Two major modifications to the existing steady state fusion neutron science facility (FNSF) concept (Kessel et al 2018 Fusion Eng. Des. 135 236–70) are investigated with the aim of determining whether or not its predicted performance can be improved. The modifications are high magnetic field and pulsed operation. We find that high field leads to major economic improvements in a steady state FNSF, although at the expense of lowering the engineering gain. Pulsed operation replaces the problems associated with low current drive efficiency, with hopefully more manageable engineering problems. Here, however, high toroidal field (TF) is not helpful, and a lower TF field is more desirable economically. Pulsed FNSFs also have a reduced engineering gain. Further modifications lead to FNSF designs satisfying the additional constraint of engineering gain equal to unity. For these designs there is a large cost penalty for the steady state FNSF but only a modest penalty for the pulsed FNSF. All of our modified designs show modest to large potential economic improvements over the existing design. Overall, our conclusion is that it may be desirable to carry out a more detailed analysis of one of our improved designs, the choice depending upon which issue in the existing design is most important.

With the funding provided by this award, we developed numerical codes for the study of magnetically confined plasmas for fusion applications. Accordingly, our work can be divided into two separate categories: 1) the design and analysis of novel numerical methods providing high accuracy and high efficiency; 2) the study of the equilibrium and stability of magnetically confined plasmas with some of these numerical codes, as well as the study of the nature of the turbulent behavior which may arise in the presence of instabilities. We first developed new numerical schemes based on integral equation methods for the computation of steady-state magnetic configurations in fusion experiments, providing high accuracy for the magnetic field and its derivatives, which are required for stability and turbulence calculations. We employed different integral formulations depending on the application of interest: axisymmetric or non-axisymmetric equilibria, force-free or magnetohydrodynamic equilibria, fixed-boundary equilibria or free-boundary equilibria. While efficient, these methods do not yet apply to plasma boundaries which are not smooth, a situation which is fairly common in magnetic confinement experiments. To address this temporary weakness, we also constructed a new steady-state solver based on the Hybridizable Discontinuous Galerkin (HDG) method, which provides full geometric flexibility. In addition to these numerical tools focused on steady-states, we also contributed to the improvement of the speed and accuracy of codes simulating the plasma dynamics of fusion plasmas, by developing a novel velocity space representation for the efficient solution of kinetic equations, which most accurately describe the time evolution of hot plasmas in fusion experiments. Using the tools discussed above, we studied several questions pertaining to the equilibrium and stability of magnetically confined plasmas. In particular, we derived a new simple model for axisymmetric devices called tokamaks, to predict how elongated a fusion plasma can be before it becomes unstable and collapses. We also looked at the effect of the shape of the outer plasma surface on key properties of the steady-state magnetic configurations, and how these properties impact turbulence in fusion plasmas, and the corresponding transport of momentum. Likewise, we studied the role of large localized flows on the steady-state magnetic configurations, and how they may influence plasma stability and turbulence. Non-axisymmetric steady-state magnetic configurations are inherently more complex than axisymmetric steady-state configurations, and the subject of ongoing controversies regarding the regularity of the equations determining such steady-states, and their solutions. Implementing an existing NYU code in a new geometry, we studied the nature of the singularity of the solutions observed in the code, and methods to eliminate them. Our main conclusion is that by appropriately tailoring the plasma boundary, it is possible to eliminate the singularities otherwise appearing in our simulations, and to obtain steady-states which appear to be smooth. To gain further insights on incompletely understood turbulence phenomena, we proposed a new reduced model capturing most of these phenomena, which is simple enough to not require expensive numerical simulations on massive supercomputers to investigate them. We demonstrated the strong similarity between our simulations and published results obtained from computationally expensive simulations, and plan to rely on our reduced model to identify the key mechanisms determining the evolution and strength turbulent driven transport in fusion plasmas. Finally, we proposed a new framework for tokamak reactor design studies, enabling us to consider the relative merits of steady-state versus pulsed fusion reactors. We found that pulsed fusion reactors may benefit most from recent advances in magnet technology, and the availability of very high field magnets. As such, they may become more desirable than steady-state tokamak reactors for cost efficient electricity generation.

