flows for turbulence saturation, the resulting flux-gradient relationship, and the coexistence of ion-temperature-gradient modes with trapped ion modes in the saturated state.
G G Plunk - One of the best experts on this subject based on the ideXlab platform.
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direct construction of optimized stellarator shapes part 2 numerical quasisymmetric solutions
Journal of Plasma Physics, 2019Co-Authors: Matt Landreman, Wrick Sengupta, G G PlunkAbstract:Quasisymmetric Stellarators are appealing intellectually and as fusion reactor candidates since the guiding-centre particle trajectories and neoclassical transport are isomorphic to those in a tokamak, implying good confinement. Previously, quasisymmetric magnetic fields have been identified by applying black-box optimization algorithms to minimize symmetry-breaking Fourier modes of the field strength $B$ . Here, instead, we directly construct magnetic fields in cylindrical coordinates that are quasisymmetric to leading order in the distance from the magnetic axis, without using optimization. The method involves solution of a one-dimensional nonlinear ordinary differential equation, originally derived by Garren & Boozer ( Phys. Fluids B, vol. 3, 1991, p. 2805). We demonstrate the usefulness and accuracy of this optimization-free approach by providing the results of this construction as input to the codes VMEC and BOOZ_XFORM, confirming the purity and scaling of the magnetic spectrum. The space of magnetic fields that are quasisymmetric to this order is parameterized by the magnetic axis shape along with three other real numbers, one of which reflects the on-axis toroidal current density, and another one of which is zero for stellarator symmetry. The method here could be used to generate good initial conditions for conventional optimization, and its speed enables exhaustive searches of parameter space.
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advances in stellarator gyrokinetics
Nuclear Fusion, 2015Co-Authors: P Helander, R Kleiber, J Riemann, F Jenko, G G Plunk, T Bird, J H E Proll, P. XanthopoulosAbstract:Recent progress in the gyrokinetic theory of stellarator microinstabilities and turbulence simulations is summarized. The simulations have been carried out using two different gyrokinetic codes, the global particle-in-cell code EUTERPE and the continuum code GENE, which operates in the geometry of a flux tube or a flux surface but is local in the radial direction. Ion-temperature-gradient (ITG) and trapped-electron modes are studied and compared with their counterparts in axisymmetric tokamak geometry. Several interesting differences emerge. Because of the more complicated structure of the magnetic field, the fluctuations are much less evenly distributed over each flux surface in Stellarators than in tokamaks. Instead of covering the entire outboard side of the torus, ITG turbulence is localized to narrow bands along the magnetic field in regions of unfavourable curvature, and the resulting transport depends on the normalized gyroradius ρ* even in radially local simulations. Trapped-electron modes can be significantly more stable than in typical tokamaks, because of the spatial separation of regions with trapped particles from those with bad magnetic curvature. Preliminary non-linear simulations in flux-tube geometry suggest differences in the turbulence levels in Wendelstein 7-X and a typical tokamak.
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collisionless microinstabilities in Stellarators iii the ion temperature gradient mode
Physics of Plasmas, 2014Co-Authors: G G Plunk, P. Xanthopoulos, P Helander, J W ConnorAbstract:We investigate the linear theory of the ion-temperature-gradient (ITG) mode, with the goal of developing a general understanding that may be applied to Stellarators. We highlight the Wendelstein 7X (W7-X) device. Simple fluid and kinetic models that follow closely from existing literature are reviewed and two new first-principle models are presented and compared with results from direct numerical simulation. One model investigates the effect of regions of strong localized shear, which are generic to stellarator equilibria. These “shear spikes” are found to have a potentially significant stabilizing affect on the mode; however, the effect is strongest at short wavelengths perpendicular to the magnetic field, and it is found to be significant only for the fastest growing modes in W7-X. A second model investigates the long-wavelength limit for the case of negligible global magnetic shear. The analytic calculation reveals that the effect of the curvature drive enters at second order in the drift frequency, co...