Publications

@article{muirNumericalIntegrationParticle2022,

title = {Numerical Integration of Particle Orbits in Discontinuous Fields Using VENUS-LEVIS and SPEC},

author = {Dean Muir and David Pfefferl\'{e} and Zhisong Qu and Matthew Hole and Markus Hegland},

doi = {10.1016/j.cpc.2021.108191},

issn = {0010-4655},

year  = {2022},

date = {2022-02-01},

urldate = {2022-02-01},

journal = {Computer Physics Communications},

volume = {271},

pages = {108191},

abstract = {The orbit code VENUS-LEVIS is adapted to follow particles in plasma equilibria with discontinuous fields generated by the Stepped Pressure Equilibrium Code (SPEC). The latter is an implementation of the Multi-Region relaxed MHD model, which efficiently computes Taylor states in a series of nested toroidal volumes and supports the formation of magnetic islands and chaotic field regions. To adapt VENUS-LEVIS, an event location procedure is implemented in the existing numerical integrator, which ensures the particle sees the correct field along its trajectory, regardless of the discontinuities present in the Stepped Pressure Equilibrium model. The algorithm is tested in the case where the magnetic field is uniform in the upper and lower half-spaces but has a discontinuity in its direction (shear) on the plane z=0. Particle drifts due to the discontinuity are studied. The convergence properties of the numerical scheme are highlighted by the numerical accuracy, and conservation of the system's invariants, such as energy and momentum. Simulations and convergence studies using the SPEC-LEVIS interface in axisymmetric geometry are then presented. Finally, illustrative particle drifts due to the discontinuity are studied and explained: we examine drifts associated with the change in Larmor radius of passing particles with small excursion from flux surfaces.},

keywords = {discontinuous fields, full-orbit, numerical method, SPEC, VENUS-LEVIS},

pubstate = {published},

tppubtype = {article}

}

@article{jeyakumarAnalysisIsotropicAnisotropic2021,

title = {Analysis of the Isotropic and Anisotropic Grad\textendashShafranov Equation},

author = {S. Jeyakumar and D. Pfefferl\'{e} and M. J. Hole and Z. S. Qu},

doi = {10.1017/S002237782100088X},

issn = {0022-3778, 1469-7807},

year  = {2021},

date = {2021-10-01},

urldate = {2021-10-01},

journal = {Journal of Plasma Physics},

volume = {87},

number = {5},

publisher = {Cambridge University Press},

abstract = {Pressure anisotropy is a commonly observed phenomenon in tokamak plasmas, due to external heating methods such as neutral beam injection and ion-cyclotron resonance heating. Equilibrium models for tokamaks are constructed by solving the Grad\textendash Shafranov equation; such models, however, do not account for pressure anisotropy since ideal magnetohydrodynamics assumes a scalar pressure. A modified Grad\textendash Shafranov equation can be derived to include anisotropic pressure and toroidal flow by including drift-kinetic effects from the guiding-centre model of particle motion. In this work, we have studied the mathematical well-posedness of these two problems by showing the existence and uniqueness of solutions to the Grad\textendash Shafranov equation both in the standard isotropic case and when including pressure anisotropy and toroidal flow. A new fixed-point approach is used to show the existence of solutions in the Sobolev space H10H_0^1 to the Grad\textendash Shafranov equation, and sufficient criteria for their uniqueness are derived. The conditions required for the existence of solutions to the modified Grad\textendash Shafranov equation are also constructed.},

keywords = {anisotropy, grad-shafranov, mathematical analysis, MHD equilibrium, PDE},

pubstate = {published},

tppubtype = {article}

}

@article{perrellaStefanSussmannTheoremNormal2021,

title = {A Stefan-Sussmann Theorem for Normal Distributions on Manifolds with Boundary},

author = {David Perrella and David Pfefferl\'{e} and Luchezar Stoyanov},

year  = {2021},

date = {2021-09-01},

urldate = {2021-09-01},

journal = {arXiv:2109.04845 [math]},

abstract = {An analogue of the Stefan-Sussmann Theorem on manifolds with boundary is proven for normal distributions. These distributions contain vectors transverse to the boundary along its entirety. Plain integral manifolds are not enough to "integrate" a normal distribution; the next best "integrals" are so-called neat integral manifolds with boundary. The conditions on the distribution for this integrability is expressed in terms of adapted collars and integrability of a pulled-back distribution on the interior and on the boundary.},

keywords = {differential geometry, foliations, Froebenius, integrability, manifolds, topology},

pubstate = {published},

tppubtype = {article}

}

@article{pfefferleGaugeFreedomMagnetostatics2021,

title = {Gauge Freedom in Magnetostatics and the Effect on Helicity in Toroidal Volumes},

author = {David Pfefferl\'{e} and Lyle Noakes and David Perrella},

doi = {10.1063/5.0038226},

issn = {0022-2488},

year  = {2021},

date = {2021-09-01},

urldate = {2021-09-01},

journal = {Journal of Mathematical Physics},

volume = {62},

number = {9},

pages = {093505},

publisher = {American Institute of Physics},

abstract = {Magnetostatics defines a class of boundary value problems in which the topology of the domain plays a subtle role. For example, representability of a divergence-free field as the~curl~of a vector potential comes about because of homological considerations. With this in mind, we study gauge freedom in magnetostatics and its effect on the comparison between magnetic configurations through key quantities such as the magnetic helicity. For this, we apply the Hodge decomposition of k-forms on compact orientable Riemaniann manifolds with smooth boundary, as well as de Rham cohomology, to the representation of magnetic fields through potential one-forms in toroidal volumes. An advantage of the homological approach is the recovery of classical results without explicit coordinates and assumptions about the fields on the exterior of the domain. In particular, a detailed construction of minimal gauges and a formal proof of relative helicity formulas are presented.},

keywords = {algebraic topology, boundary value problems, cohomology, differential geometry, exterior calculus, gauge freedom, helicity, Hodge decomposition, magnetostatics},

pubstate = {published},

tppubtype = {article}

}

@article{lazersonModelingMeasurementEnergetic2021,

title = {Modeling and Measurement of Energetic Particle Slowing down in Wendelstein 7-X},

author = {Samuel A. Lazerson and David Pfefferl\'{e} and Michael Drevlak and Hr{a}kan Smith and Joachim Geiger and Simppa \"{A}k\"{a}slompolo and Pavlos Xanthopoulos and Andreas Dinklage and Oliver Ford and Paul McNeely and Norbert Rust and Sergey Bozhenkov and Dirk Hartmann and Kian Rahbarnia and Tamara Andreeva and Jonathan Schilling and Christian Brandt and Ulrich Neuner and Henning Thomsen and Robert C. Wolf and The W7-X. Team},

doi = {10.1088/1741-4326/ac0771},

issn = {0029-5515},

year  = {2021},

date = {2021-07-01},

urldate = {2021-07-01},

volume = {61},

number = {9},

pages = {096005},

publisher = {IOP Publishing},

abstract = {The energetic particle slowing down model in the BEAMS3D stellarator neutral beam code is compared to analytic models and experimental data from the Wendelstein 7-X experiment (W7-X). Recently, the first neutral beam experiments were performed in W7-X, providing validation of neutral beam deposition codes (Lazerson S.A. et al 2020 Nucl. Fusion 60 076020). This work builds upon that work, and follows the gyro-center orbits of the neutral-beam-generated fast ions to the plasma boundary. Slowing down times based on measurements of diamagnetic energy changes are compared to simulation data. A discharge solely heated by neutral beam injection is used to compare neoclassical heat flux estimates to neutral beam fueling, heating, and current drive. Experimental estimates of electron heat diffusivity suggest that electron turbulence is destabilized by density peaking in the discharge. Neutral beam current drive dominates over bootstrap current, resulting in a reversal of the toroidal current, as seen experimentally. Particle losses and heat flux through the equilibrium boundary are described. The effects of the magnetic configuration and plasma density on such parameters are also assessed. Benchmarking based on analytic estimates and other energetic particle codes is presented.},

keywords = {benchmark, fast particles, numerical method, slowing-down distribution, stellarator, VENUS-LEVISLEVIS},

pubstate = {published},

tppubtype = {article}

}

@article{hennebergCombinedPlasmaCoil2021,

title = {Combined Plasma\textendash Coil Optimization Algorithms},

author = {S. A. Henneberg and S. R. Hudson and D. Pfefferl\'{e} and P. Helander},