We have carried out a detailed analysis that compares steady state versus pulsed tokamak reactors. The motivations are as follows. Steady state current drive has turned out to be more difficult than expected—it takes too many watts to drive an ampere, which has a negative effect on power balance and economics. This is partially compensated by the recent development of high temperature REBCO superconductors, which offers the promise of more compact, lower cost tokamak reactors, both steady state and pulsed. Of renewed interest is the reduction in size of pulsed reactors because of the possibility of higher field Ohmic transformers for a given required pulse length. Our main conclusion is that pulsed reactors may indeed be competitive with steady state reactors and this issue should be re-examined with more detailed engineering level studies.

Part 1 described a wide range of analytic tokamak equilibria modelling smooth limiter surfaces, double- and single-null divertor surfaces, arbitrary aspect ratio, elongation, triangularity and beta. Part 2 generalizes the analysis to further include edge pedestals and toroidal flow. Specifically, edge pedestals are allowed in the pressure, pressure gradient and toroidal current density. Also, an edge-localized contribution to the bootstrap current is treated. In terms of flow, analytic solutions are obtained for two cases: a $\gamma = 2$ adiabatic and a $\gamma = \infty $ incompressible energy conservation relation.

Tokamak equilibria have been derived that are analytic solutions to the Grad–Shafranov equation. This paper, Part 1, describes a wide range of such equilibria including smooth limiter surfaces, double- and single-null divertor surfaces, arbitrary aspect ratio, elongation, triangularity and beta. Part 2 generalizes the analysis to include edge pedestals and toroidal flow.

This document is the product of a stellarator community workshop, organized by the National Stellarator Coordinating Committee and referred to as Stellcon, that was held in Cambridge, Massachusetts in February 2016, hosted by MIT. The workshop was widely advertised, and was attended by 40 scientists from 12 different institutions including national labs, universities and private industry, as well as a representative from the Department of Energy. The final section of this document describes areas of community wide consensus that were developed as a result of the discussions held at that workshop. Areas where further study would be helpful to generate a consensus path forward for the US stellarator program are also discussed. The program outlined in this document is directly responsive to many of the strategic priorities of FES as articulated in “Fusion Energy Sciences: A Ten-Year Perspective (2015–2025)” [1]. The natural disruption immunity of the stellarator directly addresses “Elimination of transient events that can be deleterious to toroidal fusion plasma confinement devices” an area of critical importance for the US fusion energy sciences enterprise over the next decade. Another critical area of research “Strengthening our partnerships with international research facilities,” is being significantly advanced on the W7-X stellarator in Germany and serves as a test-bed for development of successful international collaboration on ITER. This report also outlines how materials science as it relates to plasma and fusion sciences, another critical research area, can be carried out effectively in a stellarator. Additionally, significant advances along two of the Research Directions outlined in the report; “Burning Plasma Science: Foundations—Next-generation research capabilities”, and “Burning Plasma Science: Long pulse—Sustainment of Long-Pulse Plasma Equilibria” are proposed.

A highly elongated plasma is desirable in order to increase plasma pressure and energy confinement to maximize fusion power output. However, there is a limit to the maximum achievable elongation which is set by vertical instabilities driven by the n = 0 MHD mode. This limit can be increased by optimizing several parameters characterizing the plasma and the wall. The purpose of our study is to explore how and to what extent this can be done. Specifically, we extend many earlier calculations of the n = 0 mode and numerically determine scaling relations for the maximum elongation as a function of dimensionless parameters describing (1) the plasma profile (beta(p) and l(i)), (2) the plasma shape (epsilon and delta), (3) the wall radius (b/a) and (4) most importantly the feedback system capability parameter gamma T-w. These numerical calculations rely on a new formulation of n = 0 MHD theory we recently developed (Freidberg et al 2015 J. Plasma Phys. 81 515810607, Lee et al 2015 J. Plasma Phys. 81 515810608) that reduces the 2D stability problem into a 1D problem. This method includes all the physics of the ideal MHD axisymmetric instability while reducing the computation time significantly, so that many parameters can be explored during the optimization process. The scaling relations we present include the effects of the optimal triangularity and the finite aspect ratio on the maximum elongation, and can be useful for determining optimized plasma shapes in current experiments and future tokamak designs.

