On the Edge of Magnetic Fusion Devices [1st ed.] 9783030495930, 9783030495947

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Springer Series in Plasma Science and Technology

Sergei Krasheninnikov Andrei Smolyakov Andrei Kukushkin

On the Edge of Magnetic Fusion Devices

Springer Series in Plasma Science and Technology Series Editors Michael Bonitz, Kiel, Germany Liu Chen, Hangzhou, China Rudolf Neu, Garching, Germany Tomohiro Nozaki, Tokyo, Japan Jozef Ongena, Brussel, Belgium Hideaki Takabe, Dresden, Germany

Plasma Science and Technology covers all fundamental and applied aspects of what is referred to as the “fourth state of matter.” Bringing together contributions from physics, the space sciences, engineering and the applied sciences, the topics covered range from the fundamental properties of plasma to its broad spectrum of applications in industry, energy technologies and healthcare. Contributions to the book series on all aspects of plasma research and technology development are welcome. Particular emphasis in applications will be on hightemperature plasma phenomena, which are relevant to energy generation, and on low-temperature plasmas, which are used as a tool for industrial applications. This cross-disciplinary approach offers graduate-level readers as well as researchers and professionals in academia and industry vital new ideas and techniques for plasma applications.

More information about this series at http://www.springer.com/series/15614

Sergei Krasheninnikov • Andrei Smolyakov • Andrei Kukushkin

On the Edge of Magnetic Fusion Devices

Sergei Krasheninnikov University of California La Jolla, CA, USA

Andrei Smolyakov University of Saskatchewan Saskatoon, SK, Canada

Andrei Kukushkin Kurchatov Institute & NRNU MEPhI Moscow, Russia

ISSN 2511-2007 ISSN 2511-2015 (electronic) Springer Series in Plasma Science and Technology ISBN 978-3-030-49593-0 ISBN 978-3-030-49594-7 (eBook) https://doi.org/10.1007/978-3-030-49594-7 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To our almae matres: “ΦизTех,” “MГУ,” and “Kурчатовский Институт”

Preface

Nuclear reactions provide a virtually inexhaustible source of energy, which can be harnessed in the process of either fusion of light elements (e.g., hydrogen isotopes) or splitting heavy elements (e.g., uranium, plutonium, thorium, etc.). In both cases, the total mass of the initial nuclei, Min, is larger than the total mass of the products, Mpr, of the nuclear reactions. Then, from the famous relation between the energy and mass, E ¼ Mc2 (where c is the speed of light), and the energy conservation law, the energy ΔE ¼ (Min  Mpr) c2 goes into the kinetic energy of the products of the nuclear reactions. This is somewhat similar to what happens when we burn fossil fuels, where, however, we operate with the energy of chemical bonds of fossil fuel. The fusion process requires the rapprochement of nuclei to a really small distance. Since the nuclei are positively charged, this is only possible when the initial kinetic energy (temperature) of the nuclei is large enough to overcome the strong repulsive electric force. Thus, nuclear fusion is only possible at large temperatures ~108 K which correspond to ~104 eV, where eV is the energy/temperature unit widely used in physics. This makes nuclear fusion to be very different from fission, where the release of nuclear energy occurs in the processes of the interaction of the nucleon with a neutron (having zero electric charge and, therefore, not subject to the electric force) and can proceed even at the room temperature. The harnessing of nuclear energy via the fusion process is a challenging endeavor. First, the fusion fuel must be heated by an external energy source to a very high temperature to turn the fusion reactions on. Secondly, the high temperature of the fuel can be maintained by the release of nuclear energy only when the energy leakage from the fusion fuel is relatively slow, or, as physicists usually say, energy “confinement” or thermal insulation of the fusion fuel is good enough. Thus, again, we see that fusion is similar to the process of burning fossil fuel, which usually occurs at an elevated temperature maintained by the release of the chemical energy and the proper energy confinement. Finally, the extremely high temperatures needed for fusion reactions make it impossible to use any solid material-based furnace for the “confinement” of the fusion fuel. This is in striking contrast to the “furnace” fired by fission or fossil

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fuel, having the “burning temperatures” compatible with many materials (such as metals or ceramics). Today, there are two major approaches to confining the fusion fuel at the fusionrelevant temperatures. The first one is the so-called “inertial confinement,” where the hot fusion fuel is actually not really confined, but the fuel is burned down so fast that it has not enough time to cool down. In a sense, this approach is analogous to what happens during the explosion of an H-bomb, but on a much smaller scale tolerable for the apparatus designed for the inertial confinement. The second one utilizes the fundamental feature of the matter at high temperatures, where electrons are stripped from neutral atoms, which results in the formation of plasma consisting of negatively charged electrons and positively charged ions. It should be noted that the plasma we consider in this book has virtually no net charge since due to the different signs of the charge of electrons and ions, they attract each other so that the plasma largely moves as a quasi-neutral “fluid” and its inertial confinement lifetime is determined by the thermal speed of ions which are much heavier than electrons. At the fusion temperatures ~108 K, the thermal speed of the ions (nuclei) of hydrogen isotopes is about 106 m/s. Therefore, if one makes no effort to confine such a quasi-neutral motion of plasma, the latter would be disintegrated and cooled down within a short time (this is what actually happens in the inertial confinement devices). The addition of an external magnetic field drastically changes the plasma dynamics. Indeed, the charged particles, being subject to the Lorentz force, cannot move freely across the magnetic field. Instead, they undergo gyromotion around the magnetic field lines. By applying a strong magnetic field, the gyro-radii of both electrons and ions can be made much smaller than the size of the apparatus (e.g., for the fusion-relevant temperatures and the strength of the magnetic field ~1 T, the gyro-radii of the ions of hydrogen isotopes are ~102 m). In addition, to prevent the particle leakage along the magnetic field lines, one can utilize a closed toroidal configuration of the external magnetic field so that the charged particles can only escape across the magnetic field. This is the second major approach to confining the hot fusion fuel, which is called “magnetic confinement,” Stellarator and tokamak are two, the most known and advanced, fusion devices using the magnetic confinement. Stellarator, from “stellar,” suggests the device that produces energy like stars, for example, the Sun. The “tokamak” is an acronym of Russian words “current,” “chamber,” and “magnetic coils.” The magnetic field in the magnetic confinement devices (e.g., tokamaks) does not solve automatically all issues related to harnessing the fusion energy. The fusiongrade plasma is very “fragile” and sensitive to the presence of impurities (non-hydrogenic species) since they open the plasma energy sink by strongly enhancing the radiation losses and cooling down the plasma very efficiently (e.g., the relative concentration of a heavy impurity such as tungsten in a fusion-grade plasma must be below 104). Therefore, fusion plasma, embedded into a strong magnetic field, must be insulated from the atmosphere in some vacuum chamber. The plasma-facing material (PFM) of the vacuum chamber, however, can be eroded by the impinging ions of even much smaller energy (~few tens of eV) than needed for fusion. As a result of such erosion, individual atoms, clusters, and even

Preface

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microscopic dust particles from the PFM can penetrate into the fusion grade plasma and cool it down, thus killing the fusion reactions. Therefore, the plasma temperature in the vicinity of the PFM should be below the erosion threshold (preferably ~ 1 eV) to avoid such catastrophic scenarios. As we see, we are facing a very challenging issue: to ensure a descent rate of the fusion reactions, the core plasma temperature must be ~104 eV, whereas the plasma temperature near the plasma-facing components (PFCs) situated at the distance ~ 1 m from the core should be in the range of 1 eV. An additional complication is that the magnetic confinement of plasma is not perfect. First, there is so-called “classical” transport of the plasma energy and particles across the magnetic field associated with collisions of the charged particles. Secondly, there is also the so-called “anomalous” plasma leakage across the magnetic field, which is related to the turbulent electromagnetic fields driven by different plasma instabilities, which often exceeds classical transport by orders of magnitude. As a result, both the energy and particles (in particular, the products of fusion reactions, the so-called “ash”—helium for the case of the deuterium-tritium reactions) are transported from the core to the peripheral region. Although this can heat the plasma near the PFCs above the erosion limit, it allows removing the ash and prevents the “poisoning” of the magnetic fusion device with the ash. Yet, the energy transport to the PFCs can be strongly non-uniform and cause a local overheating of the PFCs beyond the acceptable level. All these issues, associated with the exhaust of fusion energy and ash, are controlled by plasma transport processes in the peripheral region of magnetic fusion devices close to the PFCs (which is usually called the “edge plasma”). Transport processes in edge plasmas are complex and nonlinear and often result in the formation of the plasma transport barrier and the transition of the entire magnetic fusion device to a high-confinement operation regime (the so-called “Hmode”). Today, it is assumed that all future magnetic fusion reactors (such as ITER which is under construction in France, and the first electric power generating device DEMO which is under discussion) will be operating in H-mode. At the same time, steep variations of the plasma parameters within the transport barrier can trigger violent plasma instabilities causing rapid expulsion of hot (~ few keV) plasma towards the PFCs, which for the reactor-relevant parameters can have catastrophic consequences for the PFM. Thus, we see that the edge plasma plays a crucial and multifaceted role in magnetic fusion devices: on the one hand, it sets the plasma confinement which defines virtually all major parameters of the magnetic fusion reactor (e.g., the size of the apparatus and the magnetic field strength), whereas on the other hand, it determines the energy and ash exhaust as well as the plasma interaction with the PFCs, which, in addition to the potential threat of core plasma contamination by the eroded material, control the lifetime of the entire device. As a matter of fact, the latter issues were recognized by such “founding fathers” of magnetic fusion as A. D. Sakharov and L. Spitzer already at the dawn of the era of magnetic fusion research. Today, it is widely accepted that without reliable solutions to the outstanding issues

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of edge plasma physics, no magnetic fusion reactor is possible, and the edge plasma physics becomes the forefront of magnetic fusion research. In this book, we present the core of the modern understanding of the most crucial edge plasma phenomena. Since edge plasma physics is very complex and involves many different topics ranging from classical and anomalous transport of multispecies plasma to atomic physics, the energy radiation loss and radiation transport in opaque media, the plasma material interactions and even the material physics, we cannot describe here all these topics in detail. In many instances, we will emphasize the theoretical background of the underlying physics process and illustrate our conclusions with the available experimental data while referring to other books, review papers, and journal publications for the details. At this time, there are hundreds and thousands of journal publications devoted to different aspects of the edge plasma physic, so it is impossible to cite all of them. We hope that our colleagues understand this and will not be too upset if some of their publications are absent from the reference list. La Jolla, CA, USA Saskatoon, SK, Canada Moscow, Russia

Sergei Krasheninnikov Andrei Smolyakov Andrei Kukushkin

Contents

1

Edge Plasma Issues in Magnetic Fusion Devices . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 11

2

Atomic Physics Relevant to Fusion Plasmas . . . . . . . . . . . . . . . . . 2.1 Basic Quantum Mechanical Features of Atoms, Molecules, and Ions Relevant for Magnetic Fusion Research . . . . . . . . . . . 2.2 Collisional-Radiative Model . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Line Radiation Transport in Edge Plasma . . . . . . . . . . . . . . . . . 2.4 Application of CRM to Edge Plasma Relevant Species . . . . . . . 2.4.1 Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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14 19 27 32 32 38 43 43

Plasma-Material Interactions in Magnetic Fusion Devices . . . . . . . 3.1 Reflection of Plasma Particles Impinging on Material Surfaces and Sputtering of Plasma-Facing Materials . . . . . . . . . 3.2 Basic Features of Hydrogen/Helium Transport in Plasma-Facing Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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57 65 66

4

Sheath Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 85 85

5

Dust in Fusion Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Experimental Study of Dust in Magnetic Fusion Devices . . . . . . 5.1.1 Dust Particle Density, Size Distribution, and Composition in Fusion Devices . . . . . . . . . . . . . . 5.1.2 Mobilization of Dust Particles from Plasma-Facing Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.1.3

Dust Particle Dynamics in Fusion Devices, Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Theoretical Aspects and Numerical Simulations of Dust-Related Phenomena in Magnetic Fusion Devices . . . . . . 5.2.1 Dust Particle Dynamics in Fusion Devices, Theoretical Approaches . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Numerical Simulations of Dust Particle Dynamics and Dust Impact on Edge Plasma Parameters . . . . . . . . 5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

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Fluid Description of Edge Plasma Transport . . . . . . . . . . . . . . . . . 6.1 Hierarchy and Closure of the Fluid Equations . . . . . . . . . . . . . . 6.2 Collisionless Cross-Field Components of Energy and Momentum Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 21-Moment Grad Approximation . . . . . . . . . . . . . . . . . . . . . . . 6.4 The Electron Heat Transport in a Weakly Collisional Regime . . 6.5 Fluid Description of Neutrals in Edge Plasmas . . . . . . . . . . . . . 6.6 Anomalous Effects in Edge Plasma Transport Equations . . . . . . 6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anomalous Cross-Field Transport in Edge Plasma . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Linear Theory of Edge Plasma Instabilities . . . . . . . . . . . . . . . . 7.2.1 Collisionless Drift Waves . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Localized Drift Wave . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Dissipative Drift Wave Instabilities in Slab Geometry . . 7.2.4 Destabilizing Effect of Ion Temperature Gradient . . . . . 7.2.5 Plasma Instabilities Driven by Toroidal Effects . . . . . . 7.2.6 Interchange and Resistive Interchange Modes . . . . . . . 7.2.7 Electromagnetic Effects . . . . . . . . . . . . . . . . . . . . . . . 7.2.8 Effect of “Open” Magnetic Field Lines . . . . . . . . . . . . 7.2.9 Impact of Magnetic Shear . . . . . . . . . . . . . . . . . . . . . . 7.2.10 Impact of Plasma “Macro- and Mesoscale” Flows . . . . 7.3 Nonlinear Effects and Anomalous Transport . . . . . . . . . . . . . . . 7.3.1 Generation of Sheared Plasma Flow Via Plasma Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 “Blobs” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 3D Edge Plasma Turbulence Modeling . . . . . . . . . . . . 7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Computational Modeling of the Edge Plasma Transport Phenomena 8.1 Transport Modeling of the Plasma . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Model Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Parallel Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Cross-Field Transport . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Neutral Transport Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Fluid Description of Neutrals . . . . . . . . . . . . . . . . . . . . 8.2.2 Monte-Carlo Models . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Coupling to the Plasma Model . . . . . . . . . . . . . . . . . . . 8.3 Selection of Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Boundary Conditions at the Targets . . . . . . . . . . . . . . . . 8.3.2 Boundary Conditions at the Core Boundary . . . . . . . . . . 8.3.3 Boundary Conditions at the “Radial” Edges of the Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 Fueling Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Physics Results and Model Validation . . . . . . . . . . . . . . . . . . . . 8.4.1 2D Transport Modeling . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Modeling the 3D Effects . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201 202 203 204 205 207 208 209 210 211 212 212

9

Physics of Some Edge Plasma Phenomena . . . . . . . . . . . . . . . . . . . 9.1 MARFE and Poloidaly Symmetric Plasma Detachment . . . . . . . 9.2 Self-Sustained Divertor Plasma Oscillations . . . . . . . . . . . . . . . 9.3 Divertor Plasma Detachment . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

229 229 234 241 252 252

10

Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

8

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213 213 215 215 219 222 223

About the Authors

Sergei Krasheninnikov is a professor in the Mechanical Engineering Department at the University California San Diego, USA. He graduated from the Moscow Institute of Physics and Technology, known informally as PhysTech (Физтех). He worked at the Kurchatov Institute of Atomic Energy, Moscow, Russia, and Plasma Science and Fusion Center at MIT, Cambridge, USA, and was involved in INTOR and ITER projects. His scientific interests span through different topics of plasma and atomic physics in fusion devices, plasma–material, and laser–plasma interactions. Andrei Smolyakov is a professor in the Department of Physics and Engineering Physics at the University of Saskatchewan, Saskatoon Saskatchewan, Canada. He graduated from the Moscow Institute of Physics and Technology, known informally as PhysTech (Физтех), and worked at the Kurchatov Institute of Atomic Energy, Moscow, Russia. His scientific interests include waves, instabilities, nonlinear processes in magnetically confined fusion plasmas, material processing, and electric propulsion applications. Andrei Kukushkin is a leading scientist at NRC Kurchatov Institute and NRNU MEPhI, Moscow, Russia. He graduated from Lomonosov Moscow State University known informally as MSU (МГУ). Being an employee of Kurchatov Institute, he was strongly involved in the INTOR project and then spent over 20 years at ITER developing the technology of the edge plasma modeling and applying it to providing the information on expected divertor performance for the designers. His scientific interests are concentrated on numerical modeling, mostly, of the edge and divertor plasma.

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Chapter 1

Edge Plasma Issues in Magnetic Fusion Devices

Abstract This chapter gives a short overview of the history of edge plasma studies, defines the terminology used in the rest of the book, and gives some particular examples of how multifaceted edge plasma physics plays one of the key roles in the modern magnetic fusion research.

Today, there are two major classes of magnetic confinement devices: stellarators and tokamaks. The main difference between them is the way to form the helical structure of magnetic field lines, which is needed to compensate unwanted drift of charged particles in the curved and inhomogeneous magnetic field and to form nested magnetic flux surfaces necessary for plasma confinement (see the magnetic field structure in a tokamak in Fig. 1.1). In a stellarator that has no toroidally symmetric magnetic field, this is done by a complex shaping of the magnetic coils, whereas in a toroidally symmetric tokamak by toroidal current flowing through the plasma. As already mentioned in the Preface, the negative impact of the interaction of the hot plasma with the plasma-facing components (PFC) of the vacuum chamber on reactor performance was envisioned at the very beginning of the fusion era. The main identified issues were (i) contamination of the core plasma with the eroded PFC material beyond the acceptable level (which actually is very low for high-Z impurities), where the plasma radiation loss due to impurity exceeds the fusion power released in alpha-particles so that no self-sustained fusion burn becomes possible, and (ii) strong erosion of the PFC material, which can severely limit the lifetime of the fusion reactor and make it unfeasible. L. Spitzer and I.E. Tamm with A.D. Sakharov suggested two conceptually different solutions to this problem in the 1950th. The Spitzer’s idea (e.g. see [2] and the references therein) was to isolate as much as possible the region of intense plasma-wall interaction from the core plasma. For this he suggested to use special magnetic coils to divert the magnetic field lines at the edge of the magnetic fusion device into some partially closed volume – the divertor (see Fig. 1.2). As a result, the magnetic field lines in the core and in the edge become separated by the so-called separatrix, whereas the impurity flux from the divertor into the core plasma is suppressed due to both a rather narrow divertor throat and plugging with the plasma flowing into the divertor. © Springer Nature Switzerland AG 2020 S. Krasheninnikov et al., On the Edge of Magnetic Fusion Devices, Springer Series in Plasma Science and Technology, https://doi.org/10.1007/978-3-030-49594-7_1

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1 Edge Plasma Issues in Magnetic Fusion Devices Central solenoid

Toroidal magnetic field coils

Plasma Magnetic lines of force

Poloidal magnetic field coils

Plasma current, toroidal magnetic field directions

Vacuum Vessel

Fig. 1.1 Schematic view of the structure of helical magnetic field lines (produced by both magnetic coils and plasma current) which form nested magnetic flux surfaces in a tokamak. (Reproduced with permission from [1], © IAEA 2012)

VACUUM WALL

NEGATIVE COIL SOLENOIDAL COILS

LINES OF MAGNETIC FORCE

VACUUM PUMPS

VACUUM PUMPS

Fig. 1.2 Design of the divertor suggested by L. Spitzer for the stellarator. (Reproduced with permission from [2], © AIP Publishing 1958)

The idea of Tamm and Sakharov was to form a cushion of neutral gas in front of the PFCs, so that the hot plasma arriving from the core would be cooled down in the course of the interaction with the neutrals so that the temperature of the plasma

1 Edge Plasma Issues in Magnetic Fusion Devices

3

Fig. 1.3 (a) Sketch of a toroidally symmetric limiter which is designated to accommodate the most severe plasma interaction with the PFCs. The LCFS that separates the nested closed magnetic flux surfaces in the core, which are occupied by hot, fusion grade plasma, from the open ones where the magnetic field lines intersect the material targets; and (b) Schematic view of the poloidal crosssection of a tokamak with a poloidal divertor formed by the plasma current and the currents in toroidal magnetic coils. In both figures, QSOL is the power coming from the core into the SOL, the red and blue arrows show the directions of the poloidal magnetic field and the orange arrows indicate the direction of the heat flux

interacting with the material surface would be so low that virtually no material erosion would be possible [3]. Interestingly, today’s concept of handling the issue of plasma interaction with the PFCs in magnetic fusion reactors is essentially a symbiosis of these two fundamental ideas adapted to the particular features of the magnetic devices. The original Spitzer’s divertor design was focused on a stellarator magnetic configuration (e.g. see [2]), where, because of the complexity of magnetic geometry, the formation of divertor was only possible by the reversal of the strong toroidal magnetic field. Nonetheless, the implementation of such a divertor in B-65 stellarator [4] has demonstrated encouraging results showing a significant impurity reduction in the core plasma. In much simpler, toroidally symmetric magnetic geometry of a tokamak, the divertor (the so-called poloidal divertor) can be relatively easy formed by proper arrangement of the electric current( s) in additional (or even existing) magnetic coils, reversing some relatively weak poloidal magnetic field. However, experimental studies of the impact of poloidal divertors on tokamak performance have started only at the end of 1970th. Before that, the most common approach was the designation of some parts of the tokamak PFC – the “limiters” - to handle the plasma-material interaction. For example, Fig. 1.3a shows the sketch of a toroidally symmetric limiter which is designated to accommodate the most severe plasma interaction with the PFCs. The so-called “last closed magnetic flux surface” (LCFS) separates the nested, closed magnetic flux surfaces in the core, which are

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1 Edge Plasma Issues in Magnetic Fusion Devices

occupied by hot fusion grade plasma, from the open ones where the magnetic field lines intersect the PFC material. The schematic view of the poloidal cross-section of magnetic configuration in a tokamak with the simplest poloidal divertor is shown in Fig. 1.3b. Such magnetic configuration can be formed just by adding a toroidally symmetric magnetic coil under the divertor targets, which carries the electric current in the same direction as the electric current in the plasma. These two currents create the magnetic separatrix that plays the role of the LCFS for the case of the toroidal limiter and separates the closed and open magnetic flux surfaces. Under the X-point, where the total poloidal magnetic field vanishes by definition, there is a so-called “private flux” region (PFR) having a very limited connection to the core plasma. Due to cross-field plasma transport, the heat from the core comes to the “scrape-off layer” (SOL) plasma where it can reach divertor targets quickly due to fast plasma transport along the magnetic field lines. The region between the X-point and the divertor targets is called the “divertor volume” or just a “divertor” and is often used for the designation of the whole ensemble of the PFR and the “outer” and “inner” divertors located, respectively at the outer and inner sides of the torus. The footprints of the heat flux at the targets are determined by the competition of fast plasma transport along the magnetic field lines and relatively slow cross-field plasma transport. As a result, the footprints appear to be small and all estimates show that if in fusion reactor all the power QSOL coming into the SOL from the core would reach the targets, the maximum heat load on the targets would greatly exceed the tolerable level. Therefore, a large fraction of this power should be dissipated on the way to the target through the impurity and hydrogen radiation losses and this is one of the main missions of the divertors. The first poloidal magnetic divertors were implemented in tokamaks only in the 1970s (e.g. see Fig. 1.4). And, like it had been found before in the stellarators, it was demonstrated that the implementation of a divertor in a tokamak reduces the impurity content in the core plasma significantly. Apart from that, at the beginning of the 1980s, it was discovered that the divertor magnetic configuration promotes transition into new regimes of (i) improved core plasma confinement, the so-called “H-mode” [7]; and (ii) highly radiative divertor operation regimes with dense cold plasma and neutral gas cushion formed in the divertor region, resulting in a strong reduction of the heat loading on the PFCs (e.g. see [8–11]). Since that, these new regimes became the key ingredients of the tokamak reactor designs and the main topics in the tokamak research. In some sense, these divertor operation conditions are the combination of Spitzer’s divertor concept and Tamm & Sakharov’s idea of the neutral gas cushion in front of the PFCs. Such divertor regimes, called “high recycling regimes”, are characterized by a very strong recirculation of neutrals and plasma in the divertor volume via neutral ionization and plasma neutralization at the divertor targets and through the volumetric recombination processes. As a result, the neutral ionization source in the divertor region in the high recycling regimes appears to be by orders of magnitude higher than the neutral puffing and pumping rates.

1 Edge Plasma Issues in Magnetic Fusion Devices

(a)

5

(b)

Collector Plates Divertor Throat

0.6 0.4

III

III

Divertor Coil Triplet

0.2 z

Separatrix

0

–0.2 –0.4 –0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 x

II

II I +

Fig. 1.4 Poloidal divertor magnetic configurations in the PDX (a) (Reproduced with permission from [5], © IAEA 1981) and ASDEX (b). (Reproduced with permission from [6], © IOP Publishing 1982) tokamaks)

Further studies of divertor performance have shown that with increasing plasma density and divertor radiation loss, not only the heat load on the PFCs but also the plasma particle flux to divertor targets starts to decrease (e.g. see [12] and the references therein). In a way, it looks like the plasma detaches from the divertor targets and these regimes are called the “detached divertor” regimes. The reduction of the plasma flux to the targets allows the reduction of the power loading associated with the release of the ionization potential energy, which, for a reactor, can be very substantial and exceed the tolerable limit. The examples of plasma flux reduction are shown in Fig. 1.5 where the ion saturation currents, which give the specific plasma flux, measured on JET and C-Mod tokamaks are shown. As one can see from Fig. 1.5a, the plasma fluxes on both the inner and outer divertor targets increase, then saturate, and then, in the detached regime, decrease (the so-called “rollover”) with increasing plasma density. However, as seen in Fig. 1.5b, such a decrease can occur only on some (although initially the most loaded) part of the divertor target. Due to the relative simplicity of the formation of the poloidal divertor configuration in tokamaks, very different magnetic configurations have been used in experiments, whereas more and, in some cases, very exotic divertor configurations have been suggested over the history of the magnetic fusion research. The main goals pursued in these developments of the divertor design are to maintain the tolerable level of the maximum target heat load together with the low plasma temperature at the targets to minimize target erosion, as well as to enhance the core plasma performance by improving plasma confinement and reducing the core plasma contamination.

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1 Edge Plasma Issues in Magnetic Fusion Devices

(a)

(b) Pulse No’s. 35734,‘35,‘40,‘41,‘43,‘24

60 50

Jsat Profile at Outer Divertor Surface attached

40

[email protected] s

30 20 10 60

Density ramp Steady state

50

Outer divertor

Jsat (A m–2)

Ion saturation current (Acm–2)

107

Inner divertor

106

105

40

detached

30

[email protected] s

20 10 2

3 4 5 6 7 Line averaged density (1019m–3)

104 –5 0 Separatrix

5

10 ρ (mm)

15

20

Fig. 1.5 Specific plasma fluxes on (a) the inner and outer divertor targets in JET versus the line averaged plasma density (Reproduced with permission from [13], © IAEA 1998) and (b) the outer divertor target in attached and detached regimes in C-Mod tokamak (Reproduced with permission from [14], © IAEA 1999)

For example, in Fig. 1.3b, the so-called “single null” divertor configuration (having one X-point) is sketched. Apart from this, “double null” divertor configurations, having X-points above and below the core plasma, are used in some experimental studies. When the double null is exact, only one separatrix separates the core and both upper and lower divertors. In this case, the outer and inner parts of the SOL become magnetically disconnected, which has important implications for anomalous plasma transport in the inner and outer SOL regions. Such a double null configuration, along with the lower and upper single null and “near-double-null” configurations used in the C-Mod tokamak are shown in Fig. 1.6a. We notice that the very first divertors were designed to allow up to 2 X-points in ASDEX and up to 4 X-points in the PDX tokamaks (see Fig. 1.4). A standard X-point magnetic geometry, which can be formed with just two effective toroidal currents, has two divertor legs and the strength of the poloidal magnetic field in the vicinity of the X-point is proportional to the distance to the X-point, rX. However, with at least three effective toroidal currents, the X-point could produce four divertor legs and, as a result, reduce the peak heat load on divertor targets. Such a magnetic configuration can be realized with a “snowflake” divertor concept (see Fig. 1.6b). In addition, in such a case, the strength of the poloidal magnetic field in the vicinity of the X-point becomes proportional to r2X . This increases the length of the magnetic field lines in the SOL, and, therefore, could slow down parallel plasma transport and, therefore, broaden the footprint of the heat flux on the targets. It also results in an increase of the volume occupied by plasma in the vicinity of the X-point, which could help to increase the radiation loss from the divertor. Magnetic configurations of the TCV tokamak divertor, having a long outer divertor leg, are shown in Fig. 1.6c. As one can see, manipulation of the currents

1 Edge Plasma Issues in Magnetic Fusion Devices

7

Fig. 1.6 (a) Lower and upper single null, “near-double-null” and exact double null configurations used in the C-Mod tokamak (Reproduced with permission from [15], © Elseivier 2017); (b) Sketch of the “snowflake” magnetic configuration (Reproduced with permission from [16], © AIP Publishing 2007); (c) Divertor configuration in the TCV tokamak with the long outer “divertor leg” and compressed (left) and expanded (right) magnetic flux surfaces in the outer divertor (Reproduced with permission from [17], © IAEA 2017); (d) So-called “Super-X” divertor configuration with a large radial extension and expanded poloidal magnetic flux in the outer divertor leg, which is reachable in the MAST-U tokamak (Reproduced with permission from [18]); (e) X-point target divertor concept suggested for the ADX tokamak project (Reproduced with permission from [19], © IAEA 2015). The thin lines show the magnetic flux surfaces

in the magnetic coils can produce a very wide spread of the open magnetic flux surfaces in the divertor. More complex divertor designs are shown in Figs. 1.6d and e. The so-called Super-X divertor shown in Fig. 1.6d has a very pronounced extension of the outer divertor leg along the major radius, whereas the even more complex X-point target divertor (shown in Fig. 1.6e) in addition to the radial extension of the outer divertor leg has multiple X-points in the vicinity of the divertor targets. All these features could increase the radiation loss from the divertor and reduce of the peak power loading of the targets. However, the complexity of such divertors can significantly limit the flexibility of the shaping of the magnetic configuration of the core plasma, which can be necessary for obtaining the best core plasma performance and maximizing the fusion yield. In addition, such divertors

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1 Edge Plasma Issues in Magnetic Fusion Devices

(a)

(b)

12

ρextent (mm)

10

slot divertor ‘nose’ location

vertical plate slot-geometry flat-plate

vertical plate divertor nose

–0.40 N

4

N

N

–0.50

2 0

(d)

–0.30

8 6

(c)

–0.20

1

3 2 ne(1020 m–3)

4

–0.60 0.5

0.6

0.7

0.5

0.6

0.7

0.5

0.6

0.7

Fig. 1.7 (a) The flux surface extent of divertor detachment for different divertor geometries: (b) the “vertical target”, (c) the “slot geometry”, and (d) the “flat plate”. (Reproduced with permission from [20], © Taylor & Francis 2007)

occupy large volume inside the toroidal magnetic field that is expensive to generate. Therefore, the usage of such divertors in future fusion reactors requires high confidence in assessments of both divertor and core plasma performance. The high recycling and, in particular, detached divertor regimes are characterized by strongly coupled plasma-neutral interactions providing, for example, an efficient cooling channel for the plasma within ~eV temperature range where the radiation energy losses become virtually negligible. But, whereas due to fast plasma transport along the magnetic field lines, the plasma parameters can be explicitly affected by the magnetic configuration, neutral transport is not affected directly by the magnetic field (although an indirect effect, caused by the plasma parameter variation, is present). However, neutral transport can be directly impacted by the special shaping of the divertor PFCs (the so-called “closed” divertors), which can better confine the neutrals in the divertor region. Therefore, in an attempt to facilitate divertor detachment, both the magnetic configuration and the geometry of the divertor material structures should be taken into account. An impact of geometrical effects on divertor detachment can be clearly seen from Fig. 1.7a where the onset of divertor detachment in the most closed, slot-like divertor geometry (Fig. 1.7c) occurs at the plasma density which is significantly lower than for the most open “flat-plate” geometry (Fig. 1.7d). The evolution of divertor geometries from the “open” to the more “closed” ones, which were used in different time on JET and ASDEX tokamaks, is shown in Fig. 1.8, and the impact of the closed divertor geometry on the increase of the radiation loss in the divertor volume and the reduction of the power reaching the targets is demonstrated in Fig. 1.9. Both the magnetic configuration and the divertor geometry of ITER are shown in Fig. 1.10. As one can see, the single-null magnetic configuration and a rather closed divertor geometry will be used there. Summarizing this chapter, we find that the physics of the edge plasma is very complex and multifaceted. It involves (i) many different species of both neutral and charged particles, including the eroded and deliberately injected atoms, molecules, and even dust particles formed due to erosion and re-deposition of the PFC material; (ii) anomalous and classical (e.g. drifts) cross-field plasma transport that is

1 Edge Plasma Issues in Magnetic Fusion Devices

9 divertor closure

Mark I

Div I

Mark IIAP

Div II, “Lyra” Mark IIGB (Gasbox)

Fig. 1.8 The evolution of divertor geometries from the “open” to the more “closed” ones on the ASDEX Upgrade (left) and JET (right) tokamaks. (Reproduced with permission from [21], © IOP Publishing 1999)

Fig. 1.9 Radiation loss in divertor volume (left) and power coming to divertor targets (right) versus power coming into SOL in ASDEX tokamak for open (green circles) and closed (black circles) divertor geometries. (Reproduced with permission from [21], © IOP Publishing 1999)

complicated by X-point effects; (iii) plasma transport along the magnetic field lines, which is often altered by kinetic effects (e.g. kinetic effects in plasma heat conduction along the magnetic field lines); (iv) atomic physics processes playing the crucial role in the radiation losses due to impurity and hydrogenic species, plasma-neutral interactions, plasma recycling, and in establishing both the high recycling and detached divertor regimes; (v) plasma interactions with the material surfaces of the PFCs, including the formation of the so-called sheath region in a close proximity to

10

1 Edge Plasma Issues in Magnetic Fusion Devices

Fig. 1.10 Magnetic configuration and divertor geometry in ITER

the surface, reflection and desorption of hydrogenic species, erosion and re-deposition of the PFC materials; etc. In addition, the distribution of the plasma and neutral gas parameters in the edge plasma is very non-uniform. Whereas at the midplane of the SOL, the plasma density and temperature are ~1013–14 cm 3 and ~100 eV, in the divertor region the plasma density can reach ~1015 cm 3 (and even higher) and the temperature drops to ~10 eV in the attached and even to sub-eV in detached regimes. Whereas the neutral density at the midplane is well below the plasma density, in the divertor volume the neutral gas density can be comparable to the plasma one. The characteristic cross-field scale of plasma parameter variation at the midplane of the SOL is ~few mm, whereas the effective distance between the SOL midplane and divertor targets along the magnetic field (the connection length) can be ~100 m. All of these make quantitative theoretical/computational description of edge plasma phenomena very difficult. Moreover, as of today, it is not feasible to describe all processes in the edge plasma with a single “super code”. Therefore, the researchers usually separate the fast “micro” turbulence in the plasma and the slow macroscopic transport of the plasma and neutral gas and describe them with different and very complex codes. We notice that the applicability of such splitting of the processes into the turbulent and “mean-field” parts in the edge plasmas is often questionable. From the experimental side, the situation is not any simpler because of the strong non-uniformity of the plasma parameters and often a limited plasma parameter range accessible to some diagnostics (e.g. for the Langmuir probes). In addition, due to the geometrical complexity, it is often difficult to post-process the available experimental data, and incompleteness of the experimental data complicates its interpretation. Nonetheless, by interconnecting many bits of the information coming from the experimental data, simplified theoretical models, and the results of numerical simulations, the edge plasma community has been able to build up a rather complete physical picture of many important edge plasma phenomena. It is obvious that today it is not possible to describe in detail all aspects of the edge plasma physics because some topics are still poorly understood (e.g. modification of

References

11

the PFC material under fusion plasma irradiation and the related retention of the helium and hydrogenic species). Therefore, in the following Chapters, we will discuss the main components of the edge plasma physics and phenomena in the edge plasma, which are reasonably well understood. In Chaps. 2 and 3 we consider, correspondingly, the atomic physics and plasma material interaction issues relevant to the edge plasma. In Chap. 4, the basic features of the so-called sheath – a narrow region at the interface between the plasma and the material wall – are discussed. Although the sheath occupies only a tiny fraction of the whole edge plasma volume, it plays an important role in both the physics of the edge plasma and the plasma-material interaction. Chap. 5 is dedicated to the physics of the dust that is virtually ubiquitous in the edge plasmas. Chap. 6 is dedicated to classical edge plasma transport, whereas in Chap. 7, the basic ingredients of anomalous cross-field plasma transport and the available numerical tools used for the modeling of the edge plasma turbulence are considered. In Chap. 8, we consider the modern approaches to numerical modeling of the edge plasma transport. In Chap. 9, we discuss the physics of some macroscopic phenomena that are distinctive for the edge plasma. They include (i) MARFE (which stands for the Multifaceted Asymmetric Radiation From the Edge) and poloidaly symmetric plasma detachment; (ii) self-sustained edge plasma oscillations; (iii) divertor plasma detachment. In Chap. 10 we present our assessment of the current understanding of the complex and multifaceted physics of the edge plasma and discuss the main gaps remaining there.

References 1. International Atomic Energy Agency, P.K. Kaw, et al., The case for fusion, in Fusion Physics, (IAEA, Vienna, 2012), pp. 1–58 2. L. Spitzer, The Stellarator concept. Phys. Fluids 1, 253–264 (1958) 3. I.E. Tamm, A.D. Sakharov, in Proceedings of the Second International Conference on the Peaceful Uses of Nuclear Energy, ed. by M. A. Leontovich, vol. 1, (Pergamon, Oxford, 1961), pp. 1–47 4. C.R. Burnett, D.J. Grove, R.W. Palladino, T.H. Stix, K.E. Wakefield, The Divertor, a device for reducing the impurity level in a Stellarator. Phys. Fluids 1, 438–445 (1958) 5. D. Meade, V. Arunasalam, C. Barnes, M. Bell, M. Bitter, K. Bol, R. Budny, J. Cecchi, S. Cohen, C. Daughney, S. Davis, D. Dimock, F. Dylla, P. Efthimion, H. Eubank, R. Fonck, R. Goldston, B. Grek, R. Hawryluk, E. Hinnov, H. Hsuan, M. Irie, R. Jacobsen, D. Johnson, L. Johnson, H. Kugel, H. Maeda, D. Manos, D. Mansfield, R. McCann, D. McCune, K. McGuire, D. Mikkelson, S. Milora, D. Mueller, M. Okabayashi, K. Owens, M. Reusch, K. Sato, N. Sauthoff, G. Schmidt, E. Silver, J. Sinnis, J. Strachan, S. Suckewer, H. Takahashi, F. Tenney, PDX experimental results, in Plasma Physics and Controlled Nuclear Fusion Research 1980, Proceedings of an International Conference, Brussels, 1980, vol. 1, (IAEA, Vienna, 1981), p. 665 6. M. Keilhacker, K. Lackner, K. Behringer, H. Murmann, H. Niedermeyer, Plasma boundary layer in limiter and Divertor tokamaks. Phys. Scr. T2, 443–453 (1982)

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7. F. Wagner, G. Becker, K. Behringer, D. Campbell, A. Eberhagen, W. Engelhardt, G. Fussmann, O. Gehre, J. Gernhardt, G.V. Gierke, G. Haas, M. Huang, F. Karger, M. Keilhacker, Q. Kluber, M. Kornherr, K. Lackner, G. Lisitano, G.G. Lister, H.M. Mayer, D. Meisel, E.R. Müller, H. Murmann, H. Niedermeyer, W. Poschenrieder, H. Rapp, H. Röhr, F. Schneider, G. Siller, E. Speth, A. Stäbler, K.H. Steuer, G. Venus, O. Vollmer, Z. Yü, Regime of improved confinement and high beta in neutral-beam-heated divertor discharges of the ASDEX tokamak. Phys. Rev. Lett. 49, 1408–1412 (1982) 8. M. Keilhacker, U. Daybelge, Divertors and impurity control. Nucl. Fusion 21, 1497–1504 (1981) 9. E.R. Müller, K. Behringer, H. Niedermeyer, Radiation losses and global energy balance for ohmically heated discharges in ASDEX. Nucl. Fusion 22, 1651–1660 (1982) 10. M. Shimada, M. Nagami, K. Ioki, S. Izumi, M. Maeno, H. Yokomizo, K. Shinya, H. Yoshida, A. Kitsunezaki, Helium ash exhaust with single-null poloidal Divertor in doublet III. Phys. Rev. Lett. 47, 796–799 (1981) 11. Y. Shimomura, M. Keilhacker, K. Lackner, H. Murmann, Characteristics of the divertor plasma in neutral-beam-heated ASDEX discharges. Nucl. Fusion 23, 869–879 (1983) 12. G.F. Matthews, Plasma detachment from divertor targets and limiters. J. Nucl. Mater. 220–222, 104–116 (1995) 13. A. Loarte, R.D. Monk, J.R. Marítn-Solís, D.J. Campbell, A.V. Chankin, S. Clement, S.J. Davies, J. Ehrenberg, S.K. Erents, H.Y. Guo, P.J. Harbour, L.D. Horton, L.C. Ingesson, H. Jäckel, J. Lingertat, C.G. Lowry, C.F. Maggi, G.F. Matthews, K. Mccormick, D.P. O’brien, R. Reichle, G. Saibene, R.J. Smith, M.F. Stamp, D. Stork, G.C. Vlases, Plasma detachment in JET Mark I divertor experiments. Nucl. Fusion 38, 331–371 (1998) 14. ITER Physics Basis, Chapter 4: Power and particle control. Nucl. Fusion 39, 2391–2469 (1999) 15. B. LaBombard, A.Q. Kuang, D. Brunner, I. Faust, R. Mumgaard, M.L. Reinke, J.L. Terry, J.W. Hughes, J. Walk, M. Chilenski, Y. Lin, E. Marmar, G. Wallace, D. Whyte, S. Wolfe, S. Wukitch, High-field side scrape-off layer investigation: Plasma profiles and impurity screening behavior in near-double-null configurations. Nucl. Mater. Energy 12, 139–147 (2017) 16. D.D. Ryutov, Geometrical properties of a “snowflake” divertor. Phys. Plasmas 14, 064502 (2007) 17. C. Theiler, B. Lipschultz, J. Harrison, B. Labit, H. Reimerdes, C. Tsui, W.A.J. Vijvers, J.A. Boedo, B.P. Duval, S. Elmore, P. Innocente, U. Kruezi, T. Lunt, R. Maurizio, F. Nespoli, U. Sheikh, A.J. Thornton, S.H.M. van Limpt, K. Verhaegh, N. Vianello, the TCV team and the EUROfusion MST1 team, Results from recent detachment experiments in alternative divertor configurations on TCV. Nucl. Fusion 57, 072008 (2017) 18. MAST-U Research Plan v4.0, https://ccfe.ukaea.uk/wp-content/uploads/2019/12/MAST-U_ RP_2019_v1.pdf 19. B. La Bombard, E. Marmar, J. Irby, J.L. Terry, R. Vieira, G. Wallace, D.G. Whyte, S. Wolfe, S. Wukitch, S. Baek, W. Beck, P. Bonoli, D. Brunner, J. Doody, R. Ellis, D. Ernst, C. Fiore, J.P. Freidberg, T. Golfinopoulos, R. Granetz, M. Greenwald, Z.S. Hartwig, A. Hubbard, J.W. Hughes, I.H. Hutchinson, C. Kessel, M. Kotschenreuther, R. Leccacorvi, Y. Lin, B. Lipschultz, S. Mahajan, J. Minervini, R. Mumgaard, R. Nygren, R. Parker, F. Poli, M. Porkolab, M.L. Reinke, J. Rice, T. Rognlien, W. Rowan, S. Shiraiwa, D. Terry, C. Theiler, P. Titus, M. Umansky, P. Valanju, J. Walk, A. White, J.R. Wilson, G. Wright, S.J. Zweben, ADX: A high field, high power density, advanced divertor and RF tokamak. Nucl. Fusion 55, 053020 (2015) 20. B. Lipschultz, B. LaBombard, J.L. Terry, C. Boswell, I.H. Hutchinson, Divertor physics research on Alcator C-Mod. Fusion Sci. and Tech. 51, 369–389 (2007) 21. A. Kallenbach, D. Coster, J.C. Fuchs, H.Y. Guo, G. Haas, A. Herrmann, L.D. Horton, L.C. Ingesson, C.F. Maggi, G.F. Matthews, R.D. Monk, J. Neuhauser, F. Ryter, J. Schweinzer, J. Stober, W. Suttrop, ASDEX Upgrade Team and JET Team, Closed divertor operation in ASDEX upgrade and JET. Plasma Phys. Control. Fusion 41, B177–B189 (1999)

Chapter 2

Atomic Physics Relevant to Fusion Plasmas

Abstract Atomic physics processes involving electrons, ions and neutrals of hydrogenic species and impurities are playing a vital role in virtually all macroscopic edge plasma phenomena. In this chapter a brief overview of the essentials of atomic physics relevant to the edge plasma processes is given.

As demonstrated in Chap. 1, both the atomic physics and the interactions of plasma with the materials of the PFCs (in particular, the first wall of the vacuum chamber and divertor targets) play very important role in virtually all edge plasma phenomena including plasma recycling, energy dissipation, divertor detachment, erosion of the PFCs, plasma contamination with impurities, etc. In this chapter, we will focus on the atomic physics issues relevant to fusion plasmas. Although atomic physics processes at the edge of magnetic fusion devices have some similarity to those in low-temperature gas discharge plasmas (e.g. see [1–3]), which have been under intense theoretical studies over 100 years, there are also important differences. First, the fusion plasma consists mostly of hydrogenic species, having some (~10%) helium as well as a potentially controlled (deliberately injected), relatively small percentage (~1%) of impurity species (e.g. neon, argon, etc.) for plasma cooling, and some fraction of impurities originated from erosion of the PFC materials (e.g. lithium, beryllium, tungsten, etc.). Secondly, unlike most of the gas discharges that feature rather homogeneous, low temperature (~1 eV) plasmas, the edge plasma parameters in fusion devices are very non-uniform (e.g. the edge plasma temperature in the discharge can vary from sub-eV to few 100 eV). As a result, atomic processes taking place in edge plasma are very diverse and ranging from plasma recombination in low-temperature regions to both neutral hydrogen and impurity ionization at high temperatures (we notice that at temperature ~ keV, neon can be completely stripped off of all electrons). As we will see, both diversity and inhomogeneity of the edge plasma parameters increase the number of atomic processes that should be allowed for, which complicates the edge plasma description. In this section, we review basic quantum mechanical features of atomic species relevant for edge plasma studies and discuss the physics behind the CollisionalRadiative Model (CRM) widely used in fusion research for the description of the © Springer Nature Switzerland AG 2020 S. Krasheninnikov et al., On the Edge of Magnetic Fusion Devices, Springer Series in Plasma Science and Technology, https://doi.org/10.1007/978-3-030-49594-7_2

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rates of different atomic processes. We will also consider some important examples of the application of the CRM to atoms, molecules and ions for edge plasma conditions, as well as line radiation transport in edge plasma and its implication for relevant atomic processes.