doi = {10.1017/S0022377821000271},

issn = {0022-3778, 1469-7807},

year  = {2021},

date = {2021-04-01},

urldate = {2021-04-01},

journal = {Journal of Plasma Physics},

volume = {87},

number = {2},

publisher = {Cambridge University Press},

abstract = {Combined plasma\textendash coil optimization approaches for designing stellarators are discussed and a new method for calculating free-boundary equilibria for multiregion relaxed magnetohydrodynmics (MRxMHD) is proposed. Four distinct categories of stellarator optimization, two of which are novel approaches, are the fixed-boundary optimization, the generalized fixed-boundary optimization, the quasi-free-boundary optimization, and the free-boundary (coil) optimization. These are described using the MRxMHD energy functional, the Biot\textendash Savart integral, the coil-penalty functional and the virtual casing integral and their derivatives. The proposed free-boundary equilibrium calculation differs from existing methods in how the boundary-value problem is posed, and for the new approach it seems that there is not an associated energy minimization principle because a non-symmetric functional arises. We propose to solve the weak formulation of this problem using a spectral-Galerkin method, and this will reduce the free-boundary equilibrium calculation to something comparable to a fixed-boundary calculation. In our discussion of combined plasma\textendash coil optimization algorithms, we emphasize the importance of the stability matrix.},

keywords = {coil design, optimisation, SPEC, variational calculus},

pubstate = {published},

tppubtype = {article}

}

@article{hirvijokiLongevityPowerDensity2021,

title = {Longevity and Power Density of Intermediate-to-Deep Geothermal Wells in District Heating Applications},

author = {Eero Hirvijoki and David Pfefferl\'{e} and Manasvi Lingam},

doi = {10.1140/epjp/s13360-021-01094-8},

issn = {2190-5444},

year  = {2021},

date = {2021-01-01},

urldate = {2021-01-01},

journal = {The European Physical Journal Plus},

volume = {136},

number = {1},

pages = {137},

abstract = {This paper assesses the potential of intermediate-to-deep geothermal wells for district heating purposes in non-hot spot regions as a means for replacing carbon-intensive heat sources. In analysing the problem of heat transfer from the bedrock to a flowing coolant in the well, we perform parameter scans to assess the longevity and power density of different-size wells and derive analytical estimates to explain salient characteristics of the well behaviour. The results are then utilized to illustrate how intermediate-to-deep geothermal wells would compare with the requirements of typical large-scale district heating systems, by using the city of Helsinki in Finland as an example.},

keywords = {geothermal wells, heat equation, renewable energy},

pubstate = {published},

tppubtype = {article}

}

@article{pfefferleWhittleMaximumLikelihood2020,

title = {Whittle Maximum Likelihood Estimate of Spectral Properties of Rayleigh-Taylor Interfacial Mixing Using Hot-Wire Anemometry Experimental Data},

author = {David Pfefferl\'{e} and Snezhana I. Abarzhi},

doi = {10.1103/PhysRevE.102.053107},

year  = {2020},

date = {2020-11-01},

urldate = {2020-11-01},

journal = {Physical Review E},

volume = {102},

number = {5},

pages = {053107},

publisher = {American Physical Society},

abstract = {Investigating the power density spectrum of fluctuations in Rayleigh-Taylor (RT) interfacial mixing is a means of studying characteristic length, timescales, anisotropies, and anomalous processes. Guided by group theory, analyzing the invariance-based properties of the fluctuations, our paper examines raw time series from hot-wire anemometry measurements in the experiment by Akula et al. [J. Fluid Mech. 816, 619 (2017)]. The results suggest that the power density spectrum can be modeled as a compound function presented as the product of a power law and an exponential. The data analysis is based on Whittle's approximation of the power density spectrum for independent zero-mean near-Gaussian signals to construct a maximum likelihood estimator of the parameters. Those that maximize the log-likelihood are computed numerically through Newton-Raphson iteration. The Hessian of the log-likelihood is used to evaluate the Fisher information matrix and provide an estimate of the statistical error on the obtained parameters. The Kolmogorov-Smirnov test is applied to analyze the goodness of fit, by verifying the hypothesis that the ratio between the observed periodogram and the estimated power density spectrum follows a $chi$2 probability distribution. The dependence of the parameters of the compound function is investigated on the range of mode numbers over which the fit is performed. In the domain where the relative errors of the power-law exponent and the exponential decay rate are small and the goodness of fit is excellent, the parameters of the compound function are clearly defined, in agreement with the theory developed in the paper. The study of the power-law spectra in RT mixing data suggests that rigorous physics-based statistical methods can help researchers to see beyond visual inspection.},

keywords = {data analysis, data fitting, exponential decay, hypothesis testing, maximum likelihood, parameter estimation, power-law, rayleigh-taylor, spectrum},

pubstate = {published},

tppubtype = {article}

}

@article{quCoordinateParameterisationSpectral2020,

title = {Coordinate Parameterisation and Spectral Method Optimisation for Beltrami Field Solver in Stellarator Geometry},

author = {Z. S. Qu and D. Pfefferl\'{e} and S. R. Hudson and A. Baillod and A. Kumar and R. L. Dewar and M. J. Hole},

doi = {10.1088/1361-6587/abc08e},

issn = {0741-3335},

year  = {2020},

date = {2020-11-01},

urldate = {2020-11-01},

volume = {62},

number = {12},

pages = {124004},

publisher = {IOP Publishing},

abstract = {The numerical solution of the stepped pressure equilibrium (Hudson et al 2012 Phys. Plasmas 19 112502) requires a fast and robust solver to obtain the Beltrami field in three-dimensional geometry such as stellarators. The spectral method implemented in the stepped pressure equilibrium code (SPEC) is efficient when the domain is a hollow torus, but ill-conditioning of the discretised linear equations occurs in the (solid) toroid due to the artificially singular coordinate parameterisation near the axis. In this work, we propose an improved choice for the reference axis to prevent coordinates surfaces from overlapping. Then, we examine the parity and asymptotics of the magnetic vector potential near the axis and suggest the use of recombined and rescaled Zernike radial basis functions. The maximum relative error in the magnetic field of the Wendelstein 7-X geometry is shown to reach 10-9 at high resolution in a series of convergence tests and benchmarks against the boundary integral equation solver for Taylor states. The new method is also reported to significantly improve the accuracy of multi-volume SPEC calculations. A comparison between free-boundary SPEC and the analytical Dommaschk potential is presented with higher-than-usual Fourier resolution. It is illustrated that we are able to resolve low amplitude current sheets when an interface is placed where there is no flux surface in the analytic solution. This was previously concealed because of insufficient numerical resolution.},

keywords = {curvilinear coordinates, MHD equilibrium, numerical method, SPEC},

pubstate = {published},

tppubtype = {article}

}

@article{pfefferle-rigidityb,

title = {Rigidity of MHD equilibria to smooth incompressible ideal motion near resonant surfaces},

author = {David Pfefferl\'{e} and Lyle Noakes and Yao Zhou},

url = {https://doi.org/10.1088%2F1361-6587%2Fab8ca3},

doi = {10.1088/1361-6587/ab8ca3},

year  = {2020},

date = {2020-06-01},

journal = {Plasma Physics and Controlled Fusion},

volume = {62},

number = {7},

pages = {074004},

publisher = {IOP Publishing},

abstract = {In ideal MHD, the magnetic flux is advected by the plasma motion, freezing flux-surfaces into the flow. An MHD equilibrium is reached when the flow relaxes and force balance is achieved. We ask what classes of MHD equilibria can be accessed from a given initial state via smooth incompressible ideal motion. It is found that certain boundary displacements are formally not supported. This follows from yet another investigation of the Hahm\textendashKulsrud\textendashTaylor (HKT) problem, which highlights the resonant behaviour near a rational layer formed by a set of degenerate critical points in the flux-function. When trying to retain the mirror symmetry of the flux-function with respect to the resonant layer, the vector field that generates the volume-preserving diffeomorphism vanishes at the identity to all order in the time-like path parameter.},

keywords = {Hamiltonian, MHD, MHD equilibrium, perturbation theory, resonant surfaces},

pubstate = {published},

tppubtype = {article}

}

@article{lazerson-2020,

title = {Validation of the BEAMS3D neutral beam deposition model on Wendelstein 7-X},

author = {Samuel A Lazerson and Oliver P Ford and Carolin Nuehrenberg and Simppa \"{A}k\"{a}slompolo and Peter Zs. Poloskei and Mike Machielsen and Paul McNeely and Lilla Van\'{o} and Norbert Rust and Sergey Bozhenkov and Tristan W C Neelis and Jonathan P Graves and David Pfefferl\'{e} and Annabelle Spanier and Dirk Hartmann and Nikolai Marushchenko and Yuriy Turkin and Matthias Hirsch and Neha Chaudhary and Udo Hoefel and Torsten Stange and Gavin Weir and Novimir Pablant and Andreas Langenberg and Peter Traverso and Pranay Valson and Jens Knauer and Kai Jakob Brunner and Ekkehard Pasch and Marc Beurskens and Hannes Damm and Golo Fuchert and Philipp Nelde and Evan Scott and Uwe Hergenhahn and Andrea Pavone and Kian Rahbarnia and Tamara Andreeva and Jonathan Schilling and Christian Brandt and Ulrich Neuner and Henning Thomsen and Marcin Jakubowski and Adnan Ali and Yu Gao and Holger Niemann and Aleix Puig Sitjes and Ralf Koenig and Robert Wolf C and},