Pioneering plasma physicist Ronald Crosby Davidson died on 19 May 2016 in Cranbury, New Jersey, due to complications from pneumonia. Long-time editor-in-chief of Physics of Plasmas, he also directed the Princeton Plasma Physics Laboratory (PPPL) during a crucial period of its history and was a founding director of the Plasma Fusion Center at MIT.Ronald Crosby DavidsonPPT|High resolutionRon was born on 3 July 1941 in Norwich, Ontario, Canada. He grew up on a family dairy farm, where by age 11 he was driving a tractor and pitching in wherever needed. Ron learned at an early age how the annual cycle of farm activities comprised seasonal tasks, none of which could be postponed. Many years later, both as lab director and as editor-in-chief, Ron would be struck by similarities between farm tasks and administrative responsibilities. Likewise, his colleagues would be struck by Ron’s early-riser habits, incredible energy, and penchant for getting things done as early as weather would permit.Had he been the oldest child, Ron would have inherited and been expected to run the family farm. Instead, after he finished eighth grade in a one-room schoolhouse, Ron developed a passion for physics and mathematics in high school that drove his monumental impact in plasma physics.Ron earned a BSc degree in physics from McMaster University in 1963. He received his PhD in astrophysical sciences from Princeton University in 1966 under the advice of Edward Frieman. After an exceptionally distinguished career at several institutions, Ron arrived at MIT in 1978 with the challenge of organizing and expanding its plasma-science research, especially the Alcator tokamak program. He established the MIT Plasma Fusion Center with a broad scientific agenda that included toroidal and mirror field confinement experiments, theory and computation, magnet engineering, and particle-beam research. Ron returned to Princeton in 1991 to serve as a professor of astrophysical sciences and, until 1996, as director of PPPL.Ron made fundamental theoretical contributions to several areas of pure and applied plasma physics, including nonneutral plasmas, nonlinear effects and anomalous transport, kinetic equilibrium and stability properties, propagation of intense charged-particle beams in high-energy accelerators, and coherent radiation generation by relativistic electrons. He is the author of more than 450 journal articles and several books. His 1972 book, Methods in Nonlinear Plasma Theory (Academic Press), served as a key early textbook on plasma physics. He also is known for his advanced research monographs, including Theory of Nonneutral Plasmas (Benjamin Press, 1974), Physics of Nonneutral Plasmas (Addison-Wesley, 1990), and, with one of us (Qin), Physics of Intense Charged Particle Beams in High-Energy Accelerators (World Scientific, 2001).As director of PPPL, Ron oversaw the achievements of the Tokamak Fusion Test Reactor (TFTR) during its deuterium–tritium campaign. In December 1993, for the first time in the history of tokamak plasma devices, a reactor fuel mix of 50% deuterium and 50% tritium was used. Introducing that fuel mix into the TFTR released about 6 million watts of fusion power. In November 1994 the TFTR achieved 10.7 million watts of fusion power for one second, enough to momentarily power 2000 to 3000 homes.Ron gave years of distinguished service to the physics community as the chair of two divisions—plasma physics and physics of beams—of the American Physical Society and as editor-in-chief for Physics of Plasmas from 1991 to 2015. One of the longest-serving editors at the American Institute of Physics (which publishes Physics Today), Ron was an inspirational leader who conducted his editorial service with humble kindness. Shortly after becoming editor of Physics of Fluids B, he established Physics of Plasmas as its successor. Thereafter, his steady leadership, research excellence, and welcoming embrace of the international community of plasma physicists led to the expansion of Physics of Plasmas and its becoming the most highly cited journal in its field.Among his many recognitions and honors, Ron received the Kaul Foundation Award for Excellence in 1993, the IEEE Particle Accelerator Science and Technology Award in 2005, the American Physical Society’s James Clerk Maxwell Prize in Plasma Physics in 2008, and the Fusion Power Associates Distinguished Career Award in 2014.To his students and young colleagues, Ron was a gentle, exemplary mentor with high standards for scientific quality, especially in the implementation of research projects. His moral compass was so clear that the rest of us would lean over his shoulder at critical moments to see which way it was pointing. As a scientist and friend, Ron will be greatly missed.© 2016 American Institute of Physics.