2.1

Basic Quantum Mechanical Features of Atoms, Molecules, and Ions Relevant for Magnetic Fusion Research

As known from quantum mechanics (e.g. see [4]), atoms, molecules and their ions can only occupy some particular quantum energy states bounded between the so-called ground state and ionization (or dissociation) continuum of the corresponding neutral/ion (or the molecule/molecular ion), which represents a free electron and the remaining ion (or separated neutrals and ions). Such states can be related to different electronic configurations and also, in the case of molecules, to different rotationally and vibrationally excited states. It appears that the neutrals and ions occupying excited states (situated in energy space above the ground state) play important roles in virtually all atomic physics-related processes in edge plasmas even though the relative fraction of such particles is often small. The situation with excited particles is somewhat similar to that with free chemical radicals, which have low concentrations but are important in many chemical reactions (e.g. see [5, 6]). In this sub-section, we just review the main features of quantum states in the atoms, molecules, and ions relevant for edge plasma (for more details one can refer to [4] and special literature). We start with quantum states in a hydrogen-like ion having one bound electron and the charge number of nucleus Z (the case Z ¼ 1 corresponds to the hydrogen atom). Omitting all relativistic effects, we find [4] that the energy levels, En, depend only on the principal quantum number n (1  n < 1) and Z EH n ¼ 

m e Z 2 e4 1 , 2ħ2 n2

ð2:1Þ

where zero energy corresponds to the free electron (continuum), me and e are the electron mass and charge, and ħ is the reduced Planck constant (we neglect here the terms of the order of the ratio of me to the nucleon mass Mnucl). From Eq. (2.1) it follows that the ionization potential of this hydrogen-like ion from the ground state n ¼ 1 is IHZ ¼ me Z2 e4 =2ħ2 . However, it appears that the quantum states with n > 1 are not stable and decay rather quickly into states having lower principal quantum numbers. The decay time from the level n to level k (k < n) is determined by the Z Einstein coefficients, AH n!k, which in the quasi-classical Kramers approximation can be written as [3]:

2.1 Basic Quantum Mechanical Features of Atoms, Molecules, and Ions Relevant for. . .

15

Fig. 2.1 The Balmer series of lines (solid curves) obtained from a wall-stabilized arc discharge (a), Reproduced with permission from [8], © Springer 2016) and recombining divertor plasma of Alcator C-Mod tokamak (b), Reproduced with permission from [48], © AIP Publishing 1998)

Z AH n!k 

1:6  1010 Z4 1
s : n3 k n2  k2

ð2:2Þ

For a hydrogen atom, the inverse time of decay from the first excited state to 8 1 the ground one is AH 2!1  6  10 s . In some cases, it is important to have an estimate of the decay time from the level n to all lower levels, which is given by Pk¼n1 Z AH n ¼ k¼1 An!k . Then from Eq. (2.2) we have Z AH n

 3  Z4 n n  1:6  10 5 ℓn s1 : 2 n 10

ð2:3Þ

For a single hydrogen atom in vacuum, the number of available quantum states (the so-called Rydberg states) is not limited. However, in the magnetic fusion environment, this is not the case. Even intuitively it is difficult to imagine hydrogen atom with an effective radius of the electron orbit larger than the average inter-ion 1=3 distance  ne (e.g. the Inglis–Teller equation predicts that the highest observable 2=15 [7]). In practice, isolated high Rydberg states, hydrogen quantum state nmax  ne 2=3 being affected by the plasma-induced micro-electric field, Emicro  ene , eventually disappear and merge with continuum, so that the number of states in Eq. (2.3) becomes finite. One can see this from the intensities of the Balmer series of hydrogen lines obtained from a wall-stabilized arc discharge and recombining divertor plasma of Alcator C-Mod tokamak (Fig. 2.1). In the arc plasma with density ne  1017cm3, the highest distinguishable Rydberg state corresponds to n  7 (Fig. 2.1a), whereas in the divertor plasma of Alcator C-Mod, having a somewhat lower density, ne  1015cm3, the Balmer lines merge to continuum at n  11 (Fig. 2.1b). In magnetic fusion devices having strong magnetic field, an additional reason limiting the available number of Rydberg states can be related to the Zeeman splitting of excited states and their spontaneous ionization by the effective electric

16

2 Atomic Physics Relevant to Fusion Plasmas Parahelium eV

n1S0

24.47 4 3

n1P1 5 4 3

Orthohelium

n1D2 5 4 3

n1F3 5 4

2

n3P2.1

n3P0

n3D3.21

5 4

5 4 3

5 4 3

5 4

2

2

3

22 20.55 19.77

n3S1

2

2

3

23S1 (metastable) 18 Term scheme for Para - and Orthohelium with one electron in ground state 1s and one excited electron

16 14

n2S+1LJ

12

n = principal quantum number S = total spin L = angular quantum number (S=0, P=1, D=2, F=3, ...) J = total angular momentum

10 8 6 4 2 0

1

11S0

Fig. 2.2 The energy levels in the helium atom, singlets (para-) and triplets (ortho-), for the case with one electron in the ground state (1 s) and one excited electron (taken from Wikipedia, https:// en.wikipedia.org/wiki/Helium_atom)

field, EB ~ (VN/c)B, caused by neutral motion across the magnetic field (here B is the magnetic field strength, c is the speed of light, and VN is the speed of the neutral). As we see, the structure of the quantum states in a hydrogen-like ion/hydrogen atom is rather simple. This “simplicity” is due to the peculiar degeneracy of quantum states over electron orbital angular momentum. However, the situation quickly becomes much more complex for atoms/ions having few electrons. As an example, in Fig. 2.2 one can see the helium quantum energy levels, which now depend also on the total orbital angular momentum, L ¼ 0, 1, 2, . . . (which is usually denoted by the letters S, P, . . .), spin, S, and total angular momentum J ¼ L + S (see the insert in Fig. 2.2). The only exception is negative hydrogen ion, H, also having two electrons, but only one bounded state with a low “ionization” potential of ~0.75 eV. H makes (as we will see later) some contribution to divertor plasma recombination but plays an important role in the generation of MeV range neutral

2.1 Basic Quantum Mechanical Features of Atoms, Molecules, and Ions Relevant for. . .

17

beams suitable for plasma heating in magnetic fusion reactors (e.g. see [9] and the references therein). We notice that some excited quantum states of complex atoms/ions, the so-called “metastable” states (spontaneous transitions from these states to lower energy states are “forbidden” by selection rules [4]), exhibit no spontaneous decay to lower states for a time much longer than ~2109s that follows from Eq. (2.2) for the transition from the first excited to the ground state in the hydrogen atom. The examples of such metastable states in helium are the triplet electron configuration 23S1 (see Fig. 2.2), having extremely long natural life-time ~ 104 s [10], and the singlet state 21S0 which has much shorter, ~20 ms [11], natural life-time. Such long-living metastable states (e.g. 23S1 in helium) play very important role in ionization balance of low temperature, weakly ionized plasma, by providing the so-called Penning ionization channel, A + B ! A + B+ + e, where A is a particle in a metastable state with the energy higher than the ionization potential of the particle B (e.g. see [12–14] and the references therein). So far, we were discussing quantum effects related to atoms/ions. However, plasma recycling on the PFCs results partly in the formation of molecules, which play an important and somewhat peculiar role in edge plasma processes. Since hydrogen is the major component in fusion plasmas, we will consider mainly the hydrogen molecules. As before, we will distinguish molecules containing different isotopologues of hydrogen molecule (e.g. H2, D2, DT, etc.) only when it becomes important, otherwise, we will call them just hydrogen molecules and use the notation H2. Molecular hydrogen having two nuclei introduces new features in the energy spectrum of the quantum states. Using disparity in electron and nuclei dynamics, related to the difference in their masses, one could start with the analysis of electronic states assuming that the separation distance, R, between the two nuclei is fixed (e.g. see [4]). As a result, the energy of the electronic quantum states and the corresponding electrostatic potential of the interacting nuclei, Un(R), depend on R. Since the dynamics of nuclei, in zero-order approximation, is ignored, the potential curves Un(R) remain the same for all isotopologues of hydrogen molecule (e.g. H2, D2, DT, etc.). The Un(R) terms in diatomic molecules are described by the projections of the total orbital angular momentum on the axis passing through the two nuclei, Λ ¼ 0, 1, . . . (which are denoted with the Greek letters Σ, Π, . . .) and the total spin of all electrons S (in the same way as in atoms). Finally, for the case where the atoms in the molecule are the same, the Hamiltonian is invariant with respect to the change of sign of the coordinates of all electrons and we can speak of the parity of electron wave functions, which can be even, denoted as “g” or odd, denoted as “u” (from corresponding German words “gerade” and “ungerade”). Strictly speaking, we cannot apply this property for diatomic hydrogenic molecules composed of different isotopes. But in practice, the isotopic effect produces a very tiny impact (of the order of the electron to nucleon mass ratio [15]) on the energy spectrum, which, in practice, affects only the line radiation transport we will consider later. As a result, the molecular terms of all diatomic hydrogen molecules are practically identical to that of H2, shown in Fig. 2.3.

18

2 Atomic Physics Relevant to Fusion Plasmas

Fig. 2.3 Molecular hydrogen terms. (Reproduced with permission from [16], © Springer 2012)

Once we know the potential Un(R), we can take into account the dynamics of nuclei describing the rotational and vibrational quantum states [4]. For the background state of molecular hydrogen isotopologues, we can introduce the rotational e 2, M e is the reduced mass of energy as Urot(R) ¼ ErotK(K + 1), where Erot ¼ ħ2 =2MR the nuclei and K is the quantum number (K ¼ 0, 1, 2, . . .) of the total angular momentum of the molecule. Then, using the potential UK(R) ¼ Un(R) + Urot(R) and expanding UK(R) near the minimum at R ¼ Rmin we can write e 2 M ω ðδRÞ2 , ð2:4Þ 2 v  e 1 d2 UK ðRÞ=dR2  and ω2v ¼ M . The quantum mechanUK ðRÞ ¼ UK ðRmin Þ þ

where δR ¼ R  Rmin

R¼Rmin

ical motion of a particle in a quadratic potential gives the following expression for the energy of the vibrational quantum states: Ev ¼ ħωv(v + 1/2), where v is the vibrational quantum number (v ¼ 0, 1, 2, . . .). As a result, the energy terms of the diatomic molecule include three components: electronic, rotational, and vibrational, which gives Utot ðRÞ ¼ Un ðRÞ þ Erot KðK þ 1Þ þ ħωv ðv þ 1=2Þ:

ð2:5Þ

The expression (2.5), where the contributions of vibrational and rotational states to the total energy are additive, is only valid for relatively low K and v values (e.g. see [4]), and for the higher ones their contributions are mixed. Therefore, the vibrational and rotational states are often called ro-vibrational states. Because the function

2.2 Collisional-Radiative Model

19

Un(R) remains the same for all isotopologues of the hydrogen molecule, from the definition of Erot and ωv we find dependences on the reduced mass of the pffiffiffiffitheir ffi e , ωv / 1= M e , and for the H2 molecule, ħωv ¼ 0.54 eV nucleon: Erot / 1=M [4]. The vibrational energy quanta for different isotopologues of the hydrogen molecule are different but the energy terms Un(R) are the same. Therefore, the same dissociation energy of 4.48 eV results in different numbers offfi the available pffiffiffiffi e (e.g. the H2 vibrational quantum states, which can be estimated as vmax / M molecule has vmax ¼ 14). In Fig. 2.3, in addition to the molecular terms for H2, are also shown the terms for ð þÞ ðÞ the positive molecular ion, H2 , and the negative one, H2 . The latter is metastable with a relatively short natural life-time. However, it plays a crucial role in both the ðÞ excitation of vibrational levels by electron impact H2 ðvÞ þ e ! H2 ! H2 ðv0 Þ þ e ð Þ and dissociative attachment H2 ðvÞ þ e ! H2 ! H þ H (e.g. see [12, 17, 18] and the references therein).

2.2

Collisional-Radiative Model

As we already mentioned, excited states of atoms and molecules in edge plasmas play important roles in virtually all atomic physics-related processes. Here, using the example of a hydrogen atom, we consider the basic physics of the CRM which is the main approach for a quantitative description of the processes the excited states can be involved in. The population of excited states (for the simplest case of hydrogen-like ions, they correspond to n > 1) in edge plasma, depending on the plasma parameters, is determined by an interplay of electron impact excitation, electron transition from the continuum, radioactive decay, re-absorption of resonance photons, etc. We notice that for the energies relevant for the edge plasmas, the impact excitation of electronic and vibrational states by atoms and ions is usually negligible. Therefore, we start with a short review of electron-involved processes and their contributions to the rate equations governing the population of the excited states. The binary interactions of particles, including electrons and heavy particles (atoms, molecules, and ions), are usually described by the effective cross-section, σ, of the particular process. In a very general case, σ depends on the relative speed of the interacting particles. Taking into account the large mass difference between the electrons and nuclei, for the case of electron interactions with atoms, molecules, and ions, the relative speed of the interacting particles is virtually equal to the electron speed. As a result, the cross-section of such interaction only depends on the electron kinetic energy, Ee. The cross-sections of many processes relevant for the edge plasmas are available from both quantum mechanical calculations and experiments (e.g. see [19–21] and the references therein). We notice that the cross-sections of the forward and reverse processes are related by the principle of detailed equilibrium (e.g. see [22]), so there is no need to calculate them separately.

20

2 Atomic Physics Relevant to Fusion Plasmas

As an illustration, we consider here the cross-sections of some electron-hydrogen interaction processes relevant to the population of excited states. For example, the cross-section of the electron impact excitation of the hydrogen atom from a level k to level n (k < n) can be written (e.g. see [23]) as σH k!n ðEe Þ

 2 2 ℓnðεnk 128 ħ k5 n7 e þ 1Þ ¼ 3=2 4 ¼ 2 nk 5 2 εe þ 1 3 Z me e ðn2  k Þ  2 2 ℓnðεnk 32 e kn3 e þ 1Þ ¼ 3=2 , 3 nk þ 1 2 ΔE ε 2 nk 3 e ðn  k Þ

ð2:6Þ

where εnk e ¼ ðEe  ΔEnk Þ=ΔEnk , ΔEnk ¼ Ek  En is the energy difference between the levels n and k, which gives for a hydrogen-like ion ΔEnk / (k2  n2). The cross-section for electron transition from continuum to hydrogen excited state n due to radiative recombination, H+ + e ! H(n) + ħω (where Ee  EH n ¼ ħω, recall that is negative), is given by the Kramers formula (e.g. see [12]) EH n σH cont!n ðEe Þ

 2 2  1 EH 8π e n ¼ 3=2 1 : Ee 3 ð137Þ2 n3 Ee

ð2:7Þ

Applying the classical Thompson formula for the hydrogen atom ionization crosssection from the excited state n, we find  σnion ðEe Þ

/

e2 EH n

2

ðεne  1Þ ðεne Þ2

ð2:8Þ

where εne ¼ Ee =jEH n j. Equation (2.8) gives a good agreement with more sophisticated models (e.g. see [23]). Interestingly, the dependences for electron impact excitation, radiative recombination, and ionization somewhat similar to Eqs. (2.6) (2.7) and (2.8) can be found from simple dimensional arguments. Indeed, the excitation (or ionization) crosssection, σexc, should depend on the electron energy Ee and the energy difference between the initial and final quantum states (including continuum) ΔEnk. Therefore, the most general expression for σexc can be written as follows [3]  σk!n exc ðEe Þ ¼

e2 ΔEnk

2


nk  f k,n exc εe ,

ð2:9Þ


k,n  where the function f k,n exc ðxÞ f exc ðx < 1Þ ¼ 0 depends on some quantum mechanical particularities of the transition (e.g. on n and k). We notice that the scaling following from Eq. (2.9) works rather well for many inelastic processes ranging from neutral

2.2 Collisional-Radiative Model

21

and ion ionization [3] to electron excitation (e.g. the electron impact excitation of hydrogen-like ions from n ¼ 1 to k ¼ 2 follows Eq. (2.9) for a wide range of Z [24]). Equation (2.9) predicts an important feature of the excitation and ionization crosssections: their maximum values, which are ~(ΔEnk)2, increase roughly ~n4 with increasing n. In addition, both the excitation and ionization energy thresholds decrease with increasing n, which increases the number of electrons that can contribute to these transitions. The latter circumstance is particularly important for low-temperature plasma, such that T < IH. Finally, recalling Eq. (2.3), we see that the rate of spontaneous decay of highly excited states falls down. All these effects suggest an appreciable population of the excited states and emphasize the importance of these states in overall reaction rates (e.g. hydrogen ionization). In other words, the so-called multistep processes including multiple excitation/ de-excitation of atoms, molecules, and ions with subsequent ionization, radiation, recombination and other processes affecting the population of excited states can significantly alter the rates of many atomic processes. They include not only ionization, recombination, and radiation loss, but also many other processes. Moreover, as we will see later, the presence of excited particles can “switch on” some important chemical reactions, which would not be possible otherwise. To allow properly for the impact of the excited states on the rates of different atomic processes, one should consider the rate equations for the populations of these states. As an illustration, we consider here the rate equations for the excited states of a hydrogen atom. First, we should recall that the rate of a “reaction” (e.g. ionization) involving binary collisions between particles A and B can be written as KAB[A][B], where [A] and [B] are the densities of species A and B and KAB is the rate constant of this reaction, which can be expressed in terms of the cross-section of the process σAB, as follows (e.g. see [22]) Z KAB ¼

   ! ! ! ! ! ! ! ! dv A d v B σAB jv A  v B j jv A  v Bj f A v A f B v B ð½A½BÞ1 , ð2:10Þ

  ! ! where f A v A and f B v B are the distribution functions of species A and B in the R ! ! velocity space normalized to their densities [A] and [B] (e.g. dv A f A v A ¼ ½A). Being interested in binary collisions of electrons with “heavy” hydrogen atoms and protons, we can ignore the speed of the “heavy” particles in the expressions for the corresponding rate constants (2.10). As a result, the rate constants of electron“heavy” particle interactions (e.g. electron impact excitation of the hydrogen atom  ! ! ðeÞ Kk!n ) only include averaging over the electron distribution function f e v , r , t :

22

2 Atomic Physics Relevant to Fusion Plasmas

Z

ðeÞ

Kk!n ne ¼

 ! ! ! ðeÞ σk!n ðvÞvf e v , r , t d v ,

ð2:11Þ

ðeÞ

where σk!n ðvÞ is the cross-section of the process under consideration. Thus, strictly speaking, in order to find the rate constant, one needs also to allow for the evolution of the electron distribution function. However, today it is not feasible to use a full kinetic approach for the study of all processes in the edge plasmas. Therefore with some exceptions, which will be considered in Chap. 6, edge plasma transport codes use a fluid approach where the distribution functions of the plasma particles are assumed to be shifted Maxwellian (applicability and limitations of this assumption will be considered in Chap. 6). Since the electron flow velocity in the edge plasma is much lower than the thermal electron speed, one can use in Eq. (2.11) the non-shifted Maxwellian electron distribution function. As a result, the rate constants ð eÞ of binary reactions involving electrons (e.g. Kk!n ) depend on the electron temperature Te only. For the case where the population of the excited states is due to electron impact and emission and reabsorption of photons, a symbolic form for the corresponding rate equations can be written as follows ( ) X ð eÞ X ðradÞ X ðeÞ d½ H n  ¼ ½Hn  ne Kn!k þ νn!k þ Kk!n ½Hk  ne dt þ

X k6¼n

k6¼n

k6¼n

k6¼n

X ðradÞ νk!n ½Hk  þ SðnradÞ þ SðntranspÞ ,

ð2:12Þ

k6¼n

where [Hn] and ne are the densities of the hydrogen atoms in the quantum state n and ðeÞ electrons respectively; Kk!n is the rate constant of electron-induced transition from ðradÞ the state k to the state n; νk!n is the effective frequency of spontaneous decay of the state k to the state n. SðnradÞ and SðntranspÞ are the sources and sinks of the population of different quantum states, caused by photon absorption and transport of atoms, respectively (we notice that the states k can also include continuum). Examining different terms in Eq. (2.12), one finds that for the edge plasma conditions, the characteristic equilibration time of the population of excited states (described by the first term on the right-hand side of Eq. (2.12)) is usually much shorter than the characteristic times of (i) variation of the hydrogen density in the ground state, |dℓn[H1]/dt|1, and (ii) transport of the exited states described by the last term in Eq. (2.12). As a result, the population of all electronically excited states in a hydrogen atom can be considered in a local quasi-steady approximation assuming that [H1] and the plasma density and temperatures are fixed. A similar approximation is often used for chemical radicals in theoretical models of chemical reactions (see [5, 6] and the references therein). The equations following from this approximation that greatly simplifies the rate equations for the population of excited

2.2 Collisional-Radiative Model

23

states are called the Collisional-Radiative Model. Originally, it was developed for hydrogen, and then extended to helium, impurities, and molecular hydrogen (e.g. see [12, 13, 25–31] and the references therein). The CRM can be simplified further by using proximity of the states with n  nmax to continuum and describing their population by the local thermodynamic equilibrium (LTE) by implementing the Saha equilibrium, so that ½Hnmax  / ½Hþ ne , (see [5, 12, 13, 25, 26, 29] and the references therein). Moreover, since in edge plasmas n¼n Pmax ½H1  ½Hn  , we can assume that [H1] is equal to the density of atomic n¼2

hydrogen [H]. As a result, we arrive at the set of linear algebraic equations for the population of quantum states 1 < n < nmax: ( 0 ¼ ½Hn 

nmax X

ðeÞ ne Kn!k

þ

ðionÞ ne Kn!cont

þ

k¼1, k6¼n

þ

nmax X

n1 X

) ðradÞ νn!k

k¼1

þ

nmax X

ð eÞ

ne Kk!n ½Hk 

k¼1, k6¼n

X ðradÞ ðrecÞ νk!n ½Hk  þ K cont!n ½H þ ne þ SðnradÞ , k6¼n

k>n

ð2:13Þ ðionÞ

ðrecÞ

where the rate constants Kn!cont and Kcont!n describe ionization and radiative recombination from the continuum. Recalling that ½Hnmax  / ½Hþ ne , we find that the solution of Eq. (2.13) can be expressed as a linear combination of the density of atomic hydrogen [H], the product [H+]ne, and a term describing radiation-induced transitions SðnradÞ . Neglecting, for simplicity, the SðnradÞ terms, we have þ

H þ ½Hn  ¼ ξH n ½H þ ξn ½H ne ,

ð2:14Þ

þ

H where the functions ξH n and ξn can be found from the solution of the corresponding sub-sets of Eq. (2.13) in terms of the electron and hydrogen ion densities and the rate ð eÞ ðionÞ ðradÞ ðradÞ constants and the frequencies of radiative decay Kn!k , K1!cont , νn!k , and Kcont!n . Substituting expression (2.14) into the equation for the population of the ground state, from (2.12) we find

( ) nmax X d½H ðionÞ ðionÞ H ¼  K1!cont þ Kk!cont ξk ½H ne dt k¼2 ( ) nmax nmax X X ðrecÞ ðeÞ Hþ ðradÞ Hþ 1 þ Kcont!1 þ ne Kk!1 ξk þ ne νk!1 ξk ½Hþ  ne þ Stransp , H k¼2

k¼2

ð2:15Þ where the first and second terms on the right-hand side of Eq. (2.15) can be interpreted as the hydrogen ionization and electron-ion recombination (EIR) rates

24

2 Atomic Physics Relevant to Fusion Plasmas

(including both radiative and three-body recombination). Accordingly, the expressions in the first and second braces on the right-hand side of Eq. (2.15) are called the hydrogen ionization, KH ion , and electron-ion recombination (EIR, which includes both radiative and three-body recombination processes), KH rec , rate constants. As a result, we can re-write Eq. (2.15) as follows d½ H  H þ ðtranspÞ ¼ KH : ion ne ½H þ Krec ne ½H  þ Sn dt

ð2:16Þ

The CRM is widely used in the modeling of gas discharge, fusion, and astrophysical plasmas (e.g. see [12, 13, 23, 25, 27–31] and the references therein). It is much simpler than the time-dependent rate Eq. (2.12) whereas allowing for the impact of excited states on both the ionization and recombination processes. We notice that since the population of excited states is established in a competition of electron-induced transitions and spontaneous decays, both rate constants KH ion and depend on the electron density and temperature. However, in optically thick KH rec plasma, re-absorption of resonance photons (in fusion plasmas they are usually Lyα and Lyβ) can alter the population of the excited states (recall terms SðnradÞ in Eq. (2.13)) and, therefore, the ionization and recombination rate constants. We will see below that because of high ionization rates of the excited states, the multistep processes including excitation and quenching of the excited hydrogen levels result, at sufficiently high plasma density, in a significant increase of the effective hydrogen ionization rate constant. Moreover, even the constants of elastic processes involving excited states can depend on n. In particular, the rate KðcxnÞ of the so-called charge exchange process, H(n) + H+ ! H+ + H(n), increases with increasing n and is proportional, although approximately, to n4 [32]. Such elastic collisions play a vital role in plasma-neutral momentum exchange and divertor detachment physics (e.g. see [33] and the references therein). Therefore, correct assessment of the impact of excited states on overall momentum exchange is important. A crude estimate of the effective resonance charge exchange rate constant, based on a simple averaging of corresponding cross-sections over relative P population of excited states, similar to that of the ionization rate constant, Kcx ¼ n KðcxnÞ ½Hn =½H, demonstrates a significant increase of Kcx in comparison to the cross-section involving hydrogen in the ground state, Kðcx1Þ [34]. If this simple estimate held, it would have an important impact on both hydrogen transport and plasma-neutral coupling described with both Monte Carlo codes and fluid models (e.g. see [35]). However, more thorough consideration shows that such a simplified description of the contribution of the excited states to hydrogen transport and plasma-neutral momentum exchange is incorrect [36]. For a proper consideration of ion-neutral  momentum exchange, we need to ! consider the distribution functions of protons, f i v , and atoms in all excited states,  ! fH ðnÞ v . Then, since the charge-exchange cross-section of excited states is much

2.2 Collisional-Radiative Model

25

higher than that of the ground state, recall KðcxnÞ / n4, we can assume that not only the density of the excited states  (as it is considered within CRM) but also their velocity ! v distribution functions f H can be treated in a quasi-equilibrium approximation ðnÞ leaving only a relatively slow variation of density and distribution function of the ground state. Then we can write thefollowing counterparts of Eq. (2.13) and (2.15) H ! for the distribution functions f ðnÞ v allowing, in addition to electron-neutral interactions and spontaneous decays of excited states, for the charge-exchange processes: 0¼



(

! f H ðnÞ v nmax X

þ

k¼1, k6¼n

nmax X

ðeÞ ne Kn!k

ðionÞ þ ne Kn!cont

þ

k¼1, k6¼n

n1 X k¼1

) ðradÞ νn!k

þ

nmax X

 ! ðradÞ νk!n f H ðkÞ v

k>n

 n   o ! ! ðeÞ ðnÞ H ! ne Kk!n f H , ðkÞ v  Kcx ni f ðnÞ v  ½Hn f i v ð2:17Þ

and  ! df H v dt

( ¼ ne ( þ

nmax X

) ðeÞ K1!k

ðionÞ þ K1!cont

k¼2 nmax  X

ð eÞ ne Kk!1

ðradÞ þ νk!1

 ! fH v

)  n   o ! ! H ! f ðkÞ v  Kðcx1Þ ni f H v  ½Hf i v ,

k¼2

ð2:18Þ     ! ! ! H ! H ! v v v =∂t þ v  , df =dt ¼ ∂f where ni is the proton density, f H v f H ð 1Þ  ! ðnÞ ∇!r f H v . For simplicity, we assume that σcx ðvÞv KðcxnÞ does not depend on the velocity v and neglect all free-to-bound transitions (i.e. recombination processes), which assumes the electron temperature corresponding to “ionizing” plasma. We notice that integration of Eq. (2.17) and (2.18) in the velocity space brings us back R ! H ! to Eqs. (2.13) and (2.15) and, therefore, to the relation ½Hn  ¼ dv f ðnÞ v , with [Hn]  ! given by the CRM. To find f H ðkÞ v , we observe that the solution of Eq. (2.17)   ! ! depends only on f i v and f H v . Then, following [37], we can write the functions   n   o ! ! ! H ! v =½H , where δn are as f H fH ðnÞ v ðnÞ v ¼ ½Hn   δn f i v =ni þ ð1  δn Þf  ! the partition coefficients. Substituting this expression for f H ðnÞ v into Eqs. (2.17) and (2.18), we obtain algebraic equations for δn, which are somewhat similar to the Eq. (2.13) but allowing also for charge exchange processes of the atoms in excited states:

26

2 Atomic Physics Relevant to Fusion Plasmas

(

nmax X

0 ¼  ½Hn δn

ð eÞ ne Kn!k

þ

ðionÞ ne Kn!cont

k¼1, k6¼n

þ

nmax X

þ

n1 X

) ðradÞ νn!k

þ

KðcxnÞ ni

k¼1 nmax X

ðradÞ

νk!n ½Hk δk þ

ð eÞ

ne Kk!n ½Hk δk þ KðcxnÞ ni ½Hn ,

k¼1, k6¼n

k>n

ð2:19Þ

 ! and the following equation for the evolution of the function f H v :  ! df H v dt

 n   o ! H ! H ! v  KH v  ½Hf i v , ¼ KH ion ne f cx ni f

ð2:20Þ

where the first term on the right-hand side of Eq. (2.20) describes just the effective ionization rate constant, recall Eq. (2.16), and ð1Þ KH cx ¼ Kcx þ

nmax X

 ðeÞ ðradÞ n1 ne Kk!1 þ νk!1 δk ð½Hk =½HÞ, i

ð2:21Þ

k¼2

is the effective charge exchange rate constant including the contribution of the excited states. As we see, the structure of the expression (2.21) is very different from the effective charge P exchange cross-section accounting for the contribution of excited states Kcx ¼ n KðcxnÞ ½Hn =½H suggested in Ref. [34]. By adopting Grad expansion of both the neutral and ion distribution functions (see Ch. 6 for details), it is possible to extend the analysis of the role of excited states to a very general hydrogen-ion elastic collision operator (including the chargeexchange one). However, it goes beyond our simple demonstration of the possible extension of the CRM to the evaluation of an impact of the excited states on elastic collisions and neutral transport. Calculations performed in [37] have shown that in a ð1Þ contrast to the results from [34], KH cx exceeds Kcx by only 10–15% and an impact of the excited states on neutral hydrogen transport is not very important. This is because of: i) the comparability of the magnitude of Kðcx1Þ to the electron impact excitation rate constant of the hydrogen atom, and ii) fast transition from excited to the ground state. However, the conclusion of the importance of electron exchange recombination of an impurity ion in the course of the interactions with excited hydrogen atoms [34] largely holds because hydrogen in the ground state does not undergo such a “resonance” charge exchange with impurity. The application of the CRM to impurity atoms/ions and molecules results in equations more complex than Eqs. (2.13) and (2.15). This is because (i) the number of states, which should be considered within CRM for each individual atom/ion, increases; (ii) few ionization states of the same kind of atom/ion can exist for given plasma density and electron temperature; (iii) the population of excited states of impurity atom/ion in edge plasma can be affected by charge-exchange process involving hydrogen atoms (e.g. AZ + 1 + H ! AZ(n) + H+, where AZ and AZ(n) are

2.3 Line Radiation Transport in Edge Plasma

27

impurity atoms in the ionization state Z and the excited level of the ionization state Z) [21]; (iv) the so-called dielectronic recombination process (accompanied by excitation of an ion’s electron and formation of a doubly excited atom/ion with subsequent emission of the photon, AZ+1 + e ! AZ(n, n0) ! AZ + ħω) should be considered (e.g. see [38, 39] and the references therein); and v) some excited quantum states of complex atoms/ions are “metastable” or just stable (e.g. the ro-vibrational states of hydrogenic molecules). To take into account such processes, one needs to modify the rate Eq. (2.12), adding some new terms (for example, quenching of metastable states on the vacuum chamber walls, which becomes important in weakly ionized plasma (e.g. see [13]). This means that in such plasmas, one should consider the transport terms for the particle density not only in the ground state but also in the metastable states. However, in relatively high density and rather hot fusion plasma, the effective lifetime of the metastable states is determined largely by the interactions of the metastable atoms/ions with electrons and is strongly reduced in comparison to their natural lifetime. In addition, the natural lifetime of ionic metastable states is decreasing as Z8 with increasing the ion charge Z. We also remind that some quantum states within the “thin structure” of the hydrogen energy levels, caused by relativistic effects, are also metastable. But due to the strong “mixing” of these levels in a fusion plasma environment, these metastable states play no significant role. As a result, in fusion research, the transport of metastable states is usually ignored and their populations (as well as ionization/recombination balance and radiation loss) are considered in a quasi-equilibrium approximation (recall Eq. (2.13)) on the equal footage with other excited states (e.g. for details see [30] and the references therein). The results of comprehensive numerical modeling show that for the edge plasma conditions, even the metastable state 23S1 of helium, having the lifetime ~104 s, can be treated in quasi-equilibrium approximation [40]. The presently most advanced database providing the fusion-relevant impurity radiation loss, the contribution of different lines, the rate constants, etc., is the ADAS database [41]. The divertor modeling codes whose development had started before the ADAS database became the de-facto standard can use some other data sources (for example, the AMJUEL, HYDHEL and METHANE data sets [42] in SOLPS).

2.3

Line Radiation Transport in Edge Plasma

As we have seen in the previous sub-section, the populations of excited states are determined by the competition between the processes involving interactions with electrons and, playing important role, radiative decays (e.g. from level n to level k), which are accompanied by the emission of photons having the energy ħω0  ΔEnk, where ω0 is the photon frequency corresponding to the decay n ! k. This is the so-called “line radiation”, which dominates in edge plasmas. However, in our simplified analysis of the CRM for hydrogen atoms (recall Eq. (2.13)), we neglected

28

2 Atomic Physics Relevant to Fusion Plasmas

the term SðnradÞ describing the inter-state transitions stimulated by the photons. These two features of our analysis could only be reconciled for the case of the so-called “transparent” media, where the effective mean free path of a photon to absorption by a neutral/ion, ℓabs, is longer than the characteristic scale-length, L, (e.g. for the case of radiation in the divertor volume, it could be the poloidal width of the divertor). An opposite case, where ℓabs < L, is called the “opaque” or “optically thick” one. Since the photon absorption rate is proportional to the density of available absorbers, and absorption process in edge plasma largely has a resonant nature (recall the relation ħω0  ΔEnk), edge plasma can be transparent for the radiation corresponding to some lines and opaque for the other ones (the latter case is often referred to as radiation “trapping”). Because the density of hydrogen atoms (hydrogen molecules are dissociated rather quickly due to electron impact) is the largest among the radiating species in the edge plasma, the line radiation related to atomic hydrogen is the first candidate for being trapped. Moreover, since the majority of atomic hydrogen is in the ground state, the most strongly “trapped” hydrogen lines are Lyα and Lyβ, related, respectively, to the transitions n ¼ 2 ! k ¼ 1 and n ¼ 3 ! k ¼ 1 (we will see that the absorption of other lines in Lyman series is weaker due to reduction of corresponding oscillator strengths). It was shown (e.g. see [43, 44]) that the trapping effects for Lyα and Lyβ lines become important already for current tokamaks and they are expected to be much more pronounced in the future tokamak reactors (e.g. ITER [45]). As one could notice, we have stated that the frequency of the emitted photon ω0 is only approximately equal to ΔEkn/ħ. The reason for this is the so-called line “broadening”, which results in the fact that emitted photons have some frequency distribution, described by the “line shape” function, a(ω) (localized around ω0 and R having characteristic width Δω ω0), such that a(ω)dω ¼ 1. There are few reasons for line broadening in edge plasma. First, there is a “natural” broadening of the line, Γ, caused by the finite time of the radiation emission, which corresponds to the decay rate of the excited state determined by the Einstein coefficients (2.2). However, in practice Δω is, in most cases, much larger than Γ. In edge plasma both Δω and the shape of the function a(ω) are largely determined by (i) Doppler broadening related to the shift of the frequency of the radiation emitted by moving particles, so that ΔωD ~ ω0(Vth/c), where Vth is the particle thermal speed; (ii) Stark broadening due to 2=3 the micro-electric fields Emicro ∼ ene , causing a change in the energy of the quantum states and yielding, for a hydrogen atom, ΔωS ~ (ħ/mee)Emicro; (iii) Zeeman effect that results, in the presence of a strong magnetic field, in splitting the quantum states; and, finally, (iv) the so-called motional Stark effects related to the effective electric field, EB ~ (VN/c)B. (e. g. see [46, 47], and the references therein). For the case where the line width is only determined by the decay rate, the effective cross-section of absorption of a resonant photon, σabs(ω0), is proportional to the square of the photon wavelength,p i.e. σabs(ω0) ~ (c/ω0)2. We note that for the ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi edge plasma conditions, ω0 ωpe 4πne e2 =me , where ωpe is the Langmuir frequency, so one can neglect plasma effects in the dispersion of the line radiation and take ω0 ¼ k0c, where k0 is the photon wavenumber. However, broadening of the

2.3 Line Radiation Transport in Edge Plasma

29

Fig. 2.4 The dependences of the Lyβ transmission (a) and the intensity of Lyα (b) on the intensity of Dα. (Reproduced with permission from [48], © AIP Publishing 1998)

line, such that Δω Γ, results in a strong reduction of σabs(ω0), which now becomes ~(c/ω0)2(Γ/Δω). For example, for the case where the Doppler effect dominates the line broadening, the characteristic absorption length of the line corresponding to the transition between the n-th excited state and the ground one in a hydrogen atom, ℓabs ~ 1/σabs(ω0)[H], can be found from the following expression [20] n
 o1 ℓabs ¼ π 3=2 ðc=ω1n Þ2 n2 AH , n!1 =ΔωD ½H

ð2:22Þ

where ω1n ¼ ΔE1n/ħ, ΔωD ¼ ω1n(2T[H]/Mnucl)1/2/c, and T[H] is the temperature of the hydrogen atoms. For [H] ¼ 1014cm3 and T[H] ¼ 3 eV, which are rather typical for dense divertor plasma, from Eq. (2.22) we find the absorption lengths of Lyα ~ 0.2 cm and Lyβ ~ 2 cm, which is shorter than the characteristic scale-length of the variation of the neutral gas density and one can expect trapping of both Lyα and Lyβ radiation. The radiation trapping alters the partition of radiation in different line series (e.g. Lyman and Balmer series). For example, in a transparent plasma, the partition of Lyβ (transition 3 ! 1) and Hα (transition 3 ! 2) intensities depends only on the corresponding spontaneous emission coefficients and their ratio should remain constant. However, if Lyβ is trapped (Hα is not trapped in the edge plasma due to a very small ratio [Hn¼2]/[H]), this partition will change, which exhibits a clear signature of the radiation trapping effects. Such a change in Lyβ and Hα partition was, in particular, observed in the experiments on Alcator-C-Mod tokamak [48] and can be seen in Fig. 2.4a, where the Lyβ intensity decreases with increasing Dα intensity (Dα is the line 3 ! 2 in deuterium). We note that the increase of the Dα

30

2 Atomic Physics Relevant to Fusion Plasmas

intensity at the same plasma temperature implies the increase of the atomic hydrogen density. In addition, in Fig. 2.4b we see that the intensity of Lyα decreases also with increasing Dα intensity, implying Lyα radiation trapping. These results are consistent with the expression (2.22), which predicts that the trapping effects become more pronounced with increasing hydrogen density. So far, in this subsection, we did not distinguish between the hydrogen isotopes. However, in practice, the expression (2.1) has some correction ~me/Mnucl caused by the finite electron to the nucleus, Mnucl, mass ratio. It causes the isotopedependent shift, Δωnucl  ω0(me/Mnucl) ~ 3104ω0, of the resonance frequencies ω0. However, for typical neutral hydrogen temperature ~few eV we have ΔωD ~ 3105ω0 Δωnucl. This implies small overlapping of corresponding line shapes a(ω) of different isotopes and weak interference of their line radiation transport. However, at high plasma density, Stark broadening could exceed the Doppler one and overlapping of the lines could be much stronger. In our estimates related to both absorption length of line radiation and interference of line radiation emitted by different hydrogen isotopes, we have used rather crude models where the details of the line shape a(ω) were ignored. As a matter of fact, in these models, we focused on the transport of photons corresponding to the “center” of the line. However, for the case where the photons having the frequency close to the line center are trapped and, therefore, their transport is inhibited, the main contribution to energy transport via line radiation will come from the “wings” of the line shape [12, 49, 50]. To include the impact of different line broadening eÞ ¼ Vðω e, σ mechanisms, the Voigt line shape, aðω is the convolution of ω , γÞ,pwhich ffiffiffiffiffi 2 2 e , σω Þ ¼ exp e the Gaussian profile, Gðω ω =2σω =σω 2π, (accounting for Doppler  e 2 þ γ2 , (describing, to e , γÞ ¼ π1 γ= ω broadening) and the Lorentzian profile, Lðω some approximation, the Stark effect), Z1 e , σω , γÞ ¼ Vðω

 0
0 
e , σω L ω eω e 0 , γ de G ω ω,

ð2:23Þ

1

e ¼ ω  ω0 , whereas σω and γ are the characteristic widths of the Gaussian where ω and Lorentzian profiles, respectively. Although the Voight approximation is often used in simplified models, more detailed calculations show (e.g. [40, 51].) that Zeeman splitting (which is not included in the expression (2.23)) can play an important role in edge plasma radiation transport. In order to describe properly the line radiation trapping effects on both the energy loss and the atomic physic processes, one needs to consider the photon kinetic equation (e.g. see [12, 49, 50] and the references therein). Here, just for illustration, we consider the case where the radiation trapping is important for a particular line corresponding to the transition in the atom “D” from a quantum level with high energy (“u”)  to!the lower one (“d”). First, we introduce the radiant intensity per solid ! ! angle, Iω r , Ω , which depends on the spatial coordinate r and the photon

2.3 Line Radiation Transport in Edge Plasma

31 !

propagation direction  ! in the solid angle Ω. Then, assuming: i) steady-state approx! imation for Iω r , Ω and ii) that the absorption and emission line shapes are described by the same isotropic function a(ω), we arrive at the following equation: 4π ! c  ∇Iω ¼ aðωÞðAu,d ½Du  þ Iω Bu,d ½Du   Iω Bd,u ½Dd Þ, ħω

ð2:24Þ !

where [D(. . .)] are the densities of atoms in the high and low energy states; c is the photon velocity vector; Bd, u, Bu, d, and Au, d are the Einstein coefficients Bu,d ¼ Au,d

4π 3 c2 , ħω3

Bd,u ¼

gu B , gd u,d

ð2:25Þ

g(. . .) is the statistical weight of the state (. . .), and Au, d can be found from quantum mechanics (e.g. for hydrogen atom it is determined by Eq. (2.2)). The terms in the brackets on the right-hand side of Eq. (2.24) describe, respectively, spontaneous and induced emission of the photons and photon absorption. For edge plasmas, induced emission is small and can be neglected. By integrating the photon absorption term we find the source for the population of the “u” quantum state caused by absorption of the radiation by atoms in the “d” quantum state, R ! SðuradÞ ¼ ð4πÞ1 aðωÞIω dωdΩ Bd,u ½Dd , which enters in Eq. (2.13) for the population of excited states. We note that in edge plasmas, the atom densities and the line shape functions depend on the spatial coordinate. This makes it extremely difficult to find reliable estimates of radiation transport [12, 50]. In addition, the radiation trapping modifies the rates of the atomic processes (recall Eq. (2.13) and, therefore, the population densities appearing in Eq. (2.24). Thus, we see that both radiation transport and the dynamics of the population of the excited states of atomic hydrogen become coupled. Moreover, since the emission of a photon can happen in one region and its absorption in another one, the synergistic effects of radiation transport and atomic processes appear to be non-local. As a result, except very crude models that will be discussed later, realistic solutions of these complex coupled problems can only be found numerically. At this moment, the most advanced numerical package capable of treating both the atomic physics and radiation transport effects in complex edge plasma geometry is built into the EIRENE Monte Carlo code (e.g. see references [40, 52, 53, 77]). Another multi-dimensional code which was used for the radiation transport modeling in edge plasma is Cretin [51, 54–56]. The results of the simulations performed with both EIRENE and Cretin show a reasonably good agreement. Modeling of the JET, Alcator-C-Mod, and ITER plasmas with EIRENE shows that radiation transport plays a crucial role in hydrogen ionization processes in optically thick (large product of the hydrogen atom density and the spatial scale length) devices. For example, while in JET, the radiation-stimulated ionization contributes only 10–20% to the total ionization source, in more optically thick

32

2 Atomic Physics Relevant to Fusion Plasmas

Alcator-C-Mod, this number increases to 30%, and in ITER, depending on the regime, it rises to 60–90% [40]. Unfortunately, self-consistent modeling of radiation transport and the atomic physics effects is very computationally expensive, so that in many cases the radiation trapping effects are ignored. Partly it is justified by the fact that some features important for the reactor design, such as the heat load on the divertor targets, appear to be quite insensitive to the outcome of the radiation trapping effects (e.g. see [53]. However, these effects appear to be crucial for proper modeling of some particular phenomena observed in experiments (e.g. modeling of MARFE in JET tokamak where it was found that 90% of Lyα and 70% of Lyβ lines are trapped [57]) and they are also often important for interpretation of the diagnostic (e.g. spectroscopic) data [48].

2.4

Application of CRM to Edge Plasma Relevant Species

In this sub-section, we consider the results of the application of the CRM to different atomic and molecular species relevant to the edge plasma in magnetic fusion devices.

2.4.1

Hydrogen

We start with hydrogen atoms and molecules. An impact of radiation trapping on the atomic rate constants of hydrogen species, in general, depends on the particular distribution of the plasma and neutral gas parameters. However, just to taste a flavor of the radiation trapping effects, one can consider a model where hydrogen radiation in some particular lines is completely trapped. This case corresponds to a CRM where spontaneous decay from some particular quantum states is turned off, which mimics quick reabsorption of the resonance photons [26]. In Fig. 2.5, one can find the dependence of both the atomic hydrogen ionization H KH ion and the EIR Krec rate constants on the temperature for different plasma densities for the case of fully transparent plasma (Fig. 2.5a) and suppressed spontaneous decay from the levels n 2 to the ground state (Fig. 2.5b), which mimics the complete opacity conditions for the Lyman lines, obtained from the SOLPS database [42]. As one could expect, both the increase of the electron density above ~1014cm3 and the suppression of the spontaneous decay of the transition 2 ! 1, which enhance the population of excited states, are boosting the ionization rate constant. In Fig. 2.5a we also plot the charge-exchange rate constants Kðcx1Þ (which will be used for our further considerations) for different hydrogen isotopes assuming that the electron and ion/neutral temperatures are the same. However, in practice, the analysis of edge plasma and neutral hydrogen transport requires more detailed knowledge of the

2.4 Application of CRM to Edge Plasma Relevant Species

33

H Fig. 2.5 Dependence of the hydrogen ionization, KH ion, and EIR, Krec, rate constants on the electron temperature for different electron densities for the case of fully transparent plasma (Fig. 2.5a) and suppressed spontaneous decay from the levels n 2 to the ground state (Fig. 2.5b), which mimics the complete opacity conditions for the Lyman lines. In Fig.2.5a, the charge-exchange rate constant Kðcx1Þ is shown for different hydrogen isotopes assuming that the electron/ion/neutral temperatures are equal

elastic collisions involving the ions, atoms, and molecules of the hydrogenic species. The relevant cross-sections (including momentum and charge transfer) can be found from both semi-classical and fully quantal calculations (e.g. see [58, 59]). Both the ionization and recombination rates are necessary for the evaluation of the plasma recycling processes. However, the ionization of neutrals, which is accompanied by neutral gas excitation and following radiation, results in plasma energy dissipation. Thus, as it was noted in Chap. 1, to maintain plasma recycling, the recycling region must be supplied with power. To assess the energy dissipation caused by plasma recycling, it is convenient to introduce the hydrogen “ionization cost” [60], EH ion , which corresponds to plasma energy dissipation per an ionization event: ( EH ion

¼

X X ðionÞ ðradÞ ½Hn  En Kn!cont ne þ ΔEnk νn!k n

!)




½Hne KH ion

1

,

ð2:26Þ

k 1) for stationary plasma parameters [78]:  1
 1 dξZ transp Zþ1 ξ þ K ξ  ξ n τ ¼ 0, ¼  KZion þ KZrec ξZ þ KZ1 e Z1 Zþ1 Z ion rec imp ne dt ð2:31Þ where KZion and KZrec are the corresponding ionization and recombination rate constants, whereas ξZ ¼ nZ/nimp and nimp ¼ ∑ZnZ are the partition of impurity density over the ionization states and the total impurity density. Naturally, KZion ¼ 0 for the highest charge state and KZrec ¼ 0 for the neutrals. In order to sustain the total impurity density, the transport term in Eq. (2.31) for neutrals (Z ¼ 0) is replaced with the corresponding source term. Note that writing ionization balance for the impurity

2.4 Application of CRM to Edge Plasma Relevant Species 10–30 10–31 Limp(Te), (Mw/m3)

Fig. 2.14 Cooling rates Limp(Te) for different impurities calculated in “coronal approximation”. (Reproduced with permission from [79], © IAEA 1999)

41

W

10–32

Kr Ar

10–33 Ne 10–34

C Be

10–35 He

10–36 10–37 1

H 10

100 1000 Te (eV)

104

ions this way, we imply that the transport processes simply remove the charged particles, replacing them by neutrals. Although in practice impurity transport is much more complex, such an approach gives a simple and useful estimate for the impurity radiation loss. By solving the algebraic Eq. (2.31) and neglecting threebody recombination of the impurity ions (which is only important at low temperatures where the impurity radiation loss is insignificant), we find that ξz only depends on ne, Te, and ne τtransp imp . As a result, we can write the following expression for the volumetric plasma energy loss due to impurity radiation, Wrad imp :  e Z ξZ ne , Te , ne τtransp K cool imp Z  eimp ne , Te , ne τtransp :

ne nimp L imp

Wrad imp ¼ ne nimp

X

ð2:32Þ

e Taking τtransp imp ! 1 and neglecting dependence of Limp on electron density, we come to the so-called “coronal approximation” for the impurity radiation loss, eimp Limp ðTe Þ , which is often used in analytic and semi-analytic models. The L function Limp(Te) is shown in Fig. 2.14 for most common impurities in fusion plasmas. transp The finite τtransp imp implies the volumetric source of impurity, Simp ¼ nimp =τimp . By combining Wrad imp and Simp, it is useful to introduce an effective neutral impurity “ionization cost”, Eimp ion :  transp e transp rad L Eimp ¼ W =S ¼ n τ n , T , n τ , imp e imp imp e e e imp imp ion

ð2:33Þ

which is just an analog of the hydrogen ionization cost. imp In Fig. 2.15 one can find the dependencies of Wrad imp (a) and Eion (b) for N on electron temperature for the electron density ne ¼ 1014cm3 and different values of

42

2 Atomic Physics Relevant to Fusion Plasmas

imp Fig. 2.15 Dependence of Wrad imp (a) and Eion (b) for N on electron temperature for the electron 14 3 10 9 8 3 density ne ¼ 10 cm and different values of ne τtransp imp for 10 , 10 , and 10 s cm , calculated from the ADAS database

imp Fig. 2.16 Dependence of Wrad imp (a) and Eion (b) for Ne on electron temperature for the electron 10 9 8 3 density ne ¼ 1014cm3 and different values of ne τtransp imp for 10 , 10 , and 10 s cm , calculated from the ADAS database

ne τtransp imp calculated from the ADAS database. Similar dependencies, but for Ne, are imp shown in Fig. 2.16. As one can see from Figs.2.15 and 2.16, in both cases Eion ~ few 10 3 keV if the impurity ion confinement is good ne τtransp . The estimate imp ∼10 s  cm Eimp ion ∼ few keV is consistent with the experimental and computational results from Refs. [80, 81], where the so-called impurity “radiation potential” (having the physical meaning somewhat similar to Eimp ion ) was introduced.

References

43

According to [80], the “radiation potential” for carbon is about 3 keV at Te ~ 20 eV and falls to ~1 keV at Te ~ 60 eV, showing the trend consistent with Fig. 2.15. H Comparing Eimp ion ∼ few keV and Eion ∼ 30 eV, we can conclude that impurity starts to dominate the energy loss from edge plasma when the impurity fraction in the total flux of neutrals into the plasma from the PFCs exceeds ~1%. Equation (2.31) can be extended by incorporation of charge-exchange between the impurity ion and hydrogen atom Az+1 + H ! AZ(n) + H+ (e.g. see [82]), which eimp one more free parameter, the ratio [H]/ne. This effect introduces in the function L could also increase the impurity radiation loss at high electron temperatures [20, 82]. However, neutral hydrogen is not abundant where the temperature is high.

2.5

Conclusions

In conclusion to this chapter, it would be fair to say that our knowledge of atomic processes in the edge plasma is reasonable with respect to both understanding of the main physical processes and completeness of the data needed to model and diagnose the most critical processes in the edge plasma of fusion devices. It is not surprising because our studies in this area are based on the century-old effort of a few generations of scientists. Nonetheless, some additional data would be needed for the case where the transport processes in edge plasma should be described kinetically. However, incorporation of plasma kinetic processes and radiation transport effects into the edge plasma modeling tools is beyond current computer capabilities. We notice also that the addition of the radiation-induced transitions between different quantum states (e.g. for hydrogen atoms) for opaque regimes, relevant for detached divertor plasmas in fusion reactors such as ITER, makes the simulation of the edge plasma very “expensive” computationally. Therefore, there are only a few cases where such effects were taken into accounted.