url = {https://doi.org/10.1088%2F1741-4326%2Fab8e61},

doi = {10.1088/1741-4326/ab8e61},

year  = {2020},

date = {2020-06-01},

journal = {Nuclear Fusion},

volume = {60},

number = {7},

pages = {076020},

publisher = {IOP Publishing},

abstract = {The neutral beam deposition model in the BEAMS3D code is validated against neutral beam attenuation data from Wendelstein 7-X (W7-X). A set of experimental discharges where the neutral beam injection system of W7-X was utilized were reconstructed. These discharges scanned the magnetic configurations and plasma densities of W7-X. The equilibrium reconstructions were performed using STELLOPT which calculates three-dimensional self-consistent ideal magnetohydrodynamic equilibria and kinetic profiles. These reconstructions leveraged new capabilities to incorporate electron cyclotron emission and x-ray imaging diagnostics in the STELLOPT code. The reconstructed equilibria and profiles served as inputs for BEAMS3D calculations of neutral beam deposition in W7-X. It is found that if reconstructed kinetic profiles are utilized, good agreement between measured and simulated beam attenuation is found. As deposition models provide initial conditions for fast-ion slowing down calculations, this work provides a first step towards validating our ability to predict fast ion confinement in stellarators.},

keywords = {fast particles, full-orbit, guiding-centre, neoclassical transport, stellarator},

pubstate = {published},

tppubtype = {article}

}

@article{hirvijoki-2020,

title = {Energy and momentum conservation in the Euler\textendashPoincar\'{e} formulation of local Vlasov\textendashMaxwell-type systems},

author = {Eero Hirvijoki and Joshua W Burby and David Pfefferl\'{e} and Alain J Brizard},

url = {https://doi.org/10.1088%2F1751-8121%2Fab8b38},

doi = {10.1088/1751-8121/ab8b38},

year  = {2020},

date = {2020-05-01},

journal = {Journal of Physics A: Mathematical and Theoretical},

volume = {53},

number = {23},

pages = {235204},

publisher = {IOP Publishing},

abstract = {The action principle by Low (1958 Proc. R. Soc. Lond. A 248 282\textendash7) for the classic Vlasov\textendashMaxwell system contains a mix of Eulerian and Lagrangian variables. This renders the Noether analysis of reparametrization symmetries inconvenient, especially since the well-known energy- and momentum-conservation laws for the system are expressed in terms of Eulerian variables only. While an Euler\textendashPoincar\'{e} formulation of Vlasov\textendashMaxwell-type systems, effectively starting with Low’s action and using constrained variations for the Eulerian description of particle motion, has been known for a while Cendra et al (1998 J. Math. Phys. 39 3138\textendash57), it is hard to come by a documented derivation of the related energy- and momentum-conservation laws in the spirit of the Euler\textendashPoincar\'{e} machinery. To our knowledge only one such derivation exists in the literature so far, dealing with the so-called guiding-center Vlasov\textendashDarwin system Sugama et al (2018 Phys. Plasmas 25 102506). The present exposition discusses a generic class of local Vlasov\textendashMaxwell-type systems, with a conscious choice of adopting the language of differential geometry to exploit the Euler\textendashPoincar\'{e} framework to its full extent. After reviewing the transition from a Lagrangian picture to an Eulerian one, we demonstrate how symmetries generated by isometries in space lead to conservation laws for linear- and angular-momentum density and how symmetry by time translation produces a conservation law for energy density. We also discuss what happens if no symmetries exist. Finally, two explicit examples will be given\textemdashthe classic Vlasov\textendashMaxwell and the drift-kinetic Vlasov\textendashMaxwell\textemdashand the results expressed in the language of regular vector calculus for familiarity.},

keywords = {drift-kinetic, euler-poincar\'{e}, Hamiltonian, vlasov-maxwell},

pubstate = {published},

tppubtype = {article}

}

@article{PhysRevLett.124.155001,

title = {Identification of an Optimized Heating and Fast Ion Generation Scheme for the Wendelstein 7-X Stellarator},

author = {H W Patten and J P Graves and W A Cooper and J Eriksson and D Pfefferl\'{e}},

url = {https://link.aps.org/doi/10.1103/PhysRevLett.124.155001},

doi = {10.1103/PhysRevLett.124.155001},

year  = {2020},

date = {2020-04-01},

journal = {Phys. Rev. Lett.},

volume = {124},

pages = {155001},

publisher = {American Physical Society},

abstract = {A Doppler shifted resonance minority species ion cyclotron range of frequency (ICRF) scheme for heating neutral beam ions has been identified and optimized for the Wendelstein 7-X stellarator. Compared with more conventional methods, the synergetic scheme increases the normalized core collisional power transfer to the background plasma, and induces larger concentrations of energetic ions. Simulations in the intricate 3D magnetic stellarator geometry reveal an energetic distribution function that is only weakly anisotropic, and is thus relevant to fast ion and alpha particle driven Alfv\'{e}n eigenmode experimental preparation. Quasilinear theory and simulations of the Joint European Torus indicate that the excellent confinement properties are due to increased velocity diffusion from ICRF interaction along the magnetic field lines. Agreement is found between SCENIC simulations and Joint European Torus experimental measurements for the total neutron rate and the energy distribution of the fast ions.},

keywords = {fast particles, ion cyclotron resonance, neutral beam injection, stellarator},

pubstate = {published},

tppubtype = {article}

}

@article{kaye-2019,

title = {NSTX/NSTX-U theory, modeling and analysis results},

author = {S M Kaye and D J Battaglia and D Baver and E Belova and J W Berkery and V N Duarte and N Ferraro and E Fredrickson and N Gorelenkov and W Guttenfelder and G Z Hao and W Heidbrink and O Izacard and D Kim and I Krebs and La R Haye and J Lestz and D Liu and L A Morton and J Myra and D Pfefferle and M Podesta and Y Ren and J Riquezes and S A Sabbagh and M Schneller and F Scotti and V Soukhanovskii and S J Zweben and J W Ahn and J P Allain and R Barchfeld and F Bedoya and R E Bell and N Bertelli and A Bhattacharjee and M D Boyer and D Brennan and G Canal and J Canik and N Crocker and D Darrow and L Delgado-Aparicio and A Diallo and C Domier and F Ebrahimi and T Evans and R Fonck and H Frerichs and K Gan and S Gerhardt and T Gray and T Jarboe and S Jardin and M A Jaworski and R Kaita and B Koel and E Kolemen and D M Kriete and S Kubota and B P LeBlanc and F Levinton and N Luhmann and R Lunsford and R Maingi and R Maqueda and J E Menard and D Mueller and C E Myers and M Ono and J -K Park and R Perkins and F Poli and R Raman and M Reinke and T Rhodes and C Rowley and D Russell and E Schuster and O Schmitz and Y Sechrest and C H Skinner and D R Smith and T Stotzfus-Dueck and B Stratton and G Taylor and K Tritz and W Wang and Z Wang and I Waters and B Wirth},

url = {https://doi.org/10.1088%2F1741-4326%2Fab023a},

doi = {10.1088/1741-4326/ab023a},

year  = {2019},

date = {2019-06-01},

journal = {Nuclear Fusion},

volume = {59},

number = {11},

pages = {112007},

publisher = {IOP Publishing},

abstract = {The mission of the spherical tokamak NSTX-U is to explore the physics that drives core and pedestal transport and stability at high- and low collisionality, as part of the development of the spherical tokamak (ST) concept towards a compact, low-cost ST-based pilot plant. NSTX-U will ultimately operate at up to 2 MA and 1 T with up to 12 MW of neutral beam injection power for 5 s. NSTX-U will operate in a regime where electromagnetic instabilities are expected to dominate transport, and beam-heated NSTX-U plasmas will explore a portion of energetic particle parameter space that is relevant for both -heated conventional and low aspect ratio burning plasmas. NSTX-U will also develop the physics understanding and control tools to ramp-up and sustain high performance plasmas in a fully-noninductive fashion. NSTX-U began research operations in 2016, but a failure of a divertor magnetic field coil after ten weeks of operation resulted in the suspension of operations and initiation of recovery activities. During this period, there has been considerable work in the area of analysis, theory and modeling of data from both NSTX and NSTX-U, with a goal of understanding the underlying physics to develop predictive models that can be used for high-confidence projections for both ST and higher aspect ratio regimes. These studies have addressed issues in thermal plasma transport, macrostability, energetic particlet-driven instabilities at ion-cyclotron frequencies and below, and edge and divertor physics.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{lanthaler-2019,

title = {Guiding-centre theory for kinetic-magnetohydrodynamic modes in strongly flowing plasmas},