We present a numerical scheme that can be combined with any fixed boundary finite element based Poisson or Grad-Shafranov solver to compute the first and second partial derivatives of the solution to these equations with the same order of convergence as the solution itself. At the heart of our scheme is an efficient and accurate computation of the Dirichlet to Neumann map through the evaluation of a singular volume integral and the solution to a Fredholm integral equation of the second kind. Our numerical method is particularly useful for magnetic confinement fusion simulations, since it allows the evaluation of quantities such as the magnetic field, the parallel current density and the magnetic curvature with much higher accuracy than has been previously feasible on the affordable coarse grids that are usually implemented.

The analytic theory presented in Paper I is converted into a form convenient for numerical analysis. A fast and accurate code has been written using this numerical formulation. The results are presented by first defining a reference set of physical parameters based on experimental data from high performance discharges. Numerically obtained scaling relations of maximum achievable elongation versus inverse aspect ratio are obtained for various values of poloidal beta, wall radius and feedback capability parameter in ranges near the reference values. It is also shown that each value of maximum elongation occurs at a corresponding value of optimized triangularity, whose scaling is also determined as a function of inverse aspect ratio. The results show that the theoretical predictions of maximum elongation are slightly higher than experimental observations for high performance discharges as measured by high average pressure. The theoretical optimized triangularity values are noticeably lower. We suggest that the explanation is associated with the observation that high performance involves not only MHD considerations, but also transport as characterized by confinement time. Operation away from the MHD optimum may still lead to higher performance if there are more than compensatory gains in the confinement time. Unfortunately, while the empirical scaling of the confinement time with the elongation has been determined, the dependence on the triangularity has still not been quantified. This information is needed in order to perform more accurate overall optimizations in future experimental designs.

This paper attempts to bridge the gap between tokamak reactor design and plasma physics. The analysis demonstrates that the overall design of a tokamak fusion reactor is determined almost entirely by the constraints imposed by nuclear physics and fusion engineering. Virtually, no plasma physics is required to determine the main design parameters of a reactor: a, R0, B0, Ti, Te, p, n, τE, I. The one exception is the value of the toroidal current I, which depends upon a combination of engineering and plasma physics. This exception, however, ultimately has a major impact on the feasibility of an attractive tokamak reactor. The analysis shows that the engineering/nuclear physics design makes demands on the plasma physics that must be satisfied in order to generate power. These demands are substituted into the well-known operational constraints arising in tokamak physics: the Troyon limit, Greenwald limit, kink stability limit, and bootstrap fraction limit. Unfortunately, a tokamak reactor designed on the basis of standard engineering and nuclear physics constraints does not scale to a reactor. Too much current is required to achieve the necessary confinement time for ignition. The combination of achievable bootstrap current plus current drive is not sufficient to generate the current demanded by the engineering design. Several possible solutions are discussed in detail involving advances in plasma physics or engineering. The main contribution of the present work is to demonstrate that the basic reactor design and its plasma physics consequences can be determined simply and analytically. The analysis thus provides a crisp, compact, logical framework that will hopefully lead to improved physical intuition for connecting plasma physic to tokamak reactor design.