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impurities in dynamic finite density plasmas: I. The generalized collisional–radiative model for light elements. Plasma Phys. Controlled Fusion 48, 263–293 (2006) 31. K. Sawada, M. Goto, Rovibrationally resolved time-dependent collisional-radiative model of molecular hydrogen and its application to a fusion detached plasma. Atoms 4, 29 (2016) 32. R.K. Janev, C.J. Joachain, N.N. Nedeljković, Resonant electron transfer in slow collisions of protons with Rydberg hydrogen atoms. Phys. Rev. A 29, 2436–2469 (1984) 33. S.I. Krasheninnikov, A.S. Kukushkin, Physics of ultimate detachment of a tokamak divertor plasma. J. Plasma Phys. 83, 155830501 (2017) 34. V.A. Abramov, V.S. Lisitsa, A.Y. Pigarov, Changes in effective charge-exchange cross-sections in a plasma. Sov. Phys. JTP Lett. 42, 356–359 (1985) 35. P. Helander, S.I. Krasheninnikov, P.J. Catto, Fluid equations for a partially ionized plasma. Phys. Plasmas 1, 3174–3180 (1994) 36. S.I. Krasheninnikov, A.Y. Pigarov, Kinetics of atoms in multistep excitation processes in plasma. Contrib. Plasma Phys. 28, 345–346 (1988) 37. S.I. Krasheninnikov, V.S. Lisitsa, A.Y. Pigarov, Charge exchange in a divertor plasma with excited particles. Sov. J. Plasma Phys. 14, 612–616 (1988) 38. V.P. Zhdanov, Dielectronic recombination, in Reviews of Plasma Physics, ed. by M. A. Leontovich, B. B. Kadomtsev, vol. 12, (Consultants Bureau, New York, 1987), p. 103 39. Y. Hahn, Electron-ion recombination processes in plasmas, in Atomic and Molecular Processes in Fusion Edge Plasmas, ed. by R. K. Janev, (Plenum, New York, 1995), p. 91 40. V.S. Lisitsa, M.B. Kadomtsev, V. Kotov, V.S. Neverov, V.A. Shurygin, Hydrogen spectral line shape formation in the SOL of fusion reactor plasmas. Atoms 2, 195–206 (2014) 41. See website http://open.adas.ac.uk/ 42. D. Reiter, The EIRENE Code User Manual. http://www.eirene.de. (2017) 43. S.I. Krasheninnikov, A.Y. Pigarov, Superhigh density operating conditions for a poloidal divertor in a tokamak reactor. Nucl. Fusion Suppl. 3, 387–394 (1987) 44. R. Marchand, J. Lauzon, Hydrogen recycling with multistep and resonance line absorption effects. Phys. Fluids 4, 924–933 (1992) 45. D. Reiter, V. Kotov, P. Börner, K. Sawada, R.K. Janev, B. Küpers, Detailed atomic, molecular and radiation kinetics in current 2D and 3D edge plasma fluid codes. J. Nucl. Mater. 363–365, 649–657 (2007) 46. T. Fujimoto, Plasma Spectroscopy, The International Series of Monographs on Physics, 123 (Oxford University Press, Oxford, 2004) 47. V.I. Kogan, V.S. Lisitsa, G.V. Sholin, Spectral line broadening in plasma, in Reviews of Plasma Physics, ed. by B. B. Kadomtsev, vol. 13, (Consultants Bureau, New York, 1987), p. 261 48. J.L. Terry, B. Lipschultz, A.Y. Pigarov, S.I. Krasheninnikov, B. LaBombard, D. Lumma, H. Ohkawa, D. Pappas, M. Umansky, Volume recombination and opacity in Alcator C-Mod divertor plasmas. Phys. Plasmas 5, 1759–1766 (1998) 49. H.R. Griem, Principles of Plasma Spectroscopy (Cambridge University Press, Cambridge, 1997) 50. V.A. Abramov, V.I. Kogan, V.S. Lisitsa, Radiative transfer in plasma, in Review of Plasma Physics, ed. by M. A. Leontovich, B. B. Kadomtsev, vol. 12, (Consultants Bureau, New York, 1987), p. 151 51. M.L. Adams, H.A. Scott, R.W. Lee, J.L. Terry, E.S. Marmar, B. Lipschultz, A.Y. Pigarov, J.P. Freidberg, Application of magnetically-broadened hydrogenic line profiles to computational modeling of a plasma experiment. J. Quant. Spectroc. Radiat. Transf. 71, 117–128 (2001) 52. D. Reiter, M. Baelmans, P. Börner, The EIRENE and B2-EIRENE codes. Fusion Sci. Technol. 47, 172–186 (2005) 53. V. Kotov, D. Reiter, A.S. Kukushkin, H.D. Pacher, P. Bőrner, S. Wiesen, Radiation absorption effects in B2-EIRENE divertor modelling. Contrib. Plasma Phys. 46, 635–642 (2006) 54. H.A. Scott, Cretin—a radiative transfer capability for laboratory plasmas. J. Quant. Spectrosc. Radiat. Transf. 71, 689 (2001)

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55. H.A. Scott, M.L. Adams, Incorporating line radiation effects into edge plasma codes. Contrib. Plasma Physics 44, 51–56 (2004) 56. M.L. Adams, H.A. Scott, Effect of hydrogen line radiation on the divertor target plate incident heat flux. Contrib. Plasma Physics 44, 262–267 (2004) 57. V. Kotov, D. Reiter, Formation of a natural X-point multifaceted asymmetric radiation from the edge in numerical simulations of divertor plasmas. Plasma Phys. Controlled Fusion 54, 082003 (2012) 58. P.S. Krstic, D.R. Schultz, Elastic scattering and charge transfer in slow collisions: Isotopes of H and H+ colliding with isotopes of H and with He. J. Phys. B Atomic Mol. Phys. 32, 3485–3509 (1999) 59. P.S. Krstic, Inelastic processes from vibrationally excited states in slow H++H2 and H+H2 collisions: Excitations and charge transfer. Phys. Rev. A 66, 042717 (2002) 60. R.K. Janev, D.E. Post, W.D. Langer, K. Evans, D.B. Heifetz, J.C. Weisheit, Survey of atomic processes in edge plasmas. J. Nucl. Mater. 121, 10–16 (1984) 61. M.J. Brunger, S.J. Buckman, Electron–molecule scattering cross-sections. I. Experimental techniques and data for diatomic molecules. Phys. Rep. 357, 215–458 (2002) 62. D.E. Atems, J.M. Wadehra, Vibrational excitation of H2 and HCl by low-energy electron impact. An isotope scaling law. Chem. Phys. Lett. 197, 525–529 (1992) 63. M. Capitelli, R. Celiberto, F. Esposito, A. Laricchiuta, K. Hassouni, S. Longo, Elementary processes and kinetics of H2 plasmas for different technological applications. Plasma Sources Sci. Technol. 11, A7–A25 (2002) 64. F. Gaboriau, J.P. Boeuf, Chemical kinetics of low pressure high density hydrogen plasmas: Application to negative ion sources for ITER. Plasma Sources Sci. Technol. 23, 065032 (2014) 65. C.E. Treanor, J.W. Rich, R.G. Rehm, Vibrational relaxation of anharmonic oscillators with exchange-dominated collisions. J. Chem. Phys. 48, 1798–1807 (1968) 66. J.M. Wadehra, Dissociative attachment to rovibrationally excited H2. Phys. Rev. A 29, 106–110 (1984) 67. M. Bacal, M. Wada, Negative hydrogen ion production mechanisms. Appl. Phys. Rev. 2, 021305 (2015) 68. A.S. Kukushkin, S.I. Krasheninnikov, A.A. Pshenov, D. Reiter, Role of molecular effects in divertor plasma recombination. Nucl. Mater. Energy 12, 984–988 (2017) 69. S.I. Krasheninnikov, A.Y. Pigarov, D.J. Sigmar, Plasma recombination and divertor detachment. Phys. Lett. A 214, 285–291 (1996) 70. D. Reiter, C. May, M. Baelmans, P. Börner, Non-linear effects on neutral gas transport in divertors. J. Nucl. Mater. 241-243, 342–348 (1997) 71. R.K. Janev, Alternative mechanisms for divertor plasma recombination. Phys. Scr. T96, 94–101 (2002) 72. N. Ohno, N. Ezumi, S. Takamura, S.I. Krasheninnikov, A.Y. Pigarov, Experimental evidence of molecular activated recombination in detached recombining plasmas. Phys. Rev. Lett. 81, 818–821 (1998) 73. A. Tonegawa, M. Ono, Y. Morihira, H. Ogawa, T. Shibuya, K. Kawamura, K. Takayama, Observation of molecular assisted recombination via negative ions formation in a divertor plasma simulator, TPDSHEET-IV. J. Nucl. Mater. 313-316, 1046–1051 (2003) 74. A. Okamoto, S. Kado, K. Sawada, Y. Kuwahara, Y. Iida, S. Tanaka, Contribution of hydrogen molecular assisted recombination processes to population of hydrogen atom in divertor simulator MAP-II. J. Nucl. Mater. 363-365, 395–399 (2007) 75. B. Lipshchultz, Private communication, 2019 76. V. Sizyuk, A. Hassanein, Heat loads to divertor nearby components from secondary radiation evolved during plasma instabilities. Phys. Plasmas 22, 013301 (2015) 77. A.A. Pshenov, A.S. Kukushkin, E.D. Marenkov, S.I. Krasheninnikov, On the role of hydrogen radiation absorption in divertor plasma detachment. Nucl. Fusion 59, 106025 (2019) 78. R.A. Hulse, Numerical studies of impurities in fusion plasmas. Nucl. Technol. Fusion 3, 259–272 (1983)

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79. ITER, Physics basis, chapter 4: Power and particle control. Nucl. Fusion 39, 2391–2469 (1999) 80. U. Samm, P. Bogen, H.A. Claassen, H. Gerhauser, H. Hartwig, E. Hintz, Y.T. Lie, A. Pospieszczyk, D. Rusbüldt, B. Schweer, Influence of impurity radiation losses on plasma edge properties in TEXTOR. J. Nucl. Mater. 176–177, 273–277 (1990) 81. M.Z. Tokar, F.A. Kelly, The role of plasma–wall interactions in thermal instabilities at the tokamak. Phys. Plasmas 10, 4378–4386 (2003) 82. S.L. Allen, M.E. Rensink, D.N. Hill, R. Wood, D. Nilson, B.G. Logan, R. Stambaugh, T.W. Petrie, G.M. Steabler, M.A. Mahdavi, R. Hulse, R.B. Campbell, A design study for an advanced divertor for DIII-D and ITER: The radiative slot divertor. J. Nucl. Mater. 196–198, 804–809 (1992)

Chapter 3

Plasma-Material Interactions in Magnetic Fusion Devices

Abstract Diverse plasma interactions with the materials of the plasma facing components (PFCs) are very important in magnetic fusion devices. They determine the PFC erosion and, therefore, the PFC’s lifetime, plasma contamination with unwanted impurities, retention of radioactive tritium, and often result in a strong degradation of the material properties. Unfortunately, this area of research is still underdeveloped and our understanding of complex processes involving plasmamaterial interactions is still far from completeness. Here the current status of our knowledge of plasma interactions with the PFC materials is presented.

The choice of material for a fusion reactor is a complex problem, which concerns both the structural and PFC materials facing different requirements such as material resilience to high heat fluxes and neutron fluence (e.g. see [1] and the references therein). Here we only consider the PFC materials. The plasma-material interaction for fusion-relevant conditions is a relatively young area of research where our understanding of the physics involved is still incomplete. Therefore, in this chapter, we will focus on the issues that, we believe, are the most important for the description of the edge plasma processes on the one hand and, on the other hand, the physics of which is reasonably well understood and verified. In addition, we will also discuss some gaps in our understanding/description of plasma material interactions in magnetic fusion devices. Usually, it is assumed that the major issues of plasma interaction with the materials of the PFCs are related to the PFC erosion (determining the PFC’s lifetime) and tritium retention. For example, carbon had been dismissed as the PFC material for ITER based on the assessment of tritium retention [2], which was later, to some extent, confirmed experimentally [3]. However, there is still a need for both better assessment of tritium retention in the materials potentially suitable for the PFCs and understanding of other phenomena related to plasma-material interaction in future reactors. The processes occuring at the surface and within some depth beneath the surface of the PFCs in fusion devices are very complex and diverse. They determine a wide spectrum of different phenomena including erosion of the PFC material, plasma contamination with impurity, plasma recycling and tritium retention, dust formation, © Springer Nature Switzerland AG 2020 S. Krasheninnikov et al., On the Edge of Magnetic Fusion Devices, Springer Series in Plasma Science and Technology, https://doi.org/10.1007/978-3-030-49594-7_3

49

50

3 Plasma-Material Interactions in Magnetic Fusion Devices

Fig. 3.1 Schematic representation of hydrogen-helium plasma interaction with (a) virgin crystal structure of the PFM and (b) modified material structure after plasma exposure

Fig. 3.2 (a) optical image of carbon co-deposits on the leading edge of Tore Supra neutralizer. (Reproduced with permission from [4], © Elsevier 2009) and (b) SEM picture of “nano-fuzz” rods growing on tungsten surface under helium irradiation. (Reproduced with permission from [5], © IAEA 2009)

modification of the surface morphology and the physical properties of the undersurface material, etc. Many of these processes depend not only on the particular material but also on the temperature, the constituency of the plasma species and their energies. One of the complications in the study of the physics of plasma-material interactions for fusion-relevant conditions is a strong modification of the original material during its exposure to plasma in the fusion devices (see Fig. 3.1). This can be related to many factors including strong saturation of the near-surface material layer with hydrogen/helium caused by plasma bombardment, formation of co-deposited layers (due to transport of the eroded material) which have physical properties different from the original ones (e.g. amorphous rather than crystal structure), melting with further re-crystallization resulting in the change of the grain size, etc. Different experimental and theoretical techniques are used for the study of particular features of the phenomena related to the PFM in a fusion plasma environment. For example, on the experimental side, the optical and different kind of electron microscope (e.g. tunneling, TEM, and secondary emission, SEM, electron microscopes) images of material surfaces exposed to the plasma are often used to monitor the surface morphology modification (e.g. see Fig. 3.2), whereas the temperature desorption spectra (TDS) and nuclear reaction analysis (NRA) along with other methods are used both to infer the concentrations of hydrogen and helium

3 Plasma-Material Interactions in Magnetic Fusion Devices

51

Fig. 3.3 TDS data showing D release from ITER-grade tungsten irradiated by 38 eV deuterium ions with the flux of 1022 m
2/s and fluences 1026 (a) and 1027 D/m2 (b) at various temperatures. (Reproduced with permission from [6], © IOP Publishing 2014)

trapped in the plasma-facing materials and to shed some light on the possible trapping mechanisms (e.g. see Fig. 3.3). We notice that most of the experimental data on plasma-material interactions come from specially designed experiments performed on relatively small-scale linear devices with well-characterized plasma parameters. However, tokamak experiments provide the data on plasma-material interactions for the “real” tokamak plasma environment, which is often characterized by violent anomalous transport phenomena in multi-species plasma and a long-range migration of the eroded material. On the theoretical side, the approaches to study of the PFM-related physics range from the first-principle codes utilizing the density functional theory (DFT) [7] (e.g. QUANTUM-ESPRESSO [8], VASP [9]) to the simulations of particular, relatively small-scale features with different versions of Molecular Dynamic (MD) and Monte Carlo (MC) codes (e.g. LAMMPS [10], TRYDIN [11]), and to the study of macroscopic phenomena with the codes based on the continuum reaction-diffusion approximation (e.g. TMAP [12, 13], FACE [14]) and semianalytic models. Whereas the DFT-based simulations are capable of treating only a few tens of atoms and are usually used to determine the binding energies of the hydrogen and helium atoms in different lattice defects in fusion-relevant materials and to infer the inter-particle interaction potentials, the MD simulations employing complex multi-particle interaction potentials can describe phenomena associated

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3 Plasma-Material Interactions in Magnetic Fusion Devices

Fig. 3.4 MD simulations of growing helium nanobubble in tungsten emitting dislocation loops. Helium atoms are in red and displaced tungsten atoms are in grey colors. (Reproduced with permission from [15], © Elsevier 2015)

with the dynamics of millions of particles (e.g. the formation of the helium nanobubbles in tungsten, see Fig. 3.4), and the reaction-diffusion-based codes are used to simulate such macroscopic features as hydrogen and helium transport and trapping in the fusion-related materials and to interpret the TDS data. However, in practice, application of even MD simulations to the study of many important practical problems (e.g. nucleation and growth of the fuzz shown in Fig. 3.2b) is beyond both the present and near-future computer capabilities. In addition, interatomic potentials used in the MD simulations do not always result in the physically meaningful outcomes (e.g. see [16]). On the other hand, the reactiondiffusion-based codes, which can be used for the long-time, large-scale simulations, rely on the transport properties of the species of interest (e.g. hydrogen and helium) and the rate constants of different “reactions”. These hydrogen and helium transport models, as well as the rate constants, which are either taken from some ad hoc assumptions or deduced from the MD simulations, in most cases, do not allow for the effects of lattice stress, which can play an important role in many different phenomena. As a result, understanding of the physics involved in the fusion plasma-materialrelated phenomena in many cases is still rather poor. Therefore, in this chapter, we will consider only the most generic features of the plasma interactions with the PFC materials in a magnetic fusion environment and give short reviews of some interesting phenomena, even though their physics might be not entirely clear yet. Further details of the current research activities, considering both solids (beryllium and tungsten) and liquids (lithium, tin) as potential materials for the PFCs, can be found in relevant review papers (e.g. see [6, 17–25]) and original journal publications.

3.1 Reflection of Plasma Particles Impinging on Material Surfaces and Sputtering of. . .

3.1

53

Reflection of Plasma Particles Impinging on Material Surfaces and Sputtering of Plasma-Facing Materials

Both reflection and absorption of plasma particles (in particular, hydrogen and helium) impinging on material surfaces result in the edge plasma energy and particle sinks and, therefore, are crucially important for the plasma recycling process. In addition, the accompanying processes of sputtering of the plasma-facing materials are essential for plasma contamination with impurity and erosion of the PFCs. We start with the particle and energy reflection coefficients. Whereas the particle reflection coefficient, RN, is just the probability of particle impinging on the target surface to be reflected back, the energy reflection coefficient, RE, needs some clarification. The reflected particles usually have a broad energy distribution function having an average value hERi. For example, for the case where a light projectile strikes a target of heavy elements (e.g. tungsten), some reflected particles have the energy close to the initial projectile energy, Ep. Therefore, for accurate treatment of the energy of the reflected particles with MC neutral codes used in the edge plasma studies, one should consider the whole energy spectrum of the reflected particles. However, for some crude estimates one can use the average energy hERi, which can be expressed in terms of RN, RE, and Ep as follows: hER i ¼ Ep RE =RN :

ð3:1Þ

The energy dependence of the particle and energy reflection coefficients of both hydrogen and helium within the energy range of interest (from ~10 eV to ~1 keV) for the case of normal incidence onto the target can be described by the following expression [26]:

RN=E


 N=E N=E A1 ℓn A2 εp þ e ¼ , N=E N=E N=E  A N=E  A 1 þ A3 εp 4 þ A5 εp 6

ð3:2Þ

where εp is the Thomas-Fermi reduced energy of the projectile εp ¼ 3:255  10
2

Mt =Mp 1 þ Mt =Mp




Ep ½eV 2=3

Zp Zt Z2=3 p þ Zt

1=2 ,

ð3:3Þ

Mp (Mt) and Zp (Zt) are the mass and charge of the projectile (target) nuclei and Ep N=E [eV] is the projectile energy in eV. The coefficients Að...Þ taken from [26] can be found in Table 3.1. However, the expression (3.2) fails to reproduce the reflection coefficients at low Ep (below ~ a few eV). For example, from Eq. (3.2) and the data from Table 3.1, it follows that the reflection coefficient of a low energy 4He from tungsten is ~0.8.

54

3 Plasma-Material Interactions in Magnetic Fusion Devices N=E

Table 3.1 Fitting parameters Að...Þ for both the particle and energy reflection coefficients. (Reproduced with permission from [19], © Springer 2007) N=E

RN(Mt/Mp > 20) RE(Mt/Mp > 20) RN(Mt/Mp ¼ 3) RE(Mt/Mp ¼ 3)

N=E

A2 21.41 27.16 2.985 3.848

A1 0.8250 0.6831 0.3680 0.2058

N=E

A3 8.606 15.66 7.122 19.07

N=E

A4 0.6425 0.6598 0.5802 0.4872

N=E

A5 1.907 7.967 4.211 15.13

N=E

A6 1.927 1.822 1.597 1.638

However, a closer consideration shows that in order to penetrate into the tungsten lattice and have a chance to be trapped there, the helium particle has to overcome a potential barrier of ~6 eV [27]. Therefore, the helium reflection coefficient from a pure tungsten target should be equal to unity for the helium kinetic energy below this potential barrier. The situation with hydrogen impinging on the tungsten target is more complex since hydrogen can be chemically adsorbed on the tungsten surface (e.g. see [28] and the references therein). Next, we consider the so-called physical sputtering of the targets. Physical sputtering assumes no formation of chemical bonds between the projectile and the target particles. The probability of the projectile to sputter a target particle (sputtering yield) depends on both the projectile energy Ep and the incidence angle. For the normal incidence, the yield of physical sputtering Yph(Ep) is given by the following expression [19] 



Yph Ep ¼

  Asp 1 f εp



Asp Ep =Eth
1 2  Asp2 , Asp þ E =E
1 p th 3

ð3:4Þ

where Eth is the sputtering threshold energy, Asp ð...Þ are the fitting parameters and the function f(εp) is given by the following formula   f εp ¼

  0:5ℓn 1 þ 1:2288εp  0:1504 : pffiffiffiffiffi εp þ 0:1728 εp þ 0:008 εp

ð3:5Þ

We notice that the surface binding energy, Es, of Be, W, and Li has values of 3.38, 8.68 and 1.67 eV respectively, which are, as follows from Table 3.2, significantly lower than Eth, in particular, for the case of a large mass ratio of the target to projectile atoms. This is not surprising, since according to the binary collision approximation, the maximum relative energy transfer between the two particles is 4MpMt/(Mp+Mt)2, which is small for Mp  Mt (e.g. for the collisions of hydrogenic species with tungsten atoms). Both the reflection coefficients and the sputtering yields depend also on the projectile incidence angle, ϑp. Whereas the particle reflection coefficient increases with increasing ϑp, the yield of physical sputtering, having a minimum at the normal incidence, initially increases also with increasing ϑp, reaches a maximum at some

3.1 Reflection of Plasma Particles Impinging on Material Surfaces and Sputtering of. . .

55

Table 3.2 Threshold energy Eth and the fitting parameters Asp ð...Þ for the physical sputtering of Be, W and Li by H, D, T, 4He, self-sputtering, N, and Ne. (Reproduced with permission from [19], © Springer 2007) H

Y, sputtering yield (atoms/ion)

Be Eth(eV) Asp 1 Asp 2 Asp 3 W Eth(eV) Asp 1 Asp 2 Asp 3 Li Eth(eV) Asp 1 Asp 2 Asp 3

D

4

T

He

Self

N

Ne

14.3 0.0564 1.5147 0.8007

9.5 0.1044 1.9906 1.7575

9.4 0.1379 1.5660 2.0794

12.3 0.3193 1.6989 1.7545

17 0.8241 1.3437 2.0334

16.5 0.9334 2.5368 5.2833

22.8 1.8309 1.9400 2.5474

457 0.0075 1.2046 1.0087

228.8 0.0183 1.4410 0.3583

153.9 0.0419 1.5802 0.2870

120.6 0.1151 1.7121 0.1692

25 18.6006 3.1273 2.2697

45.5 1.4389 2.0225 0.0921

38.6 2.5520 1.9534 0.0828

5.56 0.0833 1.4705 0.9540

4.6 0.1321 1.2091 1.4358

4.86 0.1629 0.9741 1.8839

6.5 0.3617 1.2501 1.9370

5.5 0.5159 1.7546 8.2237

–

100

100 4 keV He → Mo

50 keV He → Mo

10–1

10–1

Rotin79a, Bay79a, Eckstein93 fit to calc. values

10–2 0

–

30

60

10–2 0 90 angle of incidence (degrees)

Bohdansky62, Eckstein93 fit to calc. values

30

60

90

Fig. 3.5 Sputtering yields of molybdenum by 4 keV (left) and 50 keV (right) helium ions as a function of the incidence angle. The dots are the experimental data and the solid lines are the fits. (Reproduced with permission from [19], © Springer 2007)

angle (ϑp)max < π/2 and then drops to a virtually zero value. Although the particular angular dependence of Yph(Ep, ϑp) is significant, rather complex and sensitive to Ep (e.g. see Fig. 3.5 and Ref. [19] for further details), in practice, for the edge plasma conditions where the distribution function of the particles impinging on the target has a broad angular and energy spread, this effect does not result in any specific feature of the average sputtering yield. In addition, the surface roughness also smears out the angular dependence of sputtering.

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3 Plasma-Material Interactions in Magnetic Fusion Devices

Fig. 3.6 Temperature dependence of the sputtering yields of Li (left) and Be (right) targets from both experimental data and theoretical model based on the “adatom” concept. (Reproduced with permission from [32], © AIP Publishing 2004)

Although physical sputtering should not depend on the target temperature, the experimental data undoubtedly show that starting with some elevated temperature, the yield of physical sputtering of different solid (e.g. Ag, Bi, Cu, Ge, Zn, [29] and the references therein, W [30], C [31], Be [32]) and liquid (e.g. Li [33, 34], Ga [35]) materials strongly increases with increasing target temperature. Theoretical explanations of this effect are ranging from different variations of an old idea of formation of a “hot spot” around the striking point of the projectile [36] to the creation of the Frenkel pairs (in a carbon target) [37] with further diffusion of the interstitial carbon atoms to the surface and their subsequent evaporation. However, for different reasons, these models are unable to fit the available experimental data. It seems that at this moment, the model based on the idea of “adatoms” (the target atoms that are “splashed” to the target surface in the course of the projectile-target interactions) [32] shows the best agreement with the experimental data for both the solid and liquid materials (see Fig. 3.6). Whereas the energy distribution of the atoms sputtered via the “standard” physical sputtering process follow the Thompson distribution [38] f Th ðEÞ /

E , ðE þ E s Þ3

ð3:6Þ

the atoms sputtered due to the temperature-induced effects have the energy dependence determined by the target temperature (e.g. see [39]). The sputtering processes discussed so far do not involve the possible formation of chemical bonds between the projectile and the target atoms. However, “chemical” effects, resulting in the so-called chemical sputtering, could be for some cases the dominant sputtering mechanism. In particular, irradiation of a carbon target with hydrogen, even at low energies, results in the formation of volatile hydrocarbon molecules, which could dominate the carbon erosion (e.g. see [19] and the references

3.2 Basic Features of Hydrogen/Helium Transport in Plasma-Facing Materials

57

therein). However, we notice that the formation of the chemical bonds between the projectile and the target atoms does not necessarily result in chemical sputtering. For example, in Ref. [40] it was found that beryllium irradiation with hydrogen results in the formation of beryllium hydride, but, unlike hydrocarbons, it is not volatile and, therefore, does not contribute to the chemical erosion mechanism. We also note that both the reflection coefficients and the sputtering yields are sensitive to the constituency of the very first atomic layers of the target, which can be altered by mixing/implantation of different target materials (e.g. Be and W) or by the saturation of these layers with hydrogen or helium.

3.2

Basic Features of Hydrogen/Helium Transport in Plasma-Facing Materials

Once the hydrogen and/or helium atoms penetrate into the PFC material lattice, they become subject to complex multi-body interactions with the lattice atoms. For illustration purposes, these interactions are often portrayed as motion of the hydrogen or helium atoms through effective potential structures (see Fig. 3.7). Such a motion of the hydrogen or helium atoms is assumed to be “strongly damped”, which implies that the total particle energy (kinetic plus potential) is not conserved because of the multi-body nature of the particle interactions. The spacing between the local, relatively small (~0.4 eV) minima of the potential energy is determined by the lattice arrangement (recall that it has 3D structure). Due to thermal effects, the particles can move from one minimum to another in a random way, so in the simplest case, dynamics of the hydrogen and helium atoms can be described by a diffusion process with the diffusivity D / exp (
Edif/T), where Edif is the “depth” of the local potential well, which depends on both the material and the diffusing particle He (e.g. for tungsten we have EH dif  0:25 eV and Edif  0:15 eV ). However, in practice, the lattice arrangement is not perfect and has some defects (e.g. vacancies, dislocations). In most cases, the effective potential well associated with these lattice defects, Etr, for both the hydrogen and helium atoms is significantly deeper than Edif. As a result, the atoms become virtually “trapped” in these deep

Fig. 3.7 Effective hydrogen (left) and helium (right) potential structures illustrating hydrogen and He helium transport in tungsten, where Edif is the diffusion activation energy ðEH dif  0:25 eV, Edif  He  2:5 eV, E  6 eV), 0:15 eV), Etr is the “trapping” energy, Eb is the surface energy barrier (EH b b e Φlayer, nano-tendrils with diameter ~10 nm with some embedded helium nano-bubbles start to grow from the surface (see Fig. 3.10), forming the so-called “fuzz” and individual trees of nano-tendrils (e.g. see [5, 72– 74]). According to the experimental data, the thickness of the fuzz is proportional to (ΦHe
Φlayer)1/2 [62]. We notice that similar growth of nano-tendrils under helium irradiation was observed on many metals including Mo, Ta, Fe, Ni, Ti, etc., which indicates that fuzz formation exhibits very general features of helium interaction with metals (see corresponding references in [24]). Although few mechanisms of nano-tendrils growth have been suggested (see related references in [24] for details), neither of them can explain the whole set of the available experimental data. Most probably, the fuzz growth is related to large stresses imposed in the bulk material by continuously growing helium bubbles, which results in plastic deformation of the lattice. Recent experimental observations

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Fig. 3.10 On the left: SEM micrograph of the fuzz grown on a single crystal after 1-hour irradiation by 40 eV helium ions. (Reproduced with permission from [71], © Elsevier 2010) and on the right: individual tree of nano-tendrils. (Reproduced with permission from [72], © Elsevier 2017) grown on a tungsten surface exposed to 50 eV helium ion irradiation

of a strong material mixture in both the tendrils and the bulk [75, 76] seem to support this idea. Let us now discuss hydrogen transport in tungsten. Even though interstitial hydrogen does not exhibit self-trapping effects, the experimental data show its rather strong accumulation in the tungsten lattice (e.g. see Fig. 3.9). Therefore, some other hydrogen trapping mechanism(-s) should exist. One of the possible candidates is hydrogen trapping in the vacancies. The DFT simulations (e.g. see [77–80] and the references therein) show that a mono-vacancy at room temperature could trap ~6 hydrogen atoms with the binding energy ~1 eV. Moreover, a combination of the DFT results and thermodynamic consideration suggests that a large amount of hydrogen embedded into a metal lattice (including tungsten) strongly promotes the vacancy formation and buildup of the so-called superabundant vacancies (SAV). Experimental results, although involving the metals with the face-centered cubic lattice such as palladium, nickel, and some others, seem to support the hydrogeninduced formation of SAV (see [81, 82]). The trapping energies ~1 eV, similar to those found from the DFT calculations for hydrogen trapping in a mono-vacancy in tungsten, are usually inferred from the TDS data (e.g. see [44, 83–85]), although in some cases, other traps, in particular, with a higher, ~2 eV, trapping energy, are needed to fit the entire TDS. Traps with Etr ~ 2 eV are usually attributed to hydrogen trapping in voids. Other possible trapping sites for hydrogen are related to dislocations. The DTF simulations show that three hydrogen atoms can be bound to the jogs of a screw dislocation (e.g. see [86]) in tungsten with the binding energy ~1.4 eV [87, 88] although the further increase of the number of the hydrogen atoms decreases the binding energy significantly. The TEM images demonstrate indeed the presence of screw dislocations decorated by hydrogen clusters [87]. Moreover, they also suggest that the formation of these dislocations, even in a monocrystalline sample, is facilitated by hydrogen [64]. In [89], the MD modeling of hydrogen interactions with both the edge and screw dislocations in tungsten demonstrated a hydrogen-

3.2 Basic Features of Hydrogen/Helium Transport in Plasma-Facing Materials

63

Fig. 3.11 Blisters produced on monocrystalline tungsten by 1.5 keV deuterium ion irradiation at the sample temperature of 400 K. (Reproduced with permission from [90], © Elsevier 2011)

induced modification of the tungsten lattice in the vicinity of the dislocations, formation of dynamic platelet-like hydrogen-reach structures with the hydrogen trapping energy Etr ~ 1 eV. The amount of hydrogen trapped in such platelet structures depends on the tungsten temperature and the hydrogen concentration. In contrast to hydrogen trapping in a vacancy, at low temperatures, hydrogen trapping in platelets related to a dislocation can greatly exceed 100 hydrogen atoms per a dislocation segment. It is conceivable that the temporal evolution of such platelets can account for the growth of the blisters observed experimentally under hydrogen irradiation of tungsten and some other metals at relatively low temperatures (see Fig. 3.11). Formation of similar platelet-like structures could also be triggered by the presence in the sample of the non-hydrostatic tensile and shear stress components. We notice that an impact of a strain applied to the tungsten lattice on the hydrogen solution energy, found from the DFT simulations, was also reported in [54]. Neutron irradiation of tungsten, unavoidable in fusion reactors, results in tungsten lattice damage and the formation of a large amount of additional hydrogen traps. However, experiments with neutron-irradiated samples can only be done just in a few laboratories (e.g. see [91, 92]). Therefore, some energetic ion (such as Si, He, Cu, W, etc.) irradiation is often used as a proxy for the neutron damage [6, 52, 63, 92–95]. The available experimental data demonstrate that the lattice damage causes a large increase of trapped hydrogen although the comparison of hydrogen retention in the neutron- and ion-damaged tungsten samples shows a significant difference in the corresponding TDS peaks [92]. This could suggest different structures of the lattice damage imposed by neutrons and energetic ions. However, annealing of the damaged samples at high temperature before hydrogen irradiation strongly reduces the impact of the lattice damage on hydrogen retention [94]. In fusion plasma, hydrogen is always accompanied by helium (the fusion ash). Possible synergistic effects of tungsten irradiation by low energy hydrogen with small admixture of helium ions on hydrogen retention were studied both experimentally and theoretically [63, 96–103]. All these studies show that an admixture of even small (~5%) of helium results in a strong reduction of the amount of retained hydrogen, which could be attributed to the binding of hydrogen to helium nanobubbles close to the surface, as seem to be indicated by some experimental data and simulations. However, this effect could also be related to possible interconnections of helium nano-bubbles resulting in a back leakage of hydrogen [104].

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Another PFM used in fusion devices is beryllium. Although the beryllium crystal structure, hexagonal close-packed, is very different from the body-centered cubic lattice of tungsten, the DFT simulations show that only up to five hydrogen atoms could be trapped in a beryllium lattice vacancy with the de-trapping energy ~1.7 eV [105, 106]. The trapping energies from 1.25 to 2 eV were also deduced from the analysis of the TDS data (e.g. see [107]). However, unlike tungsten where the amount of trapped hydrogen is increasing with ΦH [65], the amount of hydrogen retained in beryllium saturates with ΦH at a relatively low value [101, 108]. This is probably due to the formation of interconnected hydrogen nano-bubbles creating a porous structure in a rather thin sub-surface layer [108–110]. However, due to relatively high erosion of beryllium, the beryllium co-deposits (formed from the eroded beryllium atoms transported to and deposited at some particular locations in fusion devices) are the main sources of retained hydrogen in beryllium-containing tokamaks such as JET [3, 40, 101]. Similarly, for the case of carbon-based tokamaks, the main reservoir of retained hydrogen was in the carbon co-deposits (e.g. see [2] and the references therein). According to [40, 111], beryllium co-deposits formed at relatively low temperature have an amorphous structure. Interestingly, the hydrogen outgassing flux from the PFM at a constant temperature, Γout, from different tokamaks with both beryllium and carbon co-deposits exhibit (varying by orders of magnitude) similar temporal dependence Γout(t) / t
α with α  0.7 [3, 112–114]. Whereas the tokamak data could be related to some peculiarities of wall loading with hydrogen, the laboratory experiments with a constant deposition rate also demonstrate the power-law temporal dependence of Γout(t) with similar values of α [115]. We notice that the standard diffusion law would give α ¼ 0.5. The physical reason for such an unexpected dependence of Γout(t) is not clear. It is plausible that the amorphous co-deposits have a rather broad band of hydrogen trapping energies so that hydrogen transport can finally be described with fractional diffusion equations resulting in α 6¼ 0.5 [46]. However, during a discharge, both the hydrogen outgassing flux from the PFM and the wall uptake have a much more complex temporal behavior [116, 117]. As an example, from Fig. 3.12 one can see, how the dynamic particle (hydrogen) exchange between the plasma and the wall structures during a discharge in the JET tokamak actually works. Finally, we discuss briefly the usage of lithium as the material for the PFCs (e.g. see [22] and the references therein). Lithium (both solid and liquid) is used for the PFCs in fusion-related experiments (including tokamaks [118–122]) for more than a decade. One of the clear advantages of lithium is low Z, which makes it rather benign from the point of view of plasma contamination. It is difficult to predict, whether lithium will be used in future fusion devices (but see [123]), but these days lithium becomes more and more popular in the fusion community as, at least, a tool for improving tokamak performance and solving some current issues with the PFM. Liquid lithium is a subject of both Bénard-Marangoni convection (caused by ! ! temperature-dependent surface tension) and j  B force, where the electric current ! j can be driven by both the plasma and the thermoelectric effects. The latter ones

3.3 Conclusions

65

Fig. 3.12 Temporal variation of (a) number of plasma particles, (b) dynamic wall retention, (c) neutral pressure in the sub-divertor region, and (d) cumulative wall retention, in a JET discharge. (Reproduced with permission from [116], © Elsevier 2013)

could be used to induce the lithium flow [124] allowing to avoid lithium overheating by the plasma that results in the excessive lithium influx into the plasma. However, large uncontrolled plasma currents during ELMs can splash the freely flowing lithium into the plasma, which can terminate the discharge. An alternative way of introducing lithium for the PFM is to use some porous structure, so that lithium would wet up the front surface of this structure but would still be confined in the pores [118, 119]. Virtually all the available experimental data from current tokamaks using, in some way, lithium report a significant reduction of plasma contamination with impurity and improvement of core plasma confinement. However, lithium is a strong absorber of the hydrogen isotopes. Therefore, to maintain the tritium budget, the application of lithium in future reactors would require virtually complete tritium recovery from lithium. However, on the other hand, strong absorption of hydrogen can, potentially, open the way to a new, very favorable “zero-recycling” operational regime of a tokamak [125]. Recent experimental data seem to show that such a regime could indeed exist [121].

3.3

Conclusions

In conclusion to this chapter, it would be fair to say that the situation with the plasma-material interactions and material-related effects is far from being satisfactory and much more should be done in this area. Whereas we understand rather well the basic features of particle reflection from the material targets and physical

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sputtering, it is still difficult to make quantitative predictions of both reflection and sputtering for the case where the constituency of the first atomic layers of the wall material is evolving in time due to the plasma-material interactions. The situation becomes even worse if we need to make an assessment of helium and hydrogen transport in the material lattice. It seems that we understand the basics of the helium trapping mechanism at relatively low target temperatures, but we are yet unable to provide reliable models of the growth of fuzzy structures when the temperature goes up. For hydrogen transport and retention in tungsten, few primary mechanisms of hydrogen trapping were identified (e.g. SAV, dislocations). But still, no definite conclusion has been reached among the scientific community. In addition, there are some indications of a possible nonlinear, hydrogen-induced generation of the traps but there is no quantitative, predictive theoretical model that could be challenged by the experimental data yet. Interestingly, the experimental data on hydrogen retention in tungsten, for the conditions excluding such unwanted and complex effects as blistering, suggest that for large hydrogen fluence, ΦH, the amount of trapped pffiffiffiffiffiffiffi hydrogen is simply proportional to ΦH [65].

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81. Y. Fukai, N. Okuma, Formation of superabundant vacancies in Pd hydride under high hydrogen pressures. Phys. Rev. Lett. 73, 1640–1643 (1994) 82. Y. Fukai, Formation of superabundant vacancies in M–H alloys and some of its consequences: a review. J. Alloys Compd. 356–357, 263–269 (2003) 83. D.F. Johnson, E.A. Carter, Hydrogen in tungsten: Absorption, diffusion, vacancy trapping, and decohesion. J. Mater. Res. 25, 315–327 (2010) 84. J. Guterl, R.D. Smirnov, S.I. Krasheninnikov, M. Zibrov, A.A. Pisarev, Theoretical analysis of deuterium retention in tungsten plasma-facing components induced by various traps via thermal desorption spectroscopy. Nucl. Fusion 55, 093017 (2015) 85. E.A. Hodille, Y. Ferro, N. Fernandez, C.S. Becquart, T. Angot, J.M. Layet, R. Bisson, C. Grisolia, Study of hydrogen isotopes behavior in tungsten by a multi trapping macroscopic rate equation model. Phys. Scr. T167, 014011 (2016) 86. W. Cai, W.D. Nix, Imperfections in Crystalline Solids (Cambridge University Press, Cambridge, 2016) 87. D. Terentyev, V. Dubinko, A. Bakaev, Y. Zayachuk, W. Van Renterghem, P. Grigorev, Dislocations mediate hydrogen retention in tungsten. Nucl. Fusion 54, 042004 (2014) 88. V.I. Dubinko, P. Grigorev, A. Bakaev, D. Terentyev, G. van Oost, F. Gao, D. Van Neck, E.E. Zhurkin, Dislocation mechanism of deuterium retention in tungsten under plasma implantation. J. Phys. Condens. Matter 26, 395001 (2014) 89. R.D. Smirnov, S.I. Krasheninnikov, Stress-induced hydrogen self-trapping in tungsten. Nucl. Fusion 58, 126016 (2018) 90. N. Enomoto, S. Muto, T. Tanabe, Grazing-incidence electron microscopy of surface blisters in single- and polycrystalline tungsten formed by H+, D+ and He+ irradiation. J. Nucl. Mater. 385, 606–614 (2009) 91. M. Shimada, G. Cao, Y. Hatano, T. Oda, Y. Oya, M. Hara, P. Calderoni, The deuterium depth profile in neutron-irradiated tungsten exposed to plasma. Phys. Scr. T145, 014051 (2011) 92. M. Shimada, Y. Hatano, Y. Oya, T. Oda, M. Hara, G. Cao, M. Kobayashi, M. Sokolov, H. Watanabe, B. Tyburska-Püschel, Y. Ueda, P. Calderoni, K. Okuno, Overview of the US– Japan collaborative investigation on hydrogen isotope retention in neutron-irradiated and ion-damaged tungsten. Fusion Eng. Des. 87, 1166 (2012) 93. I.I. Arkhipov, S.L. Kanashenko, V.M. Sharapov, R.K. Zalavutdinov, A.E. Gorodetsky, Deuterium trapping in ion-damaged tungsten single crystal. J. Nucl. Mater. 363–365, 1168–1172 (2007) 94. B. Tyburska, V.K. Alimov, O.V. Ogorodnikova, K. Schmid, K. Ertl, Deuterium retention in self-damaged tungsten. J. Nucl. Mater. 395, 150–155 (2009) 95. M.J. Simmonds, Y.Q. Wang, J.L. Barton, M.J. Baldwin, J.H. Yu, R.P. Doerner, G.R. Tynan, Reduced deuterium retention in simultaneously damaged and annealed tungsten. J. Nucl. Mater. 494, 67–71 (2017) 96. M. Miyamoto, D. Nishijima, Y. Ueda, R.P. Doerner, H. Kurishita, M.J. Baldwin, S. Morito, K. Ono, J. Hanna, Observations of suppressed retention and blistering for tungsten exposed to deuterium–helium mixture plasmas. Nucl. Fusion 49, 065035 (2009) 97. V.K. Alimov, W.M. Shu, J. Roth, K. Sugiyama, S. Lindig, M. Balden, K. Isobe, T. Yamanishi, Surface morphology and deuterium retention in tungsten exposed to low-energy, high flux pure and helium-seeded deuterium plasmas. Phys. Scr. T138, 014048 (2009) 98. O.V. Ogorodnikova, T. Schwarz-Selinger, K. Sugiyama, V.K. Alimov, Deuterium retention in tungsten exposed to low-energy pure and helium-seeded deuterium plasmas. J. Appl. Phys. 109, 013309 (2011) 99. M.J. Baldwin, R.P. Doerner, W.R. Wampler, D. Nishijima, T. Lynch, M. Miyamoto, Effect of He on D retention in W exposed to low-energy, high-fluence (D, He, Ar) mixture plasmas. Nucl. Fusion 51, 103021 (2011) 100. N. Juslin, B.D. Wirth, Molecular dynamics simulation of the effect of sub-surface helium bubbles on hydrogen retention in tungsten. J. Nucl. Mater. 438, S1221–S1223 (2013)

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the lithium capillary-pore system in future fusion reactor devices. Plasma Phys. Controlled Fusion 48, 821–837 (2006) 119. G. Mazzitelli, M.L. Apicella, D. Frigione, G. Maddaluno, M. Marinucci, C. Mazzotta, V. Pericoli Ridolfini, M. Romanelli, G. Szepesi, O. Tudisco, FTU Team, FTU results with a liquid lithium limiter. Nucl. Fusion 51, 073006 (2011) 120. T.W. Morgan, P. Rindt, G.G. van Eden, V. Kvon, M.A. Jaworski, N.J.L. Cardozo, Liquid metals as a divertor plasma-facing material explored using the Pilot-PSI and Magnum-PSI linear devices. Plasma Phys. Controlled Fusion 60, 014025 (2018) 121. D.P. Boyle, R. Majeski, J.C. Schmitt, C. Hansen, R. Kaita, S. Kubota, M. Lucia, T.D. Rognlien, Observation of flat Electron temperature profiles in the Lithium tokamak experiment. Phys. Rev. Lett. 119, 015001 (2017) 122. H.W. Kugel, D. Mansfield, R. Maingi, M.G. Bel, R.E. Bell, J.P. Allain, D. Gates, S. Gerhardt, R. Kaita, J. Kallman, S. Kaye, B. LeBlanc, R. Majeski, J. Menard, D. Mueller, M. Ono, S. Paul, R. Raman, A.L. Roquemore, P.W. Ross, S. Sabbagh, H. Schneider, C.H. Skinner, V. Soukhanovskii, T. Stevenson, J. Timberlake, W.R. Wampler, J. Wilgren, L. Zakharov, the NSTX Team, Evaporated lithium surface coatings in NSTX. J. Nucl. Mater. 390–391, 1000–1009 (2009) 123. R.J. Goldston, R. Myers, J.A. Schwartz, The lithium vapor box divertor. Phys. Scr. T167, 014017 (2016) 124. D.N. Ruzic, W. Xu, D. Andruczyk, M.A. Jaworski, Lithium-metal trenches (LiMIT) for heat removal in fusion devices. Nucl. Fusion 51, 102002 (2011) 125. S.I. Krasheninnikov, L.E. Zakharov, G.V. Pereverzev, On lithium walls and the performance of magnetic fusion devices. Phys. Plasmas 10, 1678–1682 (2003)

Chapter 4

Sheath Physics

Abstract Although the physics of the “sheath”, a thin layer of plasma just in front of the material surface, has been studied for more than 100 years, the peculiarities of fusion devices, such as strong magnetic field, a shallow angle at which the magnetic field lines intersect the material surface, and inhomogeneity of the plasma parameters, bring some new and important features in this topic, which are discussed in this chapter. One of the most distinct processes at the edge of plasma devices is the plasma flow along the magnetic field lines to the material surface (target) where plasma is neutralized. Since for the edge plasma conditions, the electron thermal speed greatly exceeds the ion one, to maintain ambipolarity of plasma flow (for simplicity, here we assume no electric current), such plasma flow is accompanied by the formation of the so-called sheath region in the vicinity of the target. This region is characterized by a rather strong electrostatic electric field, which, usually, repels most of the free streaming electron flux and, as a result, establishes ambipolarity of the plasma flow (see Fig. 4.1). Therefore, in the absence of strong electron emission from the material surface, a monotonic electrostatic potential, φ(z), is built up between the plasma interior and the material surface (here the z coordinate goes perpendicular to the surface that is considered to be flat). The magnitude of the sheath potential drop φsh (see Fig. 4.1) is determined either from the condition of ambipolarity of the plasma flow or by the value of the electric current flowing through the plasma-surface interface. The electric field in the sheath causes energy exchange between the electrons and ions and for the target potential negative with respect to the plasma, the energy of the ions impinging on the surface can significantly exceed the thermal ion energy at the entrance to the sheath. This effect can boost erosion and degradation of the plasmafacing components and cause unwanted plasma contamination with impurities. ! The structure of the sheath depends on the angle, α, between the magnetic field, B, ! and the target (see Fig. 4.2; we notice that the direction of B is arbitrary, whereas the “parallel” coordinate, ℓ, goes toward the material surface). For the case of no magnetic field or the magnetic field being perpendicular to the surface, the thickness of the © Springer Nature Switzerland AG 2020 S. Krasheninnikov et al., On the Edge of Magnetic Fusion Devices, Springer Series in Plasma Science and Technology, https://doi.org/10.1007/978-3-030-49594-7_4

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Fig. 4.1 Electrostatic potential φ(z) within the sheath plugs most of the free streaming electron flux to maintain ambipolar plasma flow

Fig. 4.2 (left) Direction of the magnetic field and the coordinates which will be used. (right) Magnetic presheath, with the width ~ρi, and Debye sheath with the width ~λD for the case α  1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sheath, Δzsh, is of the order of the Debye length, λD ¼ Te =4πnsh e2, where Te is the electron temperature, nsh is the plasma density in the sheath and e is the elementary charge. For α 6¼ π/2pthe thickness of the sheath starts to depend on the ion gyro-radius, ffiffiffiffiffiffiffiffiffiffiffi ρi ¼ ðMc=eBÞ Ti =M (here Ti is the ion temperature, c is the speed of light, B is the strength of the magnetic field, and M is the ion mass) and for α  1, Δzsh ~ ρi since in the edge plasma, the Debye length is usually small in comparison with ρi. In this case, one can split the sheath into the magnetic presheath and the Debye sheath with the thicknesses ~ρi and ~λD respectively. What is important for many aspects of edge plasma physics is that for smooth transition of electrostatic potential from the sheath region into the plasma interior, the average ion velocity along the magnetic field lines for such plasma flow ð1Þ relatively far away from the surface, Vk , is limited by the so-called BohmChodura [1, 2] sheath criterion: ð1Þ

Vk

 Vcrit ,

ð4:1Þ

where the critical velocity Vcrit is determined by the plasma parameters (see also [3]).

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75

Fig. 4.3 Sketch of electron distribution function fe(vz) at z !  1

Interestingly, inequality (4.1) can be obtained from an analysis of the asymptotic behavior of the solution of the Poisson equation with proper perturbation of the electron and ion density by the sheath potential at z ! 1. To demonstrate this, we consider an idealized case assuming no magnetic field and monoenergetic ions streaming to the target with the velocity component perpendicular to the target at z ! 1 equal to V1. We also assume that at z ! 1, the electrons with vZ > 0 are  described  by the Maxwellian distribution function f e ðvz > 0Þ / exp mv2z =2Te . pffiffiffiffiffiffiffiffiffiffiffi Considering the case V1  Te =m and taking into account that at z ! 1 the plasma is quasi-neutral, we conclude that in order to maintain ambipolarity of the plasma flow to the target, the majority of the electrons reaching the sheath region must be reflected back by the electrostatic potential. As a result, the electron distribution function at z ! 1 is almost symmetric with respect to the sign of vz, with the only exception related to the cut-off of the tail of the reflected electrons (see Fig. 4.3), which corresponds to the absorption of the electrons that penetrate through the potential barrier eφw at the target (recall Fig. 4.1). From ion energy conservation and the ion continuity equation, we find the following expression for the ion density at z ! 1, assuming that MV21 =2  e j φðzÞj:   eφðzÞ ni ðzÞ ¼ nsh 1 þ , MV21

ð4:2Þ

where nsh is the plasma density at the entrance to the sheath (in our case, at z ! 1). Neglecting the impact of the cut-off of the tail of the electron distribution function at z ! 1, we can take the Boltzmann relation for the electron density, which for Te  e jφ(z)j gives ne(z) ¼ nsh(1 + eφ(z)/Te). Substituting both the ion and electron densities into the Poisson equation, we find   d 2 φð z Þ 1 Te ¼ 1  φðzÞ: MV21 λ2D dz2

ð4:3Þ

From Eq. (4.3) one sees that in accordance with the expression (4.1), a “smooth” (exponentially decaying at z ! 1) variation of the electrostatic potential is only possible for

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V1  Vcrit  Cs ¼

pffiffiffiffiffiffiffiffiffiffiffiffi Te =M:

ð4:4Þ

We should notice that such a flow of monoenergetic ions through almost Maxwellian electrons results in an instability associated with excitation of the sound waves (e.g. see [4, 5]). However, this instability has a convective nature and is stabilized by broadening, even small, of the ion distribution function [4]. Since in practice, the ion velocity distribution at z ! 1 is far from being monoenergetic, the simple expression (4.1) for the Bohm-Chodura criterion should be altered   to allow for the finite spread of the ion velocity distribution function, ð1Þ ! fi v , at z ! 1. For the case of no magnetic field, it was shown in Ref. [6] that the expression (4.4) can be generalized as follows: Z1 0

ð1Þ

fi

ð vz Þ M dvz  , Te nsh v2z

ð4:5Þ

which for the case of monoenergetic ions reduces to the expression (4.4). As we see from Eq. (4.5), the integral expression on the left-hand side converges only for the ð1Þ ðvz ! þ0Þ approaches zero fast enough. case where f i The Bohm-Chodura limitation on the velocity of the plasma flow onto the material surface is often viewed as a result of “pure” plasma effects, based solely on electron-ion coupling through the ambipolar electric field. However, this is not the case. The constraint similar to Eq. (4.1) does also exist for the velocity of collisional neutral gas flowing onto an absorbing surface (e.g. see [7]). Another example is the constraint on the speed of the gas flow into a standing shock wave where such speed must be supersonic. As a matter of fact, all these features have deep physical meaning. Indeed, both the plasma and neutral gas flows onto absorbing targets resemble the gas flow into a standing shock. However, the stability of the 1D standing shock wave is ensured by the fact that the gas flow velocity into it is supersonic, or, in other words, there is no wave propagating upstream away from the shock [8]. Interestingly, in [9] it was demonstrated that the expression (4.5) means that the ion sound waves cannot propagate in the direction away from the target. Therefore, the Bohm-Chodura constraint can be viewed as an extension of the Landau stability criterion for the ion sound waves, in the collisionless plasma flow onto an absorbing target. Although the expression (4.5) removes the limitation of the monoenergetic ion velocity distribution and goes beyond the simple constant (4.4), it still does not describe the effect of the magnetic field line inclination with respect to the target (see Fig. 4.2), even though this feature is ubiquitous in the edge plasma. To address both these issues, we need to consider kinetic equations for the ions and electrons, which in the stationary 1D limit can be written as follows:

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77

! !       v B e dφðzÞ ! ! ! ! v  ∇f i=e v , z e  ∇!v f i=e v , z  e z  ∇!v f i=e v , z ¼ 0, M dz Mc

!