author = {S Lanthaler and J P Graves and D Pfefferl\'{e} and W A Cooper},

url = {https://doi.org/10.1088%2F1361-6587%2Fab1d21},

doi = {10.1088/1361-6587/ab1d21},

year  = {2019},

date = {2019-05-01},

journal = {Plasma Physics and Controlled Fusion},

volume = {61},

number = {7},

pages = {074006},

publisher = {IOP Publishing},

abstract = {A kinetic-magnetohydrodynamic model with kinetic pressure closure is derived from a consistent guiding-centre framework. Higher-order (gyroviscous) corrections to the pressure tensor are derived in complex geometry from a reduced kinetic equation. The proposed model allows for flows of the order of the thermal ion velocity, taking into account important centrifugal effects due to the ExB flow, as well as the effects of diamagnetic flows associated with finite Larmor radius corrections to both ion fluid inertia and long mean-free path contributions. Wave\textendashparticle interactions, such as toroidal drift-resonance, are retained. Furthermore, the linearised model includes a quasi-neutrality equation, allowing the effects of a parallel electric field to be studied in fast rotating tokamak plasmas.},

keywords = {drift-kinetic, guiding-centre, MHD, plasma flow},

pubstate = {published},

tppubtype = {article}

}

@article{hudson-2018,

title = {Differentiating the shape of stellarator coils with respect to the plasma boundary},

author = {S R Hudson and C Zhu and D Pfefferl\'{e} and L Gunderson},

doi = {https://doi.org/10.1016/j.physleta.2018.07.016},

issn = {0375-9601},

year  = {2018},

date = {2018-09-29},

journal = {Physics Letters A},

volume = {382},

number = {38},

pages = {2732 - 2737},

abstract = {The task of designing the geometry of a set of current-carrying coils that produce the magnetic field required to confine a given plasma equilibrium in stellarators is expressed as a minimization principle, namely that the coils minimize a suitably defined error expressed as a surface integral, which is recognized as the quadratic-flux. A penalty on the coil length is included to avoid pathological solutions. A simple expression for how the quadratic-flux and coil length vary as the coil geometry varies is derived, and an expression describing how this varies with variations in the surface geometry is derived. These expressions allow efficient coil-design algorithms to be implemented, and also enable efficient algorithms for varying the shape of the plasma surface in order to simplify the coil geometry, and a numerical illustration of this is given.},

keywords = {coil design, MHD equilibrium, stellarator},

pubstate = {published},

tppubtype = {article}

}

@article{pfefferle-elastica,

title = {Non-planar elasticae as optimal curves for the magnetic axis of stellarators},

author = {D Pfefferl\'{e} and L Gunderson and S R Hudson and L Noakes},

doi = {10.1063/1.5040894},

year  = {2018},

date = {2018-09-26},

journal = {Physics of Plasmas},

volume = {25},

number = {9},

pages = {092508},

abstract = {The problem of finding an optimal curve for the target magnetic axis of a stellarator is addressed. Euler-Lagrange equations are derived for finite length three-dimensional curves that extremise their bending energy while yielding fixed integrated torsion. The obvious translational and rotational symmetries are exploited to express solutions in a preferred cylindrical coordinate system in terms of elliptic Jacobi functions. These solution curves, which, up to similarity transformations, depend on three dimensionless parameters, do not necessarily close. Two closure conditions are obtained for the vertical and toroidal displacement (the radial coordinate being trivially periodic) to yield a countably infinite set of one-parameter families of closed non-planar curves. The behaviour of the integrated torsion (Twist of the Frenet frame), the Linking of the Frenet frame, and the Writhe of the solution curves are studied in light of the C\u{a}lug\u{a}reanu theorem. A refreshed interpretation of Mercier's formula for the on-axis rotational transform of stellarator magnetic field-lines is proposed.},

keywords = {elastica, euler-lagrange equations, linking number, magnetic axis, non-planar curves, rotational transform, stellarator, twist, variational problem, writhe},

pubstate = {published},

tppubtype = {article}

}

@article{patten-2018,

title = {The effect of magnetic equilibrium on auxiliary heating schemes and fast particle confinement in Wendelstein 7-X},

author = {H Patten and J P Graves and J Faustin and W A Cooper and J Geiger and D Pfefferl\'{e} and Y Turkin},

doi = {10.1088/1361-6587/aac9ee},

year  = {2018},

date = {2018-06-01},

journal = {Plasma Physics and Controlled Fusion},

volume = {60},

number = {8},

pages = {085009},

publisher = {IOP Publishing},

abstract = {The performance of the auxiliary heating systems ion cyclotron resonance heating and neutral beam injection is calculated in three different magnetic mirror configurations foreseen to be used in future experiments in the Wendelstein 7-X stellarator: low, standard and high mirror. This numerical work is implemented with the SCENIC code package, which is designed to model three-dimensional magnetic equilibria whilst retaining effects such as anisotropy and the influence of including a finite orbit width of the particles. The ability to simulate NBI deposition in three-dimensional equilibria, the implementation of the realistic beam injector geometry, and the modification of the SCENIC package to permit the investigation of the 3-ion species heating scheme, are recent developments. Using these modifications, an assessment of the advantages and disadvantages of these two fast-ion producing auxiliary heating systems is made in the three different magnetic mirror equilibria. For NBI heating, the high mirror configuration displays the best global confinement properties, resulting in a larger collisional power transfer to the background plasma. The standard mirror has the best particle confinement in the core region, but the worst towards the edge of the plasma. The low mirror has the largest lost power and thus the lowest total collisional power. For ICRH, the displacement of the RF-resonant surface significantly impacts the heating performance. Due to the large toroidal magnetic mirror in the high mirror equilibrium, resonant particles easily become trapped and cannot remain in resonance, generating only small energetic particle populations. Despite this, global confinement is still the strongest in this equilibrium. The low mirror is the only equilibrium to produce peaked on-axis collisional power deposition, with associated peaked on-axis fast ion pressure profiles. A highly energetic particle population is then produced but this results in larger lost power as this equilibrium is not sufficiently optimised for fast ion confinement. A comparison between the two heating methods concludes that NBI produces a smaller fraction of lost to input power, and a reduced sensitivity of the performance to variations of the toroidal magnetic mirror. The main limit of NBI which does not apply to ICRH is the production of highly energetic particle populations, with predictions of energetic particles of E ~ 0.45 MeV.},

keywords = {fast particles, heating, MHD equilibrium, stellarator},

pubstate = {published},

tppubtype = {article}

}

@article{pfefferle-m3dc1,

title = {Modelling of NSTX hot vertical displacement events using M3D-C1},

author = {D Pfefferl\'{e} and N Ferraro and S C Jardin and I Krebs and A Bhattacharjee},

doi = {10.1063/1.5016348},

year  = {2018},

date = {2018-04-03},

journal = {Physics of Plasmas},

volume = {25},

number = {5},

pages = {056106},

abstract = {The main results of an intense vertical displacement event (VDE) modelling activity using the implicit 3D extended MHD code M3D-C1 are presented. A pair of nonlinear 3D simulations are performed using realistic transport coefficients based on the reconstruction of a so-called NSTX frozen VDE where the feedback control was purposely switched off to trigger a vertical instability. The vertical drift phase is solved assuming axisymmetry until the plasma contacts the first wall, at which point the intricate evolution of the plasma, decaying to large extent in force-balance with induced halo/wall currents, is carefully resolved via 3D nonlinear simulations. The faster 2D nonlinear runs allow to assess the sensitivity of the simulations to parameter changes. In the limit of perfectly conducting wall, the expected linear relation between vertical growth rate and wall resistivity is recovered. For intermediate wall resistivities, the halo region contributes to slowing the plasma down, and the characteristic VDE time depends on the choice of halo temperature. The evolution of the current quench and the onset of 3D halo/eddy currents are diagnosed in detail. The 3D simulations highlight a rich structure of toroidal modes, penetrating inwards from edge to core and cascading from high-n to low-n mode numbers. The break-up of flux-surfaces results in a progressive stochastisation of field-lines precipitating the thermalisation of the plasma with the wall. The plasma current then decays rapidly, inducing large currents in the halo region and the wall. Analysis of normal currents flowing in and out of the divertor plate reveals rich time-varying patterns.},

keywords = {disruption, M3D-C1, resistive MHD, vertical displacement event},

pubstate = {published},

tppubtype = {article}

}

@article{pfefferle-vdemagneto,

title = {Algebraic motion of vertically displacing plasmas},

author = {D Pfefferl\'{e} and A Bhattacharjee},

doi = {10.1063/1.5011176},

year  = {2018},

date = {2018-01-27},

journal = {Physics of Plasmas},

volume = {25},

number = {2},

pages = {022516},

abstract = {The vertical motion of a tokamak plasma is analytically modelled during its non-linear phase by a free-moving current-carrying rod inductively coupled to a set of fixed conducting wires or a cylindrical conducting shell. The solutions capture the leading term in a Taylor expansion of the Green's function for the interaction between the plasma column and the surrounding vacuum vessel. The plasma shape and profiles are assumed not to vary during the vertical drifting phase such that the plasma column behaves as a rigid body. In the limit of perfectly conducting structures, the plasma is prevented to come in contact with the wall due to steep effective potential barriers created by the induced Eddy currents. Resistivity in the wall allows the equilibrium point to drift towards the vessel on the slow timescale of flux penetration. The initial exponential motion of the plasma, understood as a resistive vertical instability, is succeeded by a non-linear “sinking” behaviour shown to be algebraic and decelerating. The acceleration of the plasma column often observed in experiments is thus concluded to originate from an early sharing of toroidal current between the core, the halo plasma, and the wall or from the thermal quench dynamics precipitating loss of plasma current.},

