Princeton Plasma Physics Laboratory

A method for solving the initial-value problem of the Korteweg-deVries equation is presented which is applicable to initial data that approach a constant sufficiently rapidly as $|x|\ensuremath{\rightarrow}\ensuremath{\infty}$. The method can be used to predict exactly the "solitons," or solitary waves, which emerge from arbitrary initial conditions. Solutions that describe any finite number of solitons in interaction can be expressed in closed form.

This article appears in the following collection:Letters from the Past - A PRL Retrospective2008 marked PRL's 50th anniversary. As part of the celebrations a collection of milestone Letters was started. The collection contains Letters that have made long-lived contributions to physics, either by announcing significant discoveries, or by initiating new areas of research.

A comprehensive review of zonal flow phenomena in plasmas is presented. While the emphasis is on zonal flows in laboratory plasmas, planetary zonal flows are discussed as well. The review presents the status of theory, numerical simulation and experiments relevant to zonal flows. The emphasis is on developing an integrated understanding of the dynamics of drift wave-zonal flow turbulence by combining detailed studies of the generation of zonal flows by drift waves, the back-interaction of zonal flows on the drift waves, and the various feedback loops by which the system regulates and organizes itself. The implications of zonal flow phenomena for confinement in, and the phenomena of fusion devices are discussed. Special attention is given to the comparison of experiment with theory and to identifying directions for progress in future research. © 2005 IOP Publishing Ltd.

Abstract We present the first Event Horizon Telescope (EHT) observations of Sagittarius A* (Sgr A*), the Galactic center source associated with a supermassive black hole. These observations were conducted in 2017 using a global interferometric array of eight telescopes operating at a wavelength of λ = 1.3 mm. The EHT data resolve a compact emission region with intrahour variability. A variety of imaging and modeling analyses all support an image that is dominated by a bright, thick ring with a diameter of 51.8 ± 2.3 μ as (68% credible interval). The ring has modest azimuthal brightness asymmetry and a comparatively dim interior. Using a large suite of numerical simulations, we demonstrate that the EHT images of Sgr A* are consistent with the expected appearance of a Kerr black hole with mass ∼4 × 10 6 M ⊙ , which is inferred to exist at this location based on previous infrared observations of individual stellar orbits, as well as maser proper-motion studies. Our model comparisons disfavor scenarios where the black hole is viewed at high inclination ( i > 50°), as well as nonspinning black holes and those with retrograde accretion disks. Our results provide direct evidence for the presence of a supermassive black hole at the center of the Milky Way, and for the first time we connect the predictions from dynamical measurements of stellar orbits on scales of 10 3 –10 5 gravitational radii to event-horizon-scale images and variability. Furthermore, a comparison with the EHT results for the supermassive black hole M87* shows consistency with the predictions of general relativity spanning over three orders of magnitude in central mass.

The major increase in discharge duration and plasma energy in a next-step DT fusion reactor will give rise to important plasma-material effects that will critically influence its operation, safety and performance. Erosion will increase to a scale of several cm from being barely measurable at a micron scale in today's tokamaks. Tritium co-deposited with carbon will strongly affect the operation of machines with carbon plasma-facing components. Controlling plasma wall interactions is critical to achieving high performance in present-day tokamaks and this is likely to continue to be the case in the approach to practical fusion reactors. Recognition of the important consequences of these phenomena has stimulated an internationally co-ordinated effort in the field of plasma-surface interactions supporting the engineering design activities of the international thermonuclear experimental reactor project (ITER) and significant progress has been made in better understanding these issues. This paper reviews the underlying physical processes and the existing experimental database of plasma-material interactions both in tokamaks and laboratory simulation facilities for conditions of direct relevance to next-step fusion reactors. Two main topical groups of interactions are considered: (i) erosion/re-deposition from plasma sputtering and disruptions, including dust and flake generation, (ii) tritium retention and removal. The use of modelling tools to interpret the experimental results and make projections for conditions expected in future devices is explained. Outstanding technical issues and specific recommendations on potential R and D avenues for their resolution are presented. (orig.)