ð4:6Þ

  ! ! where f i=e v , z is the ion/electron velocity distribution function and e z is the unit vector along the z-coordinate. A similar equation can be used for the electrons. Incorporating the solutions of the ion and electron kinetic equations into the Poisson equation, we can find all the necessary information. However, in practice, to find the constraint similar to Eqs. (4.4) and (4.5), we do not need to have the full solution of these equations. As we have found in the course of the derivation of Eq. (4.4), the Bohm-Chodura constraint comes from the asymptotic behavior of the solution of the Poisson equation, recall Eq. (4.3), at z ! 1 where the electrostatic potential is low. Therefore, instead of solving the complex nonlinear system of the kinetic and Poisson equations, we can consider their linearized versions. As a result, we  come  to ! the problem similar to that of finding the dielectric constant of the plasma, ε ω, k . !

Usually, it is determined as a function of the frequency, ω, and the wavenumber, k , which characterize the plasma wave. However, in our case, we are looking for the conditions of the formation of a stationary evanescent electrostatic potential that links the sheath region with the plasma interior. Therefore, we should consider the plasma dielectric constant with zero frequency and the wavenumber having the imaginary part ensuring that φ(z  ! 1) !0.      ! ð1Þ ! ð1Þ ! ð1Þ ! Following [4] we take f i=e v , z ¼ f i=e v þ f i=e v , z , where f i v , z is a small correction. Moreover, byanalogy  we  with Eq. (4.3), willassume  and  justify ð1Þ ! ð1Þ ! b e b v, z ¼ f v exp z kz , where a posteriori that φðzÞ ¼ φ0 exp z kz and f i=e

i=e

b kz is an adjustable parameter playing the role of the wavenumber of the evanescent wave, which can be found from the solution of the Poisson   equation. For a “smooth” transition from the sheath to z ! 1 we need Re b kz > 0.   ð1Þ ! v , being the solution of the stationary kinetic equation We notice that f i=e with φ(z) ¼ 0, should be expressed in terms of the integrals of motion, which gives: !2 ! ð1Þ ! f i=e ðv Þ  Fi=e ðvk , ε⊥ Þ where ε⊥ ¼ v ⊥ =2 whereas v ⊥ and vk are the velocity components perpendicular and parallel to the magnetic field. Then, from [4] we have

2 b kz ¼ 

X 4πe i=e

1 2 X

mi=e

j¼1

* J2 j



b  i cosðαÞ ΩkzBv⊥



i=e

kz vk jΩBi=e  i sinðαÞb

∂Fi=e ∂Fi=e  i sinðαÞ b kz jΩBi=e ∂ε⊥ ∂vk

!+ , ð4:7Þ

R where Jj(x) are the Bessel functions and h. . .i ¼ (. . .) dε⊥dvk. We notice that the normalization of Fi/e(vk, ε⊥) assumes hFi/e(vk, ε⊥)i ¼ nsh.

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First, we consider the case of the normal incidence of the magnetic field lines onto the target, α ¼ π/2. In this case, Eq. (4.7) is reduced to 2 b kz ¼ 

X 4πe2 1 ∂Fi=e

: mi=e vz ∂vz i=e

ð4:8Þ

Since we assume that the ions striking the target are absorbed by the material surface, we have Fi(vz) / H(vz), where H(x) is the Heaviside function, H(x > 0) ¼ 1 and H(x < 0) ¼ 0. For electrons, as we discussed above, we have largely symmetric distribution Fe(vz) ¼ Fe(|vz|). As a result, for Maxwellian Fe(vz), from Eq. (4.8) we find   2 1 T Fi b kz ¼ 2 1  e : Mnsh v2z λD

ð4:9Þ

  As we see, the condition Re b kz ∼ λ1 D > 0 can only be satisfied for the case where the generalized Bohm-Chodura criterion Eq. (4.5) is valid. From Eq. (4.9) one finds also that, in agreement with Eq. (4.3), the characteristic scale of b kz is of the order of λ1 . D Next, we consider the case 1 > α > λD =ρi

and

λD =ρe > 1,

ð4:10Þ

where ρe and ρi are the electron and ion gyro-radii. For comparable electron and ion temperatures, which is rather typical for the plasma near the targets in high recycling conditions, the following restriction on the angle pffiffiffiffiffiffiffiffiffiffiffi these inequalities impose and, taking into account that λD/ρe > 1 α: 1 > α > m=M. We assume that b kz ∼ λ1 D and α  1, we conclude that, due to the conservation of both the electron adiabatic invariant v2⊥ =B ¼ const: and the total electron energy, the electron dynamics in the sheath region is virtually adiabatic, so that mv2k =2  eφ ¼ const: [10]. Therefore, similarly to our previous cases, neglecting the small tail cut-off, we can assume an almost symmetric electron distribution function Fe(vk) ¼ Fe(|vk|). We also assume complete absorption of the ions on the target, which gives FP i(vk) / H(vk). Then, 2 taking into account the inequalities (4.10) and recalling that 1 j¼1 Jj ðςÞ ¼ 1 for arg jςj < π, from Eq. (4.7) we find that it can be reduced to * 2 b kz

1 T ¼ 2 1 e Mn sh λD

+! Fi v2k

* 1 0 ! nsh

+ Fi v2k



M , Te

ð4:11Þ

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79

which justifies our assumption b kz ∼ λ1 D . For the case of a monoenergetic ion distribution function along the magnetic field, the expression (4.11) is reduced to the Chodura inequality (4.1) [2].   ð1Þ ! v , the potential drop Once the ion flux onto the target is defined by f i between the plasma away from the target and the target can be found by equilibrating the electric currents in the plasma and through the sheath. For the case of ambipolar plasma flow, assuming that the ion flow to the target along the magnetic field is nshCs, ignoring other effects that could alter the ion flux to the target (e.g. drifts), and adopting the Maxwellian electron distribution function, we find e φsh ΛshTe, where Λsh ~ ℓn(M/m) ~ 3 4 [3]. In [5] the constrains (4.5)and (4.11) were criticized, in particular, from the point  2 2 of view of “unphysical” vz vk moment of the ion distribution function and the omission of the impact of the collision operator for small vz(vk) in the course of the derivation of Eqs. (4.5) and (4.11) from the corresponding Vlasov equations. A replacement of Eq. (4.5) based on positive powers of the velocity moments of the distribution function, which do not diverge at v ¼ 0, was suggested (see [5] for details). But in [11] it was argued that the inequality (4.5) holds also in the presence of collisions provided that Eq. (4.5) is applied at the entrance to the sheath and the sheath is considered in the limit of λD ! 0. Although this discussion is important from an academic point of view, for the practical application of the Bohm constraint as the boundary condition for edge plasma flow to the target it becomes rather meaningless. This is because in the most relevant, from the point of view of the reduction of the power loading of the target, high recycling regimes of divertor operation, strong plasma-neutral interactions become ubiquitous. As a result, in this case, all the models suggest virtually the same constraint on the plasma flow velocity hvzi Cs (for the case of normal incidence of the magnetic field lines onto the target) with some correction for the finite ion temperature. The difference between various models is within the error bar imposed by the boundary conditions used for the ion and electron heat fluxes to the target, the calculation of which should take into account spatial variation of both the electrostatic potential and the electron and ion distribution functions at the entrance to the sheath (e.g. see [12]). In our considerations, we assume that the electron distribution function along the magnetic field lines for vk > 0 (in the direction towards the target) is Maxwellian, whereas forffi vk < 0 the tail of the distribution function is cut at velocities beyond pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  eφsh =m . For the case of the ambipolar plasma flow, φsh can be found by equilibrating the electron and ion fluxes. However, such approximation is only applicable for the case where electron collisionality in the SOL plasma is relatively high and the tail can be replenished by Coulomb collisions before the electrons reach the target. In the opposite case, depletion of the tail of the electron distribution function can drive the whistler waves, which are capable of scattering the electrons and effectively populating the “gap” in the electron distribution function (see [13] and the references therein). However, in practice, in the SOL plasmas, the electrons with the energies ~ e φsh ΛshTe are weakly collisional, which makes it difficult to

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make quantitative estimates of the shape of the tail of the electron distribution function and of the subsequent impact of developing of the whistler wave instability. Usually, in the edge plasma transport codes it is assumed that the electrons impinging onto the target have the Maxwellian distribution. So far we assumed that the plasma flowing onto the target is completely collisionless within some proximity to the target. In practice, this is not the case and collisions are always present. In particular, in the detached divertor regime, the neutral density becomes high and ion-neutral collisions result in a very short ion-neutral collision mean free path, λiN. However, in a ballpark, they do not change one of the key conclusions of the sheath physics: for the case of λiN > λD, the plasma flow velocity at the distance ~λiN from the target should about Cs (see [3] for details). As an illustration, we consider an ambipolar flow of weakly ionized plasma on a material surface. We assume a constant electron temperature Te, Boltzmann electrons, no plasma sink or source, and no magnetic field. Then, the plasma flow is governed by the following equations   d n1 dφ i Mj ¼ e ni þ Mνj, dx dx 2



d2 φ ¼ 4πeðni  ne Þ, dx2

ð4:12Þ ð4:13Þ

where j ¼ const. is the plasma particle flux, ν ¼ const. is the ion-neutral collision w frequency, ne ¼ nw e exp ðeφ=Te Þ is the electron density, and ne is the electron density at the target where, for convenience, unlike Fig. 4.1, we take zero electrostatic potential. To maintain ambipolarity of the plasma flow, we adopt the following 1=2 boundary condition at the target: j ¼ βnw , where β ~ 1. e ðTe =mÞ Equation (4.12) describes two regimes of the ion flow: (i) dynamic ion acceleration, corresponding to the case where the term on the left-hand side and the first term on the right-hand side dominate (this case describes standard acceleration of ions in a collisionless sheath); and (ii) diffusive ion flow corresponding to the case where the term on the left-hand side is small. In the latter case, assuming plasma quasi-neutrality, from Eq. (4.12) and the Boltzmann relation we find ne ffi ni / x and

φ / ℓnðxÞ:

ð4:14Þ

It is more convenient to switch in Eqs. (4.12) and (4.13) from the variable φ and coordinate x to the variable fðɸÞ  ni ðɸÞ=nw e expðɸÞ , where ϕ ¼ eφ/Te, and the coordinate η ¼ (m/M)1/2β1 exp (ϕ) (we notice that η  ηmin  (m/M)1/2β1). As a result, from Eqs. (4.12) and (4.13) we find

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81



 2 1 d fη  ðfηÞ1  f 2 df=dη ¼ Pðf  1Þ, 2 dη

ð4:15Þ

where P ¼ 4πe2j/MCsν2. To shed the light on the physical meaning of the parameter P, let us calculate the ratio of the ion-neutral collision mean-free path, λiN,ptoffiffiffiffiffiffiffiffi the local Debye length, λD. After simple algebra we find ξ  λiN =λD ¼ f(η~1) ~ 1, the ion velocity is close to the f 1 P=η . We will see below that at p ffiffiffi sound speed. As a result, we find ξ P. So at P  1, Λ is large and ions in the vicinity of the target are moving virtually in the dynamic regime. In what follows, we will assume P  1. When the plasma is close to quasi-neutrality (f ffi 1), from Eq. (4.15) we find the following correction (~ P1  1) to f:    3 f f 1 ðηÞ  1 þ η η2 þ 1 =P η2  1 :

ð4:16Þ

From the Poisson equation, it is easy to show that the expression (4.16) gives the correct asymptotic dependence φ / ℓn(x) for quasi-neutral plasma (recall Eq. (4.14)). However, at η ! 1 function f1(η) diverges. On the other hand, the dynamic regime of the ion flow is described by the expression in the brackets on the left-hand side, which should be close to zero in the dynamics regime. This can only be satisfied if the right-hand side of Eq. (4.15) is e 1, which holds for P  1. Then, from the equation fη  (fη)1  f2df/ large for f > dη ¼ 0, we find f ¼ η1{1  2ℓn(η)}1/2 where, keeping in mind the expression (4.16), we take the boundary condition f(η¼1) ¼ 1. However, this boundary condition implies that the ion velocity reaches Cs at the entrance to the domain with the dynamic acceleration of ions (the Debye sheath). Once f(η) is known, the spatial dependence φ(x) can be deduced from the Poisson equation. Keeping in mind our assessment of the physical meaning of P, from the Poisson equation we find that the spatial domain occupied by quasi-diffusive, quasi-dynamic ion transport corresponding to f ~ 1 extends about few λiN from the target, whereas the dynamic acceleration of ions occurs at the scale ~ λD  λiN just in front of the material surface. The numerical solution of Eq. (4.15) shows a good agreement with the results of the analytic consideration presented here (see Fig. 4.4). As we see from Fig. 4.4, there is a smooth transition of f(η) and, therefore, φ(x) from the ion diffusion-limited to the ion dynamic-limited regimes. We notice that our fluid equation-based consideration predicting that the ion flow velocity becomes ~ Cs at the entrance to the Debye sheath agrees, in a ballpark, with the solution of the one-dimensional kinetic equation from [14, 15] where the ion-neutral collisions were described as the charge exchange process whereas the electrons were assumed to follow the Boltzmann relation with a constant temperature. The ion distribution, f i ðbεi Þ (where bεi ¼ Mv2 =2Te), at the entrance to the sheath,

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Fig. 4.4 Comparison of numerical solutions of Eq. (4.15) for different parameters P with the asymptotic analytic solution corresponding to P ! 1

Fig. 4.5 Ion distribution function at different spatial locations, found from the solution of one-dimensional kinetic equation allowing for the ion-neutral charge exchange and assuming Boltzmann electrons with a constant temperature Te. (Reproduced with permission from [15], © AIP Publishing 2006)

found from the corresponding kinetic equation, is shown in Fig. 4.5. We notice that at small bεi , we have f i ðbεi Þ / bεi , so the integral in Eq. (4.5) does not diverge [14]. Since in the edge plasma of fusion devices, rather strong ion-neutral collisions in the vicinity of the divertor target are ubiquitous, these results justify the widely used boundary condition for the plasma flow velocity, Vk Cs at the target, which, otherwise, is ill-defined by the inequalities (4.4) and (4.11). In our analysis of the Bohm-Chodura constraint of the plasma flow onto the target, described so far, we assumed that the electric field there results only from the sheath effects and is normal to the target. However, in the edge plasma of fusion

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83 !

devices, an electric field component E p, parallel to the target and hence crossing the magnetic field, can also be present (e.g. due to the electric currents or electron temperature variation along the target, etc.). As we will see, the impact of this electric field can significantly alter the Bohm-Chodura constraint. For the case where one can ignore the spatio-temporal variation of both the magnetic field and ! ! E p , the impact of E p on the Bohm-Chodura constraint can be easily found by a transition to the moving frame [16]. Indeed, considering the sheath in a slab ! approximation (see Fig. 4.2, where E p is in x-direction), we recall that the transition !

from the laboratory frame to the frame moving with nonrelativistic velocity Vf !0

!

results in the following transformation of the electric field: E p ¼ E p þ ! ! !0 Vf B =c (where E p is the electric field in the moving frame), whereas the magnetic field in the moving frame remains virtually equal to the magnetic field in ! the laboratory frame. Since we assume that both the magnetic field and E p are !0

!

constants, with a proper choice of Vf we can have E p ¼ 0. As a result, in the moving frame we can use the standard Bohm-Chodura constraint (e.g. given, in the simplest case, by Eq. (4.4), V0k > Cs ). Then, making the backward transformation into the !

laboratory frame, we find the impact of E p on the Bohm-Chodura constraint: ! ! ! !  V1 ¼ V0k B=B þ c E p B⊥ B2 ⊥ ,

ð4:17Þ

!

where B⊥ is the component of the magnetic field perpendicular to the target. The sheath properties impose important boundary conditions for such quantities as the plasma flow velocity to the target, the electric current from the plasma to the material surface and the electron and ion heat fluxes to the target (e.g. see [3, 12, 17, 18] and the references therein). These boundary conditions are used as the closures for the differential fluid plasma equations at the target in 2D fluid plasma transport codes such as SOLPS or UEDGE. For example, in the simplest case of normal incidence of the magnetic field onto the surface, the electric current flowing through the plasma to the target, jtar z , has the following relation to the electrostatic potential drop φsh:   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jtar z ¼ ensh V1  ð1  γsee Þ Te =2πm exp ðeφsh =Te Þ ,

ð4:18Þ

emission that where γsee < 1 is the effective coefficient of secondary electron pffiffiffiffiffiffiffiffiffiffiffi ffi includes also thermionic electron emission. Taking V1 ¼ Te =M , from Eq. (4.18) it follows that for ambipolar plasma flow to the target and γsee  1, we have e jφshj ~ ℓn(M/m)Te.

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Fig. 4.6 Electrostatic potential profiles for: the standard sheath, γsee  1, (black), the SCL sheath for ðcritÞ 1 > γsee  γsee (blue), and “inverse sheath”, γsee  1, (red)

ðeÞ

For the electron and ion heat fluxes to the material surface, one can find qz ¼ ðiÞ γe jðzeÞ Te and qz ¼ γi jðziÞ Ti, where jðzeÞ and jðziÞ are correspondingly the electron and ion particle fluxes to the target, whereas γe and γi are the so-called heat transmission coefficients, which, in a general case of the tilted magnetic field, depend on the ion distribution function Fi(vk, ε⊥), secondary electron emission, drifts, and potential drop φsh. However, the expressions for the fluid closures at the target (including those for the electron and ion heat fluxes) become much more cumbersome if we include grazing magnetic field, particle drifts and time dependence of the plasma parameters related, for example, to the SOL plasma turbulence. Discussion of these issues goes beyond the scope of this chapter. More details can be found in [12, 18, 19] and the references therein. As indicated in Eq. (4.18), the secondary electron emission can significantly alter ðcritÞ the magnitude of φsh. Moreover, closer consideration shows that for 1 > γsee  γsee , the structure of the sheath becomes non-monotonic. This is the so-called space-chargelimited (SCL) sheath (see Fig. 4.6) where to maintain the ambipolarity of the plasma flow, a part of the emitted electrons are reflected back to the target by the hump of the electrostatic potential [20]. In Ref. [21] it was suggested that the secondary electron emission could be used for cooling the edge plasma in magnetic confinement devices. In [22] it was shown that the secondary electron emission can result in focusing the heat flux and the formation of the hot spots on the plasma-facing components. However, in [23] it was argued that for the case of a strongly emitting surface, the so-called “inverse sheath” (IS) (see Fig. 4.6) could be formed. In this case, the positive (with respect to the plasma) potential at the target prevents ions from reaching the target and ambipolarity is maintained by equilibrating the fluxes of the plasma and emitted electrons. Moreover, further studies show that in the presence of cold neutrals, both the ionization and charge exchange processes within the hump of the electrostatic potential of the SCL sheath make the SCL sheath unstable and the latter evolves into the IS [24]. The authors of [25] speculated that the IS can promote divertor detachment. However, a more thorough investigation of the effect of the IS on divertor detachment with a 2D code UEDGE has shown that the IS per se does virtually not alter the SOL plasma parameters and does not advance divertor detachment [26]. However, since the IS conditions inhibit the ion flux to the target, the IS regime

References

18

Plasma potential obtained from cold probe

16 Floating potential (V)

Fig. 4.7 Floating potential of the probe versus the laser heating power that is used to facilitate thermionic emission from the probe. (Reproduced with permission from [27], © John Wiley and Sons 2011)

85

14 12

floating potential @ a radial position of 20,5 (a.u.) @ different heating power

10 8 6 4 2 0 0

10 20 30 40 Laser heating power (50% = 25W)

50

could be beneficial from the point of view of the strong reduction of the target erosion. The formation of the IS could also explain some, otherwise puzzling, experimental data on positive floating potential of the plasma with respect to strongly electronemitting objects observed in [27, 28], e.g. see Fig. 4.7.

4.1

Conclusions

In conclusion for this chapter, we note that the sheath plays quite a unique role in the edge plasma physics. Even though it occupies a tiny region close to the plasmafacing components, the sheath can make a large impact on the erosion of those. It imposes some constraints on the plasma flow to the material surfaces and sets the boundary conditions for both kinetic and fluid-based plasma codes, which are used to study different phenomena in the edge plasma, ranging from plasma transport to edge plasma turbulence. Finally, as we will see in Chap. VII, the effective boundary conditions at the sheath can result in specific sheath driven instabilities of the edge plasma, which might alter cross-field plasma transport and, therefore, the heat and particle fluxes on the plasma-facing components.

References 1. D. Bohm, E.H.S. Burhop, H.S.W. Massey, The Characteristic of Electrical Discharges in Magnetic Fields (McGraw-Hill, New York, 1949) Chapter 2 2. R. Chodura, Plasma-wall transition in an oblique magnetic field. Phys. Fluids 25, 1628–1633 (1982)

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3. K.-U. Riemann, Theory of the collisional presheath in an oblique magnetic field. Phys. Plasmas 1, 552–558 (1994) 4. A.B. Mikhailovskii, Theory of plasma instabilities, in Instabilities in a Homogeneous Plasma, vol. 1, (Springer, New York, 1974) 5. S.D. Baalrud, C.C. Hegna, Kinetic theory of the presheath and the Bohm criterion. Plasma Sources Sci. Technol. 20, 025013 (2011) 6. E.R. Harrison, W.B. Thompson, The low pressure plane symmetric discharge. Proc. Phys. Soc. 74, 145–152 (1959) 7. K. Aoki, Y. Sone, T. Yamada, Numerical analysis of gas flows condensing on its plane condensation phase on the basis of kinetic theory. Phys. Fluids A2, 1867–1878 (1990) 8. L.D. Landau, L.M. Lifshitz, Fluid mechanics, in Course of Theoretical Physics, vol. 6, 2nd edn., (Butterworth-Heinemann, Oxford, 1987) 9. J.E. Allen, A note on the generalized sheath criterion. J. Phys. D. Appl. Phys. 9, 2331–2332 (1976) 10. R.H. Cohen, D.D. Ryutov, Particle trajectories in a sheath in a strongly tilted magnetic field. Phys. Plasmas 5, 808–817 (1998) 11. K.-U. Riemann, Comment on ‘kinetic theory of the presheath and the Bohm criterion’. Plasma Sources Sci. Technol. 21, 068011 (2012) 12. V. Rozhansky, E. Kaveeva, P. Molchanov, I. Veselova, S. Voskoboynikov, D. Coster, G. Counsell, A. Kirk, S. Lisgo, The ASDEX-Upgrade Team and The MAST Team, New B2SOLPS5.2 transport code for H-mode regimes in tokamaks. Nucl. Fusion 49, 025007 (2009) 13. Z. Guo, X.-Z. Tang, Ambipolar transport via trapped-electron whistler instability along open magnetic field lines. Phys. Rev. Lett. 109, 135005 (2012) 14. K.-U. Riemann, Kinetic analysis of the collisional plasma–sheath transition. J. Phys. D. Appl. Phys. 36, 2811–2820 (2003) 15. S. Kuhn, K.-U. Riemann, N. Jelić, D.D. Tskhakaya Sr., D. Tskahaya Jr., Link between fluid and kinetic parameters near the plasma boundary. Phys. Plasmas 13, 013503 (2006) 16. I.H. Hutchinson, The magnetic presheath boundary condition with E B drifts. Phys. Plasmas 3, 6–7 (1996) 17. A.V. Chankin, P.C. Stangeby, The effect of diamagnetic drift on the boundary conditions in tokamak scrape-off layers and the distribution of plasma fluxes near the target. Plasma Phys. Control. Fusion 36, 1485–1499 (1994) 18. R.H. Cohen, D.D. Ryutov, Sheath physics and boundary conditions for edge plasmas. Contrib. Plasma Phys. 44, 111–125 (2004) 19. R.H. Cohen, D.D. Ryutov, Non-steady-state boundary conditions for a sheath in a tilted magnetic field. Plasma Phys. Rep. 23, 805–809 (1997) 20. G.D. Hobbs, J.A. Wesson, Heat flow through a Langmuir sheath in the presence of electron emission. Plasma Phys. 9, 85–87 (1967) 21. M.Y. Ye, S. Masuzaki, K. Shiraishi, S. Takamura, N. Ohno, Nonlinear interactions between high heat flux plasma and electron-emissive hot material surface. Phys. Plasmas 3, 281–292 (1996) 22. M.Z. Tokar, A.V. Nedospasov, A.V. Yaroshkin, The possible nature of hot spots on tokamak walls. Nucl. Fusion 32, 15–24 (1992) 23. M.D. Companell, Negative plasma potential relative to electron-emitting surfaces. Phys. Rev. E 88, 033103 (2013) 24. M.D. Companell, M.V. Umansky, Strongly emitting surfaces unable to float below plasma potential. Phys. Rev. Lett. 116, 085003 (2016) 25. M.D. Companell, G.R. Johnson, Thermionic cooling of the target plasma to a sub-eV temperature. Phys. Rev. Lett. 226, 015003 (2019) 26. R. Masline, R.D. Smirnov, S.I. Krasheninnikov, Influence of the inverse sheath on divertor plasma performance in tokamak edge plasma simulations. To appear in Contrib. Plasma Phys. (2020)

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27. C. Ionita, J. Grünwald, C. Maszl, R. Stärz, M. Čerček, B. Fonda, T. Gyergyek, G. Filipič, J. Kovačič, C. Silva, H. Figueiredo, T. Windisch, O. Grulke, T. Klinger, R. Schrittwieser, The use of emissive probes in laboratory and tokamak plasmas. Contrib. Plasma Phys. 51, 264–270 (2011) 28. B.F. Kraus, Y. Raitses, Floating potential of emitting surfaces in plasmas with respect to the space potential. Phys. Plasmas 25, 030701 (2018)

Chapter 5

Dust in Fusion Plasmas

Abstract Dust is ubiquitous in magnetic fusion devices. It comes either from plasma-induced erosion of the plasma-facing components or, in some cases, dust (powder, aerosol) is deliberately injected into edge plasma for some diagnostic or control purposes. The key processes controlling dust dynamics in the edge plasma differ significantly from those important in “thin”, low-temperature plasma in the so-called “dusty plasma” experiments. The main features of dust dynamics in fusion plasmas, including charging, forces, and ablation processes, are considered in this chapter.

The fact that dust is present in the plasma of magnetic fusion devices is known for a long time (e.g. [1–3]). Already Ohkawa [1] had argued that dust particles can be an important source of impurity in fusion plasmas and can significantly degrade the performance of fusion plasmas. Later, experimental data confirmed that in some cases, the appearance of dust particles in fusion plasmas can even result in termination of the plasma discharge (e.g. see Fig. 5.1 and Refs [4, 5]). The interest to the study of dust-related phenomena in fusion devices was boosted by the ITER project (see Refs. [7–9]). The main initial concern of the ITER staff was related to safety issues associated with the chemical activity, tritium retention and radioactivity of the dust [10] and the dust impact on the in-vessel plasma-facing diagnostics (e.g. mirrors) [11]. As of today, large amount of experimental data available on (i) in situ dust observations with Thompson scattering diagnostics and fast cameras, which give information on the dust size distribution and the dynamics of dust motion (see Fig. 5.2) through the plasma in fusion devices and (ii) post mortem analysis of the shape (see Fig. 5.3) and material composition of the dust particles collected in fusion devices (e.g. see [7, 12–18] and the references therein). In this chapter, we will consider basic processes involving a dust grain immersed into fusion plasma, which include grain charging, the forces acting on the grain and the grain dynamics. We review main dust-related experimental observations related to the dust parameters, constituency, and dynamics in fusion plasma, dust mobilization from the surfaces of the plasma-facing components, and compare some experimental data with the results of numerical simulations. © Springer Nature Switzerland AG 2020 S. Krasheninnikov et al., On the Edge of Magnetic Fusion Devices, Springer Series in Plasma Science and Technology, https://doi.org/10.1007/978-3-030-49594-7_5

89

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CIII, FeX [a.u.]

Prad [a.u.]

Te [keV]

ne [1019m–3]

Power [kW]

Spark

250 200 150 100 50 0

Shot 53775 Pfwd(3.5U) Pref(3.5U)

1 0.5 0 1.5 1 0.5 0 8 6 4 2 0 80 60 40 20 0

Te (ECE:R=3.466m)

FeX CIII

1640 1641 1642 1643 1644 1645 Time (s)

Fig. 5.1 Spontaneous ejection (“spark”) of dust particles from the first wall terminates a long pulse discharge in the LHD stellarator. The “spark” event observed with CCD camera (left) and time evolution of the plasma parameters (right). (Reproduced with permission from [6], © Elseiver 2007)

Fig. 5.2 Dust observed by a fast camera in the DIII-D tokamak in front of the neutral beam injection (NBI) port 1 ms before (a), during (b) and 2 ms after (c) an NBI pulse. (Reproduced with permission from [12], © IAEA 2009)

We also present an assessment of the impact of dust on ITER plasma performance and discuss the gaps in our understanding of dust physics in magnetic fusion devices.

5.1 5.1.1

Experimental Study of Dust in Magnetic Fusion Devices Dust Particle Density, Size Distribution, and Composition in Fusion Devices

The size of the dust particles found in fusion devices is ranging from nano-meters (e.g. see Fig. 5.4) to few hundred μm in the Alcator C-Mod tokamak [16]. Usually,

5.1 Experimental Study of Dust in Magnetic Fusion Devices

91

Fig. 5.3 Dust particles of different shapes and sizes collected from fusion devices. (Reproduced with permission from [19], © Elseiver 2002) Fig. 5.4 Size distribution of the dust particles found in LHD. (Reproduced with permission from [20], © JSPF 2009)

the median diameter of dust grains found in fusion devices is in the range of 1–10 μm [15]. The size distribution is often estimated from the dust collected from fusion devices during the ventilation events. However, the result can depend on both the collection method and counting. For example, counting with an optical microscope gives a log-normal distribution of dust particles [19], whereas counting with scanning (SEM) and transmission (TEM) electron microscopes reveals the presence of a large amount of sub-micron particles [20, 21]. As a result, the size distribution of the dust particles of small diameter in the LHD device appears to be far from the

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Fig. 5.5 Dust particle density in the lower divertor of DIII-D. (Reproduced with permission from [13], © Elseiver 2007)

log-normal distribution and close to the power-law distribution, Fdust(ℓd) ~ (ℓd)
α (where ℓd is the characteristic size of the grain) with α  2.5 (see Fig. 5.4). However, dust mobilization from the plasma-facing components with subsequent penetration of the dust particles into the plasma volume depends on many circumstances (e.g. the initial location of the dust grains, the plasma parameters, etc.) [17, 39]. Therefore, the size distribution of dust collected from fusion devices during ventilation events can be different from that present in the plasma during the discharges. Important information about the size and spatial distribution of the dust grains in the plasma of magnetic fusion devices can be obtained with laser scattering by utilizing the non-shifted (Rayleigh) channel of the Thomson scattering diagnostics used for the measurements of electron density and temperature. The first systematic studies of dust with laser scattering had been performed on the DIII-D tokamak [13, 22], and later, laser scattering was used for dust detection in FTU [23] and JET [24]. The distribution of dust particle density found with laser scattering in the lower divertor of DIII-D is shown in Fig. 5.5. Initial results on the size distribution of the dust grains, inferred from the laser scattering data using the Rayleigh theory of light scattering by particles [22], were reconsidered in [25] on the basis of the more correct Mie theory. In addition, in [25] dust grain ablation under intense laser radiation was taken into account. It was found that the size distribution function of the dust grains with radii in the range 0.01–10 μm (the average grain radius was ~ 200 nm) in the DIII-D plasma can be described by the power-law distribution with α  2.6
2.7. However, we should notice that larger grains can not be identified by laser scattering diagnostic due to the saturation of the measured reflected signal. We also note that similar power-law size distributions of the dust collected from LHD and measured in DIII-D plasma may be just a coincidence. Observations of dust with the laser scattering technique show that the dust particle density at the edge of a magnetic fusion device depends strongly on the operational mode of the device. For example, in H-mode having rather violent MHD events such as ELMs (Edge Localized Modes), dust particle density in the SOL is few times higher than that in the “quiet” L-mode, Fig. 5.6. In Fig. 5.7 one can see the relaxation of the dust density just after an ELM burst (the solid line is the exponential fit with

5.1 Experimental Study of Dust in Magnetic Fusion Devices 3.0

H-mode Particle density (104 m–3)

Fig. 5.6 Dust density in the upper SOL of DIII-D for Hand L-confinement modes. (Reproduced with permission from [28], © Elseiver 2009)

93

2.0

L-mode 1.0

0.0 0.9

2500

2000 Dust Density (m–3)

Fig. 5.7 Time evolution of dust density after an ELM burst in DIII-D. (Reproduced with permission from [28], © Elseiver 2009)

1.0 1.1 1.2 1.3 Normalized Radius (Ψ)

1500

1000

500

0

0

50

100

150

Time After ELM (ms)

the characteristic half-time ~ 60 ms). An increase of the number of dust particles (dust mobilization events) in ELMy H-mode is also detected with “video diagnostic” (based on observations with fast cameras) developed on the AUG tokamak [17]. However, one should take into account that fast cameras can only see relatively large dust grains (e.g. according to the assessment of Ref. [26], confirmed by the experimental data [27], for the case of carbon dust, the fast cameras can only see the e μm). grains with size >1 Experimental data demonstrate that density of the dust particles can depend not only on the power of auxiliary plasma heating (for example, an increase of neutral beam heating of DIII-D plasma from 2 to 13 MW increases the dust occurrence rate

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Fig. 5.8 Effective number of dust particles in a discharge as a function of the number of discharges elapsed since a disruption. (Reproduced with permission from [17], © IAEA 2017)

by a factor ten [13]) but also on the type of the auxiliary heating (e.g. with the electron- and ion- cyclotron waves, neutral beam injection, etc.) [17]. However, all available experimental data show that the largest amount of dust particles inside the vacuum chamber is observed after disruptions (see [15] and the references therein). For example, in the FTU tokamak, the dust particle density after a disruption can reach 107 m
3, whereas before the disruption, the laser scattering diagnostic detects no dust particles [23]. Moreover, the amount of dust remains higher than the average value during few shots after a disruption, Fig. 5.8. Although many elements contribute to the composition of the dust particles, the dominant one corresponds to the material of the plasma-facing and structural components used in current magnetic fusion devices, such as carbon, tungsten, molybdenum, beryllium, lithium, etc., with the contribution from the material used for first wall conditioning (e.g. boron) (see Fig. 5.9), as well as from hydrogen isotopes and impurities used for enhancing the radiation power loss from the edge plasma (e.g. nitrogen) (see Refs [16, 29–31] and the references therein). Some of the dust particles are agglomerations that have inclusions of very different materials (see Fig. 5.10). We will see later that the dynamics of such agglomerated particles in fusion plasma can have very peculiar features. Analysis of the dust particles collected in the major fusion devices [9, 16, 29–35] suggests that the main source of dust in the current magnetic fusion devices comes from melting edges of the metal tiles, arching, and exfoliation of co-deposited layers which, depending on the particular device, can consist of carbon, beryllium, boron, etc. with significant presence of hydrogen isotopes and some impurities.

5.1 Experimental Study of Dust in Magnetic Fusion Devices

95

Fig. 5.9 SEM image of dust particles (with a size range of 50–100 μm) collected from Alcator C-Mod. The energy-dispersive X-ray spectroscopy (EDX) mapping provides the composition of the dust grains. (Reproduced with permission from [16], © Elseiver 2017)

(a)

(b)

O

1

800

W 3000

600

2 O

2000

400 1000 200

Be

0 0.0

C

c

Ni

N

w W Mo

0 0.5 keV

1 keV

2

Fig. 5.10 (a) Agglomerated dust particle, (b) EDX spectra from regions 1 and 2. (Reproduced with permission from [31], © Elseiver 2017)

5.1.2

Mobilization of Dust Particles from Plasma-Facing Components

Although postmortem analysis shows a significant amount of dust particles on the plasma-facing components virtually in all magnetic fusion devices, it is obvious that not all the dust ends up in the plasma at once. The mobilization of the dust particles from the plasma-facing components is complex and is still the topic of ongoing research. Different mechanisms can be responsible for dust mobilization from the plasma-facing components: the plasma-induced drag force and dust collisions with

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Fig. 5.11 The trajectory of tungsten dust grain (observed with a fast camera) mobilized from the oblique polished plate in the Pilot-PSI device (the second image corresponds to the reflected light). (Reproduced with permission from [46], © IAEA 2015) Fig. 5.12 Photo of droplets ejected from a tungsten target melted under the exposure in the plasma accelerator QSPA-T. (Reproduced with permission from [41], © Elseiver 2009)

the PFC surface [36–39] (see Fig. 5.11), splashes from molten metal targets [40–43] (see Fig. 5.12), thermal stresses resulting in cracking of the PFC material [44, 45] (in particular, of co-deposited layers having loose thermal contact with the bulk), etc.

5.1.3

Dust Particle Dynamics in Fusion Devices, Experimental Data

The dynamics of dust particles in fusion devices is mainly studied with fast cameras. This is because the dust grains in fusion plasmas are heated up to high temperature and can start to ablate. As a result, we have two different radiation sources that can be captured by fast cameras: (i) thermal radiation of the grain itself and (ii) radiation from the ablation cloud related to the excitation of the ablated atoms by the ambient plasma electrons. Contributions of these two sources depend on both the dust material and the parameters of the ambient plasma (strictly speaking, the time history of the dust grain trajectory can also be important). However, the specifications of the fast camera can also matter. Thorough analysis [26] of carbon dust observations in the DIII-D tokamak [27, 38] shows that for the DIII-D plasmas, thermal radiation of carbon dust particles only dominates in the far SOL, whereas radiation from the ablation cloud prevails in the relatively hotter and denser plasma close to the separatrix where the grain

5.1 Experimental Study of Dust in Magnetic Fusion Devices

97

Fig. 5.13 Fast (~500 m/s) dust particle moves toward the wall in DIII-D tokamak (left) and disintegrates after collision (right). (Reproduced with permission from [47], © IOP Publishing 2008) Fig. 5.14 3D trajectory of carbon dust particles recorded in the MAST tokamak. (Reproduced with permission from [53], © IAEA 2010)

ablation rate is high. Although the analysis was performed for the carbon dust, this conclusion has robust physics arguments and seems to be very generic. As of today, there are many photos and movies from virtually all major magnetic fusion devices [4, 6, 14, 17, 27, 38, 47–52] showing dust traces in the plasma volume and, sometimes, collisions of dust particles with the PFCs (see Fig. 5.13). Some movies are recorded with few different cameras, which allows determining both the dust speed and the 3D dust particle trajectory (e.g. see [49, 53]). These data can be used, in particular, for benchmarking the codes developed to study dust-related phenomena in fusion devices. The results obtained with fast cameras show that the dust particles can acquire the speed of ~ few hundred meters per second (e.g. [38, 49, 53]) and they move largely in the toroidal direction (e.g. see Fig. 5.14). This is in agreement with the assessment made in [36, 37] where it was shown that the plasma drag force is one of the major forces exerted on a dust grain in fusion plasmas. Since the plasma in the inner and outer divertors flows in different toroidal directions, one could expect [37] that the

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Fig. 5.15 Spiral trajectory of a dust particle recorded in the DIII-D tokamak. (Reproduced with permission from [12], © IAEA 2009)

toroidal components of the dust velocities in the inner and outer divertors should be opposite, which was indeed observed in the experiments (see Refs. [12, 49]). However, in [54] it was shown that toroidal plasma rotation can also be important in dragging the dust particles. Although fast cameras show that the majority of dust particles demonstrate rather smooth trajectories, some of the dust grains exhibit jitter-like deviation from the average direction of the trajectory [55], whereas some others have spiral-like trajectories (see Fig. 5.15). We will see in the next section that such features can be explained by the non-spherical shape and agglomerate nature of the grains. Moreover, some movies recorded with fast cameras show that the dust grains can experience significant “kicks” by large blobs and ELM filaments. A pattern recognition code developed and coupled to the fast camera imaging allows monitoring the dust mobilization rate (the number of new dust particles in the tokamak volume per unit time), see [14] and the references therein. It was shown that for the carbon-based Tore Supra tokamak, the dust mobilization rate increases exponentially with the run-time (providing that no cleaning procedure is implemented), whereas in the ASDEX-U tokamak with a tungsten first wall, the dust mobilization rate shows an initial strong reduction following the last ventilation event, and then it saturates [14]. In Tore Supra, most of the mobilized dust was coming from carbon co-deposit layers. This finding looks beneficial for the ITER design, which has no carbon-based PFCs. However, much higher heat load in ITER can provide other sources of dust, which can be related to the melting of beryllium and tungsten armors. Other experimental techniques (e.g. electrostatic dust detector, the capture of dust grains with aerogel, different gravimetric dust sensors, etc.) are also used for dust studies in magnetic fusion devices and the results found from the implementation of these techniques can be found in the review [15].

5.2 Theoretical Aspects and Numerical Simulations of Dust-Related Phenomena in. . .

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99

Theoretical Aspects and Numerical Simulations of Dust-Related Phenomena in Magnetic Fusion Devices

There is a large body of literature dedicated to the theoretical study of dust charging, forces, etc. in different environments ranging from laboratory experiments to astrophysics (see Refs. [56–58] and the references therein). However, in fusion plasma, the physics of dust has some important differences from what was studied in the laboratory experiments. First of all, unlike most of the laboratory experiments, the shape of the dust particles in fusion devices usually is far from spherical (recall Fig. 5.3) and cannot be specified a priori (unless we are dealing with dedicated experiments where well-characterized grains are injected into the fusion device). Next, the grain material of the dust particles in hot and dense fusion plasmas can be heated up to a very high temperature and dust ablation effects become important (recall the observations of dust particles with fast cameras discussed in Sect. 5.1.3). As a result: (i) the dust grains can change their shapes (e.g. metallic dust can melt), (ii) the ablated material can form a “shield” altering the plasma-grain interactions, and (iii) different plasma particle reflection coefficients and evaporation rates of different materials in the dust particles formed by agglomeration (recall Fig. 5.10a) can result in a “rocket force”, which is virtually impossible to predict and characterize a priori. All of these issues make it difficult or even impossible to develop theoretical/computational tools that would describe the dust-related phenomena in the natural fusion plasma environment precisely. Nonetheless, benchmarking of the results of the dust dynamics simulations against the experimental data shows a reasonable agreement. It suggests that overall, the models used for the description of the dust-related phenomena in magnetic fusion devices capture at least the most important features of the dust-fusion plasma interactions.

5.2.1

Dust Particle Dynamics in Fusion Devices, Theoretical Approaches

For stationary conditions, the flux of charged particles onto a dust grain immersed in the plasma should satisfy the ambipolarity conditions. Then, if there is no charge emission from the grain (e.g. thermionic of secondary electron emission), the dust grain usually becomes negatively charged to repel some electrons and equilibrate the fluxes of the light (and therefore fast) electrons and the heavy (and therefore slow) ions. For a spherical dust grain of a radius Rd, the grain charge number, Zd, can be found from the following expression (e.g. see [56–58]) Zd ¼ Λd Rd T=e2 ,

ð5:1Þ

where Λd ~ 3 is the numerical coefficient only weakly (logarithmically) depending on the plasma parameters and e is the elementary charge. We assume that the

100

5 Dust in Fusion Plasmas

electron and ion temperatures are similar Te ~ Ti ~ T. For Rd ~ 1 μm and T ~ 10 eV we find Zd ~ 104. Due to the rather high plasma density, the charging time of the dust grains in fusion devices is very short, τch ~ 10
8s [36], so the dust charging can be considered in a quasi-stationary approximation. The interaction of dust with the flow of homogeneous plasma having the velocity ! Vp results in the drag force exerted upon the grain ! !  F drag ¼ ςdrag πR2d Mi ni VTi Vp
Vd ,

!

ð5:2Þ

!

where Vd is the dust grain velocity, Mi, ni, and VTi are the ion mass, density, and thermal velocity, respectively, and ςdrag ~ 10 is a numerical factor that depends on the dust charge and plasma parameters, [57, 58]. Estimates from Ref. [36] show that the drag force is one of the dominant forces acting on the dust particles with Rd ~ few μm at the edge of fusion plasmas (the divertor and SOL regions). This is because of the strong plasma flows existing in these regions due to plasma recycling and anomalous cross-field plasma transport. Under the drag force acceleration, the dust particles in fusion devices can be easily accelerated to ~100 m/s [15]. Apart from the drag force, other forces imposed dust particles in a fusion !on the ! ! plasma are the electric, eZd E , and Lorentz, eZd Vd  B =c, forces (here c is the light speed), the gravity force, the magnetic force acting on the grain having a magnetic moment, and some others. However, ferromagnetic materials are not used in the magnetic fusion devices, so the magnetic force is unlikely to be important for the dynamics of dust naturally existing in the fusion plasmas. Moreover, estimates from [36] show that even though the dust charge number can be large, the ratio Zd/Md (determining the gyro-frequency of the dust particles) for micron-size grains is by orders of magnitude smaller than that for the plasma ions. Therefore, the Lorentz force can only alter the dynamics of very small (nano-scale) grains. Comparison of the electric and drag forces shows that for the edge plasmas in magnetic fusion devices they can only be comparable in the sheath region where the electric field is much stronger than in the bulk of the edge plasma. Gravity usually becomes important for the grains with a characteristic size of over 100 μm [36]. Other forces acting on the grain, such as the thermal (or thermophoretic) forces related to the ion/neutral temperature inhomogeneity [59, 60] for the edge plasma conditions are usually smaller than the corresponding plasma and neutral gas drag forces. However, the inhomogeneity of the material on the surface of a dust grain, which is rather typical for the agglomerated dust particles (recall Fig. 5.10a), can result in the so-called “rocket force” [36] related to the non-uniformity of the coefficients of plasma particle reflection from the dust surface or of the dust material ablation rate (for the case of strongly heated grains). In both cases, a strong unbalanced momentum flux can produce both a large force (comparable to or even exceeding the plasma drag force) and a torque acting on the agglomerated dust particles. It is plausible that

5.2 Theoretical Aspects and Numerical Simulations of Dust-Related Phenomena in. . .

101

the spiral trajectory of the dust particle shown in Fig. 5.15 is the result of such “rocket force” effects. Unfortunately, such effects are impossible to predict. The expressions (which are rather cumbersome) for the dust grain charge, forces, heat flux and dust material temperature variation for spherical dust grains, relevant to the edge plasma conditions, can be found in [15]. However, all these expressions are only valid for spherical dust particles. The dynamics of non-spherical grains that are naturally present in the fusion devices (recall Fig. 5.3) is more complex. For solid dust particles one should treat the grain dynamics as the motion of a rigid body: !

Md

dVd ! ¼ F d, dt

ð5:3Þ

!

dL d ! ¼ Kd , dt

ð5:4Þ

!

!

!

where Md and L d are the grain mass and angular momentum, whereas F d and Kd are the force and torque acting on the grain. In addition, one cannot describe any more grain charging with just the charge number Zd but should consider the distribution of the charge over the grain surface, which, in particular, results in the departure of the force acting on the grain from Eq. (5.2). Therefore, strictly speaking, the exact analysis of the dynamics of non-spherical grains can be performed only numerically. Even though rather comprehensive numerical simulations of the dust dynamics in fusion devices, which will be discussed in Sect. 5.2.2, are based on the spherical dust particle approximation, it is important to have at least some estimate of the difference between the dynamics of the spherical and non-spherical grains. Under the edge-plasma-relevant conditions, the dynamics of non-spherical grains can be analyzed by using symmetry principles. As an example, we follow [61] and consider the dynamics of a non-spherical grain in plasma without magnetic field. ! First, we that the angular velocity of the grain spinning, Ωd, is relatively low, !assume    so that Ωd τch  1. Next, we will also assume that the speeds of the grain and the plasma flow are much lower than the thermal !  speed   of the plasma ! ions  (recall that we   !    consider Te ~ Ti ~ T), which means that Vp , Vd   VTi and Ωd ℓd  VTi. Under such assumptions, both grain charging and the forces imposed on the grain can be considered in a quasi-stationary approximation. For the case where the properties of the grain surface responsible for the grain-plasma interactions are homogeneous, the ! ! ! ! ! ! directions of F d and Kd will only depend on the directions of W  Vp
Vd, Ωd and !