keywords = {eddy current, inductance, vertical displacement event},

pubstate = {published},

tppubtype = {article}

}

@article{raghunathan-2017,

title = {Heavy impurity confinement in hybrid operation scenario plasmas with a rotating 1/1 continuous mode},

author = {M Raghunathan and J P Graves and T Nicolas and W A Cooper and X Garbet and D Pfefferl\'{e}},

url = {https://iopscience.iop.org/article/10.1088/1361-6587/aa896f},

doi = {10.1088/1361-6587/aa896f},

year  = {2017},

date = {2017-10-09},

journal = {Plasma Physics and Controlled Fusion},

volume = {59},

number = {12},

pages = {124002},

abstract = {In future tokamaks like ITER with tungsten walls, it is imperative to control tungsten accumulation in the core of operational plasmas, especially since tungsten accumulation can lead to radiative collapse and disruption. We investigate the behavior of tungsten trace impurities in a JET-like hybrid scenario with both axisymmetric and saturated 1/1 ideal helical core in the presence of strong plasma rotation. For this purpose, we obtain the equilibria from VMEC and use VENUS-LEVIS, a guiding-center orbit-following code, to follow heavy impurity particles. In this work, VENUS-LEVIS has been modified to account for strong plasma flows with associated neoclassical effects arising from such flows. We find that the combination of helical core and plasma rotation augments the standard neoclassical inward pinch compared to axisymmetry, and leads to a strong inward pinch of impurities towards the magnetic axis despite the strong outward diffusion provided by the centrifugal force, as frequently observed in experiments.},

keywords = {fast particles, impurities, neoclassical transport, plasma flow, VENUS-LEVIS},

pubstate = {published},

tppubtype = {article}

}

@article{kazakov2017,

title = {Efficient generation of energetic ions in multi-ion plasmas by radio-frequency heating},

author = {Ye O Kazakov and al.},

url = {http://dx.doi.org/10.1038/nphys4167},

doi = {10.1038/nphys4167},

year  = {2017},

date = {2017-06-19},

journal = {Nature Physics},

volume = {13},

abstract = {We describe a new technique for the efficient generation of high-energy ions with electromagnetic ion cyclotron waves in multi-ion plasmas. The discussed ‘three-ion’ scenarios are especially suited for strong wave absorption by a very low number of resonant ions. To observe this effect, the plasma composition has to be properly adjusted, as prescribed by theory. We demonstrate the potential of the method on the world-largest plasma magnetic confinement device, JET (Joint European Torus, Culham, UK), and the high-magnetic-field tokamak Alcator C-Mod (Cambridge, USA). The obtained results demonstrate efficient acceleration of 3He ions to high energies in dedicated hydrogen\textendashdeuterium mixtures. Simultaneously, effective plasma heating is observed, as a result of the slowing-down of the fast 3He ions. The developed technique is not only limited to laboratory plasmas, but can also be applied to explain observations of energetic ions in space-plasma environments, in particular, 3He-rich solar flares.},

keywords = {fast particles, heating, ion cyclotron resonance},

pubstate = {published},

tppubtype = {article}

}

@article{litaudon-2017,

title = {Overview of the JET results in support to ITER},

author = {X Litaudon and al.},

url = {https://iopscience.iop.org/article/10.1088/1741-4326/aa5e28},

doi = {10.1088/1741-4326/aa5e28},

year  = {2017},

date = {2017-06-15},

journal = {Nuclear Fusion},

volume = {57},

number = {10},

pages = {102001},

abstract = {The 2014\textendash2016 JET results are reviewed in the light of their significance for optimising the ITER research plan for the active and non-active operation. More than 60 h of plasma operation with ITER first wall materials successfully took place since its installation in 2011. New multi-machine scaling of the type I-ELM divertor energy flux density to ITER is supported by first principle modelling. ITER relevant disruption experiments and first principle modelling are reported with a set of three disruption mitigation valves mimicking the ITER setup. Insights of the L\textendashH power threshold in Deuterium and Hydrogen are given, stressing the importance of the magnetic configurations and the recent measurements of fine-scale structures in the edge radial electric. Dimensionless scans of the core and pedestal confinement provide new information to elucidate the importance of the first wall material on the fusion performance. H-mode plasmas at ITER triangularity (H  =  1 at β N ~ 1.8 and n/n GW ~ 0.6) have been sustained at 2 MA during 5 s. The ITER neutronics codes have been validated on high performance experiments. Prospects for the coming D\textendashT campaign and 14 MeV neutron calibration strategy are reviewed.},

keywords = {fast particles, heating, VENUS-LEVIS},

pubstate = {published},

tppubtype = {article}

}

@article{faustin-2017,

title = {Modelling of advanced three-ion ICRF heating and fast ion generation scheme for tokamaks and stellarators},

author = {J M Faustin and J P Graves and W A Cooper and S Lanthaler and L Villard and D Pfefferl\'{e} and J Geiger and Ye O Kazakov and Van D Eester},

doi = {10.1088/1361-6587/aa72a4},

year  = {2017},

date = {2017-06-13},

journal = {Plasma Physics and Controlled Fusion},

volume = {59},

number = {8},

pages = {084001},

abstract = {Absorption of ion-cyclotron range of frequencies waves at the fundamental resonance is an efficient source of plasma heating and fast ion generation in tokamaks and stellarators. This heating method is planned to be exploited as a fast ion source in the Wendelstein 7-X stellarator. The work presented here assesses the possibility of using the newly developed three-ion species scheme (Kazakov et al (2015) Nucl. Fusion 55 032001) in tokamak and stellarator plasmas, which could offer the capability of generating more energetic ions than the traditional minority heating scheme with moderate input power. Using the SCENIC code, it is found that fast ions in the MeV range of energy can be produced in JET-like plasmas. The RF-induced particle pinch is seen to strongly impact the fast ion pressure profile in particular. Our results show that in typical high-density W7-X plasmas, the three-ion species scheme generates more energetic ions than the more traditional minority heating scheme, which makes three-ion scenario promising for fast-ion confinement studies in W7-X.},

keywords = {fast particles, heating, ion cyclotron resonance, neoclassical transport, stellarator},

pubstate = {published},

tppubtype = {article}

}

@article{pfefferle-exact,

title = {Exact collisional moments for plasma fluid theories},

author = {D Pfefferl\'{e} and E Hirvijoki and M Lingam},

doi = {10.1063/1.4979992},

year  = {2017},

date = {2017-04-20},

journal = {Physics of Plasmas},

volume = {24},

number = {4},

pages = {042118},

abstract = {The velocity-space moments of the often troublesome nonlinear Landau collision operator are expressed exactly in terms of multi-index Hermite-polynomial moments of distribution functions. The collisional moments are shown to be generated by derivatives of two well-known functions, namely, the Rosenbluth-MacDonald-Judd-Trubnikov potentials for a Gaussian distribution. The resulting formula has a nonlinear dependency on the relative mean flow of the colliding species normalised to the root-mean-square of the corresponding thermal velocities and a bilinear dependency on densities and higher-order velocity moments of the distribution functions, with no restriction on temperature, flow, or mass ratio of the species. The result can be applied to both the classic transport theory of plasmas that relies on the Chapman-Enskog method, as well as to derive collisional fluid equations that follow Grad's moment approach. As an illustrative example, we provide the collisional ten-moment equations with exact conservation laws for momentum- and energy-transfer rates.},

keywords = {chapman-enskog, collisional moments, collisions, fluid theory, hermite polynomials, non-linear resistivity},

pubstate = {published},

tppubtype = {article}

}

@article{lingam-nonlinear,

title = {Nonlinear resistivity for magnetohydrodynamical models},

author = {M Lingam and E Hirvijoki and D Pfefferl\'{e} and L Comisso and A Bhattacharjee},