Formulas for the electron thermal conductivity have been derived in the collisional and collisionless limits for the case of destroyed magnetic surfaces.Received 26 September 1977DOI:https://doi.org/10.1103/PhysRevLett.40.38©1978 American Physical Society

Progress in the area of MHD stability and disruptions, since the publication of the 1999 ITER Physics Basis document (1999 Nucl. Fusion 39 2137-2664), is reviewed. Recent theoretical and experimental research has made important advances in both understanding and control of MHD stability in tokamak plasmas. Sawteeth are anticipated in the ITER baseline ELMy H-mode scenario, but the tools exist to avoid or control them through localized current drive or fast ion generation. Active control of other MHD instabilities will most likely be also required in ITER. Extrapolation from existing experiments indicates that stabilization of neoclassical tearing modes by highly localized feedback-controlled current drive should be possible in ITER. Resistive wall modes are a key issue for advanced scenarios, but again, existing experiments indicate that these modes can be stabilized by a combination of plasma rotation and direct feedback control with non-axisymmetric coils. Reduction of error fields is a requirement for avoiding non-rotating magnetic island formation and for maintaining plasma rotation to help stabilize resistive wall modes. Recent experiments have shown the feasibility of reducing error fields to an acceptable level by means of non-axisymmetric coils, possibly controlled by feedback. The MHD stability limits associated with advanced scenarios are becoming well understood theoretically, and can be extended by tailoring of the pressure and current density profiles as well as by other techniques mentioned here. There have been significant advances also in the control of disruptions, most notably by injection of massive quantities of gas, leading to reduced halo current fractions and a larger fraction of the total thermal and magnetic energy dissipated by radiation. These advances in disruption control are supported by the development of means to predict impending disruption, most notably using neural networks. In addition to these advances in means to control or ameliorate the consequences of MHD instabilities, there has been significant progress in improving physics understanding and modelling. This progress has been in areas including the mechanisms governing NTM growth and seeding, in understanding the damping controlling RWM stability and in modelling RWM feedback schemes. For disruptions there has been continued progress on the instability mechanisms that underlie various classes of disruption, on the detailed modelling of halo currents and forces and in refining predictions of quench rates and disruption power loads. Overall the studies reviewed in this chapter demonstrate that MHD instabilities can be controlled, avoided or ameliorated to the extent that they should not compromise ITER operation, though they will necessarily impose a range of constraints.

A number of problems in physics can be reduced to the study of a measure-preserving mapping of a plane onto itself. One example is a Hamiltonian system with two degrees of freedom, i.e., two coupled nonlinear oscillators. These are among the simplest deterministic systems that can have chaotic solutions. According to a theorem of Kolmogorov, Arnol’d, and Moser, these systems may also have more ordered orbits lying on curves that divide the plane. The existence of each of these orbit types depends sensitively on both the parameters of the problem and on the initial conditions. The problem addressed in this paper is that of finding when given KAM orbits exist. The guiding hypothesis is that the disappearance of a KAM surface is associated with a sudden change from stability to instability of nearby periodic orbits. The relation between KAM surfaces and periodic orbits has been explored extensively here by the numerical computation of a particular mapping. An important part of this procedure is the introduction of two quantities, the residue and the mean residue, that permit the stability of many orbits to be estimated from the extrapolation of results obtained for a few orbits. The results are distilled into a series of assertions. These are consistent with all that is previously known, strongly supported by numerical results, and lead to a method for deciding the existence of any given KAM surface computationally.

Progress, since the ITER Physics Basis publication (ITER Physics Basis Editors et al 1999 Nucl. Fusion 39 2137-2664), in understanding the processes that will determine the properties of the plasma edge and its interaction with material elements in ITER is described. Experimental areas where significant progress has taken place are energy transport in the scrape-off layer (SOL) in particular of the anomalous transport scaling, particle transport in the SOL that plays a major role in the interaction of diverted plasmas with the main-chamber material elements, edge localized mode (ELM) energy deposition on material elements and the transport mechanism for the ELM energy from the main plasma to the plasma facing components, the physics of plasma detachment and neutral dynamics including the edge density profile structure and the control of plasma particle content and He removal, the erosion of low- and high-Z materials in fusion devices, their transport to the core plasma and their migration at the plasma edge including the formation of mixed materials, the processes determining the size and location of the retention of tritium in fusion devices and methods to remove it and the processes determining the efficiency of the various fuelling methods as well as their development towards the ITER requirements. This experimental progress has been accompanied by the development of modelling tools for the physical processes at the edge plasma and plasma-materials interaction and the further validation of these models by comparing their predictions with the new experimental results. Progress in the modelling development and validation has been mostly concentrated in the following areas: refinement in the predictions for ITER with plasma edge modelling codes by inclusion of detailed geometrical features of the divertor and the introduction of physical effects, which can play a major role in determining the divertor parameters at the divertor for ITER conditions such as hydrogen radiation transport and neutral-neutral collisions, modelling of the ion orbits at the plasma edge, which can play a role in determining power deposition at the divertor target, models for plasma-materials and plasma dynamics interaction during ELMs and disruptions, models for the transport of impurities at the plasma edge to describe the core contamination by impurities and the migration of eroded materials at the edge plasma and its associated tritium retention and models for the turbulent processes that determine the anomalous transport of energy and particles across the SOL. The implications for the expected performance of the reference regimes in ITER, the operation of the ITER device and the lifetime of the plasma facing materials are discussed.