!

the orientation of the grain. Moreover, for relatively small jW j and jΩdj one can keep ! ! ! ! only linear dependence of F d and Kd on W and Ωd . As a result, we have

102

5 Dust in Fusion Plasmas ðWÞ

ðΩÞ

ð5:5Þ

ðWÞ

ðΩÞ

ð5:6Þ

ðFd Þα ¼ Φαβ Wβ þ Φαβ ðΩd Þβ , ðKd Þα ¼ Tαβ Wβ þ Tαβ ðΩd Þβ , ð...Þ

ð...Þ

where the tensors Φαβ and Tαβ are determined only by the shape of the grain and the plasma parameters. For a general case, these tensors can only be calculated numerically. However, for the grain having rotational symmetry around some axis, ð...Þ ð...Þ the structure of the tensors Φαβ and Tαβ can be found from geometrical arguments [61]. Indeed, for this case, the spatial orientation of the grain can be characterized by ! ! ! ! a dimensionless vector D. Then, taking into account that F d , D, and W are vectors, ! ! ! whereas L d , Kd , and Ωd are pseudo-vectors, the most general form of the equations of motion of the grain can be written as follows !   ! ! dVd ðWÞ ! ðWÞ ! ! ! ¼ Φ1 W þ Φ2 D D  W þ ΦðΩÞ Ωd  D , dt !  ! !   dL d ð ΩÞ ! ð ΩÞ ! ! ! ¼ TðWÞ W  D þ T1 Ωd þ T2 D D  Ωd , dt

Md

ð5:7Þ ð5:8Þ

where (Ld)α ¼ Iαβ(Ωd)β and Iαβ ¼ I0δαβ + I1DαDβ is the inertia tensor of the grain, where δαβ is the Kronecker delta and I0 and I1 describe the components of the inertia ðWÞ ðWÞ ðΩÞ ð ΩÞ tensor. The scalars Φ1 , Φ2 , Φ(Ω), T(W), T1 , and T2 are determined by particular properties of the grain material and shape and, for the grain shape not ðWÞ ðWÞ Fd ¼ ςdrag πℓ2d Mi ni VTi , too far from spherical, can be estimated as Φ1 ∼ Φ2 ∼ b ð Ω Þ ð Ω Þ ΦðΩÞ ∼ TðWÞ ∼ b Fd ℓd and T1 ∼ T2 ∼ b Fd ℓ2d . As we see from Eq. (5.7), the force acting on a non-spherical grain is no longer ! aligned with the direction of the relative velocity W, unlike the case of a spherical grain in Eq. (5.2). The departure is due to both the grain orientation and grain spinning (correspondingly the second and third terms on the right-hand side, RHS, of Eq. (5.7)). We notice that the second term was also derived in [62] from the direct calculation of the force acting on the grain for the combined Coulomb and dipole grain-plasma interaction potentials. !  ! !   For the case Vp  jVd j, jΩd j ℓd, Eqs. (5.7) and (5.8) can be simplified and we have

Md

!   dVd ðWÞ ! ðWÞ ! ! ! ¼ Φ1 Vp þ Φ2 D D  Vp , dt ! ! ! dL d ¼ TðWÞ Vp  D , dt

ð5:9Þ ð5:10Þ

5.2 Theoretical Aspects and Numerical Simulations of Dust-Related Phenomena in. . .

103

where Eq. (5.10) is identical to the equation describing the motion of a symmetrical ! top in an effective gravity field / Vp [63]. The solution of Eq. (5.10) gives the !

oscillation of the vector D in time, which causes time oscillation of the force in Eq. (5.9) and, therefore, oscillations of the grain trajectory on a spatial scale Δd, ! which for the case of a fast top and relatively slow precession around the vector Vp can be large, Δd ~ 1 cm  ℓd (see [61]). It is plausible that the jittering of the grain trajectory observed in some cases by fast cameras is due to the non-sphericity of the grains. Eqs. (5.6), (5.7), (5.8), (5.9) and (5.10) describe the dynamics of non-spherical dust grains under the impact of the drag force associated with the plasma flow but with no effects of the magnetic field that is ubiquitous in the magnetic fusion devices. However, the impact of the magnetic field on the dust dynamics/spinning can be significant due to the dust grain interactions with the plasma. For example, in [64, 65] it was shown that gyration of the plasma particles and synergistic effects ! ! of the electric, E , and magnetic, B, fields can result in specific torques spinning up the dust particles. In [66], the approach developed in [61] was extended to the dynamics of non-spherical grains in the presence of a magnetic field and in [67] to the grains having some helical (propeller-like) features. Overall, based on the available results on the dynamics of non-spherical grains, we can conclude that in the absence of the “rocket force” effects, apart from some jittering (on the scale ~1 cm) of the dust particle trajectory, the dynamics of the spherical and non-spherical grains in fusion devices is rather similar. This justifies the applicability of the spherical grain approximation in numerical simulations. However, the dynamics of agglomerated dust particles can have a significant deviation from the predictions made with the spherical approximation. As we mentioned, the dust grains in fusion plasmas can be quickly heated up to high temperatures and start to ablate. We notice that the heat flux coming to the grain depends on the grain charge that can be affected by thermionic emission sensitive to the grain temperature. Such a nonlinear dependence of the heat flux to the grain on the grain temperature in some cases can cause a bifurcation phenomenon causing a sudden jump of the heat flux to the grain, the grain temperature and charge [68]. Dust ablation/evaporation is the mechanism of the reduction of the dust particle mass/size, which, in total, usually significantly exceeds the impact of dust material sputtering by the plasma ions impinging onto the grain. The plume of the ablated material can work as a shield reducing the heat flux coming to the grain from the ambient plasma in a way similar to the case of shielding of pellets injected into the core of the fusion plasma for fuelling purposes (e.g. see [69] and the references therein). However, it appears that significant shielding can only be formed for relatively large dust grains, e ℓshield , where ℓshield depends on both the dust material and the plasma paramℓd > eters [70, 71]. The reason for this is the fast initial expansion of the plume, which prevents the formation of the shield for small ℓd. For ℓd > ℓshield one should take into account the shielding effects. Numerical simulations show that an ad hoc reduction of the heat flux to the dust particles, imitating the shielding effect, has a very pronounced impact on plasma

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contamination with impurity because the shielded dust grains can penetrate deeper into the plasma which results in the increasing impurity concentration and radiation loss [72]. However, the models developed for the description of pellet shielding e 1 keVÞ effects are focused on the interactions of the ablated material with hot ðT > core plasma (see [69, 73, 74] and the references therein). The shielding effects in these models are described by the “stopping power” of the energetic electrons by the ablated material and no energy loss due to radiation of the ablated material is taken into account. This concept can hardly be justified for shielding of high-Z dust particles in e 100 eVÞ edge plasma, where impurity radiation can efficiently relatively cold ðT < cool the ambient plasma. Therefore, in [75–77] a model was developed, focused specifically on the shielding of high-Z (e.g. tungsten) dust grains in relatively cold edge plasmas, where impurity radiation is one of the main ingredients resulting in the reduction of the heat flux coming to the grain. The dependence of the shielding factor, γshield, which is the ratio of the heat flux to the grain with and without shielding effects, on the plasma parameters can be found in the corresponding references. Another important issue related to the dust dynamics in fusion devices is the dust collisions with the plasma-facing components, which are quite often observed with fast cameras (recall Fig. 5.13). Such collisions of negatively charged grains are only possible when the grain kinetic energy normal to the surface exceeds the repulsive electrostatic sheath potential barrier (which is usually not the case in the laboratory dusty plasma experiments [58]). Estimations show that to overcome such a potential barrier for a micron-size particle, the normal component of its velocity should exceed (depending on the dust material) ~1–5 m/s [15, 36], which is much lower than the typical speed of the dust in fusion devices ~100 m/s. Different models of dust-wall collisions are used in numerical simulations of the dust dynamics. These models range from simple reflection coefficients [37, 78] to the rather sophisticated Thornton-Ning model [79] for the sticking and bouncing of adhesive, elastic-plastic spheres (e.g. see [80]). Potentially, the dust collisions with the PFCs can have a large impact on both the dust dynamics and wall erosion. In particular, the dust grain can be disintegrated in the course of the collision, as observed in experiments (recall Fig. 5.11), and also can contribute significantly to erosion and surface morphology modification of the PFC materials. Numerical simulations of the collisions of dust grains with PFCs can be performed by using the commercial finite element code for structural analysis, LS-DYNA [81]. The LS-DYNA code solves three-dimensional transient multiphysics problems including solid mechanics, deformation, contacts, fragmentation, heat transfer, etc., and implements a large variety of material models and simulation techniques. Therefore, this code can provide, presumably, the most accurate assessment of the results of the dust collision with the PFCs. The results of such simulations of the collisions of beryllium and tungsten dust particles of 0.5 μm radius, impinging onto a beryllium target at 45 , are presented in Fig. 5.16. As one can see, at a relatively low speed, both the beryllium and tungsten grains are bouncing off the target with very little impact on both the grains and the target. However, at high

5.2 Theoretical Aspects and Numerical Simulations of Dust-Related Phenomena in. . .

105

Fig. 5.16 Simulated impact of beryllium and tungsten dust particles of 0.5 μm radius (blue) on beryllium target (red) at speeds 102 (a, c) and 103 m/s (b, d), and impact angle of 45 . (Reproduced with permission from [82], © Elseiver 2009)

speed, the beryllium grain disintegrates completely during the collision but still produces very modest target erosion, whereas the fast tungsten particle creates quite a deep crater in the target and produces a large amount of debris.

5.2.2

Numerical Simulations of Dust Particle Dynamics and Dust Impact on Edge Plasma Parameters

The dynamics of dust particles in fusion edge plasma having strongly inhomogeneous parameters is very complex. Therefore, the study of the dust dynamics in real fusion devices and assessment of a self-consistent impact of the dust on the edge plasma can only be done numerically. Up to now, few codes DUSTT [68, 78], DTOKS [83, 84], MIGRAINE [80, 85], and DUMBO [52] have been developed for these purposes. All these codes implement more or less similar models of dust grain charging and forces imposed by the dust-plasma interactions, which are considered by using the spherical grain approximation. However, some details of these codes have significant differences, which can alter some features of the dust dynamics. For example, currently, only the DUSTT code allows for the shielding effects, which slow down the ablation/evaporation of a dust grain and affect the dust penetration depth into the edge plasma and, therefore, the impurity concentration and radiation loss. On the other hand, MIGRAINE employs a rather sophisticated model describing the dust collisions with the plasma-facing components, which goes far beyond

106

5 Dust in Fusion Plasmas 4000 3000 2000 1000 0

0.8

Z, m

D 0.6

C

B

A

1.00 0.75 0.50 0.25 0.00

0.4

0.2 1.2

1.4

1.6 R, m

1.8

2.0

A

C

D

B

Temperature Td, K 0.0

0.2

0.4

0.6

0.8 C

B

A

1.0 D

Mass 0.0

0.2

0.4 0.6 Lp/Lpmax

0.8

1.0

Fig. 5.17 Left: poloidal projection of 1 μm carbon dust particle trajectories. Dust grains were launched into DIII-D from outer strike point with velocities 10 (A), 102 (B), 103 (C), and 104 (D) cm/s. Right: dust grain temperature and relative mass as a function of poloidal distance traveled. (Reproduced with permission from [78], © AIP Publishing 2005)

the simple reflection coefficients used in other codes (e.g. in DUSTT) and, therefore, is better suitable for the study of dust mobilization from the material surfaces. All of these codes were developed for dust studies in tokamaks and, therefore, assume the toroidal symmetry of the plasma parameters. However, the DUSTT code has been modified recently and is now used for the study of dust dynamics in the helical device LHD [51]. The plasma parameters used in all of these codes focused on dust dynamics studies come either from experimental measurements or, in most cases, from 2D edge plasma transport codes (e.g. see [86]) such as UEDGE [87], different versions of the SOLPS code (e.g. see [88] and the references therein), and some others. A few examples of dust trajectories found from the numerical simulations of the dust dynamics for prescribed edge plasma parameters are shown in Figs. 5.17 and 5.18. We notice that a sharp zigzag in the middle of the divertor volume of particle B in Fig. 5.17 is just a visual effect of the 3D trajectory projected on 2D poloidal coordinates. Similar effects are also seen in the poloidal projection of dust trajectories in Fig. 5.18. However, the reversal of the toroidal direction of dust propagation seen in Fig. 5.18 is due to the different directions of the toroidal components of the plasma flow in the divertor regions of the outer and inner SOL [37]. A similar change of the toroidal direction of the dust motion was also observed in numerical simulations (e.g. see [68]) and by fast cameras for the case where the dust grain moves from one divertor to the other (e.g. see Refs [12, 49]). However, a considerable amount of dust injected into a fusion device can significantly alter the plasma parameters and even cause termination of the discharge (recall Fig. 5.1 and see Refs [4, 5]). The numerical simulations performed in Ref. [78] for two cases of impurity injection into the plasma of (a) neutral atoms and (b) dust particles have shown a large difference in the edge plasma parameters even though the rate of impurity mass injection was the same in both cases. However, for accurate assessment of the dust impact on the edge plasma for the case of a relatively

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Fig. 5.18 Poloidal (left) and toroidal (right) projections of 1 μm carbon dust particle trajectories. Grains were launched into MAST from outer strike point with velocity 30 m/s under different angles. (Reproduced with permission from [83], © IOP Publishing 2008)

high dust injection rate, one should take into account the self-consistent variation of both the plasma parameters, caused by the impurity provided by the dust, and the dust dynamics and transport in dust impurity-modified plasma environment. Such self-consistent consideration became possible after the coupling of the dust and edge plasma transport codes DUSTT and UEDGE into DUSTT-UEDGE package [55]. This package allows considering different scenarios of dust injection into plasma. For example, all designated dust grains can be injected within a short time, simulating a “spark” event shown in Fig. 5.1, or the grains can be injected into the plasma continuously. We notice that the DUSTT-UEDGE package allows choosing the distribution of dust injection location over the PFC surfaces and distribution of the dust grains in the injection velocity and size. In Figs. 5.19 and 5.20 one can see the results of simulations with the DUSTTUEDGE package of continuous injection of tungsten dust into the ITER plasma [72]. As one can see from Fig. 5.19b, the tungsten radiation loss strongly increases at the core-edge interface as a result of deeper penetration of dust into the core plasma when the shielding effects are taken into account. Another illustration of shielding effects can be found in Fig. 5.20. Taking into account that ITER assumes to use impurity seeding to reach the semi-detached state in the outer divertor, the data from Fig. 5.20 suggest that to avoid possible termination of the discharge due to thermal collapse, the mass rate by continuous injection of tungsten dust into ITER plasma should not exceed ~30 mg/s. However, to this moment we have no simulation results on the tolerable amount of the tungsten dust injected into the ITER plasma on a short (e.g. ~1 ms) time scale. The reason for this is the computational challenge related to the fast and spatially localized change of the plasma parameters, which accompanies such a dust injection scenario.

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Fig. 5.19 Tungsten impurity radiation distribution in the ITER divertor for 10 μm dust grain injection with the mass rate of 60 mg/s for different shielding factors. (Reproduced with permission from [72], © AIP Publishing 2015)

Fig. 5.20 Normalized tungsten radiation power (left) and fraction of the tungsten-radiated power from the core-edge region (right) as functions of the dust mass injection rate for different grain radius and shielding factor. (Reproduced with permission from [72], © AIP Publishing 2015)

5.3

Conclusions

As we have seen, the study of the dust in fusion devices brings together the edge plasma physics and the material and surface physics of the PFC materials. In the last 10–15 years, the study of the dust physics in fusion devices has developed into a separate research area, which is recognized by the plasma fusion community as important for future magnetic fusion reactors. Over that time, many new diagnostic tools for in situ dust studies have been developed and implemented on the magnetic fusion devices (e.g. laser scattering, pattern recognition with fast cameras, etc.). Theoretical study of the dust dynamics in fusion devices greatly benefits from the models of the dust-plasma interactions developed previously for different applications (e.g. see Refs. [56–58] and the references therein). These models have been

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partly extended to non-spherical grains, which allowed assessing the impact of complex shapes of the dust particles in fusion devices on the dust dynamics. The results of numerical simulations (performed with sophisticated codes developed from scratch) of the dust particle dynamics and transport in fusion devices show decent agreement with the experimental observations. However, there are still some gaps in our understanding/description of the dust physics in fusion devices. Some of them are related to the plasma physics (e.g. the description of an impact of large turbulent plasma fluctuations, often observed in the edge plasma, on dust transport), but probably the most important and the most complex one, the dust generation and injection rate into the plasma, is related to both the material science and the plasma-material interactions. Still, a lot should be done in this direction.

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Chapter 6

Fluid Description of Edge Plasma Transport

Abstract The comprehensive description of all features of plasma dynamics requires kinetic consideration, which, however, is extremely complex and currently non-tractable in its entirety. In many cases, simplified fluid approximation based on the moments of the distribution functions of plasma particles gives rather accurate results. The main approach to the derivation of the fluid equations and some particular results of the so-called “classical” plasma transport, used in the book, are considered in this chapter. Tokamak edge plasma is especially difficult for modeling – it is the interface between the hot core plasma and relatively low-temperature divertor region where multiple impurity species and neutral gas dynamics are important. The presence of material surfaces, e.g. divertor plates, with additional surface interactions and reactions, complex magnetic field geometry, such as separatrix, and strong electric field and plasma flows make it even more complex. Additional complications are introduced by the presence of non-neutral regions (boundary sheath) near the plasma-wall boundaries. In general, the plasma edge is the region where multiple collision and atomic processes having very different characteristic times and lengths are equally important and need to be considered self-consistently and simultaneously with the anomalous turbulent transport phenomena. Clear separation of the time and length scales, e.g. between the equilibrium and fluctuating quantities, is often impossible in the edge region, which creates additional challenges. The plasma turbulence and anomalous transport remain the biggest challenge for the physics of edge plasmas. Currently, it is not feasible to simulate plasma turbulence in the edge and fully include all possible kinetic effects, neutrals and atomic physics, sheath, boundaries, etc. Such simulations are still outside the modern computer capabilities. Therefore, a number of simplifications and reductions of the problem are usually performed. Presently, one can identify two major directions in the theoretical description and modeling of the plasma edge. In one approach, which is conditionally called here the first principle turbulence modeling, the focus is on formulating adequate physics models, which would include relevant physics at small scales to describe properly instabilities, turbulence and anomalous transport (see Chap. 7 for further discussions). The global turbulence codes are also being © Springer Nature Switzerland AG 2020 S. Krasheninnikov et al., On the Edge of Magnetic Fusion Devices, Springer Series in Plasma Science and Technology, https://doi.org/10.1007/978-3-030-49594-7_6

115

116

6 Fluid Description of Edge Plasma Transport

developed, which aim to characterize nonlinear plasma fluctuations and transport at the edge of the magnetic confinement devices, including 3D effects and open magnetic field geometries [1–9]. Because of limitations noted above, such codes are often based on fluid (moment) formulation and neglect interactions with neutrals and many kinetic effects such as parallel transport. In the alternative approach (e.g. see [10, 11], and Chap. 8), the emphasis is on the characterization of large spatiotemporal scale equilibria and flows of particles and energy in complex divertor geometries including coupling to neutrals, sheath boundaries, atomic physics, plasma surface interactions, etc. Such codes, conditionally called here transport codes, include the effects of small-scale fluctuations and anomalous transport by using mostly empirical anomalous transport coefficients. The exact structure of anomalous transport, i.e. the form of the transport matrix and thermodynamical forces responsible for anomalous transport, e. g. the pinch effects, the role and the form of the residual stress, etc. is the subject of intense studies and debates. The most common approach is to modify the classical coefficients for perpendicular transport with some empirical anomalous values (see [10–12] and Chap. 8). In this chapter, we highlight the description of plasma dynamics based on the fluid (moment) approach, the main assumptions, the validity limits, and the discussion of how the moment approach is used in the transport codes. The strong magnetic field of fusion devices allows certain classification of the cross-field particle, momentum and energy fluxes, as well as some simplifications of the resulting equations governing these quantities. In particular, the collisionless cross-field fluxes play an important role in defining the electric field, the parallel current and the flows in edge plasmas and are currently included in the transport codes used to model the plasma edge [10–12]. We discuss also plasma transport driven by the inhomogeneity of the plasma parameters along the magnetic field, which are of particular importance in the SOL region where the magnetic field lines intersect material surfaces.

6.1

Hierarchy and Closure of the Fluid Equations

We introduce here the basic moment (fluid) equations to fix the notations and define the relevant variables. In the most general form, the dynamics of electrons, ions, and neutral species can be described with the kinetic Boltzmann equations ∂f α ! e þ v  ∇f α þ α mα ∂t

!! X   v B Eþ C fα, fβ , ∇!v f α ¼ c β

!

!

ð6:1Þ

  ! ! for the distribution functions f α v , r , t of all species α (characterized by the charge eα and mass mα), including the neutral particles. This would be the most comprehensive approach for plasma modeling. The collision integrals C(fα, fβ) on the righthand side of Eq. (6.1) must include all inter-particle and self-collisions as well as

6.1 Hierarchy and Closure of the Fluid Equations

117

various atomic processes such as ionization, charge-exchange, and others. The ! ! electric, E , and magnetic, B , fields in Eq. (6.1) need to be determined selfconsistently from the Maxwell equations with the electric charges and current sources found from the solution of the kinetic equations (6.1). Solving all these equations can only be possible numerically. However, in the near future, this is not feasible without a significant reduction of these equations. In general, the fluid equations themselves are the examples of such a reduction when the time evolution of the six-dimensional distribution function is replaced by a truncated   set of ! b nonlinear equations for space and time evolution of the moments, M r , t , of the     ! ! b ! distribution function f α v , r , t , defined as the integrals M r,t ¼     R b ! ! !  ! ! b b v and M b ! b v f α v , r , t dv . We notice that in general case, both M r,t M   ! ! can be tensors. The first moments of a distribution function F v , r , t are the basic   ! fluid variables such as the particle density, n r , t , the average (fluid) velocity,   !!  ! V r , t , and the pressure, p r , t , that have simple macroscopic meaning:   Z   ! ! ! ! n r , t ¼ F v , r , t dv ,

ð6:2Þ

 !  Z   ! ! ! ! ! ! n r,t V r,t ¼ vF v, r , t dv,

ð6:3Þ

Z   !  m ! ! ! 2 v0 F v0 , r , t d v , p r,t ¼ 3

ð6:4Þ

! ! !!  where v0 ¼ v  V r , t . As one can see from Eq. (6.4), the pressure is expressed in ! ! !!  terms of the “random” particle velocity v0 ¼ v  V r , t , which corresponds to the ! !  particle velocity in the reference frame of the fluid velocity V r , t . We will see that this random velocity will be used in other moments of the distribution function. ! Therefore, it is useful to re-write the kinetic equation (6.1) by using the variable v0 ! instead of v . After some algebra we find ∂f α !0 þ v  ∇f α þ ∂t

( eα mα

!

!!

VB Eþ c

!

!)

dV  dt

 ∇!0 f α v

! ! X   ∂Vi 0 ∂f α e v0  B  ∇!0 f α ¼  vk 0 þ α C fα, fβ , ∂xk ∂vi mα c c v β !  ! ! where f α ¼ f α v0 , r , t , and dð. . .Þ=dt ¼ ∂ð. . .Þ=∂t þ V  ∇ð. . .Þ.

ð6:5Þ

118

6 Fluid Description of Edge Plasma Transport

The evolution equations for the fluid variables are obtained in a standard way [13] ! by taking the moments of Eq. (6.5) with appropriate weights, 1, m v0 , and mv0 2 =2, which leads to the sequence of the equations  ! ∂n þ ∇  nV ¼ Cn , ∂t  ! ! !!  ! ! ∂ mnV ! ! $ VB þ ∇  mnVV ¼ en E þ  ∇p  ∇  Π þ CV , c ∂t  !  ! $ ! ∂ 3 3 ! p þ ∇  pV þ q þ p∇  V þ Π : ∇V ¼ Cp , 2 ∂t 2

ð6:6Þ ð6:7Þ ð6:8Þ

!

where the terms Cn, CV, and Cp are the respective moments of the collision operators resulting in the sources and sinks of the particle density, momentum and energy, which need to be specified for each species, and   Z !   ! m 2 !! q r,t ¼ v0 v0 f v0 , r , t d v0 , 2   Z !0 !0 v0 2 $ !0 !  !0 $ Π¼ m v v  I f v , r , t dv , 3 ! !

ð6:9Þ ð6:10Þ

  $ ! ! where I is the identity tensor. As we see from Eqs. (6.8) and (6.9), vector q r , t $!  and tensor Π r , t describe, correspondingly, the particle energy and momentum fluxes in the moving frame. Hereafter we omit for simplicity the indices α, β, . . . defining different species. $ ! To “close” the system of Eqs. (6.5), (6.6) and (6.7), we need to express q and Π ! (as well as the moments of the collision operator Cn, CV , and Cp) in terms of the density, average velocity and pressure. Within the fluid description, this is only possible by assuming that the Coulomb collisions of charged !particles  of the same 0 ! species are fast enough, so that the distribution function f v , r , t is close to the Maxwellian with the temperature T (which can be different for the species with a large mass difference). Quantitatively, this feature should be related to the smallness of some parameter(s). For the case with no magnetic field, such small parameters, ε  1, are λC/L p and ω/ν ffiffiffiffiffiffiffiffiffi ffi C, where νC is the frequency of the Coulomb collisions of species α, λC ¼ T=m=νC is the mean free path between such collisions, L is the spatial scale of the inhomogeneity of the plasma parameters and ω is the characteristic frequency of their temporal variation. However, the dynamics of plasma embedded in a strong magnetic field with ΩB  νC, where ΩB ¼ eB/mc is the cyclotron frequency, becomes very anisotropic. As a result, the small parameters allowing for the spatial inhomogeneity of magnetized plasma parameters within the

6.1 Hierarchy and Closure of the Fluid Equations

119

pffiffiffiffiffiffiffiffi fluid approximation become (e.g. see [13]) λC/Lk and λC ρ=L⊥ , where ρ ¼ pffiffiffiffiffiffiffiffiffi ffi T=m=ΩB is the particle Larmor radius whereas Lk and L⊥ correspond to the plasma parameter inhomogeneity along and across the magnetic field (we assume here that L⊥ defines the inhomogeneity of both the plasma parameters and the magnetic field). We notice that for ρ/L⊥  1, the cross-field plasma dynamics is reasonably well described by fluid type equations even for collisionless plasmas (e.g. see [14]). !   !!  ! In what follows we will assume that ε  1 and f v0 , r , t ffi FMxw n, T, v0 , r , t . In this case, there are two major approaches to utilize this small parameter in the ! ! $ derivation of q , Π and the moments of the collision operator Cn, CV , and Cp. Both of them are based on the representation of the distribution function in the moving frame as  !  !  ! !  ! ! f v0 , r , t ¼ FMxw n, T, v0 , r , t 1 þ Φ v0 , r , t ,

ð6:11Þ

 !!  where FMxw n, T, v0 , r , t is the dynamic Maxwellian distribution function whereas Φ is small, jΦj  1, and, in addition, does not contribute to the particle density, average velocity and pressure (temperature, T ¼ p/n). In both approaches, the expression (6.11) is substituted in Eq. (6.5). However, the further steps in “closing” the Eqs. (6.6), (6.7) and (6.8) are different. In the Chapman-Enskog approach [15] (adopted for “simple”, one ion species plasma by Braginskii [13]), in zero-order approximation in small parameter ε, an  ! impact of Φ on particle transport is completely ignored and the evolution of n r , t ,   !!  ! ! $ V r , t , and p r , t is described by Eqs. (6.6), (6.7) and (6.8) with no q , Π, and the moments of the This expressing the zero-order time  collision  ! operator.   allows  ! ! ! derivative of n r , t , V r , t , and T r , t in terms of their spatial derivatives. Then, in the first-order approximation, Φ is only retained in the largest terms describing gyro-rotation and collision operators (linearizedover Φ), whereas all ! !  ! other terms are expressed via the spatial derivatives of n r , t , V r , t , and   ! T r , t . Finally, this non-uniform linear integrodifferential equation for Φ is solved by representing Φ in the series of tensorial expansion   v0 2 δik þ . . . , Φ ¼ Φ0 þ Φi v0i þ Φik v0i v0k  3 whereas the coefficients of this expansion are written as infinite series

ð6:12Þ

120

6 Fluid Description of Edge Plasma Transport 1   X     ! 2 2 ðjÞ ! ð1=2Þ Φ 0 r , v0 ¼ a0 r Lj v0 =v2T ,

ð6:13Þ

j¼2 1   X     ! 2 2 ðjÞ ! ð3=2Þ ai r Lj v0 =v2T , Φ i r , v0 ¼

ð6:14Þ

j¼1

1   X     ! 2 2 ðjÞ ! ð5=2Þ aik r Lj v0 =v2T , Φik r , v0 ¼

ð6:15Þ

j¼0 ðαÞ

where Lj ðxÞ are the generalized Laguerre (or Sonin-Laguerre) polynomials and v2T ¼ 2T=m (e.g. see [13, 16, 17] and the references therein). We notice that all the terms in the expansion (6.12), (6.13), (6.14) and (6.15) are orthogonal to each other !0 !  (with the weight function proportional to FMxw n, T, v , r , t ). Moreover, taking   ! into account such orthogonality, we find that Φ does not contribute to n r , t and   ! T r , t whereas implementing Eq. (6.12) into Eq. (6.9) and (6.10) it is easy to show that the components of the energy flux qi and tensor Πik can be expressed through the ð1Þ ð0Þ vector ai and tensor aik :   Z  !   ! m 2 ð3=2Þ 0 2 2 2 pT ð1Þ ! ! qi r , t ¼ v0i v0 L1 a , v =vT f v0 , r , t d v0 ¼  2 5 m i

ð6:16Þ

and Πik ¼

pT ð0Þ a : m ik

ð6:17Þ

!  ! However, the orthogonal functions used for the decomposition of Φ v0 , r , t in (6.12), (6.13), (6.14) and (6.15) are not eigenfunctions of the linearized collision   ! 02 operators. Therefore, the components of Φi r , v contribute not only to the heat   ! ! ! flux q r , t but also to CV which results in the so-called thermal forces depending on ∇T (see [13] for the discussion of the physical nature of the thermal force). Thus, ðjÞ ðjÞ we arrive at the set of infinite number of equations for the coefficients ai and aik . The solutions of these equations can only be found by keeping only a finite number ð1Þ of these coefficients. But the accuracy of physically meaningful parameters ai and ð0Þ aik , determining the energy and momentum fluxes (6.16) and (6.17) and contribut!

ing to Cn, CV , and Cp depends on the number of the terms kept in these sets of ð1Þ ð0Þ equations. In [13, 18] it was found that sufficient accuracy of ai and aik can be

6.1 Hierarchy and Closure of the Fluid Equations

121

reached by considering only two extra terms in Eqs. (6.14) and (6.15) proportional, ð2Þ ð1Þ respectively, to ai and aik , which define the following quantities 

! !

q



r,t ¼

Z

!0 m 0 2 ð3=2Þ  0 2 2  !0 !  !0 v v L2 v =vT f v , r , t d v , 2

ð6:18Þ

and   Z !!  !   ! v0 2 $ ð5=2Þ 0 2 2 ! r , t ¼ m v0 v0  v =vT f v0 , r , t d v0 : I L1 3

$  !

Π

ð6:19Þ

We notice that in strongly magnetized plasmas, sufficient accuracy of collisionless $ ! components of q and Π, which are associated with the drift of particles caused by ð1Þ ð0Þ inhomogeneity of the magnetic field, can be reached by keeping only ai and aik . Although the Chapman-Enskog approach works rather well for “simple” plasma, it becomes too cumbersome for multispecies plasma typical for edge plasmas. In addition, within the Chapman-Enskog approach, some additional effort is needed to $ ! recover the contribution of the heat flux q to the viscosity tensor Π , which is important for both edge plasma transport and turbulence studies [19–21]. An alternative to the Chapman-Enskog method is the Grad approach [22]. In the Grad approach, the component of the distribution function proportional to Φ, recall Eq. (6.11), is kept in both linearized collision operators and all other terms in Eq. (6.5). Then from Eq. (6.5) one can obtain the hierarchy of the evolution ! equations the higher-order moments by multiplying Eq. (6.5) by v0 mv0 2 =2 , !! for $ m v0 v0  v0 2 I =3 , etc. and integrating it over the velocity space. Such a set of the evolution equations can be truncated at some high moment, thus resulting in a closed set of the fluid equations. For example, we find the following evolution $ ! equations for the heat q and momentum Π fluxes neglecting all higher moments: !  !  Tr ! ! ∂q 7 ! 7 ! 2 ! e! $ þ q∇  V þ q ∇ Vþ q∇ V  E  Π 5 5 m ∂t 5 ! $ 7 $ T 5p ! Π  ∇T þ ∇  Π þ ∇T  q  ΩB ¼ C!q , þ 2m m 2m

ð6:20Þ

$  ! $ ! ! ! $ ! $  !Tr $ ∂Π $ 2$ þ Π∇  V þ Π  ∇V þ Π  ∇V  Π : ∇V  Π  ΩB  ΩB  Π 3 ∂t      !Tr   ! $ ! 2 2 2$ ! ! Tr ! ∇q þ ∇q  I∇V   I ∇  q þ CΠ$ , ¼ p ∇V þ ∇V 3 5 3

ð6:21Þ

122

6 Fluid Description of Edge Plasma Transport !

!

where ΩB ¼ ΩB B=B whereas C!q and C$Π are the corresponding moments of the collision operators [17, 22, 23]. We note, however, that for the calculation of the collision-driven energy and momentum flux with sufficient accuracy, theevolution ! !  equation of the higher-order Laguerre polynomials, corresponding to q r , t and $ !  Π r , t , should be considered [13].

6.2

Collisionless Cross-Field Components of Energy and Momentum Fluxes

For strongly magnetized plasmas, ΩB  νC, the leading terms in the fluid equations (6.6), (6.7) and (6.8) and (6.20) and (6.21) are those proportional to the magnetic field strength B. This allows solving the momentum balance Eq. (6.7) for the cross! ! field component of V by expanding V⊥ in the powers of 1/B and neglecting the impact of the collision operators. Observing Eq. (6.7), one finds that in the first order in 1/B, there is a contribution from both the electric field and the pressure gradient: !ð1Þ

!

!

V⊥ ¼ VE þ Vp ,

ð6:22Þ

where ! ! ! VE ¼ c E  B =B2 !

and

!  ! Vp ¼ c B  ∇p =enB2

ð6:23Þ

!

describe, respectively, the E  B and diamagnetic drift velocities. In the second ! order, we find the components of V⊥ related to inertial and collisionless viscosity polarization !ð2Þ

!

!

V⊥ ¼ VI þ VΠ ,

ð6:24Þ

where !

VI ¼ and

!

1 B  ΩB B



 ! !  ! !  ∂ þ VE þ Vp  ∇ VE þ Vp , ∂t

ð6:25Þ

6.2 Collisionless Cross-Field Components of Energy and Momentum Fluxes

123

!

!

VΠ ¼

$ 1 B  ∇  Π: ΩB B

ð6:26Þ

!

Similar expansion for q ⊥ gives the following first-order expression. !ð1Þ q⊥

¼

  5 cp ! B  ∇T : 2 eB2

ð6:27Þ $

The first order term for collisionless (gyro-viscous) momentum flux Π can be found from the cross-field components of Eq. (6.21), which gives the following equation  $ $ ! ! $ b Π Π  ΩB  ΩB  Π K       2$ ! ! Tr ! * * Tr 2 $ * ¼ p ∇V þ ∇V  I ∇  V þ ∇q þ ∇q  I∇ q : 3 3 ð6:28Þ  $ b Π gives the following expression for the gyroThe inversion of the operator K viscous momentum flux $

Πg ¼ !

n  $ !! !! $ !o 1 ! $ $ b  W  I þ 3 b b  I þ 3b b  W  b , 4ΩB

ð6:29Þ

!

where b ¼ B=B and    !Tr   $ ! 2 ! ! Tr ∇q þ ∇q W ¼ p ∇V þ ∇V þ : 5

ð6:30Þ

The gyro-viscous momentum flux (6.29) corresponds to the collisionless components of the momentum flux in [13] with additional terms due to the heat flux gradients, obtained by Mikhailovskii [19]. From Eqs. (6.23) and (6.27) it is easy !

!ð1Þ

to see that p∇Vp ∼ ∇ q ⊥ so that the contributions to the collisionless momentum !ð1Þ

!

flux (6.29) from q ⊥ and diamagnetic velocity Vp are of the same order. However, the direct usage of the diamagnetic velocity (6.23) and the heat flux in (6.27) is not practical because the corresponding components in the balance equations (6.6), (6.7) and (6.8) contain large but divergence-free terms. These divergence-free terms can be removed in a low plasma pressure  ! case  by re-writing  !  the corresponding terms in the continuity equation as ∇  nVp ¼ ∇  nVD , where

124

6 Fluid Description of Edge Plasma Transport

  cT ! 1 VD ¼  B  ∇ : e B2 !

ð6:31Þ

!

Here VD has a simple meaning of the guiding center velocity, which is different for ! ! the electrons, VDe , and ions, VDi . Similarly, the contribution of the divergence-free terms to the energy balance equation (6.8) can be removed by noticing that  !  ! 3! 5 5 !ð1Þ Vp  ∇p þ p∇  Vp þ ∇  q ⊥ ¼ ∇  pVD : 2 2 2

ð6:32Þ

In addition, one can also observe that the diamagnetic contributions to the convective and gyro-viscous terms in the momentum balance equation (6.7) are of the same order and, similarly to the particle and energy balance equations, some of these terms cancel (this is the so-called gyro-viscous cancellation of the contributions of the diamagnetic terms [24–26]). We notice that the contribution of the diamagnetic heat !ð1Þ

$

flux q ⊥ to Πg plays an important role in such cancelation [25]. However, in a non-uniform magnetic field (e.g. in a tokamak), the cancellation is not complete [26]. A somewhat similar cancelation of the collisionless terms occurs in the parallel momentum balance equation, which finally can be written as [27]: 

 !   i ! h! ∂Vk 4p mn þ Vk b  ∇ Vk þ VE  ∇ Vk þ b  ∇ℓnðBÞ  ∇ Vk mnΩB ∂t  2V h!  i ! ! ! ! k  mnVk VE  ∇ℓnðBÞ  b  ∇p  ∇ℓnðBÞ ¼ enb  E  b  ∇p: ΩB ð6:33Þ However, note that Eq. (6.33) contains only the first-order (in 1/B) collisionless cross-field velocities. ! ! Overall, the contributions of the collisionless cross-field E  B and diamagnetic particle and energy fluxes to the particle balance equations (6.6), (6.7) and (6.8) can play an important role, in particular for the cases where anomalous cross-field plasma transport is weak (e.g. in H-mode). In addition to the collisionless particle, energy, and momentum fluxes, the moment equations used in the Grad approach contain also the terms associated ! with the collision operators (e.g. Cn, CV , Cp, etc.). These terms result in both the energy exchange and the forces (e.g. the thermal forces) between different species and provide collision-driven particle, energy, and momentum fluxes. Although the cross-field components of such fluxes are proportional to (ρ/L⊥)(νC/ΩB) and in most cases can be ignored, the components along the magnetic field (e.g. heat flux components) can play the key role in the balance equations (6.6), (6.7) and (6.8). However, careful calculation of these fluxes within the framework of the Grad approach requires the implementation of the so-called 21-moment Grad approximation.

6.3 21-Moment Grad Approximation

6.3

125

21-Moment Grad Approximation

As we have already mentioned above, accuracy of the calculations of  for sufficient ! $ !  ! ! the energy and momentum fluxes q r , t and Π r , t , as well as CV and Cp, one   $  !  ! ! needs also to consider the q r , t and Π r , t moments determined by the expressions (6.18) and (6.19). This increases the number of independent moments $ that must be found. However, from the definition of Π, recall Eq. (6.10), it is easy to $ see that Πik ¼ Πki and Πikδki ¼ 0, so Π is determined by five independent moments. $

! !

!

The same is applicable for Π . Then, allowing also for n, p (or T), V, q , and q , we find that we need to determine 21 moments. To find these moments, we consider Eqs. (6.20) and (6.21), keeping only the highest order terms. Then, for the multi-component plasma, we come to the follow$ ! ing equations for q α and Πα : X α,β ! 5 pα ! ∇Tα  q α  ΩBα ¼ C!q , 2 mα β

ð6:34Þ

  $  ! Tr ! ! $  ! ! 2$ Πα  ΩBα  ΩBα  Πα ¼ pα ∇Vα þ ∇Vα  I ∇  Vα 3   X  Tr 2 2$ ! ! ! þ  I ∇  qα  Cα,β ∇qα þ ∇qα $ , 5 3 Π β ð6:35Þ !

!

!

! !

$

!

$

α,β where Cα,β ! depends on Vα, Vβ, q α, q α, q β, and q β , whereas C$ depends on Πα, Πα, q $ $ Πβ and Πβ . Somewhat similar   $  !  ! !

for q

r , t and Π

Π

equations can be found from the evolution equations

α,β r , t with the collisional terms corresponding to Cα,β ! and C$ .

! ! ! qα, qα, qβ,

! qβ,

! CVα

q

Π $

Having found and one can calculate that, together with Πα , closes the balance equations (6.6), (6.7) and (6.8) completely. However, such calculations for multi-component plasma are extremely cumbersome and go beyond the scope of our consideration. The detail of such derivation can be found in [17]. Nonetheless, just for illustration, we consider here the derivation of the electron energy flux parallel to the magnetic field, qke, assuming that the plasma has one kind of ions with charge Z.

126

6 Fluid Description of Edge Plasma Transport !

Then from Eq. (6.34) and similar equation for q α we find (e.g. see [17])     5 3 7  p ∇ T ¼ νei p V  Vki  α11 qke  α12 qke , 2 e k e 2 e ke 2   15 7 0 ¼  pe Vke  Vki  α21 qke þ α22 qke : 4 2

ð6:36Þ ð6:37Þ

We notice that the terms on the right-hand-sidepof these ffiequations come from ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 e,i e,e e,i e,e C!q þ C!q and C! þ C! . Here νei ¼ ð4=3ÞZ 2πTe =me ðe2 =Te Þ ne ΛC is the q

q

electron-ion collision frequency and ΛC is the Coulomb logarithm, whereas pffiffiffi  pffiffiffi  2 2 13 1 3 2 69 α11 ¼ þ , α12 ¼ α21 ¼ þ , 5Z 10 7 5Z 20 pffiffiffi 9 2 433 α22 ¼ þ , 14Z 280

and ð6:38Þ

are the dimensionless matrix elements of the electron-ion and electron-electron Coulomb collision operators with respect to qke and qke . In addition, we find the following expression for the electron-ion friction force CVk α : CVk α ¼ me ne νei ðVke  Vki 

3 3  q  q Þ: 5pe ke 4pe ke

ð6:39Þ

As one can see from Eq. (6.39), the electron-ion friction force depends not only on the difference between the electron and ion velocities, Vke  Vki, but also on the electron temperature gradient ∇kTe (this part of the friction force is called the “thermal force”). Similarly, the electron energy flux qke depends not only on the electron temperature gradient but also on the difference between the electron and ion velocities. After some algebra, from Eqs. (6.37), (6.38) and (6.39) we find: qke ¼ 

  α  ð5=2Þα 5 pe ∇k Te α22 3 12 þ pe Vke  Vki 22 , 2 2 νei α0 2 α20

ð6:40Þ

where α20 ¼ α11 α22 þ α12 α21 .

6.4

The Electron Heat Transport in a Weakly Collisional Regime

Both the Chapman-Enskog and Grad approaches to the derivation of the closed system of the fluid equations from the kinetic theory assume a rather slow spatiotemporal variation of particle density, average velocity and temperature. In

6.4 The Electron Heat Transport in a Weakly Collisional Regime

127

particular, the characteristic length of spatial variation of these parameters, L, (in the ! absence of the magnetic field or in the direction parallel to B) should be larger than the mean free path ofthe thermal particles, λC. Then, as we have seen above, the  ! ! distribution function f v , r can be expanded in the series of the integer powers of the parameter γ λC/L < 1 (here for simplicity we consider a stationary process) and the closed system of fluid equations is derived. In most cases, only linear, / γ, terms are held in this expansion. However, such an approach poses some questions. First, a practical one: how small the parameter γ should be to ensure that the linear approximation describes the transport properties of the gas/plasma well? And the second, somewhat more academic question: are we sure that the distribution function does not have some terms (e.g. / exp (Cγr), where C and r are some positive constants) which cannot be expanded in the series of any powers of γ, but which can still be important for some range of γ (see also [16, 28, 29]). We start our consideration with the simpler, first issue. Historically, in plasmarelated applications, this issue was first raised with respect to the validity range of the Spitzer-Härm expression [30] for the electron conductive heat flux. According to ! b 0 ) / γ, which Eq. (6.16), the heat flux q is determined by the function Φi(v0 ) ¼ γΦ(v can be found either from expansion (6.14) or from [30], where Φi(v0) was obtained from the numerical solution of linearized electron kinetic equation. In [31, 32] it was pointed out that the integral expression (6.16) for the heat flux contains high powers of the electron velocity. As a result, the main contribution to this integral comes from the velocities v0 ∼ vcond ∼ 2 3  vTe . However, the magnitude of Φi(v0) increases with increasing v0 > vTe (e.g. see both the expression (6.14) and the results from b cond)  102 [30]. Thus, the applicability of the [30]) and for v0 ~ vcond we have Φ(v linearized solution for electron heat conduction, which requires Φðvcond Þ b ðvcond Þ < 1 gives the limitation for the validity of the Spitzer-Härm expression γΦ e 102 . Such a severe limitation can be for the electron conductive heat flux: γ < understood by recalling that the Coulomb mean-free path, λv, of a particle with velocity v scales as λv ¼ λT  ðv=vTe Þ4 . Therefore, the linear approximation for the electron conductive heat flux, which assumes that the electron distribution function is close to the Maxwellian, can only be valid if the electrons with velocities e 1, which, finally, results in γ < e 102 . v0 ∼ vcond > vTe are collisional, λvcond =L < 2 e 10 usually does not hold and However, in the edge plasmas, the inequality γ < the classical expression for the electron heat flux is not applicable. A similar problem often occurs in the plasmas related to inertial confinement experiments (e.g. see [32]). For the case of γ > 102, the electron distribution function starts to deviate significantly from the Maxwellian distribution at v0 ~ vcond. To describe this “nonlocal” effect for heat conduction along the magnetic field (in the z-direction), the expansion of the distribution function in the powers of the parameter γ becomes impractical and an integral expression for the heat flux

128

6 Fluid Description of Edge Plasma Transport

Fig. 6.1 Ratio of the effective, κeff, to the SpitzerHärm, κSH, electron heat conductivities as a function of kλT, where k is the wavenumber of the initial electron temperature perturbation. The filled circles are the results of numerical simulations [37], the curves A, K, and L are from the references [33– 35]. (Reproduced with permission from [35], © AIP Publishing 1993)

100 K

A

κeff / κSH

10–1

10–2 L 10–3

10–4 10–3

Z qð z Þ ¼ 

Kðz, z0 Þ

10–2

∂Te 0 dx , ∂z0

10–1 kλT

100

101

ð6:41Þ

can be considered with different kernels K(z, z0) which were suggested over the years (e.g. see [33–36]). A comparison of the outcomes of these nonlocal models with the results of numerical simulations of the decay of a small amplitude, harmonic electron temperature perturbation is shown in Fig. 6.1. However, in spite of the reasonable agreement of the results coming from some non-local models based on the integral expression (6.41) and numerical simulations, in the edge fluid plasma transport codes, a much simpler approach is usually employed. It is based on the so-called “flux limiting” expression for the heat flux suggested in [38]. This expression simply constrains the Spitzer-Härm heat flux, qSH(z), by the fraction, frFS  0.2  0.4, of the free streaming electron heat flux, qFS(z) ¼ n(z)Te(z){2Te(z)/πm}1/2, so that qð xÞ ¼

qSH ðzÞfrFS qFS ðzÞ : jqSH ðzÞj þfrFS qFS ðzÞ

ð6:42Þ

As we see, such expression describes the reduction of q(x) in comparison with qSH(z). For a monotonic temperature profile Te(z), which is the case for both the inertial fusion applications and the edge plasmas, this reduction has clear physical meaning in the high-temperature region, where, at γ > 102, the tail of the electron distribution function is depleted because of the runaway of the weakly collisional electrons into the low-temperature region. However, in the low-temperature region, these suprathermal electrons result in the increase of q(x) beyond qSH(z), but such an effect is not captured by Eq. (6.42). Nonetheless, expression (6.42) and similar ones for ion heat conduction and viscosity along the magnetic field are often used in edge plasma transport codes.

6.4 The Electron Heat Transport in a Weakly Collisional Regime

129

We notice that integral expressions for the heat flux, resembling Eq. (6.41), were suggested to emulate the effects of the Landau resonances in fluid turbulence codes (e.g. see [39] and the references therein). Next, we discuss the issue of non-expandable terms in the distribution function and their potential impact on the electron heat flux. It is unlikely that this issue has a universal answer valid for any setting of the electron density and temperature profiles. Therefore, following [40], we consider plasma parameter profiles which resemble those typical for the SOL plasma in the high recycling conditions. In [40] it was shown that neglecting the electron-ion energy exchange, the stationary electron kinetic equation allowing for both electron-electron, Cee(fe, fe), and electron-ion scattering, Cei(fe), as well as for the electric field, E(z), effects:

vz

  ! ∂f e v , z ∂z

  ! ∂f v , z e eEðzÞ  ¼ Cee ðf e , f e Þ þ Cei ðf e Þ, m ∂vz !

ð6:43Þ

!

allows solution in the self-similar variable w ¼ v ½m=2TðzÞ 1=2 by using the ansatz     ! ! f e v , z ¼ NF w =Tα ðzÞ,

ð6:44Þ

(where T(z) is the effective electron temperature, α is an adjustable parameter, and N is the normalization constant) providing that the T(z) satisfies the equation γ¼

λ / Tðα1=2Þ dT=dz ¼ const: L

ð6:45Þ

We notice that Eq. (6.45) gives the following relations for the electron density, n(z) / T(3/2α)(z), and the electron energy flux, q(z) / T(3α)(z). Although for α 6¼ 3 q(z) is not a constant, the relative magnitude of the corresponding energy source/ sink, jdq(z)/dxj (nTνee)1  (3α)γ, is small for γ < 1. As a result, to maintain energy balance, a relatively small energy source/sink localized at the electron energy ~T can be added into Eq. (6.43), which does not alter the kinetics of the energetic electrons we are mostly concerned about. Interestingly, the case α ¼ 3 corresponds to the electron temperature profile describing a constant electron heat flux for the 5/2 Spitzer-Härm electron heat conduction   coefficient / T . ! For γ  1, the equation for F w was solved in [40] analytically by considering different ranges of the dimensionless velocity w and then matching the corresponding solutions (something similar was done in [41, 42] for theproblem ! of runaway electrons). It was found that the distribution function F w can be represented as a series in the integer powers of γ (which is the basic assumption in e γ1=3 only, whereas for both the Chapman-Enskog and Grad approaches) for w2 <   ! e γ1=2 , F w is described by the following “unexpandable” expression w2 >

130

6 Fluid Description of Edge Plasma Transport

  n ob FðϑÞ ! F w / exp ð2=3Þγ1=2 , w2α

ð6:46Þ

where b FðϑÞ is a function of the angle ϑ between the coordinate axis z and the vector ! w (only the terms of the highest order in the small parameter γ are left in Eq. (6.46)). Recalling that the main contribution to the electron conductive heat flux is due to electrons with the normalized electron velocity ðv=VTe Þcond ¼ wcond  2 3 , we e γ1=3 results virtually in the same limitation for γ as it find that the condition w2cond < e 102. Numerical solutions of the electron kinetic equation was found in [31, 32]: γ < in self-similar variables [43] confirm the analytic results of [40]. From Eq. (6.46) we see that for the electron heat flux written in the self-similar variable w, R the expression ! 2 q ¼ mwz w FðwÞdw , diverges at large w for α 3. However, for α ¼ 3 this divergence is logarithmically weak and can be moderated by assuming that in practice, the maximum electron energy is always limited by some value. Thus, from the analysis of both the Spitzer-Härm solution of the electron kinetic equation and the solution of the electron kinetic equation in a self-similar variable, we find that applicability of both the Spitzer-Härm expression for electron heat conduction and the solution of kinetic electron equation in the form of expansion of the electron distribution function in integer powers of γ are limited by relatively e 102 . small γ : γ
e 1 for Danom  104cm2/s, T ~ 10 eV, and n ~ 1014cm3 we find that D 3 e N=n > 10 . Note that the ratio of the neutral to plasma densities for low-temperature, high recycling divertor plasma can be ~101. However, we should keep in mind that the impact of neutrals on cross-field plasma diffusion is not described by Eq. (6.48) since the effective displacement of the ion in the course of the elastic collision process is of the order of the ion gyro-radius. Nonetheless, the contributions of the neutrals to the momentum and heat transport can be very important for dumping the plasma flows (including the shear flow, which is an important ingredient in anomalous plasma transport, see Chap. 7) and for cooling the

132

6 Fluid Description of Edge Plasma Transport

divertor plasma to sub-eV temperature, promoting plasma recombination effects important in the divertor plasma detachment process (see Chap.9). To avoid the unphysical contribution of neutrals in the edge plasma regions where e Li, the neutral diffusive fluxes are majorized by corresponding free-streaming λCX > expressions similar to Eq. (6.42). Unfortunately, fluid description of the hydrogen molecules, which have a mean free path for the collisions with hydrogen ions much longer than that of the hydrogen atoms, strictly speaking, cannot be used for the plasma parameters of interest. In addition, vibrational excitation of molecules can play an important role in both plasma energy dissipation at low (~1 eV) temperatures and in plasma recombination processes (see Chap. 2). The incorporation of vibrational excitation of molecules in fluid models would significantly complicate them.