doi = {10.1063/1.4980838},

year  = {2017},

date = {2017-04-20},

journal = {Physics of Plasmas},

volume = {24},

number = {4},

pages = {042120},

abstract = {A new formulation of the plasma resistivity that stems from the collisional momentum-transfer rate between electrons and ions is presented. The resistivity computed herein is shown to depend not only on the temperature and density but also on all other polynomial velocity-space moments of the distribution function, such as the pressure tensor and heat flux vector. The full expression for the collisional momentum-transfer rate is determined and is used to formulate the nonlinear anisotropic resistivity. The new formalism recovers the Spitzer resistivity, as well as the concept of thermal force if the heat flux is assumed to be proportional to a temperature gradient. Furthermore, if the pressure tensor is related to viscous stress, the latter enters the expression for the resistivity. The relative importance of the nonlinear term(s) with respect to the well-established electron inertia and Hall terms is also examined. The subtle implications of the nonlinear resistivity, and its dependence on the fluid variables, are discussed in the context of magnetized plasma environments and phenomena such as magnetic reconnection.},

keywords = {collisional moments, collisions, fluid theory, hermite polynomials, non-linear resistivity},

pubstate = {published},

tppubtype = {article}

}

@article{lanthaler-2017,

title = {Higher order Larmor radius corrections to guiding-centre equations and application to fast ion equilibrium distributions},

author = {S Lanthaler and D Pfefferl\'{e} and J P Graves and W A Cooper},

url = {https://iopscience.iop.org/article/10.1088/1361-6587/aa5e70},

doi = {10.1088/1361-6587/aa5e70},

year  = {2017},

date = {2017-03-15},

journal = {Plasma Physics and Controlled Fusion},

volume = {59},

number = {4},

pages = {044014},

abstract = {An improved set of guiding-centre equations, expanded to one order higher in Larmor radius than usually written for guiding-centre codes, are derived for curvilinear flux coordinates and implemented into the orbit following code VENUS-LEVIS. Aside from greatly improving the correspondence between guiding-centre and full particle trajectories, the most important effect of the additional Larmor radius corrections is to modify the definition of the guiding-centre's parallel velocity via the so-called Ba\~{n}os drift. The correct treatment of the guiding-centre push-forward with the Ba\~{n}os term leads to an anisotropic shift in the phase-space distribution of guiding-centres, consistent with the well-known magnetization term. The consequence of these higher order terms are quantified in three cases where energetic ions are usually followed with standard guiding-centre equations: (1) neutral beam injection in a MAST-like low aspect-ratio spherical equilibrium where the fast ion driven current is significantly larger with respect to previous calculations, (2) fast ion losses due to resonant magnetic perturbations where a lower lost fraction and a better confinement is confirmed, (3) alpha particles in the ripple field of the European DEMO where the effect is found to be marginal.},

keywords = {drift-kinetic, guiding-centre, MHD equilibrium, perturbation theory, VENUS-LEVIS},

pubstate = {published},

tppubtype = {article}

}

@article{wenninger-2017,

title = {The DEMO wall load challenge},

author = {R Wenninger and al.},

url = {https://iopscience.iop.org/article/10.1088/1741-4326/aa4fb4},

doi = {10.1088/1741-4326/aa4fb4},

year  = {2017},

date = {2017-02-09},

journal = {Nuclear Fusion},

volume = {57},

number = {4},

pages = {046002},

abstract = {For several reasons the challenge to keep the loads to the first wall within engineering limits is substantially higher in DEMO compared to ITER. Therefore the pre-conceptual design development for DEMO that is currently ongoing in Europe needs to be based on load estimates that are derived employing the most recent plasma edge physics knowledge.



An initial assessment of the static wall heat load limit in DEMO infers that the steady state peak heat flux limit on the majority of the DEMO first wall should not be assumed to be higher than 1.0 MW m−2. This compares to an average wall heat load of 0.29 MW m−2 for the design EU-DEMO1 2015 assuming a perfect homogeneous distribution. The main part of this publication concentrates on the development of first DEMO estimates for charged particle, radiation, fast particle (all static) and disruption heat loads. Employing an initial engineering wall design with clear optimization potential in combination with parameters for the flat-top phase (x-point configuration), loads up to 7 MW m−2 (penalty factor for tolerances etc not applied) have been calculated. Assuming a fraction of power radiated from the x-point region between 1/5 and 1/3, peaks of the total power flux density due to radiation of 0.6\textendash0.8 MW m−2 are found in the outer baffle region.



This first review of wall loads, and the associated limits in DEMO clearly underlines a significant challenge that necessitates substantial engineering efforts as well as a considerable consolidation of the associated physics basis.},

keywords = {confinement, DEMO, fast particles, magnetic ripple, MHD equilibrium, neoclassical transport, perturbation theory, VENUS-LEVIS, wall load},

pubstate = {published},

tppubtype = {article}

}

@article{hirvijoki-2017,

title = {Differential formulation of the gyrokinetic Landau operator},

author = {E Hirvijoki and A J Brizard and D Pfefferl\'{e}},

editor = {

},

doi = {10.1017/S0022377816001203},

year  = {2017},

date = {2017-01-05},

journal = {Journal of Plasma Physics},

volume = {83},

number = {1},

publisher = {Cambridge University Press},

address = {Cambridge, UK},

abstract = {Subsequent to the recent rigorous derivation of an energetically consistent gyrokinetic collision operator in the so-called Landau representation, this paper investigates the possibility of finding a differential formulation of the gyrokinetic Landau collision operator. It is observed that, while a differential formulation is possible in the gyrokinetic phase space, reduction of the resulting system of partial differential equations to five dimensions via gyroaveraging poses a challenge. Based on the present work, it is likely that the gyrocentre analogues of the Rosenbluth\textendashMacDonald\textendashJudd potential functions must be kept gyroangle dependent.},

keywords = {collisions, gyrokinetics, rosenbluth potential},

pubstate = {published},

tppubtype = {article}

}

@article{hirvijoki-2016,

title = {Fluid moments of the nonlinear Landau collision operator},

author = {E Hirvijoki and M Lingam and D Pfefferl\'{e}},

url = {http://dx.doi.org/10.1063/1.4960669},

doi = {10.1063/1.4960669},

year  = {2016},

date = {2016-08-09},

journal = {Physics of Plasmas},

volume = {23},

number = {8},

pages = {080701},

abstract = {An important problem in plasma physics is the lack of an accurate and complete description of Coulomb collisions in associated fluid models. To shed light on the problem, this Letter introduces an integral identity involving the multivariate Hermite tensor polynomials and presents a method for computing exact expressions for the fluid moments of the nonlinear Landau collision operator. The proposed methodology provides a systematic and rigorous means of extending the validity of fluid models that have an underlying inverse-square force particle dynamics to arbitrary collisionality and flow.},

keywords = {collisional moments, collisions, fluid theory, hermite polynomials},

pubstate = {published},

tppubtype = {article}

}

@article{faustin-2016a,

title = {Fast particle loss channels in Wendelstein 7-X},

author = {J M Faustin and W A Cooper and J P Graves and D Pfefferl\'{e} and J Geiger},

url = {https://iopscience.iop.org/article/10.1088/0029-5515/56/9/092006},

doi = {10.1088/0029-5515/56/9/092006},

year  = {2016},

date = {2016-07-29},

journal = {Nuclear Fusion},

volume = {56},

number = {9},

pages = {092006},

abstract = {One of the main goals of Wendelstein 7-X (W7-X) is to demonstrate the fast particle confinement properties of the quasi-isodynamic stellarator concept. Fast particle populations will be produced either by Neutral Beam Injection (NBI) or by minority Ion Cyclotron Resonant Heating (ICRH). A fraction of these particles are expected to be lost (even without collisions), despite the optimisation procedure used for the W7-X design. Confinement properties of NBI particles in W7-X were presented in the paper of Drevlak et al (2014 Nucl. Fusion 54 073002). A detailed study is presented here where the loss patterns of an NBI population are described. In particular, focussing on a high-mirror equilibrium, the confinement of fast ions with varying energy injection is studied under collisional conditions. It is found that collisions are not only responsible for classical transport losses but also enhance drift induced losses caused by trapped particles. Moreover, an asymmetry is found in the toroidal position of particle losses which can be explained by local variation in the equilibrium field. The effects of a neoclassically resolved radial electric field are also investigated. Fast particle confinement is significantly improved by the associated ExB drift. In particular, an increasing radial electric field helps to reduce and even stop the losses due to the 3D equilibrium structure for times comparable to slowing down time.},

keywords = {fast particles, heating, ion cyclotron resonance, MHD equilibrium, neoclassical transport, neutral beam injection, stellarator},

pubstate = {published},

tppubtype = {article}

}

@article{pfefferle-demo,

title = {Effects of magnetic ripple on 3D equilibrium and alpha particle confinement in the European DEMO},