The fundamental physics of magnetic reconnection in laboratory and space plasmas is reviewed by discussing results from theory, numerical simulations, observations from space satellites, and recent results from laboratory plasma experiments. After a brief review of the well-known early work, representative recent experimental and theoretical works are discussed and the essence of significant modern findings are interpreted. In the area of local reconnection physics, many findings have been made with regard to two-fluid physics and are related to the cause of fast reconnection. Profiles of the neutral sheet, Hall currents, and the effects of guide field, collisions, and microturbulence are discussed to understand the fundamental processes in a local reconnection layer in both space and laboratory plasmas. While the understanding of the global reconnection dynamics is less developed, notable findings have been made on this issue through detailed documentation of magnetic self-organization phenomena in fusion plasmas. Application of magnetic reconnection physics to astrophysical plasmas is also discussed.

Three-dimensional gyrokinetic simulations of microturbulence in magnetically confined toroidal plasmas with massively parallel computers showed that, with linear flow damping, an asymptotic residual flow develops in agreement with analytic calculations. Nonlinear global simulations of instabilities driven by temperature gradients in the ion component of the plasma support the view that turbulence-driven fluctuating zonal flows can substantially reduce turbulent transport. Finally, the outstanding differences in the flow dynamics observed in global and local simulations are found to be due to profile variations.

The predictions of gyrokinetic and gyrofluid simulations of ion-temperature-gradient (ITG) instability and turbulence in tokamak plasmas as well as some tokamak plasma thermal transport models, which have been widely used for predicting the performance of the proposed International Thermonuclear Experimental Reactor (ITER) tokamak [Plasma Physics and Controlled Nuclear Fusion Research, 1996 (International Atomic Energy Agency, Vienna, 1997), Vol. 1, p. 3], are compared. These comparisons provide information on effects of differences in the physics content of the various models and on the fusion-relevant figures of merit of plasma performance predicted by the models. Many of the comparisons are undertaken for a simplified plasma model and geometry which is an idealization of the plasma conditions and geometry in a Doublet III-D [Plasma Physics and Controlled Nuclear Fusion Research, 1986 (International Atomic Energy Agency, Vienna, 1987), Vol. 1, p. 159] high confinement (H-mode) experiment. Most of the models show good agreements in their predictions and assumptions for the linear growth rates and frequencies. There are some differences associated with different equilibria. However, there are significant differences in the transport levels between the models. The causes of some of the differences are examined in some detail, with particular attention to numerical convergence in the turbulence simulations (with respect to simulation mesh size, system size and, for particle-based simulations, the particle number). The implications for predictions of fusion plasma performance are also discussed.

With extensive use of the nonlinear transformations presented in Paper I of the series, a variety of conservation laws and constants of motion are derived for the Korteweg-de Vries and related equations. A striking connection with the Sturm-Liouville eigenvalue problem is exploited.

The resistive tearing mode is analyzed in the nonlinear regime; nonlinearity is important principally in the singular layer around k·B = 0. In the case where the resistive skin time τs is much longer than the hydromagnetic time τ H, exponential growth of the field perturbation is replaced by algebraic growth like t2 at an amplitude of order (τ H / τ S )4/5. Application of the theory to the unstable tearing modes of a tokamak with a shrinking current channel yields good agreement with the observed amplitudes of the m ≥ 2 oscillations. The analysis excludes the very long wavelength mode, and m = 1 in the tokamak, for which the “constant-Ψ” approximation is invalid.

An explicit nonlinear transformation relating solutions of the Korteweg-de Vries equation and a similar nonlinear equation is presented. This transformation is generalized to solutions of a one-parameter family of similar nonlinear equations. A transformation is given which relates solutions of a ``forced'' Korteweg-de Vries equation to those of the Korteweg-de Vries equation.