6.6

Anomalous Effects in Edge Plasma Transport Equations

Here we overview the basic structure of the transport equations used in edge plasma simulations. We should note that existing formulations of the transport equations often differ in various details and effects included (e.g. see [10–12]). Here we only describe the most essential elements and comment on various additional effects. Technically, either electron or ion continuity equations can be used to describe the evolution of the plasma density. Most often, the ion continuity equation is used. Keeping only the first-order terms in the ion velocity and adding an ad hoc anomalous density transport, one has    !   !  !  ∂n þ ∇k nVk þ ∇  nVE þ ∇  nVDi þ ∇  Γ an ¼ Sn , ∂t

ð6:49Þ

!

where n is the plasma density and Γ an is the anomalous plasma density flux. In the ion continuity equation, we have neglected the second-order drift terms, such as the inertial and viscous drifts described by Eq. (6.24). This approximation is based on the assumption that the first-order electric and diamagnetic drifts, as in Eq. (6.23), are dominant. Note that some formulations include these higher-order drifts into the density evolution equation [10, 11]. The anomalous density flux in (6.49) is usually defined by correlations between !

!

!

the density and the lowest order particle velocity due to the E  B drift, Γ an ¼

 ! ! e e e n and VE are the turbulence-driven fluctuating plasma density and nV E , where e velocity, and h. . .i means statistical averaging. It is assumed here that the density fluctuation is small, j e n j n. This flux should be determined from the first-principle turbulence simulations. In transport codes, the anomalous density flux is parameterized by an empirical anomalous diffusion coefficient. In addition, besides the purely

6.6 Anomalous Effects in Edge Plasma Transport Equations

133 !

diffusive term, in some models the anomalous particle flux Γ an includes the pinch (thermo-diffusion) terms that depend on temperature gradients (e.g. [10]): ! Γ an

¼ D⊥ ∇⊥ n  D⊥Te n∇⊥ ℓnðTe Þ  D⊥Ti n∇⊥ ℓnðTi Þ,

ð6:50Þ

where D⊥, D⊥Te , and D⊥Ti are the anomalous “diffusivities”. The electrostatic potential φ, governing the electric field effects, is determined ! ! from conservation of the electric current J : ∇  J ¼ 0. This equation can be written as the evolution of the generalized vorticity ϕ¼

  ∇2 p mi ∇⊥  ðn∇⊥ φÞ þ ⊥ i : B Ze !

ð6:51Þ

!

Since the contributions of the large E  B drift terms of electrons and ions cancel, the second-order drift terms are usually added, which gives: ∂ϕ ! þ VE  ∇ϕ þ ∇  ∂t

!

Γ ϕ n

! ¼

n ! ! o 1 * B  ∇Jk þ ∇⊥  ðμ⊥i ∇⊥ ϕÞ þ ∇k ðμki ∇k ϕÞ þ ∇  n VDi  VDe , e ð6:52Þ

where μ⊥i and μki are the anomalous viscosity coefficients and  Jk ¼ σ k

 ∇k pe αT þ ∇k T e ,  ∇k φ þ en e

ð6:53Þ

is the parallel electric current, σk is the plasma conductivity along the magnetic field and αT is the thermal force coefficient that depends on the effective ion charge Zeff (for Zeff ¼ 1, αT ¼ 0.71). We omitted collisional viscosity in Eq. (6.52) and did not include the turbulent Reynolds stress effects of the negative viscosity type, nor did we consider any effects of turbulent residual stresses that may result in the generation of sheared flow velocity (see Chap. 7). The equation for plasma momentum balance can be obtained as a sum of the corresponding ion and electron equations n o ! ! ! !  ! ∂ðnVk Þ þ ∇  nVk b þ nVE þ 2nVDi þ Γ an Vk  nVk VE  ∇ℓnðBÞ ¼ ∂t     ∇k ðpe þ pi Þ  þ ∇⊥  μ⊥i ∇⊥ Vk þ ∇k  μki ∇k Vk þ SiN , mi n ð6:54Þ

134

6 Fluid Description of Edge Plasma Transport

where SiN describes the impact of the plasma-neutral interactions and includes both the ion-neutral elastic collisions and neutral ionization. The anomalous terms in the momentum conservation originate from several terms involving averaging of the ! ! plasma density, parallel velocity, and the E  B drift fluctuations, and may also involve pressure fluctuations, e.g. see Eq. (6.33). Eq. (6.54) includes also the ! ! collisionless first order E  B and diamagnetic fluxes shown in Eq. (6.33). Overall, turbulent transport is parameterized by anomalous density transport and anomalous viscosity. The turbulent momentum transport may also involve the pinch and “residual” terms, which do not depend on the velocity gradients and velocity but are rather driven by gradients of other plasma parameters. Alternatively, only the ion momentum balance equation can be considered (see [10, 11]) although for this case, one should also solve the vorticity equation (6.52) to find the electrostatic potential. The electron and ion energy balance equations are most commonly written as 3 ∂ðnTi Þ þ∇ 2 ∂t



! ! ! 3 5 3 5! !ðiÞ nT V þ nT V þ nT V b þ Γ an Ti þ q an 2 i E 2 i Di 2 i k 2



!

þ nTi ∇k Vk  2nTi VE  ∇ℓnðBÞ ¼ Spi , 



! ! Jk 3 ∂ðnTe Þ 3 5 3 þ∇ nT V þ nT V þ nT V  2 ∂t 2 e E 2 e De 2 e k en   ! Jk  2nTe VE  ∇ℓnðBÞ ¼ Spe , þ nTe ∇k Vk  en



!

ð6:55Þ

5! !ðeÞ b þ Γ an Te þ q an 2

ð6:56Þ !ðe,iÞ

ðe,iÞ

where q an ¼ nχ⊥ ∇⊥ Tðe,iÞ describe anomalous electron and ion heat conduction ðe,iÞ determined by the anomalous heat diffusivities, χ⊥ ; Spe and Spi are the electron and ion energy sinks/source terms describing the Joule heating, electron-ion energy exchange, ion and electron interactions with neutrals, etc. Strictly speaking, Eqs. (6.49), (6.50), (6.51), (6.52), (6.53), (6.54), (6.55) and (6.56) should be accompanied by corresponding equations for the impurities, which are ubiquitous in edge plasmas. However, the impurity equations are very cumbersome (e.g. see [17]) and their consideration goes beyond the scope of this chapter. We note, however, that in many cases, the analysis of the experimental data suggests that the anomalous convective cross-field energy transport should enter with the coefficient 3/2 rather than with 5/2 as it is written in Eqs. (6.55) and (6.56). Such a conclusion is also supported by some theoretical arguments valid for the case where the plasma parameters can be separated into the mean and the small, turbulence-driven fluctuating parts (see

[53,  54]) when the anomalous

 particle and ! ! ! e e ! e VE . In this case, heat fluxes can be defined as Γ an ¼ e n VE and q an ¼ ð3=2Þn T

References

135

n ! o ! one more term, 2 TΓ an þ ð2=3Þq an  ∇ℓnðBÞ, should be added on the left-hand side of the pressure balance equation (6.55) and (6.56).

6.7

Conclusions

As of today, modeling of edge plasma transport, which incorporates the particle, momentum, and energy fluxes, the atomic physics, the plasma-wall interactions, etc., relies on fluid plasma models. Although such models have some issues with their applicability (e.g. an impact of nonlocal effects on electron heat transport) and use a crude and, actually, ad hoc description of anomalous cross-field transport, they reproduce many features observed in experiments (see Chaps. 8 and 9).

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Chapter 7

Anomalous Cross-Field Transport in Edge Plasma

Abstract So-called “anomalous” plasma transport due to micro- and meso-scale fluctuations of electromagnetic fields caused by different plasma instabilities is one of the backbones of plasma physics in magnetic fusion devices. In this chapter the main mechanisms driving and stabilizing the most typical edge plasma instabilities are considered and the underlying physics is discussed.

7.1

Introduction

As we discussed in Chap. 6, classical cross-field plasma transport is determined, roughly speaking, by two main components: charged particle motion in virtually stationary magnetic and electric fields and Coulomb collisions. Cross-field drift of charged particles related to inhomogeneity of the magnetic field, the direction of which depends on the sign of the charge, causes global polarization of the tokamak ! ! plasma column, accompanied by the E  B plasma convection, and, in addition, can result in a significant departure of the particles from their initial magnetic flux surfaces. On the one hand, the Coulomb collisions control the electric current along the magnetic field lines, which balances plasma polarization due to the ! ! magnetic drift and, therefore, settles the intensity of the E  B plasma convection. On the other hand, the collisions “erase memory” of the charged particles on their “initial” magnetic flux surface and introduce stochasticity in the charged particle motion. As a result, even though the classical plasma energy and particle fluxes through magnetic flux surface are determined by the local plasma parameters and ! ! their gradients on the flux surface, the processes governing these fluxes (e.g. E  B plasma convection) happen on a “global” length-scale of the order of the tokamak minor radius a. However, due to inhomogeneity of the density and temperature, the tokamak plasma, as it is discussed below, is often unstable. These instabilities result in the
 ! formation of electrostatic potential φ r , t having filamentary structure extended along the magnetic field lines over some distance λk and a relatively small characteristic cross-field size, λ⊥  a, λk (for simplicity, we neglect perturbations of the © Springer Nature Switzerland AG 2020 S. Krasheninnikov et al., On the Edge of Magnetic Fusion Devices, Springer Series in Plasma Science and Technology, https://doi.org/10.1007/978-3-030-49594-7_7

139

140

7 Anomalous Cross-Field Transport in Edge Plasma

magnetic field). Therefore, as an illustration, we consider the motion of a charged ! ! particle in a constant magnetic field B0 ¼ B0 e z and a 2D electrostatic potential, φ(x, y, t), with a characteristic magnitude, φ0, and a spatial scale length, λ⊥, which varies on the time-scale τφ ~ ω1. Assume that λ⊥ is larger than the particle gyroradius whereas ω is smaller than the particle gyrofrequency. In this case, the charged ! ! particle will mostly experience E  B drift along the equipotentials of φ(x, y, t) with a characteristic speed V!E B! ∼ cφ0 =ðλ⊥ B0 Þ. However, the equipotentials can be e τφ and in this time, the particle would considered “fixed” only for the time t < move by the distance δ ∼ V!E B! τφ along the equipotentials, which we assume to be smaller than λ⊥. If the “landscape” of the φ(x, y, t) equipotentials is completely e τφ, then the particle at t > e τφ will drift along an equipotenchanged by the time t > e τφ . As a result, the distance δ can tial completely different from what it was for t < be considered as a “jump” of the particle in the (x,y) plane, which occurs in random directions within the time ~τφ, and particle transport in the φ(x, y, t) potential has a e λ⊥, and estimating diffusive nature. Taking into account that we assumed δ < eφ0 ~ T (where T is the plasma temperature), we find that the particle diffusion e DB ¼ cT=eB0 , where DB is the so-called Bohm diffusion coefficient Dφ < coefficient. This physical picture shows that the presence of electric field fluctuations having relatively small spatiotemporal scales can result in cross-field plasma transport exceeding, in practice, the classical one which is governed by quasi-stationary and large-spatial-scale processes. However, in reality, anomalous cross-field plasma transport is much more complex. Nonlinear interactions of plasma fluctuations result, on the one hand, in some sort of self-regulation and even suppression of anomalous transport. On the other hand, they can change transport from relatively slow diffusive to very fast convective. Today we have no full understanding of all processes governing anomalous plasma transport. Therefore, in what follows, we present just very basic ingredients of plasma instabilities and features of anomalous plasma transport, with the emphasis on the processes more typical for the edge plasmas. Comprehensive review of instabilites in inhomogeneous plasmas is given in [148]. In this chapter, we adopt the following notations: all parameters with “tilde” are considered to be perturbations small in comparison with the stationary (or quasistationary) parameters having no “tilde” sign, or resulting in such small perturbations: e.g. a perturbation of the plasma density e n is much smaller than the background ! e plasma density n ðje n j=n  1Þ ; V is a small perturbation of the velocity, which ! either is small in comparison with the background velocity V, or produces a small variation of such plasma parameters as the pressure, density, etc.

7.2 Linear Theory of Edge Plasma Instabilities

7.2 7.2.1

141

Linear Theory of Edge Plasma Instabilities Collisionless Drift Waves

We start our consideration with the simplest physical picture of collisionless drift ! ! ! waves in plasma embedded into a constant magnetic field, B ¼ B e z (where e z is the unit vector in the z-direction). These waves are characterized by the frequency, ω, which is much lower than the ion gyrofrequency, ΩBi, so the charged particle motion ! ! across the magnetic field is largely described by the E  B drift. In addition, we ! consider such wave vectors, k , that the phase velocity of the wave along the magnetic field lines, ω/kk, satisfies the following inequalities. VTi < ω=kk < VTe , !

ð7:1Þ

!

B is the wave vector component along the magnetic field, where kk ¼ k  B p=ffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi whereas VTe ¼ Te =m and VTi ¼ Ti =M are the electron and ion thermal velocities, respectively. As a result, in this case, we largely can ignore the effects of the Landau resonances of the wave with both electrons and ions, which can play an important role in collisionless or weakly collisional plasmas. First, we assume that the stationary plasma density, n(x), is inhomogeneous in the x-direction, the electron temperature, Te, is constant and an impact of the ion temperature can be neglected (the “cold” ion approximation). Let us now consider the evolution of a plasma slab, inclined at some small angle to the direction of the magnetic field, which is shifted in the x-direction from its original position (as shown in Fig. 7.1).

Fig. 7.1 Sketch of a 3D plasma density protrusion in the x-direction resulting in plasma polarization caused by the electron motion along the magnetic field lines (a); Advection of the plasma !

!

density contours at z ¼ z0 in the y-direction due to E  B drift in the x-direction (b)

142

7 Anomalous Cross-Field Transport in Edge Plasma

Since initially the plasma density was not homogeneous in the x-direction, such a protrusion will cause a weak plasma inhomogeneity along both z- and y-directions. As a result, the electrons, which are much lighter and, therefore, much faster than the ions, will try to escape from the plasma density bump, leaving the bulky and slow ions alone. However, this charge separation along the magnetic field can only go until the electric field that emerges from such plasma polarization stops further electron escape, see Fig. 7.1a. We notice that due to the inclination of the shifted plasma slab, the resulting electric field has a component in the y-direction (see ! ! Fig. 7.1b), which will cause the E  B plasma drift in the x-direction. The direction ! ! of this E  B plasma drift will be different on the different slopes of the density protrusion: on one side it will move the plasma protrusion toward its initial position (and decrease the protrusion), whereas on the other one it will move the plasma out from its initial position (and increase the protrusion) as shown in Fig. 7.1b. As a result, the plasma protrusion will be “advected” in the y-direction, exhibiting a wavelike motion. However, we note that even though such a wave of plasma density perturbation propagates in the y-direction, the displacement of the plasma density per se occurs only in the x-direction. This simplest physical picture of collisionless drift waves can be easily supplemented by a quantitative description (e.g. see [1–4]). For this, we will assume that the perturbations of the plasma density, e n , are small ðje n j  nÞ and all nonlinear effects can be ignored. In addition, we will consider the characteristic wavelength of the perturbations much larger than the Debye length, and the plasma can be considered quasi-neutral, e ne ffi e ni  e n, even though some charge polarization effects exist. We will ignore the ion motion along the magnetic field lines and assume that parallel dynamics of fast electrons reaches equilibrium virtually instantaneously so the gradient of the electron pressure along the magnetic field is balanced by the electric force, which gives the Boltzmann relation for the perturbed electron e: density and electrostatic potential φ e φ e ne ee  ϕ: ¼ Te n

ð7:2Þ

Finally, we assume that the cross-field plasma (both electron and ion) velocity is ! ! determined by E  B drift: !

V!E B! ¼ 


! c ∇φ  B : B2

ð7:3Þ !

Then, taking into account that for a straight constant magnetic field ∇  V!E B! ¼ 0, we have the following form of the plasma continuity equation

7.2 Linear Theory of Edge Plasma Instabilities

143


!  ! ∂n ∂n þ ∇  V!E B! n  þ ∇n  V!E B! ¼ 0: ∂t ∂t

ð7:4Þ

As a result, from expressions (7.2 and 7.4) we find e ∂e ∂e n ∂ϕ n ∂e n þ Udw ¼ þ Udw ¼ 0, ∂t ∂y ∂t ∂y

ð7:5Þ

where Udw ðxÞ ¼ 

cTe dℓnðnÞ cTe  Λ ðxÞ: eB dx eB n

ð7:6Þ


 ! It is easy to see that the general solution of Eq. (7.5) can be written as e n r,t ¼ f ðy  Udw tÞ and describes, in agreement with our physical picture, the wave propagating along the y coordinate (here f(y) is an arbitrary function, which can also depend on both the x and z coordinates). !

!dia

We notice that Udw is equal to the electron diamagnetic velocity, Ve , which occurs due to inhomogeneity of the electron Larmor circles and can be found from the cross-field electron momentum balance equation where the electron pressure gradient (the density gradient, for the case of constant temperature) is balanced by the Lorentz force:   dnðxÞ ! eB0 !dia ! ex  Ve  e z nðxÞ ¼ 0: Te dx c

ð7:7Þ

However, the expression for the evolution of the perturbed plasma density (7.5) does not take into account some crucially important effects. First, in the derivation of Eq. (7.5) we assumed that the cross-field motion of the charged particles is only due ! ! to E  B drift, which is the same for both electrons and ions. However, the mass difference causes an important disparity in the dynamics of the electrons and ions. It results in a more complex governing equation for the perturbed plasma density, which becomes important for both the linear stability and nonlinear interactions of the drift waves. Secondly, and more importantly, Eq. (6.5) shows that the amplitudes of both the perturbed plasma density and electrostatic potential, linked to the density perturbation through the Boltzmann relation (7.2), remain the same and do not grow in time. Therefore, it cannot describe instabilities resulting in large plasma density/potential fluctuations and strong cross-field anomalous transport observed in experiments. ! To address the first issue we consider the equation for the ion velocity, Vi, which follows from the ion momentum balance equation and in the cold ion approximation reads:

144

7 Anomalous Cross-Field Transport in Edge Plasma

M

!
 dVi eB ! ! ¼ e∇φ þ Vi  e z , dt c

ð7:8Þ

where M is the ion mass and ! !
! ! dVi ∂Vi þ Vi  ∇ Vi  dt ∂t

ð7:9Þ

is the so-called material derivative. Since we consider the characteristic frequency ω  ΩBi, the leading cross-field term depending on the ion velocity in Eq. (7.8) is the last one and we can find the ! solution for Vi⊥ through successive approximations in the small parameter ω/ΩBi. Keeping only two terms in such an expansion and allowing for a linear approximation in the amplitude of the electrostatic potential, from Eqs. (7.8 and 7.9) we find
! e

V i⊥ ¼ c

!

∇e φ  ez B

 

  1 ∂ ∇e φ c , ΩBi ∂t B

ð7:10Þ

where the second term is from the ion inertia term in Eq. (7.8). We notice that the inertial term on the right-hand side of Eq. (7.10) is smaller than ! ! the first one which is already the familiar E  B drift velocity. However, unlike the ! ! E  B drift, the second term is not divergence-free. ! ~ In the linear approximation of the parallel component of the ion velocity, Vi k , from Eq. (7.8) we have ! ~ ~ ∂Vi k e∇k φ ¼ : M ∂t

ð7:11Þ

Then, substituting the expressions (7.10 and 7.11) into the ion continuity equation, we have 2

∂ ∂t2



 2 ~ni ~ ¼ 0, ~  C2 ∇2 ɸ ~ þ Udw ∂ ɸ  ρ2s ∇2⊥ ɸ s k n0 ∂t∂y

ð7:12Þ

pffiffiffiffiffiffiffiffiffiffiffiffi where Cs ¼ Te =M is the ion sound speed and ρs is an effective ion Larmor radius, defined as ρ2s ¼ Te = MΩ2Bi . From Eq. (7.12), using the Boltzmann relation (7.2) for the electron density and assuming the quasi-neutrality condition, we obtain the following equation for the evolution of a small plasma density perturbation:

7.2 Linear Theory of Edge Plasma Instabilities

145

2 2 ∂ ∂ ~n 2 2 2 2 ~ ~ f~ n  ρ ∇ ∇ ¼ 0: n g  C n þ U dw ⊥ s s k ∂t2 ∂t∂y

ð7:13Þ

As we can see, unlike Eq. (7.5), Eq. (7.13) does not describe the advection
ofthe ! perturbed plasma density in the y-direction as a whole anymore, e n t, r ¼ f ðy  Udw tÞ. The reason for this is the so-called dispersion of the drift wave
! frequency ! (i.e. the dependence of the frequency ω on the wave vector k , ω k ). Indeed, assuming that the wavelength in the x-direction is much smaller than the characteristic scale length of the inhomogeneity of the plasma density, |dℓn(n)/dx|1, we can assume that Λn(x) ¼ const. and use the eikonal approximation [5] taking e n¼
! ! b nω,!k exp iω þ ik  r , where b nω,!k is the amplitude of the corresponding wave !

packet. To simplify notations, hereafter we omit both the “hat” and the ω and k indices over and at the Fourier harmonics. Then, from Eq. (7.5) we find
! ω k ¼ Udw ky  ω :

ð7:14Þ


! ! ! This simple relation gives the wave group velocity, ∂ω k =∂ k ¼ Udw e y , !

independent of k , which means that all spatial scale lengths will be advected in the y-direction
with  the same speed and, therefore, the spatial shape of δn will be ! preserved: e n t, r ¼ f ðy  Udw tÞ . But, Eq. (7.13) yields a more complex dispersion, 2 2 ω C s kk þ 2 ¼ 1 þ ρ2s k2⊥ , ω ω

ð7:15Þ

!

where k ⊥ and kk are the components of the wave vector perpendicular and parallel to ! the magnetic field lines.
Thus,  now the group velocity depends on k and, therefore, ! the spatial shape of e n r , t will change in time (e.g. in the y-direction). The solution of Eq. (7.15) has two important branches. For k2k ! 0, Eq. (7.15) gives the dispersion of the drift waves modified, in comparison with Eq. (7.14), by the cross-field ion inertia:
! ω k ¼

ω C s kk , 1 þ ρ2s k2⊥

whereas for k2k ! 1, we have the ion sound waves with

ð7:16Þ

146

7 Anomalous Cross-Field Transport in Edge Plasma

Cs kk ω ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω : 1 þ ρ2s k2⊥

ð7:17Þ

pffiffiffiffiffiffiffiffiffiffiffiffi We notice that the phase velocity of the ion sound waves is ∼Cs ¼ Te =M . Therefore, inequalities (7.1) for the ion sound waves only hold for Te Ti, which is compatible with the cold ion approximation we are using here. However, in the tokamak plasmas, where usually Te Ti, the ion sound waves are strongly damped due to the Landau resonance with ions. We notice that in Eq. (7.12) we neglect the x-component of the inertial part of the ! cross-field ion velocity in the term Vi⊥  ∇n since its contribution is much smaller ! ! than that of the corresponding E  B drift velocity component. However, this term becomes important for the evolution of the amplitude of the wave packet. This effect can also be seen for the case of non-constant Λn(x) where the drift wave can be localized within some range along the x-coordinate.

7.2.2

Localized Drift Wave

In the previous consideration of the drift waves, we used the eikonal approximation where all perturbations are proportional to exp(ikxx), which is valid for j kx j j Λn j ¼ const.. However, taking into account the x-dependence of Λn, we can also consider a drift wave where the perturbations are described by bfðxÞ expðiωt þ iky y þ ikk zÞ, where bf ðxÞ is some function localized in the x-direction. As an example, we consider a case of nðxÞ ¼ n 


 Δn x tanh , 2 w

ð7:18Þ

where n and Δn are constants and w is some scale-length. We assume that Δn  n, and using the Boussinesq approximation that omits the density variation in the inertial term, neglecting the parallel ion dynamics in Eq. (7.13), we arrive at the e ð xÞ following differential equation for ϕ
e e e b  =ωÞ ðω d2 ϕ ϕ 2 2 ϕ þ ¼ 1 þ ρ k , s y ρ2s dx2 cosh 2 ðx=wÞ ρ2s

ð7:19Þ

cTe Δn e ðxÞ, ω should be considered an effective . For a localized ϕ eB 2wn eigenvalue of the solution of Eq. (7.19). We notice that Eq. (7.19) is similar to the
 Schrödinger equation for an electron with effective electron energy /  1 þ ρ2s k2y b ¼ where ω

in the potential well / cosh2(x/w) [6]. Using the results of [6], after some algebra, we find that the solution of Eq. (7.19) is characterized by an integer number m

7.2 Linear Theory of Edge Plasma Instabilities

147

(m ¼ 0, 1, 2, . . .). For the m ¼ 0 mode, we have the following frequency and the eigenfunction ωm¼0 ¼

b ω , 1 þ ρ2s k2y þ K1

e m¼0 ðxÞ ¼ cosh K ðx=wÞ, ϕ

ð7:20Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e m¼0 ðxÞ can be approximated as where K ¼ ðw=ρs Þ 1 þ ρ2s k2y . For w ρs, ϕ follows n o e 0 ðxÞ / exp Kðx=ρs Þ2 , ϕ

ð7:21Þ

pffiffiffiffiffiffiffiffi which shows that the “localization width” of the m ¼ 0 mode is ∼ ρs w  w. Although the solution for a localized drift wave we consider here is somewhat idealized, it will allow us making some illustration of the impact of plasma flow velocity shear on drift wave plasma instability later.

7.2.3

Dissipative Drift Wave Instabilities in Slab Geometry


! All drift wave dispersion equations we considered so far show that ω k is real and, therefore, no growth of initially small perturbations is possible. However, we will see that allowing for dissipative effects in the electron dynamics results in destabilization and growth of the drift wave amplitude. Such dissipative effects can be caused by both the Landau resonance of the wave with electrons (even though we assume that ω/kk < VTe, some small, but finite effect of such a resonance is still present) and electron-ion collisions. To allow for the impact of the Landau resonance on the drift wave (7.16), we need to describe the electron dynamics kinetically. For
our case this can be done by ! ! introducing the electron distribution function, f e v , r , t , and employing the electron drift-kinetic equation (e.g. see [1, 2]), which reads:
! ∂f e ∂f c e ∂f þ vk e  2 ∇φ  B  ∇f e þ ∇k φ e ¼ 0: m ∂t ∂z B ∂vk

ð7:22Þ



 ! ! ! Since we consider linear perturbations, we take f e v , r , t ¼ f ðe0Þ v , x þ

 ! ! ef e ! v , r , t , where f ðe0Þ v , x is the initial electron distribution function and

!! ! ef e ! v , r , t / exp iωt þ ik r is a small correction. Then from Eq. (7.22) we

148

7 Anomalous Cross-Field Transport in Edge Plasma

find the following expression for the electron density perturbation, e ne ¼ R
! !  ! ef e v , r , t dv : Z  ~ ~ ne ¼ φ

ekk ∂f ð0Þ cky ∂f ð0Þ e e þ  B0 ∂x m ∂vk



!

dv : ω  kk vk

ð7:23Þ



 ! ð0Þ ! Assuming that f ðe0Þ v , x is Maxwellian, f e,M v , x  pffiffiffiffiffi 3

nðxÞ 2πVTe exp v2 =2V2Te , we find that Eq. (7.23) can be re-written as ~ ~ne ¼ ɸ


! Z ðω  k v Þf ð0Þ ! , x dv v k k e,M ω  kk vk

:

ð7:24Þ !

!

From Eq. (7.24) we see that due to inhomogeneity of the plasma density and E  B drift, the sign of the effective derivative of the electron distribution function at the resonance condition ω ¼ kkvk changes from being negative for the Maxwellian function to positive in Eq. (7.24) for ω < ω. Therefore, according to the standard interpretation of Landau mechanism of the wave damping/growth (e.g. recall “the bump on tail instability” caused by the Landau resonance, see Ref. [7]), in our case, the amplitude of the drift wave having ω < ω (recall expression 7.16) will grow. It also tells us that the energy needed for such a growth comes from the electron kinetic (thermal) energy. To find a quantitative expression for the growth rate, from Eq. (7.24) we find ( !) ~ne ~ ω  ω ω Z pffiffiffi ¼ ɸ 1 þ pffiffiffi , n 2kk VTe 2kk VTe

ð7:25Þ

where Z(ς) is the plasma dispersion function: 1 ZðςÞ ¼ pffiffiffi π

Z1 1

0 1

Zς 2 pffiffiffi 2

dξ exp ξ2 ¼ exp ς @i π  2 dξ exp ξ A: ð7:26Þ ξς 0

For ω  kkVTe, from Eq. (7.25) we obtain the following expression for the electron density perturbation ( ) rffiffiffi ~ne e~ φ π ω  ω ¼ 1þi , Te 2 kk VTe n

ð7:27Þ

7.2 Linear Theory of Edge Plasma Instabilities

149

which, due to the Landau resonance, is slightly different from the Boltzmann relation we have used so far. Then, recalling Eq. (7.12) and using the quasi-neutrality condition, we arrive at the following dispersion equation 2 2 ω Cs kk þ 2 ¼ 1 þ ρ2s k2⊥ þ i ω ω

rffiffiffi π ω  ω : 2 kk VTe

ð7:28Þ

Since we assume ω  kkVTe, the last term on the right-hand side of Eq. (7.28) is small, but this is the term that makes the solution of the dispersion Eq. (7.28) complex. As a result, for the drift waves, this gives ω ω¼ 1 þ k2⊥ ρ2s

(

) rffiffiffi π ω 2 2 2 2 2 k ρ ð1 þ k⊥ ρs Þ 1þi , 2 kk VTe ⊥ s

ð7:29Þ

and the frequency has a positive imaginary part, Im(ω) > 0, which implies that the
! ! amplitude of the perturbation, / exp iω þ i k  r / exp ðImðωÞtÞ , will grow exponentially with time. At the same time, for the ion sound waves having ω > ω, the last term in Eq. (7.28) results in collisionless damping. Note that the growth rate is proportional to the difference between the mode frequency ω and the drift frequency ω, and therefore requires the account of the dispersive corrections due to the ion inertia. The electron temperature gradient directly leads to the collisionless drift instability if the temperature gradient is opposite to the density gradient [148]. Next, we consider the impact of electron-ion collisions on destabilization of the drift waves described by Eq. (7.14). For this purpose we will use the plasma fluid equations (e.g. see [8–11]). As we already found, the drift waves cause some perturbation of the electron distribution function. In terms of the fluid equations, in the presence of collisions, the drift waves will result in the perturbation not only of the plasma density but also of the electron temperature (we will still consider the cold ion approximation and ignore electron energy dissipation due to electron-ion collisions). Then, from the electron fluid momentum and energy equations, omitting rather cumbersome algebra, we find (e.g. see [12]) ~ φ ne e~ ¼ Te n

(

ðω þ iνk Þð3ω=2 þ iκνk Þ þ ið1 þ αT Þω νk ðω þ iνk Þð3ω=2 þ iκνk Þ þ ið1 þ αT Þωνk

) ,

ð7:30Þ

where νk ¼ k2k ðTe =m νei Þ plays the role of the inverse characteristic time of electron diffusion on the spatial scale  k1 k , νei is the electron-ion collision frequency, κ ¼ 1.61 and αT ¼ 0.71. We notice that the terms proportional to κ and 1 + αT come, respectively, from the contribution of the electron temperature perturbation to heat conduction and from the momentum balance equations along the magnetic field lines where αT describes the electron thermal force effect. For the case νk ω, when the electron density perturbation can be described with the Boltzmann relation, from Eq. (7.30) we have

150

7 Anomalous Cross-Field Transport in Edge Plasma

   ~ne ~ ω  ω 1 þ αT ¼ɸ 1þi : 1þ n νk κ

ð7:31Þ

As one can see, similarly to the collisionless Landau dissipation (7.27), the imaginary part of the expression (7.31) is also proportional to the difference ω  ω, which results in instability for the case of the drift wave: ω

   ω ω 2 2 1 þ αT 2 2 2 ρ k ð1 þ ρ k Þ , 1 þ i 1 þ s ⊥ νk s ⊥ κ 1 þ ρ2s k2⊥

ð7:32Þ

where the frequency, similarly to Eq.(7.29), has a positive imaginary part proportional to ρ2s k2⊥ . We notice that from Eq. (7.32) one can see that the omission of the electron temperature variation (which formally corresponds to the case of κ ! 1) would only give an order of unity correction for the growth rate. However, we should keep in mind that the applicability of the fluid equations for the study of the dissipation effects caused by electron-ion collisions on the stability of the drift waves requires a relatively slow spatiotemporal variation of plasma parameters: kkVTe/νei  kkλCe  1 and ω/νei  1, where λCe is the electron mean free path. Therefore, the inequality νk ω that we used to derive Eq. (7.31) requires the following conditions: ω/νei  (kkVTe/νei)2  1. For a more general case, the contributions of both the Landau resonance and electronion collisions to the growth-rate can be comparable even for ω/νei < 1 [12]. For νk ! 0, which corresponds to slow relaxation of the electron density perturbation along the magnetic field lines, from Eq. (7.30) we have    ~ne ~ ω iνk ω ¼ɸ þ 1 : n ω ω ω

ð7:33Þ

Then, neglecting in Eq.(7.12) the ion dynamics along the magnetic field, from the quasi-neutrality condition we find the following dispersion equation ρ2s k2⊥ þ

  iνk ω 1   ¼ 0, ω ω

ð7:34Þ

which for νk ! 0 has an unstable solution with ω ¼ ð1 þ iÞ

sffiffiffiffiffiffiffiffiffiffiffiffi ν k ω 2ρ2s k2⊥

:

ð7:35Þ

The fact that ω(νk ! 0) ! 0 is not surprising if we recall that the wave is driven by the electric field, which is due to plasma polarization related to electron mobility along the magnetic field lines (see Fig. 7.1) and such mobility is strongly suppressed for the case of νk ! 0.

7.2 Linear Theory of Edge Plasma Instabilities

(a)

151

(b) 10 10

wR × 10–3 (sec)–1

2

m=1

L = 60 cm nO = 6.7×1010 cm–3 rO = 1.4 cm

10 6 2

6 2

m=1

10

m=2

wq × 10–2 (sec)–1

6

m=2

6 2

10

10 6 6 2 2

m=3 1.2

1.6 B (kG)

2.0

–2

m=3 1.2

1.6 B (kG)

2.0

Fig. 7.2 Comparison of experimental data (dots) and theoretical calculations (solid curves) for collisional drift wave frequency (left) and growth rate (right) for different azimuthal wave numbers m. (Reproduced with permission from [18], © AIP Publishing 1970)

The basic features of both the collisionless and collisional drift waves in a non-tokamak environment were extensively studied in the 1960th–1970th (e.g. see Ref. [13–17] and the references therein). Due to relatively quiescent and controllable plasmas in the devices used in those studies (in many cases these were the Q-machines, with the straight magnetic field lines where the plasma was created by ionization of cesium or potassium atoms at the surface of a hot plate), a reasonable agreement was found between the theoretical expectations and the experimental data for the case of low wave amplitudes, where the linear wave theory is valid. For example, in Fig. 7.2 one can see a good agreement between the experimental data and theoretical calculations for the collisional drift wave frequency and growth rate for different azimuthal wavenumbers m. The situation with experimental studies of the drift waves in toroidal devices, where the plasma waves can be simultaneously driven by different mechanisms, is not so obvious, and largely only qualitative agreement between the results of the drift wave theory and the experimental observations is reported (e.g. see [4, 19–21] and the references therein).

7.2.4

Destabilizing Effect of Ion Temperature Gradient

However, it appears that the presence of a cross-field ion temperature gradient can result in plasma instability with no dissipation effects. In this case, another kind of

152

7 Anomalous Cross-Field Transport in Edge Plasma

drift wave, the so-called ion temperature gradient (ITG) drift mode [22] can become unstable. As an example, we consider plasma with homogeneous both density and electron ! ! temperature embedded into a constant magnetic field B0 ¼ B0 e z but having the ion temperature depending on the x-coordinate, ∂Ti/∂x 6¼ 0. ! ! ei , Then, the ion E  B drift will result in a perturbation of the ion temperature, T similar to that of the plasma density described by Eq. (7.4):   e ei ∂ T ∂ϕ ¼ 0, þ Udw,i ∂t Ti ∂y

ð7:36Þ

cTe dℓnðTi Þ : eB0 dx

ð7:37Þ

where Udw,i ¼ 

ei results in the For the case of kk 6¼ 0, the perturbation of the ion temperature T perturbation of the plasma pressure along the magnetic field lines, which is somewhat similar to the plasma density inhomogeneity along the magnetic field shown in Fig. 7.1. Standard wisdom would suggest that the perturbation of plasma pressure should result in some sort of a “sound” wave, similar to the ion sound waves, Eq. (7.17), with ω ~ Cpkk, where Cp is the plasma sound speed. Although this option exists, we consider a very different regime of the ion dynamics, which will finally bring us to the unstable ITG mode. Following Eq. (7.1) we assume that ω VTi kk. We will see that such an inequality becomes possible because in inhomogeneous plasma, among different characteristic frequencies of the waves, there are drift frequencies such as ω ¼ Udwky (Eq. (7.12)) and, as we will see, ω,i ¼ Udw,i ky ,

ð7:38Þ

which do not depend on kk. Then, the ion dynamics can be considered as a motion under a “fast” ðω VTi kk Þ applied force caused by the ion temperature perturbation. As a result of such forced ion motion, the perturbations of both the plasma density and electrostatic potential will be relatively small but vital for the instability: e T ee φ e n ∼ i  i: Ti n Ti

ð7:39Þ

We assume here that Te ~ Ti. Then, from ion momentum balance along the magnetic field and the continuity equations we have

7.2 Linear Theory of Edge Plasma Instabilities 2

M ∂ Ti ∂t2



~ni n

153





 ~i T , Ti

ð7:40Þ

~i ~ T T  i, Ti Ti

ð7:41Þ

¼ ∇2k

which demonstrates that for ω VTi kk , e~ φ ~ni  ¼ Ti n



VTi kk ω

2

and justifies the inequalities (7.39). As a result, using the Boltzmann relation for the electron density perturbation and re-writing all our expressions in the Fourier representation, from Eqs. (7.36) and (7.40) we find ω3 ¼ ω,i k2k V2Ti :

ð7:42Þ

This third-order equation for ω has one real and two complex conjugate solutions, which ensures the existence of the solution with the positive imaginary part of ω, which implies the instability of this ITG mode. Note that the ion drift velocity here is due to the ion temperature gradient. We notice that in our derivation, among other assumptions we assumed that ω VTi kk , which, as one can see from Eq. (7.39), is satisfied for relatively small kk : ω,i VTi kk , so the growth rate of the instability described by Eq. (7.42) is significantly below ω, i.

7.2.5

Plasma Instabilities Driven by Toroidal Effects

So far we considered plasma in a straight constant magnetic field. However, in a ! tokamak, the magnetic field B has a helical structure winding around toroidally symmetric magnetic flux surfaces (see Fig. 1.1). As a result, charged particles with ! a small Larmor radius will experience cross-B drift motion associated with both the curvature of the magnetic field lines and the gradient of the magnetic field strength. In a tokamak having a large aspect ratio (the ratio of major to minor tokamak ! radii), both these drifts are largely determined by the toroidal magnetic field, Btor , and the velocities of these drifts are directed along the major tokamak axis. For this case, the average magnetic field-related drift velocity of an ensemble of the particles having a charge q and a Maxwellian distribution function with the temperature Tq can be written as

154

7 Anomalous Cross-Field Transport in Edge Plasma

Fig. 7.3 The magnetic drift-related polarization of plasma pressure protrusion !

!

and associated E  B plasma drift that can result in further radial advection of the protrusion. Here, the cylindrical R cordinate is along the x direction




!

!

!

VB,q ¼ 2

cTq qB

eR  b R

:

ð7:43Þ

Here we use cylindrical coordinates (R, Z, ϕ) where the Z coordinate goes along ! the major tokamak axis, whereas e R is the unit vector along the R coordinate, ! ! b ¼ B=B, and assume Btor / 1/R. ! We notice that the direction of VB depends on the sign of the charge, so the electrons and ions drift in opposite directions. As a result, a radial protrusion of the plasma parameters can cause plasma polarization not only due to the electron motion along the magnetic field as shown in Fig. 7.1, but also due to the magnetic fieldrelated drifts of the electrons and ions. As a result of such plasma polarization, new types of instabilities become possible. As an example, in Fig. 7.3 we show magnetic drift-related polarization of a plasma pressure protrusion which, unlike those in Fig. 7.1, is parallel to the direction of the magnetic field lines. As one can see, in this particular case, a dipole-like polarization of the plasma pressure protrusion due to the magnetic drift and associ! ! ated E  B plasma drift can result in further radial advection of the protrusion. However, we notice that the drift velocities Eq. (7.43) appear only in the motion of test particles (guiding centers). Within the fluid picture of the plasma dynamics, these velocities are “hidden” in the diamagnetic velocities of the plasma compo!dia

nents, Ve=i , which, in the presence of toroidal effects, can result in non-divergencefree perturbations of the electron/ion fluxes. Indeed, the cross-field diamagnetic flux, !dia

j q , of an ensemble of particles with charge q and pressure Pq is !dia

jq

!

c B  ∇Pq ¼ : q B2

Then, assuming B / 1/R, from Eq. (7.44) we find

ð7:44Þ

7.2 Linear Theory of Edge Plasma Instabilities

155


 ! b  e  ∇P R q 2c ¼ : qB R !

!dia

∇ jq

ð7:45Þ !

A similar result can be found from the divergence of the flux nq VB,q .

!

!

Another implication of the spatial variation of the magnetic field is that the E  B drift velocity is also no longer divergence-free even for the potential electric field ! E ¼ ∇φ. Indeed, assuming that B / 1/R, for this case from Eq. (7.3) we find
 ! b  e  ∇φ R 2c ! 2c ¼ b  ð∇ℓnðBÞ  ∇φÞ ¼  : B B R !

!

∇  V!E B!

ð7:46Þ

Although in large aspect ratio (R a) tokamaks j VB, q j is relatively small j VB,q j/

1 1  j Udw j , j Udw,i j/ , R a

ð7:47Þ

!

the cross-field drift velocity VB can cause plasma polarization (recall Fig. 7.2), which, as we will see, results in a new class of plasma instabilities. In what follows, ! ! along with the frequencies ω and ω, q, we will use frequencies ωB,q ¼ VB,q  k , which, according to Eq. (7.47), have the following ordering j ωB,q j ω , ω,q

ð7:48Þ

As an example, we consider the impact of a weakly toroidal magnetic field on the ITG mode. We introduce a local coordinate system (x,y,z), where x, y, and z are in ! the radial, poloidal, and B directions respectively and, as we did before, take constant plasma density and electron temperature and assume that the ion temperature is varying in the “radial” direction: ∂Ti/∂x 6¼ 0. We recall that for the case of a constant magnetic field, the only reason for establishing the perturbations of the plasma density and corresponding electrostatic potential was the ion motion along the magnetic field caused by the ion temperature (pressure) perturbation. However, in the presence of toroidal effects, perturbation of the ion temperature (pressure) results in a finite divergence of the diamagnetic ion flux (7.45) and, therefore, in the ion density perturbation. To emphasize the toroidal effects, we completely neglect here the impact of the ion dynamics along the magnetic field lines and the effects of the ion inertia. We will be interested in plasma fluctuations with the frequency ω such that j ωB,q j< ω < ω , ω,q :

ð7:49Þ

156

7 Anomalous Cross-Field Transport in Edge Plasma

Then we will see that in our case, the ion temperature has the largest relative e∼e ei =Ti ϕ perturbation and, similarly to Eq. (7.41), T ni =n0 (we assume here Ti ~ Te). As a result, the ion temperature perturbation can still be described by Eq. (7.36), which gives ei ωdw,i T e ¼ ϕ: Ti ω

ð7:50Þ

Finding e ni we can allow for the ion temperature perturbation in the ion diamagnetic ! ! flux (7.44) only and neglect the compressibility of the E  B drift flow since e So we have ei =Ti ϕ. T ei e ni ωB,i T ¼ : n0 ω Ti

ð7:51Þ

Assuming the Boltzmann relation for the electron density perturbation and electrostatic potential together with the plasma quasi-neutrality, from Eqs. (7.50 and 7.51) we find ω2 ¼ ω,i ωB,i  2

 2 2 Ti cTe ky dℓnðTi Þ , Te eB0 R dx

ð7:52Þ

which satisfies the inequality (7.49) and all other assumptions made in the course of our derivation of Eq. (7.52) and shows the instability of the toroidal ITG for ∂Ti/∂x < 0. We notice that x is the local coordinate and for the “standard” temperature distribution over the minor radius, ∂Ti/∂x < 0 region corresponds to the outboard side of the torus (or to the so-called “bad” curvature region), whereas at the inboard side (where the curvature is “good”) ∂Ti/∂x > 0 and the mode is neutrally stable (within our simplified treatment). Equation (7.52) gives the growth rate of the ITG instability in the eikonal approximation. More complex numerical simulations go beyond the eikonal approximation and treat the solution of the corresponding differential equations as an eigenfunction-eigenvalue problem. Such solutions provide information about the mode structure (somewhat similar to what we did for a nonlocal solution of the drift wave arriving at Eqs. (7.20) and (7.21). As an example, in Fig. 7.4 one can find the spatial structure of the eigenfunction of the electrostatic potential at some toroidal angle found for the ITG mode. We notice that in agreement with our simplified consideration, the amplitude of the electrostatic potential shows a strong enhancement at the outer side of the torus, where, according to Eq. (7.52), the driving mechanism is localized. However, at relatively high radial gradients of the plasma parameters, the spatial structure of the most unstable ITG eigenmode can be very different (e.g. see [24, 25] and the references therein).

7.2 Linear Theory of Edge Plasma Instabilities Fig. 7.4 Contour plot of the eigenfunction of electrostatic potential at some toroidal cross-section found for ITG mode. (Reproduced with permission from [23], © AIP Publishing 2017)

157

150 100

Z/ρs

50 0 –50 –100 –150 350 400 450 500 550 600 650 R/ρs

7.2.6

Interchange and Resistive Interchange Modes

Apart from the impact on the ITG instability, the magnetic drift also brings new features to the instability of the collisional drift waves we have considered for the case of a constant magnetic field (see Eqs. (7.32 and 7.35). To asses them, we will again use a local coordinate system (x,y,z), where x, y, and z are in the radial, ! poloidal, and B directions respectively. However, this time, instead of finding expressions for the electron and ion density perturbations and then using the quasi-neutrality condition, we will employ the so-called vorticity equation, which is widely used, in particular, in nonlinear simulations of plasma turbulence. The vorticity equation actually follows from the quasi! ! neutrality condition written in the form ∇  J ¼ 0, where J is the electric current in the plasma. In highly magnetized plasmas, the cross-field plasma current produced ! e inert by fluctuating plasma parameters is only due to the ion inertia, J ⊥ (e.g. recall the ! e dia expression (7.10)), and diamagnetic current, J ⊥ , associated with the electron/ion ! ! diamagnetic fluxes (7.44). We notice that i) the E  B drift velocities, which are the same for electrons and ions, do not contribute to cross-field electric current in a ! ! e dia quasi-neutral plasma, and ii) ∇  J ⊥ 6¼ 0 only for the case where B is not constant (e.g. recall expression (7.45)). For the case of the cold ion approximation, neglecting the variation of the magnetic field, from Eqs. (7.8 and 7.9) we find ∇

!inert J⊥

    ∂ ! c∇e φ þ V!E B!  ∇ ¼ e∇  n , BΩBi ∂t

whereas from Eq. (7.45) we have

ð7:53Þ

158

7 Anomalous Cross-Field Transport in Edge Plasma


 ! e b  e  ∇ P R tot 2c ¼ , B R !

∇

!dia J⊥

ð7:54Þ

where e Ptot is the perturbation of the total plasma pressure. Quite often Eq. (7.53) is considered in the Boussinesq approximation where the expression (7.53) is simplified as follows: !inert

∇  J⊥



  ∂ ! c∇2 φ þ V!E B!  ∇ : BΩBi ∂t

¼ en

ð7:55Þ

For linear theory, such an approximation is equivalent to the omission of the inertial ! part of the cross-field ion velocity in the term Vi⊥  ∇n in Eq. (7.12). Then, the linear vorticity equation reads
 ! ~tot b  e  ∇ P R ~ 2c ∂ c∇ φ ∇  J  en  þ ∇k  ~Jk ¼ 0: B R ∂t BΩBi ! ~

!

2

ð7:56Þ

To simplify our algebra, we will assume that the electron temperature is constant (which gives an order of unity correction for the growth rate, recall Eq. (7.32)) so that for the perturbed electron pressure we take e Pe ¼ e nTe . Finding the parallel component of the electric current, ~Jk , we will ignore the ion dynamics along the magnetic field lines. Then, assuming ω/νei < 1, from the parallel electron momentum balance equation in linear approximation we have   ~ n ~ ~ ∇k  Jk ¼ enνk ɸ  : n

ð7:57Þ

First, we consider the case of νk ! 0, which corresponds to reduced electron mobility along the magnetic field lines and, in the absence of toroidal effects, results in a relatively slow instability (7.35). However, toroidal effects, causing magnetic drifts, provide plasma polarization which is not related to electron mobility along the magnetic field lines and, therefore, is not bounded by the small magnitude of νk. Therefore, in the simplest case, we can take Jk ¼ 0 and relax Eq. (7.56) to ρ2s k2⊥

∂ ee φ 2cTe ∂ e n  ¼ 0: T eBR ∂t e ∂y n

ð7:58Þ

The variation of plasma density can be found from the ion continuity equation. Considering the case ρ2s k2⊥ < 1 and recalling the inequality (7.48), we can neglect ! ! the compressibility in both the inertial and E  B ion drift flows and use Eq. (7.4) for the plasma density perturbation. As a result, we arrive at the following equation

7.2 Linear Theory of Edge Plasma Instabilities

e k2⊥ ϕ

2 2Te ky ∂ℓnðnÞ e ϕ ¼ 0, MR ω2 ∂x

159

ð7:59Þ

which gives the growth rate of the so-called ideal interchange instability γI (e.g. see [2]): ω2 ¼ γ2I  

ω ωB,e 2Te dℓnðnÞ ∼ , MR dx ρ2s k2⊥

ð7:60Þ

which, similarly to the toroidal ITG mode, can only be unstable at the outboard side of the torus. The physics of the ideal interchange instability is simple: the magnetic drift causes polarization of a plasma protrusion similar to that shown in Fig. 7.2, which, in the absence of charge relaxation along the magnetic field lines, results in a continuous build-up of the electric field and increasing amplitude of the protrusion. We notice that the ideal interchange mode has a deep analogy with the RayleighTaylor instability of a stratified fluid in a gravity field (e.g. see [26]). Indeed, considering an incompressible fluid situated in a gravity field characterized by the ! ! acceleration g ¼ g e x (where g > 0) and having the mass density ρ(x), we can find e ðx, y, tÞ, which defines the the following equation describing the stream function, ψ ! e ! perturbation of the fluid velocity, V ¼ e z  ∇ψ, 2 e d2 ψ 1 dρ gky eþ e ¼ 0:  k2y ψ ψ 2 2 ρ dx ω dx

ð7:61Þ

Here we adopt the Boussinesq approximation and use the Fourier expansion of ψ in time and y-coordinate. Then, using the eikonal approximation in the x-direction, from Eq. (7.61) we obtain e k2⊥ ψ

2 dℓnðρÞ gky e ¼ 0, ψ dx ω2

ð7:62Þ

which is similar to Eq. (7.59), so we can see that the factor 2Te/MR plays the role of effective gravitational acceleration for the plasma in a toroidal magnetic field. By specifying the plasma density (the fluid mass density) profile from Eq. (7.61), one can find the localized solutions of the Rayleigh-Taylor (interchange) unstable modes. For example, similarly to Eq. (7.18), we consider the case ρðxÞ ¼ ρ 


 Δρ x tanh , 2 w

ð7:63Þ

160

7 Anomalous Cross-Field Transport in Edge Plasma

where Δρ=ρ  1. Here again, we use the analogy of Eq. (7.61) to the Schrödinger equation for an electron in a potential well / cosh2(x/w) [6] with the energy / k2y , and can find the Rayleigh-Taylor (interchange) instability growth rate, γRTm , versus the integer mode number m. For the fastest-growing mode m ¼ 0, we have the following dependence of γRTm¼0 and the corresponding eigenfunction, ψm¼0(x), on ky: γ 2RTm¼0 ¼ γ 2RT

w j ky j , w j ky j þ1

ψm¼0 ðxÞ ¼ coshwjky j ðx=wÞ,

ð7:64Þ

where γ 2RT ¼

Δρ g : 2ρ w

ð7:65Þ

Although this solution for the Rayleigh-Taylor (interchange) instability was considered for somewhat idealized conditions, we will use it later to illustrate an impact of fluid (plasma) flow velocity shear on the Rayleigh-Taylor (interchange) instability. Coming back to the plasma and analyzing the dissipation from Eqs. (7.56) and (7.57) for ρ2s k2⊥ < 1 and finite, although still rather small νk, νk < ω, we arrive at the following modification of Eq. (7.34): ω2 þ γ2I þ

iνk ρ2s k2⊥

ðω  ω Þ ¼ 0:

ð7:66Þ

For the case of νk > ωB,e , ρ2s k2⊥ ω , this gives the so-called resistive interchange mode ω ¼ ω þ i

ωB,e ω , νk

ð7:67Þ

which can be considered as a proxy for the “Resistive Ballooning Mode” (RBM) (e.g. see Refs. [27–31] and the references therein). The eigenmode structure of the RBM, found from numerical simulation for the DIII-D magnetic configuration, is shown in Fig. 7.5. Once again, one can see that the mode is largely localized at the outboard side of the torus (the “bad” curvature side). We notice that in the context of our consideration of collisional drift waves in a tokamak-like magnetic field, the term “ideal interchange mode” may sound strange. However, as we have already seen (and will see later), some dispersion equations can be rather general and, in different limits, describe different waves (e.g. recall Eq. (7.15) which describes both the ion sound and drift waves). Similarly, considering different magnitudes of νk (which depends not only on the plasma collisionality but also on the parallel wavelength), our dispersion equation can describe different modes ranging from an unstable drift wave for large νk (recall Eq. (7.32)) to the “ideal interchange mode” Eq. (7.59) for small νk.