author = {D Pfefferl\'{e} and W A Cooper and A Fasoli and J P Graves},

url = {https://iopscience.iop.org/article/10.1088/0029-5515/56/11/112002},

doi = {10.1088/0029-5515/56/11/112002},

year  = {2016},

date = {2016-07-22},

journal = {Nuclear Fusion},

volume = {56},

number = {11},

pages = {112002},

abstract = {An assessment of alpha particle confinement is performed in the European DEMO reference design. 3D MHD equilibria with nested flux-surfaces and single magnetic axis are obtained with the VMEC free-boundary code, thereby including the plasma response to the magnetic ripple created by the finite number of TF coils. Populations of fusion alphas that are consistent with the equilibrium profiles are evolved until slowing-down with the VENUS-LEVIS orbit code in the guiding-centre approximation. Fast ion losses through the last-closed flux-surface are numerically evaluated with two ripple models: (1) using the 3D equilibrium and (2) algebraically adding the non-axisymmetric ripple perturbation to the 2D equilibrium. By virtue of the small ripple field and its non-resonant nature, both models quantitatively agree. Differences are however noted in the toroidal location of particles losses on the last-closed flux-surface, which in the first case is 3D and in the second not. Superbanana transport, i.e. ripple-well trapping and separatrix crossing, is expected to be the dominant loss mechanism, the strongest effect on alphas being between 100\textendash200 KeV. Above this, stochastic ripple diffusion is responsible for a rather weak loss rate, as the stochastisation threshold is observed numerically to be higher than analytic estimates. The level of ripple in the current 18 TF coil design of the European DEMO is not found to be detrimental to fusion alpha confinement.},

keywords = {confinement, DEMO, fast particles, magnetic ripple, MHD equilibrium, neoclassical transport, perturbation theory},

pubstate = {published},

tppubtype = {article}

}

@article{faustin-2016b,

title = {ICRH induced particle losses in Wendelstein 7-X},

author = {J M Faustin and W A Cooper and J P Graves and D Pfefferl\'{e} and J Geiger},

url = {https://iopscience.iop.org/article/10.1088/0741-3335/58/7/074004},

doi = {10.1088/0741-3335/58/7/074004},

year  = {2016},

date = {2016-05-31},

journal = {Plasma Physics and Controlled Fusion},

volume = {58},

number = {7},

pages = {074004},

abstract = {Fast ions in W7-X will be produced either by neutral beam injection (NBI) or by ion-cyclotron resonant heating (ICRH). The latter presents the advantage of depositing power locally and does not suffer from core accessibility issues (Drevlak et al 2014 Nucl. Fusion 54 073002). This work assesses the possibility of using ICRH as a fast ion source in W7-X relevant conditions. The SCENIC package is used to resolve the full wave propagation and absorption in a three-dimensional plasma equilibrium. The source of the ion-cyclotron range of frequency (ICRF) wave is modelled in this work by an antenna formulation allowing its localisation in both the poloidal and toroidal directions. The actual antenna dimension and localization is therefore approximated with good agreement. The local wave deposition breaks the five-fold periodicity of W7-X. It appears that generation of fast ions is hindered by high collisionality and significant particle losses. The particle trapping mechanism induced by ICRH is found to enhance drift induced losses caused by the finite orbit width of trapped particles. The inclusion of a neoclassically resolved radial electric field is also investigated and shows a significant reduction of particle losses.},

keywords = {fast particles, ion cyclotron resonance, MHD equilibrium, neutral beam injection, stellarator, VENUS-LEVIS},

pubstate = {published},

tppubtype = {article}

}

@article{cooper-2016,

title = {Saturated ideal kink/peeling formations described as three-dimensional magnetohydrodynamic tokamak equilibrium states},

author = {W A Cooper and D Brunetti and B P Duval and J M Faustin and J P Graves and A Kleiner and H Patten and D Pfefferl\'{e} and L Porte and M Raghunathan and H Reimerdes and O Sauter and T M Tran},

doi = {10.1063/1.4945743},

year  = {2016},

date = {2016-04-06},

journal = {Physics of Plasmas},

volume = {23},

number = {4},

pages = {040701},

abstract = {Free boundary magnetohydrodynamic equilibrium states with spontaneous three dimensional deformations of the plasma-vacuum interface are computed for the first time. The structures obtained have the appearance of saturated ideal external kink/peeling modes. High edge pressure gradients yield toroidal mode number n = 1 corrugations for a high edge bootstrap current and larger n distortions when this current is small. Deformations in the plasma boundary region induce a nonaxisymmetric Pfirsch-Schl\"{u}ter current driving a field-aligned current ribbon consistent with reported experimental observations. A variation in the 3D equilibrium confirms that the n = 1 mode is a kink/peeling structure. We surmise that our calculated equilibrium structures constitute a viable model for the edge harmonic oscillations and outer modes associated with a quiescent H-mode operation in shaped tokamak plasmas.},

keywords = {internal kink, MHD equilibrium},

pubstate = {published},

tppubtype = {article}

}

@article{cooper-2015,

title = {Free boundary equilibrium in 3D tokamaks with toroidal rotation},

author = {W A Cooper and D Brunetti and J M Faustin and J P Graves and D Pfefferl\'{e} and M Raghunathan and O Sauter and T M Tran and I T Chapman and C J Ham and N Aiba and The MAST team and JET contributors},

url = {http://stacks.iop.org/0029-5515/55/i=6/a=063032},

doi = {10.1088/0029-5515/55/6/063032},

year  = {2015},

date = {2015-05-22},

journal = {Nuclear Fusion},

volume = {55},

number = {6},

pages = {063032},

abstract = {The three-dimensional VMEC equilibrium solver has been adapted to numerically investigate the approximate toroidal rotation model we have derived. We concentrate our applications on the simulation of JET snakes and MAST long-lived modes under free boundary conditions. Helical core solutions are triggered when 〈β〉 exceeds a threshold value, typically 2.7% in JET-like plasmas. A large plasma current and edge bootstrap current can drive helical core formations at arbitrarily small 〈β〉 in which the ideal saturated internal kink coexists with an ideal saturated external kink structure of opposite phase. The centrifugal force linked with the rotation has the effect of displacing the plasma column away from the major axis, but does not alter significantly the magnitude of the edge corrugation of the plasma. Error field correction coil currents in JET-like configurations increase the outer midplane distortions by 2 cm. The edge bootstrap current enhances the edge modulation of the plasma driven by the core snake deformations in MAST.},

keywords = {internal kink, MHD equilibrium, plasma flow},

pubstate = {published},

tppubtype = {article}

}

@article{pfefferle-hybrid,

title = {Hybrid guiding-centre/full-orbit simulations in non-axisymmetric magnetic geometry exploiting general criterion for guiding-centre accuracy},

author = {D Pfefferl\'{e} and J P Graves and W A Cooper},

url = {http://stacks.iop.org/0741-3335/57/i=5/a=054017},

doi = {10.1088/0741-3335/57/5/054017},

year  = {2015},

date = {2015-04-15},

journal = {Plasma Physics and Controlled Fusion},

volume = {57},

number = {5},

pages = {054017},

abstract = {To identify under what conditions guiding-centre or full-orbit tracing should be used, an estimation of the spatial variation of the magnetic field is proposed, not only taking into account gradient and curvature terms but also parallel currents and the local shearing of field-lines. The criterion is derived for general three-dimensional magnetic equilibria including stellarator plasmas. Details are provided on how to implement it in cylindrical coordinates and in flux coordinates that rely on the geometric toroidal angle. A means of switching between guiding-centre and full-orbit equations at first order in Larmor radius with minimal discrepancy is shown. Techniques are applied to a MAST (mega amp spherical tokamak) helical core equilibrium in which the inner kinked flux-surfaces are tightly compressed against the outer axisymmetric mantle and where the parallel current peaks at the nearly rational surface. This is put in relation with the simpler situation B(x, y, z) = B0[sin(kx)ey + cos(kx)ez], for which full orbits and lowest order drifts are obtained analytically. In the kinked equilibrium, the full orbits of NBI fast ions are solved numerically and shown to follow helical drift surfaces. This result partially explains the off-axis redistribution of neutral beam injection fast particles in the presence of MAST long-lived modes (LLM).},

keywords = {fast particles, full-orbit, guiding-centre, neoclassical transport, neutral beam injection, VENUS-LEVIS},

pubstate = {published},

tppubtype = {article}

}

@article{romanelli,

title = {Overview of the JET results},

author = {F Romanelli and JET on behalf of Contributors},

url = {http://stacks.iop.org/0029-5515/55/i=10/a=104001},

doi = {10.1088/0029-5515/55/10/104001},

year  = {2015},

date = {2015-03-22},

journal = {Nuclear Fusion},

volume = {55},

number = {10},

pages = {104001},

abstract = {Since the installation of an ITER-like wall, the JET programme has focused on the consolidation of ITER design choices and the preparation for ITER operation, with a specific emphasis given to the bulk tungsten melt experiment, which has been crucial for the final decision on the material choice for the day-one tungsten divertor in ITER. Integrated scenarios have been progressed with the re-establishment of long-pulse, high-confinement H-modes by optimizing the magnetic configuration and the use of ICRH to avoid tungsten impurity accumulation. Stationary discharges with detached divertor conditions and small edge localized modes have been demonstrated by nitrogen seeding. The differences in confinement and pedestal behaviour before and after the ITER-like wall installation have been better characterized towards the development of high fusion yield scenarios in DT. Post-mortem analyses of the plasma-facing components have confirmed the previously reported low fuel retention obtained by gas balance and shown that the pattern of deposition within the divertor has changed significantly with respect to the JET carbon wall campaigns due to the absence of thermally activated chemical erosion of beryllium in contrast to carbon. Transport to remote areas is almost absent and two orders of magnitude less material is found in the divertor.},