<p>Journal of Physics D: Applied Physics published the first Plasma Roadmap in 2012 consisting of the individual perspectives of 16 leading experts in the various sub-fields of low temperature plasma science and technology. The 2017 Plasma Roadmap is the first update of a planned series of periodic updates of the Plasma Roadmap. The continuously growing interdisciplinary nature of the low temperature plasma field and its equally broad range of applications are making it increasingly difficult to identify major challenges that encompass all of the many sub-fields and applications. This intellectual diversity is ultimately a strength of the field. The current state of the art for the 19 sub-fields addressed in this roadmap demonstrates the enviable track record of the low temperature plasma field in the development of plasmas as an enabling technology for a vast range of technologies that underpin our modern society. At the same time, the many important scientific and technological challenges shared in this roadmap show that the path forward is not only scientifically rich but has the potential to make wide and far reaching contributions to many societal challenges.</p>

The continuous operation of a tokamak fusion reactor requires, among other things, a means of providing continuously the toroidal current. Such operation is preferred to the conventional pulsed operation, where the plasma current is induced by a time-varying magnetic field. A variety of methods have been proposed to provide continuous current, including methods that utilize particle beams or radio-frequency waves in any of several frequency regimes. Currents as large as half a mega-amp have now been produced in the laboratory by such means, and experimentation in these techniques has now involved major tokamak facilities worldwide.

A nonlinear gyrokinetic formalism for low-frequency (less than the cyclotron frequency) microscopic electromagnetic perturbations in general magnetic field configurations is developed. The nonlinear equations thus derived are valid in the strong-turbulence regime and contain effects due to finite Larmor radius, plasma inhomogeneities, and magnetic field geometries. The specific case of axisymmetric tokamaks is then considered and a model nonlinear equation is derived for electrostatic drift waves. Also, applying the formalism to the shear Alfvén wave heating scheme, it is found that nonlinear ion Landau damping of kinetic shear-Alfvén waves is modified, both qualitatively and quantitatively, by the diamagnetic drift effects. In particular, wave energy is found to cascade in wavenumber instead of frequency.

The understanding and predictive capability of transport physics and plasma confinement is reviewed from the perspective of achieving reactor-scale burning plasmas in the ITER tokamak, for both core and edge plasma regions. Very considerable progress has been made in understanding, controlling and predicting tokamak transport across a wide variety of plasma conditions and regimes since the publication of the ITER Physics Basis (IPB) document (1999 Nucl. Fusion 39 2137-2664). Major areas of progress considered here follow. (1) Substantial improvement in the physics content, capability and reliability of transport simulation and modelling codes, leading to much increased theory/experiment interaction as these codes are increasingly used to interpret and predict experiment. (2) Remarkable progress has been made in developing and understanding regimes of improved core confinement. Internal transport barriers and other forms of reduced core transport are now routinely obtained in all the leading tokamak devices worldwide. (3) The importance of controlling the H-mode edge pedestal is now generally recognized. Substantial progress has been made in extending high confinement H-mode operation to the Greenwald density, the demonstration of Type I ELM mitigation and control techniques and systematic explanation of Type I ELM stability. Theory-based predictive capability has also shown progress by integrating the plasma and neutral transport with MHD stability. (4) Transport projections to ITER are now made using three complementary approaches: empirical or global scaling, theory-based transport modelling and dimensionless parameter scaling (previously, empirical scaling was the dominant approach). For the ITER base case or the reference scenario of conventional ELMy H-mode operation, all three techniques predict that ITER will have sufficient confinement to meet its design target of Q = 10 operation, within similar uncertainties.

While the results of early work on the density limit in tokamaks from the ORMAK and DITE groups have been useful over the years, results from recent experiments and the requirements for extrapolation to future experiments have prompted a new look at this subject. There are many physical processes which limit the attainable densities in tokamak plasmas. These processes include: (1) radiation from low Z impurities, convection, charge exchange and other losses at the plasma edge; (2) radiation from low or high Z impurities in the plasma core; (3) deterioration of particle confinement in the plasma core; and (4) inadequate fuelling, often exacerbated by strong pumping by walls, limiters or divertors. Depending upon the circumstances, any of these processes may dominate and determine a density limit. In general, these mechanisms do not show the same dependence on plasma parameters. The multiplicity of processes leading to density limits with a variety of scaling has led to some confusion when comparing density limits for different machines. The authors attempt to sort out the various limits and to extend the scaling law for one of them to include the important effects of plasma shaping, i.e. ;e = k, where ne is the line average electron density (1020 m−3), κ is the plasma elongation and (MA·m−2) is the average plasma current density, defined as the total current divided by the plasma cross-sectional area. In a sense, this is the most important density limit since, together with the q-limit, it yields the maximum operating density for a tokamak plasma. It is shown that this limit may be caused by a dramatic deterioration in core particle confinement occurring as the density limit boundary is approached. This mechanism can help explain the disruptions and Marfes that are associated with the density limit.