7.2 Linear Theory of Edge Plasma Instabilities Fig. 7.5 Eigenmode structure of the RBM for the DIII-D magnetic configuration found from numerical simulation. (Reproduced with permission from [32], © Elsevier 2011)

161 3.0 2.5 2.0

9.6e–01 7.2e–01 4.8e–01 2.4e–01 0.0e+00

1.5 1.0 0.5 0.0 1.0

1.5

2.0

2.5

Fig. 7.6 Electron temperature variation at the edge of tokamak HL-2A tokamak (upper panel) switches back and fourth the ITG and resistive ballooning modes (low panel). (Reproduced with permission from [33], © IAEA 2018)

As we see, different modes of plasma waves, coming virtually from the same set of the equations, correspond to different relations of the plasma and wave parameters. In the tokamak environment, these modes often either co-exist or are “separated” by a relatively small variation of the plasma parameters. As a result, a much more sophisticated theoretical analysis and numerical simulations, which go well beyond our basic survey, are needed to adequately describe them. Needless to say that in experiments it is often difficult to distinguish the impacts of different modes on anomalous plasma transport. Nonetheless, as an example, in Fig. 7.6, one can see how electron temperature variation at the edge of the HL-2A tokamak switches excitation of the ITG and resistive ballooning modes (here GAM

162

7 Anomalous Cross-Field Transport in Edge Plasma

(b)

140 120 100 80 60 40 20 0

1.2 kHz δn/n

Fluctuation spectrum (a.u.)

(a)

0

1

2

3 4 f (kHz)

5

6

0.8 0.7 0.6 0.5 0.4 0.3 0.2

LFS

HFS

0 20 40 60 80 100 120 140 Lc(m)

Fig. 7.7 (a) Fluctuation frequency spectra at HFS (of resistive drift wave, blue curve) and LFS (of resistive interchange mode, red and black curves) of the Helimak device; and (b) Relative density fluctuation at both LFS and HFS versus the connection length along helical magnetic field between the end plates. (Reproduced with permission from [21], © AIP Publishing 2006)

stands for the Geodesic Acoustic Mode, which is a specific, toroidally symmetric mode in a tokamak, see [34, 35] and the references therein). Another example can be found in [21], where the resistive drift waves and the resistive interchange mode limit of the resistive drift mode (driven by magnetic drifts) were observed correspondingly at the High Field Side (HFS) inboard and Low Field Side (LFS) outboard sides of the Helimak toroidal device, see Fig. 7.7.

7.2.7

Electromagnetic Effects

As we can notice, all plasma waves we have considered so far were accompanied by electric currents. These currents generate fluctuating electromagnetic fields, which in plasma are manifested as the Alfven waves. However, we neglected the electromagnetic effects in all preceding considerations. This can be justified for the case where the characteristic frequencies of the waves considered are much lower than the frequency of the Alfven waves or, topbe more ffiffiffiffiffiffiffiffiffiffiffi ffi precise, the frequency of the shear Alfven wave, kkVA, where VA ¼ B= 4πMn is the Alfven speed. As an example, we consider the impact of electromagnetic effects on the collisionless drift wave described by Eq. (7.14). Recall that to avoid the Landau damping on electrons when deriving Eq. (7.14), we assumed ω < kkVTe. Therefore, the inequality ω < kkVA is “automatically” satisfied for the case where VTe < VA (βe  nTe/(B2/4π) < m/M), which gives ω < kkVTe < kkVA. However, for larger βe (βe > m/M), the impact of electromagnetic effects can be important. To address this issue, in to the electrostatic potential φ we introduce a vector potential 
addition ! ! ! A ¼ e z Ak r , t and will assume that both of them are small. Adopting the eikonal approximation for the perturbed quantities, we find the following contributions from the electromagnetic term to the amplitudes of fluctuating electric and magnetic

7.2 Linear Theory of Edge Plasma Instabilities

163


!  ! ! ~ ~ ! ~ ! ~ fields: E ¼ iω e z A k =c and B ¼ i k  e z Ak . In addition, we have the following expression for the fluctuating electric current associated with the vector potential ! ~ Jk

  ! ~ ck2 ~ ! c ∇B ¼ ⊥A ¼ e : 4π 4π k z

ð7:68Þ !

The component of the fluctuating electric current perpendicular to B is described by Eq. (7.55). As a result, using expressions (7.55) and (7.68), from the condition ! e ~ k: e and A ∇  J ¼ 0 we find the relation between φ ~ k =kk c ¼ ðω=kk VA Þ2 φ ~: ωA

ð7:69Þ

For ω < kkVTe, the electron density perturbation can be found from the stationary parallel electron momentum balance equation where electron temperature can be assumed constant. However, now we need to allow for small bending of the magnetic field lines, which is caused by electromagnetic effects. As a result, we have ~n T  e kk e þ e n0



 ~ ωA k ω ~ ¼ 0: 1 þ kk φ ω c

ð7:70Þ

Then, using Eqs. (7.69) and (7.70), taking the ion density perturbation from Eq. (7.11) and assuming the quasi-neutrality condition, we arrive at the following dispersion equation n o ðω =ω  1Þ 1  ðω=kk VA Þ2  ρ2s k2⊥ ¼ 0:

ð7:71Þ

Equation (7.71) describes the so-called drift-Alfven wave. In particular, for the large and small values of the ω/kkVA ratio, it gives, respectively, the drift wave Eq. (7.14) and the shear Alfven wave: (

ω ¼ ω =ð1 þ ρ2s k2⊥ Þ, 2

ω ¼ ðkk VA Þ ð1 þ 2

for

ρ2s k2⊥ Þ,

ω =kk VA  1 for

ω =kk VA 1

:

ð7:72Þ

In our evaluation of electromagnetic effects, we assumed so far that the magnetic ! ! field is constant and straight, B ¼ B e z . As a result, the non-divergence-free crossfield plasma current still appears only due to ion inertia, recall Eq. (7.55). However, in a tokamak magnetic configuration, we should allow for the contribution of the diamagnetic current (7.54). Considering low β plasma and being interested in the ~ kk VA , using Eq. (7.12) for ion density waves with the characteristic frequency ω > perturbation, we can neglect the ion dynamics along the magnetic field lines (which

164

7 Anomalous Cross-Field Transport in Edge Plasma

gives a contribution ~ β  1). Then, from the vorticity Eq. (7.56) and the electron parallel momentum balance Eq. (7.64), keeping in mind the relation (7.68) and the plasma quasi-neutrality condition, after some algebra, we come to the following dispersion equation n o ð1  ω =ωÞ ω2 þ γ2I ð1  ρ2s k2⊥ ðω =ωÞÞ  ðkk VA Þ2  ρ2s k2⊥ ðkk VA Þ2 ¼ 0:

ð7:73Þ

For ρ2s k2⊥  1, Eq. (7.73) is reduced to
 ð1  ω =ωÞ ω2 þ γ2I  ðkk VA Þ2 ¼ 0,

ð7:74Þ

which describes the drift wave, ω ¼ ω, and a proxy for the ideal ballooning mode [36] with ω2 ¼ γ2I þ ðkk VA Þ2 :

ð7:75Þ

For simplicity, we considered the case where only plasma density has a cross-field gradient, whereas electron temperature was assumed to be constant and the ions were “cold”. However, as we discussed, the mechanism of the interchange mode is related to the polarization of plasma protrusion due to the magnetic drifts (recall Fig. 7.2), which, according to Eq. (7.54), is determined by the total plasma pressure. As a result, a more complete consideration shows that instead of the expression (7.59), γI should be defined, assuming k2y =k2⊥ 1, as γ2I 

2 dℓnðPtot Þ , dx MnR

ð7:76Þ

where Ptot is the total equilibrium plasma pressure. Then Eq. (7.69) shows that for the “bad” curvature case, dPtot/dx < 0, and large parallel wavelength, the magnetic drift of the charged particles can destabilize plasma perturbations. Estimating kk ~ 1/qR, where q is the safety factor, from Eqs. (7.75) and (7.76) we find that the instability starts for the so-called “MHD ballooning parameter” α exceeding unity: α  q2 R

dβ > 1: dx

ð7:77Þ

The eigenfunction of the perturbed plasma pressure for the ballooning mode in ITER, found from the numerical simulation in [37], is shown in Fig. 7.8.

7.2 Linear Theory of Edge Plasma Instabilities

165

Fig. 7.8 The eigenfunction of perturbed plasma pressure for the ballooning mode in ITER. (Reproduced with permission from [37], © IAEA 2011)

6 4 2 0 –2 –4 –6 4 5 6 7 8 9

7.2.8

Effect of “Open” Magnetic Field Lines

Here we consider how a contact of the plasma with material surfaces (e.g. limiters, divertor targets) in the SOL region can affect plasma stability. As we found, plasma instabilities, one way or another, result in a fluctuating electric current along the magnetic field lines. The volumetric resistive effects associated with this current can cause dissipative plasma instabilities (e.g. recall Eq. (7.32)). However, in Ch. IV it was shown that the plasma current into a material surface (which we will, for simplicity, assume perfectly conducting) is related to some variation of the electrostatic potential through the sheath, φsh, bridging the material surface and the plasma. Then the expression for the low magnitude of electric current perturbation to the tar target, ~Jk , can be found by linearizing the expression (4.18), which, ignoring electron emission from the surface and assuming a normal incidence of the magnetic field onto the target, gives (see also [38, 39]): (




φsh 1 ~Jtar ¼ ensh Cs ðTe Þ e~ þ Λsh  k 2 Te
 ! ~ ~nsh 1 Te sh tar þ þ Jk , nsh 2 Te


 ) ~e  T sh

Te

ð7:78Þ

where Λsh ¼ eφsh/Te. The “boundary” condition Eq. (7.78) can result in a new type of instability and alter some modes we have considered so far. First, we consider an impact of this boundary condition on the interchange mode described by Eqs. (7.59 and 7.60). For simplicity, we analyze the SOL plasma at the outer side of the torus for a double null magnetic configuration (see Fig. 1.6a). We take the cold ion approximation and

166

7 Anomalous Cross-Field Transport in Edge Plasma

assume that the background electron temperature is homogeneous and rather high so that the plasma can be considered collisionless. For such a case, both the electrostatic potential and plasma density are virtually constant along the magnetic field lines. We ee ¼ 0 . will also ignore the toroidality-induced compressibility effects and take T Finally, we consider the case with no unperturbed plasma current to the material surface. As a result, from Eq. (7.78) we have φsh ~Jsurf ¼ ensh Cs ðTe Þ e~ : k Te

ð7:79Þ

Then, integrating Eq. (7.56) along the magnetic field lines and approximating those as straight lines along the major tokamak axis, using Eq. (7.5) for the density evolution and the boundary condition (7.79), we find [40] ωþ

γ2I 2i Cs þ ¼ 0: ω ρ2s k2⊥ Lcl

ð7:80Þ

Here Lcl is the “connection length” – the length between the divertor plates along the magnetic field line. Comparing Eqs. (7.60) and (7.80) we see that the boundary condition (7.79) plays the role of effective “sheath resistivity”, which exceeds the volumetric resistivity caused by the Coulomb collisions for λCe e > Lcl

rffiffiffiffiffi m : M

ð7:81Þ

Apart from the modification of existing instabilities, the boundary condition (7.78) can result in a new type of instability. In particular, it appears that the interplay of the radial gradient of electron temperature in the SOL and the sheath boundary conditions can drive instability that is not related to the interchange drive considered in Eq. (7.80) [41]. The reason for such instability is the phase shift between the volumetric and surface dynamics of the electrostatic potential. To demonstrate the underlying physics of this instability, we take the cold ion approximation, assume that the background plasma density is homogeneous, the electron temperature is rather high so that the plasma can be considered collisionless, and the electrostatic potential and electron temperature along the magnetic field lines are virtually constant. Then, integrating Eq. (7.56) along the magnetic field lines (approximating them as straight lines along the major tokamak axis) and ignoring toroidality effects we find iωρ2s k2⊥


e  T ee φ Cs ee φ 1 þ2  þ Λsh e ¼ 0: Te 2 Lcl Te Te

ð7:82Þ

ee from the equation similar to Finding the electron temperature perturbation T Eq. (7.36), we find the following dispersion equation

7.2 Linear Theory of Edge Plasma Instabilities

167

Fig. 7.9 Plasma polarization due to electric current along the magnetic field and cross-field inhomogeneity of plasma conductivity

ωþ

n
ω o 2i Cs 1 ,e þ Λ 1  ¼ 0, sh 2 2 ω ρ2s k⊥ Lcl

ð7:83Þ

where ω,e ¼ 

cTe dℓnðTe Þ ky : eB0 dx

ð7:84Þ

One can see that the growth rate of the instability described by Eq. (7.83) can be of the order of ω, e. So far we consider the plasma waves and instabilities related to plasma polarization caused by the electron dynamics along the magnetic field and the cross-field magnetic drift of the charged particles. However, plasma polarization can also be related to the interplay of the electric current along the magnetic field lines and crossfield inhomogeneity of the plasma conductivity (e.g. see [27, 42]). Indeed, for the case where the plasma current flows only along the magnetic field lines, it should be maintained constant. However, displacement of a plasma slab inclined to the magnetic field lines, similar to that shown in Fig. 7.9, causes a perturbation of the plasma conductivity along the magnetic field lines. Therefore, to keep the electric current constant, some additional electric field appears inside the slab, caused by charge accumulation at the boundaries of the slab. But due to the inclination of the slab, a ! ! cross-field electric field and corresponding E  B plasma drift emerge, which can displace this fluid element even more. Indeed, from the conservation of parallel current, j ¼ σ(x)E ¼ const., where σ(x) is the plasma conductivity and E is the electric field, we find E

dσðxÞ e x þ ikz σe φ ¼ 0: dx

ð7:85Þ

e the perturbation of the electrostatic potential. Here e x is the displacement and φ However, on the other hand, we have

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7 Anomalous Cross-Field Transport in Edge Plasma

Vx 

de x c ! ¼  ∇e φ  e z: dt B0

ð7:86Þ

As a result, from Eqs. (7.85 and 7.86) we find de x ky cE dℓnðσÞ e x, ¼ dt kz B0 dx

ð7:87Þ

and for the proper sign of the ky/kz ratio (which defines the inclination angle of the slab with respect to the direction of the magnetic field), we have the so-called current-convective instability with a characteristic growth rate   cE dℓnðσÞ ky  : γ ¼  B0 dx kz 

ð7:88Þ

We should recall that the electric conductivity of plasma is σ / T3=2 e , therefore the perturbation of conductivity, which drives the current-convective instability, is associated with inhomogeneity of the electron temperature along the magnetic field lines. However, electron temperature perturbations along the magnetic field can be washed away by very fast parallel electron heat conduction, κe / T5=2 e , and no instability will be possible. However, recently it was shown [43] that in asymmetric “detached divertor” regimes, where the plasma temperature in the inner divertor falls to ~eV range but the outer divertor is still relatively hot, Thot ~ 10 eV), currentconvective instability can be very “active” and important in plasma transport in the inner divertor. There are two reasons for this: i) the electron thermal conductivity effects are suppressed in the inner divertor and ii) the asymmetry of the electron temperatures in the inner and outer divertors results in onset of a large electrostatic potential drop, U ~ few  Thot, through the inner divertor leg, which boosts the growth rate of the current-convective instability there, see Eq. (7.88).

7.2.9

Impact of Magnetic Shear

So far, considering the slab approximation of a magnetic confinement device, we assumed that all magnetic field lines are parallel to each other. However, in practice, ! ! this is not the case and the direction of the vector b ¼ B=B turns around the minor ! radius r with increasing r, similar to that shown for the slab geometry in Fig. 7.10a where the x-coordinate plays the role of the minor radius. Therefore, if we specify the poloidal (y-direction) component of the wavenumber (ky), the effective parallel component of the wave vector will depend on the minor radius. One can see this from Fig. 7.10b, where the blue and orange stripes along the toroidal (z) coordinate correspond to different phases of the perturbed plasma parameters with given ky and ! the vector b is shown for different radial (x-coordinate) locations.

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169

Fig. 7.10 Schematic view of the variation of the vector !

b (a) and effective kk (b) along “radial” direction x

Fig. 7.11 Magnetic shear suppression of plasma density fluctuations, caused by the excitation of collisionless drift waves. (Reproduced with permission from [45], © American Physical Society 1970)

In simplified analytic consideration of the magnetic shear effects, the following model for the magnetic field it is often used:   ! ! x B ¼ B ez þ ey , Ls

!

ð7:89Þ

where Ls determines the “strength” of the magnetic shear and the second term describes the change of the direction of the magnetic field lines with varying “radial” coordinate. As a result of the magnetic shear, the wavenumber kk along the magnetic field lines varies within the eigenfunction of a particular mode of the wave packet. Therefore, growth of the waves, the dispersion of which depends strongly on the magnitude of kk (e.g. the drift waves, recall Eqs. (7.15) and (7.28), can be significantly restricted (e.g. see [44–47], and the references therein). In addition, the magnetic shear can also shrink the radial extent of the mode eigenfunction, which can imply the reduction of the contribution of this mode to anomalous transport (e.g. see [27, 48, 49], and the references therein). As an example, in Fig. 7.11 one can see that the increasing magnetic shear (decreasing Ls) results in the reduction of the relative amplitude of plasma density fluctuations caused by the excitation of collisionless drift waves.

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Fig. 7.12 Poloidal projection of magnetic flux tube, having circular crosssection at the “mid-plane”, in different toroidal locations

However, the simple model of the magnetic field (7.83) can more or less adequately describe the effect of the magnetic shear for the case where the shear is not varying much on a magnetic flux surface. This is not the case for the magnetic flux surfaces close to the separatrix which contains at least one X-point where the magnitude of the poloidal magnetic field, Bp, is zero. In the vicinity of the X-point, the magnitude of the poloidal magnetic field is proportional to the distance from the X-point, ℓX (for the case of the first-order X-point), so Bp(ℓX) / ℓX. We notice that the similarity of such dependence of Bp(ℓX) with the model (7.89) is illusory. The z-direction in Eq. (7.89) is the direction of the total magnetic field at some effective “magnetic flux surface” corresponding to x ¼ 0, whereas for the case of the magnetic field in the vicinity of the X-point, we are dealing with the “exact” poloidal and toroidal magnetic fields. Such a structure of the magnetic field near the X-point results in a very strong magnetic shear localized there. One can see this by considering how the poloidal projection of a magnetic flux tube cross-section evolves along the tube, Fig. 7.12. The magnetic flux in the tube is mainly determined by the large toroidal magnetic field that does not vary much, so we can consider the area of the poloidal crosssection of the tube to be almost constant if the pitch angle of the magnetic field line Bp/B  1, which is always the case near the X-point. Assume that the magnetic flux tube has a circular poloidal cross-section closer to the mid-plane (see the contour (1) in Fig. 7.12). The shape of this cross-section evolves along the tube, becoming a thin oval close to the X-point (contours 2 and 3) since the magnetic surfaces that determine the radial extent of the cross-section diverge there. The variation of the poloidal width of the tube cross-section, δℓp, can be estimated taking into account that Bp(ℓX) / ℓX [50]: δℓp ℓ

X : δℓp ð0Þ ℓX ð0Þ

ð7:90Þ

Fig. 7.13 Poloidal correlation length of blobs (coherent filamentary structures) in outer and inner divertor legs as the functions of the distance to divertor targets is in agreement with the mapping of the magnetic flux tubes. (Reproduced with permission from [55], © IAEA 2018)

Correlation length (m)

7.2 Linear Theory of Edge Plasma Instabilities

171

0.04 0.03 0.02 0.01 0.00 0.00

Outer divertor leg Inner divertor leg 0.05

0.10

0.15 0.20 0.25 Elevation (m)

0.30

0.35

where ℓX(0) and ℓX are the initial and current distances from the X-point, ℓX(0) ℓX (the position 0 corresponds to the location farther from the X-point. So for the magnetic flux coming close to the X-point, we have δℓp  δℓp(0). In practice, the effect of “poloidal compression” of the flux tubes can be so strong that even flux tubes originated in the midplane at a distance ~ centimeter from the separatrix and having a ~ centimeter cross-field radius, are squeezed poloidally to the scale below the ion gyro-radius in the vicinity of the X-point [50]. Similarly, any wavy structure originated at the midplane and having a long wavelength parallel to the magnetic field will experience a strong reduction of the effective poloidal wavelength in the vicinity of the X-point, which will result in strong dissipation effects and effectively stop the wave penetration through the X-point region from the midplane into the divertor region and vice versa [50–52]. Therefore, turbulent processes in the divertor region and SOL become disconnected. However, strong dissipative effects near the X-point can play a role somewhat similar to volumetric dissipation and result in a new type of instabilities (e.g. see [53, 54] and the references therein). The evidence of turbulence disconnection between the divertor region and SOL was found in the tokamak experiments [55], where no correlation between midplane and divertor fluctuations was observed for a rather high poloidal mode number. The poloidal correlation length found in these experiments (see Fig. 7.13) was in agreement with the mapping of the magnetic flux tubes. However, the perturbations with a low poloidal wavenumber at the outer midplane can “survive” the fierce squashing of the magnetic flux tube in the vicinity of the X-point and show a strong correlation between the fluctuation measurements in the midplane and in the divertor volume [56]. Further experimental details of the X-point effects on plasma turbulence can be found in [57, 58]. We notice that strong squeezing of the magnetic flux tube caused by the X-point effects poses a substantial challenge to both theoretical and numerical studies of these effects. First of all, for the case where the effective poloidal wavelength in the vicinity of the X-point becomes comparable to or even smaller than the ion gyroradius ρi, neither fluid nor gyro-kinetic models of plasma dynamics become applicable, whereas full 3D3V (three-dimensional in both the coordinate and velocity space) kinetic description of plasma turbulence is not feasible. In addition, the small

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7 Anomalous Cross-Field Transport in Edge Plasma

spatial scale that has to be resolved near the X-point brings another complication for numerical modeling of the X-point effects. As a result, the applicability and validity of modern numerical studies of the X-point effects on the edge plasma instabilities and, in particular, turbulence are somewhat questionable. In analytic theory, the X-point effects are often described with some effective boundary conditions for the “standard” differential equations for the edge plasma waves at the “entrance” to the X-point region (see [52–54], and the references therein). These boundary conditions assume that X-point dissipation results in a fast decrease of the electrostatic potential in the direction of the X-point, which is usually described as an evanescent wave. However, even in this case, the models used for such effective closures only cover extreme cases where the poloidal scale of the electrostatic potential in the wave is either still larger than ρi or much smaller than that.

7.2.10 Impact of Plasma “Macro- and Mesoscale” Flows So far, we considered the waves and instabilities in plasma at rest. However, quite often some specific, macro- and mesoscale, plasma flows can develop. Such flows can emerge due to different inherent, nonlinear processes associated with plasma turbulence, or can be driven by outside effects such as, for example, injection of neutral beams used for plasma heating. In tokamaks, flows having very low effective poloidal wave numbers virtually do not contribute to anomalous cross-field plasma transport since such flows mostly have only poloidal and/or toroidal components of the plasma velocity. However, the characteristic radial scale length of such flows can be rather small (see Fig. 7.14). As a result, the poloidal component of the plasma flow velocity may have a large radial shear, V00 , which can drastically modify both development of the plasma instabilities and anomalous plasma transport (e.g. see Refs. [4, 59] and the references therein). Although both poloidal and toroidal flows can be important, here, for simplicity, we will discuss mostly pure poloidal plasma flows driven by the radial electric field. Fig. 7.14 Schematic view of a poloidal sheared flow of plasma (red lines) with a small radial scale length in a tokamak

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173

Fig. 7.15 Stretching of a turbulent eddy, caused by the plasma velocity shear

Very often, an impact of the velocity shear is illustrated as continuous
stretching  ! in time of plasma turbulent eddies – the contours of equipotential φ r , t , see Fig. 7.15. It is also often presumed that the plasma instability is quenched when jV00 j becomes larger than the growth rate of the instability, γinst, in the absence of the velocity shear (e.g. see [4, 60]). However, in practice, the situation is more complex and in general case, the velocity shear can even increase the growth rate of plasma instability. Just for an illustration we consider the Rayleigh-Taylor instability of a stratified fluid in a gravity field and take into account the impact of the velocity shear (e.g. see [26, 61–64, 66] and the references therein). Recall that the Rayleigh-Taylor instability can be considered as a proxy for the interchange plasma instability. We take !
! ! the unperturbed fluid velocity as V0 r ¼ V0 ðxÞ e y and assume that gravitational acceleration is in the x-direction. Then, in the Boussinesq approximation, small e ðx, y, tÞ are described by the perturbations of the fluid velocity stream function ψ following partial differential equation   2 e ∂e ψ d2 V0 1 dρ ∂ ψ b Ω∇ b 2ψ bRT&V0 ðψ eÞ ¼ Ω e L ¼ 0, þ 2 ρ dx ∂y2 ∂y dx

ð7:91Þ

bRT&V0 ð. . .Þ is an operator describing the linear phase of the evolution of the where L stratified fluid velocity perturbation in the presence of gravity and unperturbed b ð. . .Þ  ∂ð. . .Þ=∂t  V0 ðxÞ∂ð. . .Þ=∂y. Looking for the horizontal fluid flow, and Ω solution of Eq. (7.91) in the form

of a combination of the eigenfunctions, e ðx, y, tÞ ¼ ψ e ðxÞ exp iωt þ iky y , from Eq. (7.91) we arrive at the following ψ generalized version of Eq. (7.61): 2 e eψ e d2 ψ 1 dρ gky d2 ω 2 e e  k ψ þ ψ  ¼ 0, y 2 2 2 ω ρ dx dx dx e e ω

ð7:92Þ

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7 Anomalous Cross-Field Transport in Edge Plasma

Fig. 7.16 Impact of V00 on the growth rate of the Rayleigh-Taylor instability for different b κ ¼jkyj w found numerically for the density profile given by Eq. (7.63). (Reproduced with permission from [65], © AIP Publishing 2020)

e ¼ ω  ky V0 ðxÞ . This equation is usually solved as an eigenfunctionwhere ω eigenvalue problem, where the role of the eigenvalues goes to ω. We notice that the last term in Eq. (7.92) can drive the Kelvin-Helmholtz [26] and facilitate the Rayleigh-Taylor [66] instabilities. Assuming that in the fluid at rest, the Rayleigh-Taylor instability develops, the impact of the velocity shear can be characterized by the effective Richardson number 2 which in our case we define as Ri ¼ g jdℓnðρÞ=dxj V00 . The case where a stratified fluid is bounded by two horizontal walls separated by a distance h was considered in [61] in the Boussinesq approximation. It was assumed that dℓn(ρ)/dx and V00 are constants and zero perturbed fluid velocity at the walls was used as the boundary conditions. It was shown that for such settings, no unstable eigenfunction-eigenvalue solutions of Eq. (7.92) exist for Ri < Ricrit 1 2, where Ricrit depends on jky j w and stabilization of smaller ky requires a somewhat lower Ricrit. Thus, these findings are consistent with the simplified physical picture of the impact of shear stabilization on the instability e γinst . However, the results of the numerical described by the inequality jV00 j > solution of Eq. (7.92), shown in Fig. 7.16 for the case of the density profile given by Eq. (7.63), constant V00 , and the m ¼ 0 mode, portray a different picture. They demonstrate that for jkyj w > 1, where γRTm¼0 γRT, stabilization occurs at

2 Ri0  γRT =V00 > 1 , whereas for small jkyj w unstable solutions of Eq. (7.92) persist even though in this case γRTm¼0 0) ¼ 1. Then, from Eq. (7.92) we e ðjxj> 0Þ ¼ exp jky xj and integrating Eq. (7.92) around x ¼ 0, we find have ψ that the growth rate is γ2RTm¼0 ¼ γ2RT jky j w and the velocity shear does not change it. The eddies corresponding to the eigenfunctions found from the numerical solution of Eq. (7.92) are shown in Fig. 7.17. As one can see, in accordance with some expectations (recall Fig. 7.15), the velocity shear indeed causes some stretching of the eddies. We notice that at large velocity shear, jV00 j b κ > 1, the localized solution of the RT instability ceases to exist (see [65] for details). However, we should keep in mind that even though we use a slab model of the Rayleigh-Taylor as a proxy for the curvature-driven instabilities in a tokamak, in practice, this model does not allow for many important effects including both the poloidal and toroidal periodicities of the tokamak geometry, the centrifugal and Coriolis forces, electromagnetic and other effects. As a result, theoretical assessment of the role of the plasma flows, both poloidal and toroidal, on different instabilities becomes more complex (e.g. see [69–76] and the references therein). Nonetheless, it appears that the plasma flow shear is very efficient in reducing the growth rate of some plasma instabilities related to the impact of effective “gravity” associated with the magnetic drifts (e.g. see Fig. 7.18). In addition to the impact on the growth rate of the instabilities, both the poloidal and toroidal velocity shear can significantly alter the eigenfunctions of the modes (see Figs. 7.19 and 7.20), which also affects anomalous transport of plasma. We should notice that many studies of linear plasma instabilities rely on treating the corresponding partial differential equations, which describe different instabilities, as the eigenfunction-eigenvalue problems (e.g. recall our derivation of the

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7 Anomalous Cross-Field Transport in Edge Plasma

Fig. 7.18 The growth rate of the ITG instability versus plasma flow shear rate γE / V00 for different values of magnetic shear bs. (Reproduced with permission from [76], © AIP Publishing 2012)

γE = 0

γE = 0.1

γE = 0.2

γE = 0.3

Fig. 7.19 Eigenfunctions of the electrostatic potential of toroidal ITG modes in a tokamak with increasing shear of the plasma flow, γE / V00 , [77]. V. I. Dagnelie, private communication, 2020 1.0

dΩ /dq=0.0

dΩ /dq=0.1

1.0

0.5

0.5

0.5

0.0

0.0

0.0

–0.5

–0.5

–0.5

–1.0 1.0

1.5

2.0

2.5

3.0

–1.0 1.0

1.5

2.0

dΩ /dq=0.5

1.0

2.5

3.0

–1.0 1.0

1.5

2.0

2.5

3.0

Fig. 7.20 Eigenfunctions structure of the ideal ballooning mode in a tokamak with no toroidal velocity shear (left); medium shear dΩ/dq (middle) and high shear (right), were Ω and q are the toroidal angular velocity of the plasma and the safety factor. (Reproduced with permission from [73], © American Physical Society 2004)

expressions (7.20) and (7.64)). However, the general solution of these differential equations can be represented as a combination of eigenmodes only for the case where the operators defining these equations are Hermitian. And this is usually not the case when the unperturbed flow velocity is included. For example, the operator

7.2 Linear Theory of Edge Plasma Instabilities

177

bRT&V0 ð. . .Þ is not Hermitian for a finite fluid velocity. As a result, in the L non-Hermitian case, the combination of the eigenmodes (even if they exist) cannot describe the entire linear evolution of fluid parameter perturbations (e.g. see [78, 79] and the references therein). In some, although rather limited, cases (in particular, constant velocity shear), this issue can be overcome in analytic or quasi-analytic considerations by implying the so-called non-modal approach, where after some transformation of the variables, including usage of the variable ζ ¼ y  V00 xt describing effective squashing of a fluid element by the sheared flow, the problem of interest can be solved as an initial value problem. Usually the perturbations described by non-modal approach could increase with time only as tp, where p is some constant. Therefore they become important for the case where the localized modes either are stable or cease to exist. For further discussion of this approach see [63, 64, 78, 80] and the references therein. e γinst , works, Whereas the “rule of thumb” of velocity shear stabilization, jV00 j > in a ballpark, for the plasma instabilities related to effective “gravity” associated with the magnetic drifts (e.g. toroidal ITG, ballooning instability), it appears that the shear of poloidal plasma velocity makes a very mild impact on the resistive drift wave instability. By adding a poloidal plasma flow with a constant velocity shear, so that eðx, y, tÞ ¼ e ðxÞ ¼ ω  V00 ky x , from Eqs. (7.12) and (7.31), assuming that ϕ ω

eðxÞ exp iωt þ iky y , in the Boussinesq approximation and for the plasma density ϕ b  cosh 2 ðx=wÞ , we obtain the following profile (7.18), which gives ω ðxÞ ¼ ω equation ρ2s

  e e ð xÞ  ω  ð xÞ e ω  ð xÞ ω d2 ϕ 2 2  1 þ k y ρs  ϕ ¼ 0, þi νk e ð xÞ ω dx2

ð7:93Þ

where for simplicity we omit the unimportant here parallel electron heat conduction and thermal force effects. Some particular results found from the numerical solution of Eq. (7.93), which show the impact of V 00 on the growth rate, are demonstrated in Fig. 7.21. As one can see, unlike the Rayleigh-Taylor instability (recall Fig. 7.16), there is a very mild impact of V00 on the growth rate of the dissipative drift wave instability, even though jV00 j γinst. However, similar to the RT mode, at relatively large velocity shear, jV00 j>jV00 jloc, no localized solution of the resistive drift wave was found [65]. All of these observations have a rather simple explanation. First, we discuss the key difference between the RT/interchange modes and drift waves. The Rayleigh-Taylor instability is associated with the dynamics of density protrusions, which can be directly altered by the sheared flow. Similarly, the interchange plasma instability is associated with the dynamics of plasma density perturbations with embedded electric charges originated from almost “irreversible” cross-field magnetic drift effects. Spatial distribution of these charges produces ! ! E  B drifts, which, finally, drive the instability. Similar processes are relevant to all plasma instabilities driven by magnetic drift effects (e.g. the toroidal ITG and ballooning modes). Therefore, advection of plasma density perturbations with

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7 Anomalous Cross-Field Transport in Edge Plasma

Fig. 7.21 Growth rate and the V00 =γ ratio of the dissipative drift wave instability found from the numerical solution of Eq. (7.93), as a function of V00 . Other parameters used in these simulations are: kyρs ¼ 0.5, w/ρs ¼ 30, and ω ¼ 50. (Reproduced νk =b with permission from [65], © AIP Publishing 2020)

embedded electric charges by the sheared plasma flow inevitably alters such instabilities (e.g. see Figs. 7.16 and 7.18). For the case of drift waves, the situation is very ! ! different. In this case, the electric field and related E  B drifts are due to the largely “reversible” response of the fast parallel electron dynamics on plasma density perturbations. Even though the advection of plasma density perturbations by the sheared flow changes the “landscape” of density perturbations, the distribution of the electric charges has virtually no “memory” and, therefore, the sheared flow makes a very mild impact on the growth rate of the drift wave instabilities. We notice that the manifestation of the different impact of the velocity shear on the drift wave- and the magnetic drift-driven plasma instabilities can be also seen in the dependence of the corresponding eigenfunctions. For example, comparing the impact of the velocity shear on the eddies related to the eigenfunctions of the RT (Fig. 7.17) and the drift wave (Fig. 7.22) instabilities, one can see that unlike the RT instability, the eddies corresponding to the drift wave instability are not stretched by the sheared flow at all. However, the number of eddies is reduced and their center is shifted along x-coordinate. Nonetheless, even though the impact of the velocity shear on the growth rate of the resistive drift wave instability is mild, a significant reduction of the radial extent of the drift wave eddy, caused by the velocity shear, can also result in a reduction of anomalous cross-field transport. Next, we discuss the absence of solutions of Eq. (7.93) at a rather large velocity shear, jV00 j>jV00 jloc . We can interpret this effect within the eikonal approximation, where the wave packet can be considered as an effective “particle”, dynamic of which is described with the “Hamiltonian” ωðkx , xÞ ¼
1 ω ðxÞ 1 þ k2x ρ2s þ k2y ρ2s þ V00 ky x and the canonical variables x and kx. To have a localized solution of the wave packet, the motion of the “particle” should be bounded by two turning points corresponding to kx ¼ 0. To make it happen, the

Fig. 7.22 Eddies corresponding to the eigenfunctions found from numerical solutions of Eq. (7.93) for V00 ¼ 0 (left) and jV00 j = jV00 j loc ¼ 0:8 (right). Other parameters used in these simulations are: kyρs ¼ 0.5, w/ρs ¼ 30, ω ¼ 50. and νk =b (Reproduced with permission from [65], © AIP Publishing 2020)

x/ρs

7.2 Linear Theory of Edge Plasma Instabilities

179

20

20

10

10

0

0

–10

–10

–20 0

5 y/ρs

10

–20 0

5 y/ρs

10

function ω(kx ¼ 0, x) must have at least one extremum. For the case where ω ðxÞ ¼ b  cosh 2 ðx=wÞ, it is easy to show that this is only possible for ω jV00 j > H”) and H to L (“H> > L”) transitions. (Reproduced with permission from [82], © IAEA 2010)

Mode (GAM), see [34, 35] and the references therein. The physics of the GAM is ! ! rather simple: the compressibility of the toroidally symmetric E  B plasma flow causes a poloidaly asymmetric plasma pressure perturbation, e P , and the ! e ! e corresponding diamagnetic current across magnetic flux surfaces, J / B  P . This current is not divergence-free due to toroidal effects. It reverses the sign of the electric field and, finally, results in plasma oscillations, GAM. For a relatively large safety factor q, which is typical for the edge plasma, the plasma dynamics along the magnetic field lines can be ignored and, in the simplest case, the expression for the GAM frequency reads: ω2GAM ¼ 2γP0 =ρ0 R2 ,

ð7:95Þ

where P0 and ρ0 are the unperturbed plasma pressure and density, R is the tokamak major radius and γ is the ratio of the specific heats.

7.3

Nonlinear Effects and Anomalous Transport

In the previous section, we considered some plasma instabilities which can be important for plasma transport at the edge and in the SOL of a tokamak. We also considered possible stabilizing effects, which are the magnetic and velocity shear. However, in some cases, strong magnetic shear (e.g. in the vicinity of the X-point) can facilitate instabilities. In this section, we discuss some features of anomalous plasma transport associated with these instabilities as well as the available stabilizing effects.

7.3 Nonlinear Effects and Anomalous Transport

181

Fig. 7.24 Motion of a blob, seen as the bright spot, in the C-Mod tokamak [94], observed with the Gas-Puff Imaging (GPI) diagnostic [95]. The GPI is based on a higher radiation intensity of neutral hydrogen in the regions with enhanced plasma density and temperature. J. P. Terry, private communication, 2018

To be more precise, we will not discuss any particular scaling of anomalous crossfield transport coefficients in the edge plasmas (e.g. see [90]). Instead, we consider the main features governing anomalous transport in the edge plasma. The reason for this can be explained as follows. First, the most common approach to estimating analytically the impact of a particular unstable mode on anomalous transport is based on a local (on a given flux surface) diffusive approximation where the transport coefficients (say, the particle diffusion coefficient, D) are described by the expression D γk2 ⊥ ,

ð7:96Þ

where γ and k⊥ are the characteristic growth rate and cross-field wave number of the mode (e.g. see [90, 91] and the references therein). However, as we found in the previous section, in edge plasma, different modes can be unstable simultaneously and it is virtually impossible to find their contribution to the anomalous cross-field transport coefficients, which depend not only on plasma parameters and their radial derivatives but also on the shear of plasma flow velocity. We will see that sheared plasma flow can be generated by plasma turbulence itself (e.g. see [59, 88, 89, 92] and the references therein). Moreover, the interplay of the sheared plasma flow generation by turbulence and the impact of such a flow on the turbulence itself can result in time-dependent fluctuations of the amplitudes of plasma turbulence and shear of plasma flow velocity [93]. In addition, experiments show that a large contribution of edge plasma transport is from radial advection (predominantly, at the outboard side of the torus) of coherent filamentary structures with plasma density and temperature higher than those in the ambient plasma, the so-called “blobs” (see Fig. 7.24, [96, 98] and the references therein). It is widely accepted that blobs are ! ! propelled by E  B drift due to plasma polarization caused by magnetic drifts (the so-called ballooning effect) [99, 100]. We notice that such “blobby” anomalous cross-field plasma advection cannot be described a priory by a local theory describing the plasma parameters on a particular flux surface. As a result of such plasma advection on the outboard side of the torus, plasma turbulent transport and plasma parameters at the edge become strongly

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7 Anomalous Cross-Field Transport in Edge Plasma

(a)

Inner Midplane Plasma Pressure

(b)

eV/m3

eV/m3

21

10

20

10

1019

Outer Midplane Plasma Pressure

10

21

10

20

10

19

0.0 0.2 0.4 0.6 0.8 1.0 1.2 R–RSEP (cm) Inner Midplane Fluctuation Levels 105

104

(d)

δn

δn

(c)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 R–RSEP (cm)

103

Outer Midplane Fluctuation Levels 105

104 103

0.0 0.2 0.4 0.6 0.8 1.0 1.2 R–RSEP (cm)

USN DN LSN

0.0 0.2 0.4 0.6 0.8 1.0 1.2 R–RSEP (cm)

Fig. 7.25 Distribution of plasma pressure and fluctuation level at the inner and outer midplanes of the SOL for upper- and low-single null (USN, LSN) and double-null (DN) magnetic configurations. (Reproduced with permission from [101], © IAEA 2004)

dependent on the magnetic configuration and for the case of double null configuration (which effectively disconnects the inboard and outboard sides of the torus due to the X-point effects), plasma transport and the magnitude of the turbulent fluctuations on the inboard side become much weaker than on the outboard one (see Fig. 7.25). Finally, it was found experimentally that the SOL midplane width, and, therefore, midplane plasma transport, can also strongly depend on divertor conditions and for the detached divertor case, the width is about two times larger than for the attached one (see Fig. 7.26). We note that so far there is no clear theoretical explanation of such an effect, although there is an indication that neutrals, density of which strongly increases in the detached divertor regime, can play some role in the modification [103] and even enhancement of plasma transport, in particular, due to reduction of the shear of the plasma flow [104]. Thus, taking into account so many different factors which can alter anomalous edge plasma transport, we can conclude that the only plausible way to estimate anomalous transport with some confidence is to use 3D edge plasma turbulent codes such as BOUT++ [105], XGC1 [106], GENE [107], Gkeyll [108], JOREK [109] and some others. These codes are based on different plasma models (e.g. BOUT++ and JOREK codes are based, respectively, on the fluid and MHD plasma equations,

7.3 Nonlinear Effects and Anomalous Transport 14 H-mode attached

Measured λTe,u [mm]

Fig. 7.26 Outer SOL electron temperature decay length for the attached and detached outer divertor conditions. (Reproduced with permission from [102], © IOP Publishing 2015)

183

12

partially detached

10 8 completely detached

6 4

0.4

0.5

0.6 n/nGW

0.7

0.8

0.9

whereas XGC1, GENE and Gkeyll are gyrokinetic codes which, however, also have some differences). As a result, their application limits are different (for example, JOREK is usually used to simulate ELM, whereas the others are usually used for the simulation of edge plasma turbulence). We will discuss some results coming from these codes later. However, we notice that the codes used for the simulation of edge plasma turbulence, in most cases, cannot describe edge plasma transport on the relevant time scale, due to both the lack of some important physics needed for such a description and the limitation of current computational resources. Therefore, they often use the plasma parameter profiles taken either from experimental data or from the results of simulation of the edge plasma parameters with 2D edge plasma transport codes such as SOLPS or UEDGE. Now we consider the main features, outlined above, which govern anomalous transport in the edge plasma.

7.3.1

Generation of Sheared Plasma Flow Via Plasma Turbulence

As we found in the previous section, in strongly magnetized plasma, the unstable modes are characterized by a strong anisotropy, which is characterized by the ! inequality jkk j  j k ⊥ j . Therefore, within some limitations, the interaction of the plasma waves and plasma turbulence can, in general, be considered two-dimensional (2D). This feature makes the dynamics of magnetized plasma somewhat similar to that of geophysical fluids [110]. However, it is known since a long time ago that the main features of the turbulence in 2D fluids (including the magnetized plasma) are very different from the predictions following from such a cornerstone of threedimensional (3D) fluid turbulence as the Kolmogorov turbulence model. The main

184

7 Anomalous Cross-Field Transport in Edge Plasma

reason for this is that in addition to the energy, in 2D fluids, there is an extra invariant, enstrophy (e.g. see [111–113]), which prevents the energy cascade to and the energy dissipation at small scales, which happens in the Kolmogorov 3D turbulence model. As an illustration, we consider the so-called Charney-Hasegawa-Mima (CHM) model [113, 114], which is the simplest nonlinear model describing both the atmospheric Rossby and the magnetized plasma drift wave dynamics. We consider plasma embedded into a straight constant magnetic field (in the z-direction) and having constant electron temperature and cold ions. We will assume that the perturbation of the electron density, e ne , obey the Boltzmann relation which we will approximate as follows n
 o e  1 nðxÞϕ, e e ne ¼ nðxÞ exp ϕ

ð7:97Þ

where n(x) is the unperturbed plasma density depending on the “radial” coordinate x. The ion velocity can be found from Eqs. (7.8) and (7.9). Keeping nonlinear terms, we find  e ∂ ! cTe ∇⊥ ϕ ! ! þ V E B  ∇ , Vi ¼ V E B  BΩBi ∂t !



!

!

!

ð7:98Þ

!

! e Substituting the expression (7.98) into ion the where V!E B! ¼ ðcTe =eBÞ e z  ∇ϕ. continuity equation and assuming quasi-neutrality, we arrive at the CHM equation



∂ ! þ V!E B!  ∇ ∂t




 e  ρ2 ∇ 2 ϕ e ¼ 0, e  cTe dℓnfnðxÞg ! ϕ e y  ∇ϕ s ⊥ eB dx

ð7:99Þ

which in the linear case gives the drift wave frequency (7.16). Similar to the 2D Euler equation [112], the CHM equation has two exactly conserved integrals: energy, E, and enstrophy, En, which can be expressed in the continuum and spectral forms as follows: E¼

Z 


e ρs ∇ ⊥ ϕ

2

 Z  2  !

 2 ! 2 2 e  e ! þϕ dr⊥  1 þ ρs k ⊥  ϕ k  d k ⊥ , ⊥

En ¼

Z 
2
2  ! e þ ρ s ∇⊥ ϕ e ρ2s ∇2⊥ ϕ dr⊥ Z 



ð7:100Þ



1þ

ρ2s k2⊥





2 

 e!  ρ2s k2⊥  ϕ  k⊥

!

d k ⊥:

ð7:101Þ

From Eqs. (7.100) and (7.101) it follows that unlike the Kolmogorov model of 3D fluid turbulence, to conserve both integrals in the CHM model, the energy must be

7.3 Nonlinear Effects and Anomalous Transport

185

!

cascaded to large spatial scales (small j k ⊥ j) whereas the enstrophy to the small ones ! (large j k ⊥ j). Thus, we see that the generation of large-scale structures from smallscale fluctuations is inherent for 2D turbulence. However, these conservation laws tell us nothing about the generation of the zonal flows, which can suppress the plasma instabilities. Generally speaking, these large-scale structures can be largescale convective cells facilitating anomalous cross-field plasma transport [115]. For this reason, the topic of zonal flow generation from drift wave turbulence have received so much attention from both theory (e.g. see [59, 92, 116, 117] and the references therein) and experiment (e.g. see reviews [88, 89]). Just to give an idea of theoretical approaches used in these studies, we will follow [92] and consider the CHM equation modified by the presence of a weak zonal flow. For this purpose, the potential into two
 we separate
 perturbation of the electrostatic
 ! ! ! e e e e e zf ðx, tÞ parts: ϕ r , t ¼ ϕdw r ⊥ , z, t þ ϕzf ðx, tÞ , where ϕdw r ⊥ , z, t and ϕ describe respectively the drift waves and zonal flow. Such a separation is needed edw justifies the Boltzmann relation for the perturbed because the z-dependence of ϕ e plasma density and ϕdw , even though the CHM equation per se contains no direct ezf ðx, tÞ, which has no z-dependence and, z-dependence. This is to the contrary to ϕ therefore, does not enter into the Boltzmann relation, although it contributes to the ! ! e dw in Eq. (7.97) but total E  B plasma flow. Keeping this in mind and using only ϕ e ϕ in Eq. (7.98), we arrive at the following modified Hasegawa-Mima equation [92, 116]:

!
 !  ∂ e e þ Vzf þ Vdw  ∇ ϕ e edw  ρ2 ∇2 ϕ ϕdw  ρ2s ∇2⊥ ϕ s ⊥ ∂t cT dℓnfnðxÞg ! e ¼ 0, e y  ∇ϕ  e dx eB !

ð7:102Þ

!

! ezf and Vdw ¼ ðcTe =eBÞ ! e dw . where Vzf ¼ ðcTe =eBÞ e z  ∇ϕ e z  ∇ϕ !ð1Þ e ð1Þ, wavenumber k and For the case where one drift wave with the amplitude ϕ dw frequency given by expression (7.16) dominates, Eq. (7.102) describes modulation, or in a more general case, parametric instability of this wave, which describes the e zf. The growth rate of such instability, γzf, for the case where kzfρs  1 excitation of ϕ (here kzf is the x-component of the wavenumber of the zonal flow) is [92, 116]:

e ð1Þ j kzf kð1Þ cTe j ϕ y dw qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γzf ¼ eB 1 þ k21 ρ2s

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 
 ffi 2

2 , 1 þ ρ2s kðy1Þ  3 kðx1Þ

ð7:103Þ

186

7 Anomalous Cross-Field Transport in Edge Plasma

which, in particular, shows that for the case of k21 ρ2s  1, the drift wave is always unstable and generates a zonal flow. Analysis of the dynamics of uncorrelated drift wave packets performed in [92] also demonstrates the possibility of the generation of zonal flows. In [118] it was shown that in addition to the exact integrals of the CHM equation (the energy and the enstrophy) there is also a third “approximate” integral, I, which, however, becomes exact for the case of the resonant triad interactions (see also [119] and the references therein). This integral can only be expressed in a spectral form:

I¼


 Z η ! k⊥ ky

2 !