In this and the accompanying paper the problem of the maximally achievable elongation in a tokamak is investigated. The work represents an extension of many earlier studies, which were often focused on determining the elongation limits due to (1) natural elongation in a simple applied pure vertical field or (2) axisymmetric stability in the presence of a perfectly conducting wall. The extension investigated here includes the effect of the vertical stability feedback system which actually sets the maximum practical elongation limit in a real experiment. A basic resistive wall stability parameter (gammatau) is introduced to model the feedback system which although simple in appearance actually captures the essence of the feedback system. Elongation limits in the presence of feedback are then determined by calculating the maximum elongation against n=0 resistive wall modes for fixed gammatau. The results are obtained by means of a general formulation culminating in a variational principle which is particularly amenable to numerical analysis. The principle is valid for arbitrary profiles but simplifies significantly for the Solov'ev profiles, effectively reducing the 2-D stability problem into a 1-D problem. The accompanying paper provides the numerical results and leads to a sharp answer of how much elongation is too much?

The MIT Plasma Science and Fusion Center and collaborators are proposing a high-performance Advanced Divertor and RF tokamak eXperiment (ADX)-a tokamak specifically designed to address critical gaps in the world fusion research programme on the pathway to next-step devices: fusion nuclear science facility (FNSF), fusion pilot plant (FPP) and/or demonstration power plant (DEMO). This high-field (>= 6.5 T, 1.5 MA), high power density facility (P/S similar to 1.5 MW m(-2)) will test innovative divertor ideas, including an 'X-point target divertor' concept, at the required performance parameters-reactor-level boundary plasma pressures, magnetic field strengths and parallel heat flux densities entering into the divertor region-while simultaneously producing high-performance core plasma conditions that are prototypical of a reactor: equilibrated and strongly coupled electrons and ions, regimes with low or no torque, and no fuelling from external heating and current drive systems. Equally important, the experimental platform will test innovative concepts for lower hybrid current drive and ion cyclotron range of frequency actuators with the unprecedented ability to deploy launch structures both on the low-magnetic-field side and the high-magneticfield side-the latter being a location where energetic plasma-material interactions can be controlled and favourable RF wave physics leads to efficient current drive, current profile control, heating and flow drive. This triple combination-advanced divertors, advanced RF actuators, reactor-prototypical core plasma conditions-will enable ADX to explore enhanced core confinement physics, such as made possible by reversed central shear, using only the types of external drive systems that are considered viable for a fusion power plant. Such an integrated demonstration of high-performance core-divertor operation with steady-state sustainment would pave the way towards an attractive pilot plant, as envisioned in the ARC concept (affordable, robust, compact) (Sorbom et al 2015 Fusion Eng. Des. submitted (arXiv: 1409.3540)) that makes use of high-temperature superconductor technology-a high-field (9.25 T) tokamak the size of the Joint European Torus that produces 270 MW of net electricity.

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Ideal MHD is the simplest model that describes the macroscopic equilibrium and stability of high-temperature fusion plasmas. A self-consistent derivation of the model has been presented in Chapter 2. The derivation requires that one restrict attention to the MHD length and time scales. The main assumptions for validity of ideal MHD are (1) small ion gyro radius, (2) high collisionality, and (3) negligible resistive diffusion. As pointed out, the high collisionality assumption is never satisfied in fusion-grade plasmas, which makes it perhaps surprising how accurate and reliable the model is in predicting experimental behavior.

Two-dimensional configurations with toroidal axisymmetry have been investigated in Chapter 6. Many fusion concepts fall into this class – tokamaks of all types, the reversed field pinch, the levitated dipole, the spheromak, and the field reversed configuration. One common feature in each of these concepts is the need for a toroidal current to provide toroidal force balance, either using a perfectly conducting shell or a vertical field.