keywords = {VENUS-LEVIS},

pubstate = {published},

tppubtype = {article}

}

@article{pfefferle-rmp,

title = {Impact of RMP magnetic field simulation models on fast ion losses},

author = {D Pfefferl\'{e} and C Misev and W A Cooper and J P Graves},

url = {https://iopscience.iop.org/article/10.1088/0029-5515/55/1/012001},

doi = {10.1088/0029-5515/55/1/012001},

year  = {2014},

date = {2014-12-19},

journal = {Nuclear Fusion},

volume = {55},

number = {1},

pages = {012001},

abstract = {Two opposing approaches to include resonant magnetic perturbations (RMPs) in fast ion simulations are compared, one where the vacuum field caused by the RMP current coils is added to the axisymmetric MHD equilibrium, the other where the MHD equilibrium includes the plasma response within the 3D deformation of its flux-surfaces. The first model admits large regions of stochastic field-lines that penetrate the plasma without alteration. The second assumes nested flux-surfaces with a single magnetic axis, which excludes stochastic field-lines, and embeds the RMPs within a 3D saturated ideal MHD state. The two descriptions of RMPs have been implemented in the VENUS-LEVIS guiding-centre orbit code. Simulations of fast ion populations resulting from MAST neutral beam injection have been applied to MAST n = 3 RMP coil configuration. At low beam energies, particle losses are dominated by parallel transport due to the stochasticity of the field-lines (vacuum-RMP model), whereas at higher energies, losses are accredited to the 3D structure of the perturbed plasma and the resulting drifts (equilibrium-RMP model).},

keywords = {fast particles, neutral beam injection, resonant magnetic perturbations, VENUS-LEVIS},

pubstate = {published},

tppubtype = {article}

}

@article{pfefferle-levis,

title = {VENUS-LEVIS and its spline-Fourier interpolation of 3D toroidal magnetic field representation for guiding-centre and full-orbit simulations of charged energetic particles},

author = {D Pfefferl\'{e} and W A Cooper and J P Graves and C Misev},

doi = {10.1016/j.cpc.2014.08.007},

issn = {0010-4655},

year  = {2014},

date = {2014-08-16},

journal = {Computer Physics Communications},

volume = {185},

number = {12},

pages = {3127 - 3140},

abstract = {Curvilinear guiding-centre drift and full-orbit equations of motion are presented as implemented in the VENUS-LEVIS code. A dedicated interpolation scheme based on Fourier reconstruction in the toroidal and poloidal directions and cubic spline in the radial direction of flux coordinate systems is detailed. This interpolation method exactly preserves the order of the RK4 integrating scheme which is crucial for the investigation of fast particle trajectories in 3D magnetic structures such as helical saturated tokamak plasma states, stellarator geometry and resonant magnetic perturbations (RMP). The initialisation of particles with respect to the guiding-centre is discussed. Two approaches to implement RMPs in orbit simulations are presented, one where the vacuum field is added to the 2D equilibrium, creating islands and stochastic regions, the other considering 3D nested flux-surfaces equilibrium including the RMPs.},

keywords = {cubic spline, curvilinear coordinates, fast particles, numerical method, VENUS-LEVIS},

pubstate = {published},

tppubtype = {article}

}

@article{cooper-2014,

title = {An approximate single fluid 3-dimensional magnetohydrodynamic equilibrium model with toroidal flow},

author = {W A Cooper and S P Hirshman and I T Chapman and D Brunetti and J M Faustin and J P Graves and D Pfefferl\'{e} and M Raghunathan and O Sauter and T M Tran and N Aiba},

url = {http://stacks.iop.org/0741-3335/56/i=9/a=094004},

doi = {10.1088/0741-3335/56/9/094004},

year  = {2014},

date = {2014-08-13},

journal = {Plasma Physics and Controlled Fusion},

volume = {56},

number = {9},

pages = {094004},

abstract = {An approximate model for a single fluid three-dimensional (3D) magnetohydrodynamic (MHD) equilibrium with pure isothermal toroidal flow with imposed nested magnetic flux surfaces is proposed. It recovers the rigorous toroidal rotation equilibrium description in the axisymmetric limit. The approximation is valid under conditions of nearly rigid or vanishing toroidal rotation in regions with significant 3D deformation of the equilibrium flux surfaces. Bifurcated helical core equilibrium simulations of long-lived modes in the MAST device demonstrate that the magnetic structure is only weakly affected by the flow but that the 3D pressure distortion is important. The pressure is displaced away from the major axis and therefore is not as noticeably helically deformed as the toroidal magnetic flux under the subsonic flow conditions measured in the experiment. The model invoked fails to predict any significant screening by toroidal plasma rotation of resonant magnetic perturbations in MAST free boundary computations.},

keywords = {internal kink, MHD equilibrium, plasma flow},

pubstate = {published},

tppubtype = {article}

}

@article{pfefferle-nbi,

title = {NBI fast ion confinement in the helical core of MAST hybrid-like plasmas},

author = {D Pfefferl\'{e} and J P Graves and W A Cooper and C Misev and I T Chapman and M Turnyanskiy and S Sangaroon},

url = {https://iopscience.iop.org/article/10.1088/0029-5515/54/6/064020},

doi = {10.1088/0029-5515/54/6/064020},

year  = {2014},

date = {2014-05-23},

journal = {Nuclear Fusion},

volume = {54},

number = {6},

pages = {064020},

abstract = {Energetic ions are found to be transported strongly from the core of MAST hybrid-like plasmas during long-lived mode (LLM) magnetohydrodynamic activity. The resulting impact on the neutral beam ion deposition and concurrent current drive is modelled using the guiding-centre approximation in the internal kinked magnetic topology. General coordinate guiding-centre equations are extended for this purpose. It is found that the kinked core spirals around the position of strongest ionization, which remains geometrically centred, so that a large fraction of the population is deposited in the high shear external region where the plasma is almost axisymmetric. Those particles ionized in the low shear region exhibit exotic drift motion due to the strongly non-axisymmetric equilibrium, periodically passing near the magnetic axis and then reflected by the boundary of the kinked equilibrium, which in this respect acts as a confining pinch. Broad agreement is found against experimental measurement of fast ion particle confinement degradation as the MAST LLM amplitude varies.},

keywords = {confinement, fast particles, guiding-centre, internal kink, MHD equilibrium, neoclassical transport, neutral beam injection, VENUS-LEVIS},

pubstate = {published},

tppubtype = {article}

}

@article{cooper-2013,

title = {Bifurcated helical core equilibrium states in tokamaks},

author = {W A Cooper and I T Chapman and O Schmitz and A D Turnbull and B J Tobias and E A Lazarus and F Turco and M J Lanctot and T E Evans and J P Graves and D Brunetti and D Pfefferl\'{e} and H Reimerdes and O Sauter and F D Halpern and T M Tran and S Coda and B P Duval and Labit B and A Pochelon and M R Turnyanskiy and L Lao and T C Luce and R Buttery and J R Ferron and E M Hollmann and C C Petty and M van Zeeland and M E Fenstermacher and J M Hanson and H L\"{u}tjens},

url = {http://stacks.iop.org/0029-5515/53/i=7/a=073021},

doi = {10.1088/0029-5515/53/7/073021},

year  = {2013},

date = {2013-06-11},

journal = {Nuclear Fusion},

volume = {53},

number = {7},

pages = {073021},

abstract = {Tokamaks with weak to moderate reversed central shear in which the minimum inverse rotational transform (safety factor) qmin is in the neighbourhood of unity can trigger bifurcated magnetohydrodynamic equilibrium states, one of which is similar to a saturated ideal internal kink mode. Peaked prescribed pressure profiles reproduce the 'snake' structures observed in many tokamaks which has led to a novel explanation of the snake as a bifurcated equilibrium state. Snake equilibrium structures are computed in simulations of the tokamak \`{a} configuration variable (TCV), DIII-D and mega amp spherical torus (MAST) tokamaks. The internal helical deformations only weakly modulate the plasma\textendashvacuum interface which is more sensitive to ripple and resonant magnetic perturbations. On the other hand, the external perturbations do not alter the helical core deformation in a significant manner. The confinement of fast particles in MAST simulations deteriorate with the amplitude of the helical core distortion. These three-dimensional bifurcated solutions constitute a paradigm shift that motivates the applications of tools developed for stellarator research in tokamak physics investigations.},

keywords = {helical core, internal kink, MHD equilibrium},

pubstate = {published},

tppubtype = {article}

}