2  e!  d k ⊥ , 1 þ ρ2s k2⊥  ϕ k ⊥

ð7:104Þ

e ! is the Fourier component of drift wave fluctuations and where ϕ k ⊥

pffiffiffi  pffiffiffi   
!  kx þ 3ky kx  3ky η k ⊥ ¼ arctan  arctan : ρs k 2 ρs k 2

ð7:105Þ

It was shown that even approximate conservation of I ensures that the energy of the turbulence described by the CHM equation is transferred to a very anisotropic zonal flow with kx ky [118]. We notice that GAM, which in the edge plasma can be as efficient for the damping of plasma turbulence as the zonal flows, can also be excited by nonlinear processes associated with plasma turbulence [35]. The turbulence-induced generation of a zonal flow it now routinely observed in large-scale 3D plasma turbulence simulations (e.g. see [59] and the references therein). As an example, in Fig. 7.27 one can see the ion heat diffusivity and the shearing rate of the zonal flow found from the numerical simulation of ITG turbulence in ITER. One can clearly see both the variation of the sign of the shearing rate along with the normalized poloidal magnetic flux and the reduction of the ion heat diffusivity for the case of a fully developed zonal flow at a later time. There is also a significant body of experimental data supporting the generation of both zonal flows and GAM due to nonlinear processes associated with plasma turbulence (e.g. see [88, 89] and the references therein). As an example, in 2 Fig. 7.28 one can see summed cross- and auto-bicoherences, b b ðf, f GAM  f Þ , of the electric field fluctuations measured by a probe array at the edge of the HL-2A tokamak. Very distinct peaks at the frequency fGAM 7 kHz, exhibited by all three curves, demonstrate a strong coupling of GAM to the broadband plasma turbulence. Finally, Fig. (7.14) shows that the zonal flow goes over the entire poloidal circuit. However, recent experimental data [122] suggest that this might be not always the case and the co- and counter-wise streams can gradually close on each other forming, as a result, a poloidally extended convective cell at the outer side of the torus, which, nonetheless, can still be rather efficient in turbulence suppression.

7.3 Nonlinear Effects and Anomalous Transport χ/χGB0

s

Fig. 7.27 Ion heat diffusivity (top) and the shearing rate of zonal flow (bottom) found from numerical simulation of ITG turbulence in ITER as the function of time and normalized poloidal magnetic flux. (Reproduced with permission from [120], © IOP Publishing 2013)

187

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

5 4 3 2 1

1

2

3

4

5 6 t [Ω i–1]

7

9 ×105

8

0

ω E×B [cs/a] 2G extfilt

Fig. 7.28 The summed cross- and autobicoherences of electric field fluctuations measured by probe array at the edge of HL-2A tokamak. (Reproduced with permission from [121], © IOP Publishing 2008)

0.5 0 –0.5

1

2

3

4

5 6 t [Ω i–1]

7

8

9 ×105

–1

0.03

Σb 2 (f )

s

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.02



0.01

0.00

1

10

100 f(kHz)

7.3.2

“Blobs”

Meso-scale plasma structures which are now called blobs were occasionally observed experimentally for a long time as, in particular, large spikes of the ion saturation current collected by electrostatic probes at the edge of tokamaks (e.g. see [123] and Fig. 7.29). However, only after a very large plasma particle flux to the main chamber wall was discovered in the C-Mod tokamak [125] and it became clear that such a flux is

188

7 Anomalous Cross-Field Transport in Edge Plasma 1.0 0.8

Is(A)

0.4 0.0

Fig. 7.29 Time dependence of the ion saturation current on the probe situated in the SOL of the DIII-D tokamak. (Reproduced with permission from [124], © AIP Publishing 2001) Fig. 7.30 Schematic view of the filament’s polarization and advection !

!

due to magnetic and E  B drifts. (Reproduced with permission from [100], © Elsevier 2001)

wall δ B +

V∇ B ~ Cs

ρi R

VE –

plasma blob

incompatible with the diffusive nature of plasma transport [99, 125], the physics of blobs became one of the central topics of the edge plasma studies. As of today, the blobs are observed in virtually all magnetic fusion devices including both tokamaks and stellarators [126–129] and there is vast amount of literature dedicated to different experimental and theoretical aspects of blobby transport (e.g. see review papers [96, 98, 126]). As a matter of fact, the physical reason for the radial advection of blobs (which are the filamentary structures extended along the magnetic field, see Fig. 7.29) is simple [100]. Consider an isolated plasma filament situated in a vacuum (or in very low density plasma) sketched in Fig. 7.30. Then magnetic drifts of the electrons and ions will result in the polarization of the filament and the formation of a vertical (along the major tokamak axis) electric field (for simplicity, we ignore the effect of the magnetic shear). Such an electric field will cause outward advection of the filament ! ! (see Fig. 7.24) due to E  B drift. Therefore, in tokamaks, blobs are mainly observed at the outer side of the tours. The strength of the vertical electric field is altered by the parallel electric current. Different theoretical models for such a current, giving different scalings for the blob speed, were used over the years including the sheath- and X-point-limited current, and the “inhibited” current (see the review papers [96, 98] for details). In the latter case, the blobs can reach the highest speed, which, according to the numerical simulations [130], pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi for nb na can be estimated as Umax

ffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðTe þ Ti Þ=M ðnb =na Þðδb =RÞ , where nb and na are plasma density in the blob and in the ambient plasma, δb is the initial cross-field size of the blob.

7.3 Nonlinear Effects and Anomalous Transport

189

Fig. 7.31 Visible light image of blob filaments from MAST tokamak. (Reproduced with permission from [133], © IAEA 2007)

For typical tokamak edge plasma parameters we find Umax ~ 1 km/s, which is, in a ballpark, consistent with the experimental observations (e.g. see [124, 131]). For sheath-limited parallel electric current, the scaling for the blob velocity reads Ub / δ2 b [100]. The available experimental data seem to support both these scalings [132]. The filamentary structure of the blob was confirmed by direct observations with fast cameras (e.g. see Ref. [133–136]). As an example, in Fig. 7.31 one can see a snapshot made in an L-mode discharge in the MAST tokamak, which reveals multiple filamentary structures. We notice that blobs are seen in both the L- and H-modes (e.g. see [135]) (in between ELMs) and in both cases, the blobs dominate far SOL plasma transport [124]. However, recent experimental data show that blobs exist not only at the outer boundary of tokamaks but also in the divertor volume [57] although the impact of these blobs on plasma transport is not yet clear. Numerical simulations show that the shape of the blobs, in the course of their radial advection, can be significantly deformed due to effects associated with the Rayleigh-Taylor and Kelvin-Helmholtz instabilities [96]. Also, 2D and 3D simulations demonstrate that blobs can be effectively disintegrated by sheared background plasma flow [136] and the onset of the resistive drift wave instability [137]. Usually, blobby transport is enhanced when the plasma density approaches the density limit. One of the typical manifestations of such enhancement is the formation of the so-called “shoulder” on averaged plasma density profile in the far SOL region (see Fig. 7.32). Modeling of edge plasma transport with turbulence codes also shows both blob formation and advection. As an example, in Fig. 7.33 one can see the snapshots of the distribution of edge plasma parameters, having clear features of blobs, found with the gyrokinetic code XGC1 [139] (white dashed line is the effective separatrix). Although the dynamics of individual blobs in the SOL is understood rather well, the formation mechanism of large density blobs, in particular, those which are observed inside the separatrix, recall Fig. (7.24), is not clear. In [140] it was shown that the 1D version of the modified Hasegawa-Mima equation allows the

190

7 Anomalous Cross-Field Transport in Edge Plasma

Fig. 7.32 Formation of the “shoulder” on averaged plasma density in the SOL at high plasma density. (Reproduced with permission from [138], © IAEA 2017)

Fig. 7.33 Snapshots of plasma parameters at the outer midplane found from numerical simulations. (Reproduced with permission from [139], © AIP Publishing 2019)

6

8 10 12 14 16 20

25

ne

30

40 60 80 100 120 f

Te

0.1

y (m)

0.05 0 –0.05 –0.1

1.3

1.35 x (m)

1.4 1.3

1.35 x (m)

1.4 1.3

1.35 x (m)

1.4

solution in the form of a train of plasma density “blobs” propagating in poloidal direction, which resembles the experimental data, see Fig. (7.34), on the dynamics of nonlinear drift waves [14]. However, as of today, there is no direct experimental confirmation that plasma density blobs can be formed in the course of nonlinear evolution of drift waves. We notice that strong blobby transport poses a serious problem for the application of 2D edge plasma transport codes like SOLPS or UEDGE for interpretation of the experimental data [97]. The issue is that these codes deal with average plasma parameters and their results are compared with average experimental data on plasma density, temperature, etc. However, for strongly nonlinear functions such as the dependence of the rate constants of atomic processes on electron temperature, K(Te), we have hK(Te)i 6¼ K(hTei), where h. . .i means time averaging. As a result, strong intermittent fluctuations of the plasma parameters, associated with blobs, will inevitably cause a departure of the averaged experimental data from the simulation results.

7.3 Nonlinear Effects and Anomalous Transport

191

Fig. 7.34 Density oscillation in a nonlinear drift wave. The arrow on the left corresponds to zero plasma density. (Reproduced with permission from [14], © Springer 1967) Fig. 7.35 Comparison of experimental data and simulation results on the parallel Mach number of the plasma flow in the SOL of the COMPASS tokamak. Here HFS and LFS stand for the high- and low- field sides of the torus. (Reproduced with permission from [141], © IAEA 2017, experimental data from Asakura N. et al., J. Nucl. Mater. 365 41–51)

7.3.3

3D Edge Plasma Turbulence Modeling

Today, quite a few codes of different sophistication are available for edge plasma turbulence simulation: BOUT++ [147], XGC1 [106], TOKAM3X [141], GBS [142], GDB [31], GRILLIX [143], Gkeyll [108], and some others. These codes are used for both modeling some particular experiments and for studying the general characteristics and dependences of edge plasma turbulence. For example, in Fig. (7.35) one can see a very good agreement of experimental data on the parallel Mach number of the plasma flow in the SOL of the COMPASS tokamak with the results of the modeling of the impact of plasma turbulence and macroscopic ! ! E  B drifts on parallel plasma flows, performed with the TOKAM3X fluid turbulence code [141]. Another example of a comparison of modeling results and experimental data is shown in Fig. (7.36). Here one can see the probability density function for the density fluctuations from the Helimak toroidal device and the results of the numerical simulations performed with the Gkeyll gyrokinetic code. Even though all the curves in Fig. (7.36) exhibit non-Gaussian features (typical for blobby transport), the simulations do not reproduce the long tail of the density fluctuations observed in the experiment. Another important area of the application of 3D plasma turbulence codes is related to the simulation of the width, λq, of the part of the SOL, where the heat flux is transported to the divertor target. This parameter is of particular importance for ITER because it largely determines the heat load on the

192

7 Anomalous Cross-Field Transport in Edge Plasma

Fig. 7.36 Probability density functions for density fluctuations. (Reproduced with permission from [108], © AIP Publishing 2019)

divertor target. Recently established experimental scaling predicts that in H-mode in 1 between ELMs, λq / I1 p / Bpol, where Ip is the tokamak plasma current and Bpol is the strength of the poloidal magnetic field [144]. In [145] this scaling was attributed to ion drifts in the tokamak magnetic field (with no turbulent impact on the ion dynamics) so that λq becomes of the order of the poloidal gyroradius of ions. Recently, this scaling was reproduced for current tokamaks by the fluid BOUT++ and gyrokinetic XGC1 plasma turbulence codes (see Fig. 7.37). Interestingly, both codes predict a large departure of λq from the experimental scaling for ITER (see Fig. 7.37). However, the physics of this is not clear yet and further studies are needed to confirm these results. In [147] it was speculated that the transition from drift- to turbulence-dominated processes that set λq occurs in next step tokamaks due to the larger size and stronger magnetic field.

7.4

Conclusions

In this chapter, we reviewed the basic theory of plasma waves responsible for anomalous plasma transport, considered their main destabilizing mechanisms and presented some experimental data confirming the theoretical and simulation results. The situation with theoretical analysis and predictions of anomalous cross-field transport is more complex and as of today, we only have a basic theoretical understanding of the processes governing anomalous plasma transport, although there is a large amount of experimental data and simulation results supporting these ideas. Nonetheless, at present, practically all results of edge plasma transport simulation performed with 2D codes such as SOLPS or UEDGE, are based either on the usage of the anomalous transport coefficients fitting the edge plasma parameter profiles observed in experiments or on the scoping studies of an impact of the

References

193 λq [mm] (exp.) 8

6 Exp. Scaling law Exp. Error bars BOUT++ C-Mod BOUT++ EAST BOUT++ DIIID BOUT++ CFETR BOUT++ ITER Exp. C-Mod Exp. DIIID

5

λq (mm)

4 3

experimental data points corresponding to XGC1 simulations

XGC1 DIII-D C-Mod NSTX

7 6 5 4 3

2

MAST NSTX C-Mod AUG DIII-D JET

ITER→ O

2 1 0

1 0 0.2

0.4

0.6 0.8 1.0 Bpol,MP (T)

1.2

1.4

R 2=0.86 0

0.2

0.4

0.6 0.8 Bpol,MP (T)

1.0

1.2

Fig. 7.37 Predictions of λq for different existing tokamaks and ITER found (left) with the BOUT+ +. (Reproduced with permission from [146], © IAEA 2018) and (right) XGC1. (Reproduced with permission from [106], © IAEA 2013) plasma turbulence codes

transport coefficients on edge plasma performance for the cases where there is no available data yet (e.g. ITER simulations, see Chap. 9).

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140. S.I. Krasheninnikov, On the origin of plasma density blobs. Phys. Lett. A 380, 3905–3907 (2016) 141. D. Galassi, P. Tamain, H. Bufferand, G. Ciraolo, P. Ghendrih, C. Baudoin, C. Colin, N. Fedorczak, N. Nace, E. Serre, Drive of parallel flows by turbulence and large-scale EB transverse transport in divertor geometry. Nucl. Fusion 57, 036029 (2017) 142. P. Paruta, P. Ricci, F. Riva, C. Wersal, C. Beadle, B. Frei, Simulation of plasma turbulence in the periphery of diverted tokamak by using the GBS code. Phys. Plasmas 25, 112301 (2018) 143. A. Stegmeir, A. Ross, T. Body, M. Francisquez, W. Zholobenko, D. Coster, O. Maj, P. Manz, F. Jenko, B.N. Rogers, K.S. Kang, Global turbulence simulations of the tokamak edge region with GRILLIX. Phys. Plasmas 26, 052517 (2019) 144. T. Eich, A.W. Leonard, R.A. Pitts, W. Fundamenski, R.J. Goldston, T.K. Gray, A. Herrmann, A. Kirk, A. Kallenbach, O. Kardaun, A.S. Kukushkin, B. LaBombard, R. Maingi, M.A. Makowski, A. Scarabosio, B. Sieglin, J. Terry, A. Thornton, ASDEX Upgrade Team and JET EFDA Contributors, Scaling of the tokamak near the scrape-off layer H-mode power width and implications for ITER. Nucl. Fusion 53, 093031 (2013) 145. R.J. Goldston, Heuristic drift-based model of the power scrape-off width in low-gas-puff H-mode tokamaks. Nucl. Fusion 52, 013009 (2012) 146. X.Q. Xu, N.M. Li, Z.Y. Li, B. Chen, T.Y. Xia, T.F. Tang, G.Z. Deng, B. Zhu, V.S. Chan, Simulations of tokamak boundary plasma turbulent transport in setting the divertor heat flux width, 27th IAEA Fusion Energy Conference, IAEA-CN-123/45/TH/P7-21 147. Z.-Y. Li, X.Q. Xu, N.-M. Li, V.S. Chan, X.-G. Wang, Prediction of divertor heat flux width for ITER using BOUT++ transport and turbulence module. Nucl. Fusion 59, 046014 (2019) 148. A.B. Mikhailovskii, Theory of plasma instabilities, in Instabilities in a Homogeneous Plasma, vol. 2, (Springer, New York, 1974)

Chapter 8

Computational Modeling of the Edge Plasma Transport Phenomena

Abstract Both nonlinear plasma dynamics due to plasma instabilities and macroscopic edge plasma transport, involving different atomic physics processes, impurity effects and plasma-material interactions are extremely complex. As a result, only limited cases can be treated analytically, whereas a thorough study requires numerical simulations. The main approaches to computational modeling of different edge plasma phenomena, their assumptions and limitations are considered in this chapter.

Given the broad variety of the physical processes occurring in the edge plasma on very different time scales, it would be impractical to attempt producing a single, comprehensive model describing all the processes in the plasma edge from the first principles. Due to the large disparity of the characteristic time and length scales involved, such a model would require computer resources orders of magnitude larger than available at present or in the near future. Moreover, even if such a model were developed, it would probably require a special theory describing the detail of the model itself and the “second-order” modeling of the processes involved in order to understand the results. Most of the collision processes involving the edge plasma and neutral gas species occur on sub-microsecond time scales. The small-scale turbulence develops in a fraction of a millisecond. The time scale of equilibration of the plasma parameters along the magnetic field is determined by the sound speed and connection length and is typically several tens milliseconds, which is also the time scale for the cross-field transport outside the separatrix. Finally, the evolution of the state of the material surfaces surrounding the plasma, which determines the recycling conditions and impurity production rate, can take 100’s to 1000’s seconds until the saturation (if any) is achieved. Therefore, there is a more than nine orders of magnitude span of the time scales involved in the physical processes in the edge plasma, so resolving all of them in one model looks unrealistic. Nevertheless, computational modeling plays the crucial role bridging the theoretical understanding and experimental observations. The equations describing the edge plasma are very complex, so analytical treatment is only possible with introducing strong simplifying assumptions that are not always strictly justified. The experimental data are difficult to obtain and the measurements are not straightforward to interpret. Despite all limitations and many approximations, computational © Springer Nature Switzerland AG 2020 S. Krasheninnikov et al., On the Edge of Magnetic Fusion Devices, Springer Series in Plasma Science and Technology, https://doi.org/10.1007/978-3-030-49594-7_8

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modeling becomes a vital tool guiding the experiments and furthering the development of theoretical models. Today one can identify two main directions in the edge plasma modeling. One of them focuses on fast, relatively small-scale plasma instabilities and turbulence, which govern anomalous cross-field plasma transport (see Chap. 7 for further discussions). The other one is aimed at characterizing the large spatiotemporal scale quasi-equilibria and flows of particles and energy in complex divertor geometries, including coupling to neutrals, sheath boundaries, atomic physics, plasmasurface interactions, etc. The codes used for this second direction, the so-called “edge plasma transport codes”, include the effects of small-scale fluctuations by using some “anomalous” transport coefficients. These coefficients could be provided by the plasma turbulence codes, but most often, they are obtained by fitting the results of experimental measurements in the edge plasma. In this chapter, we discuss the main approaches to the edge plasma modeling with the transport codes. Generally, the edge plasma of a tokamak or, especially, a stellarator is a 3D object. Correspondingly, a 3D transport model would be desirable. However, even in the transport approximation, a full 3D model becomes too complex and its numerical realization is too slow for practical use. Therefore, 2D and even 1D plasma transport codes are typically used for the interpretation of experimental results and comparison with theoretical models [1]. There is significant progress in the development of the 3D codes [2] oriented primarily to the description of the plasma edge in stellarators where the toroidal symmetry approximation is not applicable. However, we focus here on the 2D transport models since they are most developed and widely used for tokamak modeling.

8.1

Transport Modeling of the Plasma

In the edge plasma, where the flow patterns that determine the distribution of the plasma parameters and wall loading form, the neutral particles, such as atoms and molecules, are abundant and play an important role in the physical processes occurring there. Since the neutrals, unlike the charged plasma particles, are not magnetized, their description may require a different approach. If the distribution functions of the plasma components are not far from Maxwellian (the assumption used in the transport models), then the plasma state is characterized by the density, fluid velocity and temperature of all sorts of the charged particles involved in the model. The spatial profiles of these quantities and their time evolution are described with a set of equations for the particle and energy densities and parallel momentum of the different plasma species. Different forms of such transport equations are used in different codes (see Chap. 6). One of the forms of these equations is ! ∂Us þ ∇  Γ Us ¼ SUs , ∂t

ð8:1Þ

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203

where Us corresponds to the particle and energy densities and parallel momentum of ! the species s, whereas Γ Us and SUs describe the flux of these quantities and the effective “source” terms which cannot be written in the form of divergence (e.g. the sink/source of the particles, the friction force, etc.). Since the plasma is magnetized and the modeling is aimed at resolution of the effects occurring on a time scale much longer than the ion gyration time, the momentum component parallel to the magnetic field is only retained in the momentum equations and a simplified description of the perpendicular flux components (usually, the diffusive approximation) is used in all the equations. Due to the mass difference, temperature relaxation between different sorts of ions is much faster than between the ions and electrons, so the energy transport equations for all the sorts of ions are often combined in one equation, assuming the common temperature Ti for all the ions. Because of small electron mass and plasma quasi-neutrality, no separate electron parallel momentum and continuity equations are used. The electric field, which develops in the edge plasma, is described by the Eq. (5.52), which is derived from the condition of zero divergence of the electric current. The principal sources for these currents are the electron and ion magnetic drifts and anomalous cross-field plasma transport, as well as the difference in the sheath potential at the two targets connected by the magnetic field lines. Plasma transport in a strong magnetic field is anisotropic. It is fast along the magnetic field and relatively slow, by diffusion or intermittent convection, across (see Chap. 7). Correspondingly, the coordinate system used for the representation of Eq. (8.1) in modeling is usually aligned with the magnetic field. This helps one to avoid “contamination” of the weak cross-field transport terms with the strong parallel transport by discretization of the equations.

8.1.1

Model Geometry

In the toroidal symmetry approximation, the 2D model geometry is represented with a radial cross-section of the tokamak (the poloidal plane), Fig. 8.1. The computational grid for the plasma transport description is aligned with the magnetic surfaces and the flows parallel or normal to the magnetic field lines are projected onto the poloidal plane, thus translating the anisotropy with respect to the magnetic field into anisotropy with respect to the magnetic surfaces and making the problem two-dimensional. The shape of the grid reflects the magnetic field topology, see the example in Fig. 8.1. In most major codes, such as EDGE2D [3], SOLPS [4] and UEDGE [5], the radial extent of the grid is limited by the first intersection with a material surface other than a target. This allows projection of the curvilinear grid onto a topologically equivalent, rectangular one, Fig. 8.1, and simplifies coding. In the equations written on the rectangular grid, the real geometry is included through the metric coefficients and the transport anisotropy appears distinctly in the boundary conditions. For a single-null divertor configuration, there are two sides of the

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a

Computational Modeling of the Edge Plasma Transport Phenomena

C

b C

B

H A

X F

D

G

E

X

H G

F

c C

B X

G

D X

B X A

D

F

A

FG

H

GF

E

E

Fig. 8.1 Typical grid used for discretization of the edge plasma transport equations and the topologically equivalent rectangular grid. Arrows indicate correspondence between the fluxes on the grid cuts. The grid transformation can be presented as (a) cutting the grid along the FG line, (b) unfolding it and (c) distorting it to make rectangular, hiding the curvilinearity in the metric coefficients in the equations. (Reproduced with permission from [10], © Cambridge University Press 2017)

rectangle corresponding to the targets and two others where the plasma fluxes only have the component normal to the magnetic field. This simplification can be justified for modeling plasma interaction with the targets since most of the power going through the SOL is concentrated in a narrow layer just outside the separatrix [6, 7], which is not strongly affected by interaction with the side walls. However, when the plasma profiles near, or fluxes onto the sidewall come into question, the grid needs to be extended to cover the whole chamber. Grids of this kind are implemented in the SOLEDGE2D-Eirene [8] and SOLPS [9] code packages, but they are not used widely yet.

8.1.2

Parallel Transport

Plasma transport along the magnetic field is usually described with Braginckii-type [11] terms which are valid if the ratio, γ, of the Coulomb mean-free path λC of the charged particles to the scale length L of variation of the plasma parameters along the field line is small. However, the validity of the expressions for the high order moments of the distribution functions, e.g. those describing the heat fluxes, requires this parameter to be really small, γ < 102 (see Sect. 6.4), which in practice does not

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hold. At the values of γ > 102, which is the typical situation in the SOL plasma, the heat is mostly transported by supra-thermal particles having a longer mean-free path and the parallel energy transport becomes non-local – the flux is determined by the whole temperature profile, not by the local values of the plasma parameters and their gradients. Physically, this means that the hot tails of the distribution functions are depleted in the hot SOL and enhanced in the colder divertor region, therefore reducing the heat flux upstream (see Sect. 6.4 for details). The empirical way of taking this reduction of the conducted energy flux into account is the introduction of the so-called flux-limit factors that effectively limit the conducted flux to some fraction of the free-streaming one, see Eq. (5.42). As noted in Chap. 6, this approach describes reasonably well the reduction of the heat flux upstream because of the depletion of the hot tails of the distribution functions there but does not take into account the appearance of the supra-thermal particles in the cold plasma. This may be not so important since, in the divertor regions, where intensive particle recycling occurs, the heat fluxes are dominated by convective transport [12]. In addition, low electron temperature and high plasma density in the recycling region cause strong dissipation of the electron tail so that it does not affect the ionization rates strongly [13]. The heat flux correction with the flux limiting factors is implemented in virtually all major 2D modeling codes together with similar treatment of the parallel viscosity coefficient. In principle, non-local transport requires a full kinetic description that allows a significant deviation of the velocity distribution functions of the charged particles from the Maxwellian ones. However, full kinetic treatment of a problem including many different interactions occurring on different time scales with adequate spatial resolution does not look realistic at the present state of computer development. Given the complexity of the full kinetic treatment, the attempts have been made to combine a simplified kinetic model with a fluid model of the edge plasma, treating the Coulomb collisions in kinetics and leaving the slower processes for the fluid description. In [12], a simplified kinetic model in the BGK approximation [14] was combined with a 2D fluid model. The solution was obtained in iterations where the plasma parameters from the fluid model were taken as the background for the kinetic calculations for electrons and ions, then the effective heat conductivities evaluated from the kinetic fluxes and fluid gradients were applied in the fluid code – and so on, until convergence. In a recent study [13], a Fokker-Planck kinetic model for electrons was coupled to a 1D fluid model in a similar way. However, these attempts are rather exotic and are not being used in the massive calculations with the 2D models.

8.1.3

Cross-Field Transport

Presently, there is no concise, credible theory that would describe anomalous plasma transport across the magnetic field (see Chap. 7 for the present state of the theory in this area). In this situation, the most common approach is using the diffusive ansatz that meets at least the first law of thermodynamics. The “classical” cross-field diffusion of particles, parallel momentum and energy, which is related to the binary

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collisions between the charged particles that follow different orbits in the magnetic field, yields the values of the transport coefficients that are far too low to explain the radial profiles of the plasma parameters observed experimentally. There must be so-called “collective” effects related to some relatively small-scale turbulence, which are responsible for cross-field transport. These effects appear in the edge plasma models through the “anomalous” transport coefficients that are constructed to meet some empirical expectations, mostly, the radial profiles at the mid-plane of the plasma temperature and density or of the width of the power-carrying layer close to the separatrix in the SOL [15, 16]. Given the lack of detailed understanding of the processes causing the cross-field transport, the cross-field diffusivities are often set piecewise constant in the edge plasma. There have been attempts to adjust their profiles to reach a better match to the experimental measurements (see e.g. [17–19]). However, such an adjustment depends on the assumptions made for the parallel transport, the neutral transport, drifts and so on [20, 21] and is therefore not universal. The cross-field flow of particles naturally conveys the energy and parallel momentum. Besides these diffusion-like flows, there are regular cross-field flows related to the ! ! macroscopic E  B drifts of the charged particles [22–24] and intermittent convection [25–27]. The drift-related flows are described rather strictly in the transport equations [15], but their inclusion in the computer models is still not routine. The complex picture of the magnetic drifts that cause electric currents that affect the ! ! distribution of the electric fields that cause the E  B drift flows requires special effort to make the computations stable. We will discuss the drift effects in some more detail below. The intermittent transport is less well understood. Experimentally, it is observed as a radial motion of some plasma structures (“blobs” or “filaments”) aligned with the magnetic field [26–28]. There is some insight from theory into the nature of the radial transport of such coherent structures in the magnetic field [29], which predicts the velocity of their propagation, but no reliable model of formation of these “blobs” is available at present (see also Chap. 7). In the transport models, the intermittency is usually treated as time-average outward convection with a prescribed velocity [30, 31]. However, a simple model of the outward pinch has two principal drawbacks. First, the filaments (or “blobs”) forming this flow contain the plasma with parameters close to those at the separatrix, which are very different from those of the background plasma in the far SOL, and this plasma does not mix with the background. On the contrary, the description of this flow by adding the convection velocity (or with the enhancement of the diffusivity) effectively mixes the blobs with the background. Whereas the fast-moving “blobs” can deliver a significant amount of the hot particles, before they spread along the field line and sink to the target, to the wall, thus forming some intermittent wall loading pattern, the slowly diffusing, average plasma has enough time to deposit it all onto the targets. This does also replace the heterogeneous plasma background for neutral transport with the homogeneous, averaged one. Given the non-linear, threshold nature of the dependence of the neutral penetration depth on the background plasma parameters, such

8.2 Neutral Transport Models

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averaging can introduce significant errors in the neutral penetration through the SOL [32]. Besides this, the blobs perturb the distribution of the electric potential in the ! ! edge plasma and can affect the E  B -driven flows significantly [32]. The second drawback is that applying a prescribed outward velocity to all the species of the multi-component plasma, one over-estimates the impurity screening since the impurities are flushed away with the convection. Moreover, applying any prescribed convection velocity to the impurity ion transport can be physically wrong. Indeed, given the apparent interchange nature of the blobs, their propagation outwards must cause correspondingly enhanced inward transport of the background plasma. One can easily assume that the bulk, hydrogenic ions are more abundant closer to the separatrix, so their net “convective” flow is directed outwards. However, for the impurities, this is not always the case. Therefore, for physically correct modeling of intermittent, “blobby” transport in 2D transport models, one needs to take into account both the heterogeneity of the plasma parameters and the presence of the effective backflow of the background plasma. A first step to producing a model of this kind was made in [33]. An approach of including the blob dynamics in a transport model by averaging over the ensemble of the blobs was developed there; however, the size and velocity of the blobs, as well as their starting location are still external parameters in the model. The blobs are 3D structures that occupy the full poloidal extent of the outboard SOL when mapped onto the (R, Z) plane. Combining multiple single blobs (which can be interpreted as time averaging) into a single “macro-blob” that travels across the SOL plasma without interaction allows one to solve the equations describing the propagation of the 3D filament in the framework of the 2D plasma solver (UEDGE in [33]). Such an approach provides a description of the intermittent wall loading and introduces to some extent a heterogeneous plasma background for interaction with neutrals, as well as the plasma backflow that is mimicked by the “bypass” that transfers the plasma from the blob front immediately to its wake. However, the effect of this ! ! approach on the electric fields – and hence on the E  B drifts – is not yet clear. Indeed, in this approach, a single macroscale circulation around the macro-blob replaces a set of fluctuating mesoscale circulations around every single blob. A similar concern arises regarding the heterogeneity of the plasma background in this model. Here the macro-blob acts like a piston effectively screening the neutrals that recycle off the sidewalls, whereas the ensemble of the single, poloidally localized blobs leaves a gap for neutral penetration.

8.2

Neutral Transport Models

The neutral particles play an important role in the processes occurring in the edge plasma [34]. They are not affected by the magnetic field, so their transport is different from that of the charged components of the plasma. Eqs. (8.1) contain the source terms describing also the interactions between the charged and neutral

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particles and in order to calculate these terms, one needs a proper model for neutral transport. The distribution of the neutrals in the plasma edge depends, in particular, on the plasma parameters - hence the equations describing the neutrals must be solved together with Eqs. (8.1) and the corresponding blocks of the code realizing the model must be coupled. From the coupling viewpoint, the fluid equations for neutral transport similar to Eqs. (8.1) would be the most natural choice. However, this implies short mean-free-path of neutrals with respect to neutral-ion or neutralneutral collisions, which, except for the regions of high plasma density, can be longer than the scale length of the profiles of the plasma parameters. Furthermore, because of the absence of interaction with the magnetic field, the neutral transport has no preferential direction and should be described in full 3D, which is computationally demanding. Therefore, kinetic, Monte-Carlo type modeling is mostly used in the major codes for edge plasma modeling for high fidelity simulations.

8.2.1

Fluid Description of Neutrals

Because of relative simplicity – compared with the kinetic model – the fluid-like models for neutral transport in the divertor received considerable attention from the developers of the edge modeling codes [35–40]. These models rely on the relaxation of the distribution function of hydrogen isotope atoms towards the ion distribution function in charge-exchange collisions with the plasma ions [35] since the neutralneutral collisions are usually too seldom to establish the Maxwellian distribution of neutral species (see Sect. 4.5 for more detail). The models are of different complexity; some of them take into account wall reflection and volumetric recombination and provide sources of particles, momentum and energy for Eq. (8.1). Their comparison with kinetic, Monte-Carlo models shows reasonable agreement, but for restricted Monte-Carlo models that describe the hydrogenic atoms only [41]. However, the importance of molecule transport in the description of divertor performance was clearly demonstrated using the full kinetic Monte-Carlo neutral model (see e.g. [42]) and the results obtained with a fluid neutral model differ, sometimes even qualitatively, from those obtained with the kinetic model [37]. Besides the lack of molecules in the fluid models, there are several other reasons for this difference. The radial profiles of the plasma parameters outside the separatrix are quite narrow, so the condition of smallness of the ratio of the neutral mean-free-path to the scale length of variation of the plasma parameters, necessary for validity of the fluid closure of the transport equations, is violated for neutrals in most of the edge plasma, except for the dense divertor regions close to the targets. It is difficult to describe correctly in a fluid model all the vast variety of physical processes that occur in collisions involving the neutral particles in the plasma and on the wall, as well as the geometry detail of the particular divertor configuration. In addition, the validity of the assumption of charge-exchange relaxation is not obvious for the impurity atoms. There have been efforts to produce a hybrid, fluid-kinetic model for the neutral transport in the edge plasma [43–45], which would either apply kinetic corrections to

8.2 Neutral Transport Models

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the transport coefficients or use either the fluid or the kinetic description in different regions of the calculation volume, but the applicability of such a model is still to be shown.

8.2.2

Monte-Carlo Models

The mainstream approach to describing neutral particle transport in the edge plasma is presently direct Monte-Carlo modeling [46, 47]. In this, stochastic, approach, instead of solving the kinetic equations, Eq. (6.1), directly, one models trajectories of the test particles, playing their interactions with the fixed background with the use of the pseudo-random numbers generated by the computer. The source terms for Eq. (8.1) are evaluated by calculating their intensity along the particle trajectory and summing it over the trajectories in each grid cell. This method is versatile with respect to the geometry of the problem, the composition of the particles considered and the choice of different reactions they participate in by collisions with the background particles. The latter represent, first of all, the electrons and ions of the plasma. In non-linear models, where the neutral-neutral interactions or radiation transport are included, the corresponding background is calculated in iterations on a grid of trilateral cells, which covers the regular grid used for solving Eqs. (8.1) plus the space between this grid and the material walls [42, 48–50]. The geometry can be arbitrarily complex, including sometimes 3D objects such as downpipes [51] or treating the full 3D configuration in a combination with a 3D plasma solver [52]. The selection of the test particles and background species, as well as of the reactions between them, is virtually unlimited – provided that the necessary cross-section or rate data are available. These data are normally imported from external databases [53] that are expandable. All this makes the Monte-Carlo approach very flexible and convenient. However, the Monte-Carlo approach has three principal drawbacks. First, the numerical noise is always present in the solution because of a finite number of the test particles traced, and this noise reduces but slowly with an increase of this number. Therefore, Eqs. (8.1) become a set of differential equations with noisy sources, which creates certain problems for the numerics. In particular, the highorder discretization of Eqs. (8.1) [5] becomes inefficient when a Monte-Carlo package is used to compute the sources. Given the non-linear nature of Eq. (8.1), even a purely random noise on the source terms there can produce bias on the solution [54]. There are different sampling methods proposed [55], which reduce the noise by using some information from previous iterations, either correlating the pseudo-random number sequences between the iterations or by averaging the results over a certain number of iterations. These methods are still under development and are not routinely used in the 2D modeling applications, although some studies show their efficiency, see e.g. [56]. Secondly, the calculations become slow, especially when the geometry or reaction detail need to be resolved [57]. This problem can be alleviated by the use of

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Computational Modeling of the Edge Plasma Transport Phenomena

parallel computing, but this still requires considerable computer resources. Apart from this, parallelization is efficient when one wants to suppress the noise by a significant increase of the number of the test particle histories followed. However, in the coupled fluid-kinetic calculations, the optimal strategy usually relies on keeping the noise at a certain level, so that it does not prevent reducing its effect by reducing the time step in the fluid code iterations. Pretty often it is the run time that determines the level of detail set up in the model: the runs may take months [57]. The third problem is related to the different treatment of the time dependence of the solution. Whereas Eqs.(8.1) describe the evolution of the edge plasma parameters, the standard Monte-Carlo approach assumes a steady state. It can be justified if the evolution of the plasma parameters is much slower than the relaxation of the neutral distribution, but this is not always the case. This inconsistency is the source of difficulties by coupling the neutral and plasma models, which is discussed in the next section. A modification of the Monte-Carlo algorithm that includes time dependence in the particle tracing was proposed [58]. In some sense, it includes the time as one more dimension in modeling [47]. However, the practical realization of this approach [53, 58] leads to a significant increase of the numerical noise if the particle mean-free path in this “time” direction is longer than the time step in the iterations solving Eq. (8.1).

8.2.3

Coupling to the Plasma Model

Coupling the neutral model based on differential equations to the plasma transport model has no principal problem. One simply adds to the system (8.1) some more equations of a similar structure, which requires no special measures to ensure compatibility between the two parts. However, when the Monte-Carlo approach is in effect, a problem arises. Whereas an implicit numerical scheme is normally used for advancing the discrete analog of Eqs. (8.1) in time, the Monte-Carlo description is explicit. This means that the neutral-related sources in Eqs. (8.1) are consistent with the plasma parameters before each time step, when the Monte-Carlo algorithm is applied, but the plasma parameters consistent with these sources are only available after the time step. Such discrepancy results in the appearance of parasitic sources in Eqs. (8.1) [57]. These sources, at a percent level, are not very important in the energy equations since the global energy balance in the edge plasma is sustained by equilibration of strong terms – the power input, volumetric losses and power delivered to the targets. They are probably not so important in the momentum equations also. However, in particle balance, these tiny sources may become comparable with the primary players, the fueling and pumping fluxes. Indeed, in the high recycling or detached divertor regime, see Chap. 9, the particle sources related to recombination of plasma ions and ionization of recycling neutrals can be much stronger than the fueling and pumping fluxes. Therefore, a percent level error in the recycling sources can act as an order of unity error in balance of pumping and fueling, which determines the density level or the particle content of the edge

8.3 Selection of Constraints

211

Fig. 8.2 Time traces of different terms (fluxes in 1020/s) in integral particle balance for helium, without (a) and with (b) the correction (note different vertical scale on the two plots). (Reproduced with permission from [57], © Elsevier 2011)

plasma, which, in turn, controls the divertor performance. A straightforward way from this problem would be reducing the time step in the iterations with Eqs. (8.1); however, this is time-consuming and often impractical. A correction scheme proposed in [57], which yields the correction factors close to 1 for the ionization sources, which are found by solving the non-linear algebraic equations for global particle balance in the internal iterations foreseen anyway on each time step to cope with the non-linearity of the coefficients that appear in Eqs. (8.1), allows one to resolve the pumping and fueling fluxes in high recycling conditions within reasonable computational time, Fig. 8.2. Note that neglect of these iterations for speeding up the computations may lead to a solution “converging” numerically well to very different profiles of the plasma parameters, which do not satisfy particle balance [59].

8.3

Selection of Constraints

The selection of the constraints for solving the transport equations is of primary importance when performing the modeling. Whereas the equations describe the interactions within the transport model, the constraints specify its interaction with the world external to the model, such as the plasma-wall interaction or external sources of energy, particles and momentum. In order to facilitate interpretation of the modeling results, the boundary conditions must be physically meaningful. Computationally, the most efficient constraints would be the first-type boundary conditions that specify the values of the quantities described by the transport equation – such as the temperature or density. However, the results obtained this way can be difficult to interpret. The fluxes on the boundary surfaces, which characterize interaction with the material of the plasma-facing elements or with the core plasma, can have very peculiar values that correspond e.g. to heating the plasma by the contact with the wall or to the particle outflow from the core well beyond the realistic values of the

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core fueling. This would require adjusting the boundary values and re-running the code until it converges to something reasonable.

8.3.1

Boundary Conditions at the Targets

At the targets, where the plasma flows onto the surface and is neutralized, the electrostatic sheath is formed (see Chap. 4), which corresponds to the electrical current closing through the surface (zero in the simplest case of the ambipolar flow). The boundary conditions discussed in Chap. 4 are applicable here. Usually, one specifies the flow Mach number of 1 or greater than 1 and the sheath transmission factors γi,e that relate the power and particle fluxes to the surface with the plasma temperature at the sheath entrance qi,e ¼ γi,e ji,e Ti,e ,

ð8:2Þ

where qi,e are the energy fluxes carried by the ions and electrons and ji,e the corresponding particle fluxes. The sheath transmission factors are calculated using simplified distribution functions at the sheath entrance, see Chap. 4. Their typical values for the case of Eq. (8.1) written for the full energy, absence of the secondary electron emission and the plasma flow Mach number of unity at the sheath are γi ~ 3 and γe ~ 5 (note that these values should be taken with caution). Although Eq. (8.2) appears to specify the power flux depending on γi,e, it is used to find the temperature since the fluxes are determined by the sources.

8.3.2

Boundary Conditions at the Core Boundary

This boundary describes the interaction with the core plasma. The primary role of the core in the edge modeling is supplying the power since there is no other source of energy in the edge. The most natural choice for the boundary conditions there is specifying the power flux across this boundary in the ion and electron channels. The total power flux is determined by the heating of the core plasma from the external sources (the plasma current, neutral beams, electron cyclotron waves, etc.) or from the fusion reactions, and the radiation power losses from the core. In principle, its distribution over the core boundary is non-uniform – one can expect higher fluxes on the outboard side due to ballooning effects. In practice, this flux is usually specified uniform and the ballooning effects are simulated to a certain extent by non-uniform radial transport between this boundary and the separatrix, either because of the Shafranov’s shift of the magnetic flux surfaces outwards due to a finite plasma pressure, or by prescribing the spatially non-uniform diffusivities.

8.3 Selection of Constraints

213

The core plasma can also be a source of the plasma particles that originate either from the external sources (beams, pellets) or from the neutrals penetrating the core boundary from the edge, or from the time evolution of the core plasma density. (In the latter case, the core boundary source of particles can become negative if the core density grows, reflecting core fueling by the plasma influx from the edge). Here also, specification of the particle fluxes across the core boundary is the best choice from the physics viewpoint, since these fluxes are usually well determined by the particle sources in the core. For the parallel momentum transport equation, either the zero parallel velocity (no core rotation) or zero flux of the parallel momentum across the core boundary (core rotation not related to the edge) is usually applied if no drifts are taken into account. This should create no problem since in most cases, the parallel velocity of all plasma components near the core boundary is low – the flows are well subsonic. However, plasma rotation can affect the electric fields forming there, so one may need to think some more if working with the drifts and currents [15].

8.3.3

Boundary Conditions at the “Radial” Edges of the Grid

The boundary conditions at the grid edges facing the first wall or the PFR (lines BCD and AFE in Fig. 8.1) have not received much attention when the grid does not reach the wall. There is not so much power reaching there, so the conditions at these boundaries should have no strong impact on the first target of the modeling: the power loading of the divertor targets. Usually, one sets a third type boundary condition here, which relates the radial energy or particle flux at the boundary to the particle or energy density there via prescribing the convective flux with specified velocity. Sometimes this is done through specifying the “decay length” of the temperature and density profiles [57], sometimes by specifying the effective convective velocities directly [15]. For the parallel momentum, either the slippage (zero radial flux) or sticking (zero flow velocity) condition is usually applied at this boundary.

8.3.4

Fueling Constraints

The way of specifying the level of plasma density in the computational model requires special attention. The most natural parameter to use for this purpose would be the total, ion plus neutral, particle content outside the separatrix [60, 61]. This quantity, Ntot, can be described by the particle balance equation that is only weakly related to the edge plasma solution and so can be considered a truly external, controllable parameter. Typically, Ntot changes slowly and smoothly by variation of the fueling rate and it is the parameter directly affected by gas puffing and pumping in the experiment. However, in the experiment, Ntot is practically not

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Computational Modeling of the Edge Plasma Transport Phenomena

Fig. 8.3 Schematics of “bifurcation” seen when the edge plasma density is characterized by the separatrix density nsep. A small change in nsep from level A to B corresponds to a significant increase of Ntot, which leads to a considerable change of the detachment state (represented here by the divertor plasma temperature Td)

measurable. Therefore, one needs to find a constraint that would represent Ntot and be physically meaningful and measurable, in order to allow comparisons with the experiment. In practice, the electron density at some point at the separatrix nsep is often chosen as such a constraint [19, 62, 63]. Since nsep cannot be used directly as the boundary condition (the separatrix lies inside the computational area), the constraint nsep ¼ nsep where nsep is the required value of nsep, is met by adjusting some other parameters, such as the particle flux from the core, the gas puffing rate or the cross-field transport coefficients. If the latter are fixed, the density control in the code run looks similar to that in the experiment. However, having nsep as the target in the control system can result in artificial “bifurcations” in the code or sudden jump of the detachment front to the x-point by increasing nsep. Indeed, nsep can be non-monotonic by the density ramp towards detachment [10]. It increases first with Ntot, then saturates and rolls over, and finally increases again, Fig. 8.3. If the density is raised by controlling Ntot, the detachment evolves smoothly – a small variation of Ntot causes a small variation of the other parameters. However, if one controls the fueling rate trying to realize a smooth raise of nsep, a “bifurcation” appears once nsep reaches the rollover. At this point, a further increase of nsep is only possible by a significant increase of Ntot and in terms of nsep, a small variation of its value causes a significant change of the solution. This happens in modeling [62] and probably in the experiment when the upstream density is feedback-controlled. Therefore, using Ntot as the density control parameter is preferable for the clearer physical interpretation of the modeling results [61]. However, this quantity is not very useful for analysis of the effects of coupling the edge to the core models. Using the neutral pressure pn in front of the pumping duct entrance in the divertor may help in this case [64]. On the one hand, its relation to Ntot is usually monotonic. On the other one, it is related directly to the pumping throughput, which is one of the major parameters controlling the performance of the whole machine.

8.4 Physics Results and Model Validation

8.4

215

Physics Results and Model Validation

Historically, modeling and experiment with a poloidal divertor have gone in parallel all the time, with modeling closely following the experiment and helping to interpret the experimental data. A comparison of the modeling results with experimental data serves as code validation and different groups pursue this activity permanently, following the improvements in the experimental diagnostics and development of the model. However, given the obvious lack of comprehensive physical description of the edge plasma, see Chaps. 6 and 7, this comparison can be rather tricky. In this chapter, we give examples of the application of modeling tools to several problems of the plasma edge physics, aiming at the qualitative understanding of relative importance of different processes involved and at confronting the experiments to validate the models. This is not a comprehensive review and we apologize for having not touched upon many other applications – in particular, the kinetics or turbulence codes.

8.4.1

2D Transport Modeling

Looking broadly, one can see a good qualitative agreement between the general trends in the edge plasma in experiment and modeling, even without fine-tuning of the models. Such features are, for example, evolution of the power loading and particle flux onto the divertor target along with an increase of the edge density (fueling) or impurity level (seeding) [65], appearance of plasma detachment at sufficient density or radiation level [66–68], or reduction of the upstream plasma density at the separatrix by impurity seeding [19, 69]. In particular, the effect of the ! ! E  B and ∇B drifts on the detachment asymmetry in H-mode was confirmed by comparing the UEDGE code results with DIII-D data from Thomson scattering diagnostics by different directions of the toroidal magnetic field [62]. However, a successful quantitative comparison of a modeling run with a single experimental shot, which is usually termed “model validation”, is not easy. As was noted in [70], the codes can reproduce the experimental data satisfactorily for Ohmic and L-mode discharges with no significant plasma detachment from the targets. Since then, a considerable effort has been made to model experiments with H-mode plasma and detachment (see e.g. [19, 21, 62, 71–73]). The most severe challenges in modeling the experiment are the in-out asymmetry of detachment and the appearance of the high-density region at the top of the inner divertor target in the far SOL. An example of modeling-to-experiment comparison is shown in Fig. 8.4, where the density and temperature profiles in two DIII-D H-mode shots with different orientation of the toroidal magnetic field, obtained experimentally and modeled with UEDGE, are shown. All four cases feature the same electron density at the separatrix in the outer mid-plane. For the experimental profiles, the data from the Thomson scattering diagnostics covering well the divertor region were

Computational Modeling of the Edge Plasma Transport Phenomena

Experiment Electron density Electron temperature

0.01

Te

(e)

0.1 1.0 (1020m–3)

10

0.1

DIII-D: 174273

(b)

(f)

1.0

10 (eV)

100

Forward BT Detached

(a)

DIII-D: 174148

ne

Te

Electron temperature

(c)

(d)

(g)

(h)

Reversed BT

ne

Modeling Electron density

uedge:rd_b1n28rm05preloadn

8

uedge:rd_bm1n30rm05n

216

0.01 0.1 1.0 (1020m–3)

10

0.1

1.0 10 (eV)

100

Fig. 8.4 Density (a, c, e, g) and temperature (b, d, f, h) profiles in the DIII-D divertor, measured !

from Thomson scattering (a, b, e, f) and calculated with UEDGE for the “forward” (B  ∇B drift directed downwards) (c, d) and “reverse” (g, h) orientation of the toroidal magnetic field. The gray dots on the experimental figures indicate the locations of the measurements. (Reproduced with permission from [74], © Elsevier 2019) !

!

used. In the calculations, the E  B and ∇B drifts were switched on, which yielded a pronounced effect on the in-out asymmetry of density, qualitatively similar to the experimental observations. However, the high density in the far SOL near the inner target by the forward field is not reproduced and high density in the outer divertor appears by the reverse field in modeling but is not seen in the experiment. In this study aimed at qualitative demonstration of the drift effects, there was no attempt of tuning the model parameters to fit the experiment better. In particular, the cross-field diffusivities for particles and energy were taken spatially uniform and the impurity radiation was emulated by assuming a fixed relative concentration of C in the plasma. For getting closer to the experiment, the usual practice is to adjust the radial profiles of the cross-field transport coefficients to fit the upstream profiles of n and T. This procedure allows one to reach a reasonable agreement between the experimental measurements and the plasma profiles at the mid-plane, Fig. 8.5, and in divertors, Fig. 8.6, but it often involves strong enhancement of the diffusivities in the far SOL [21, 75, 76] and sometimes the introduction of outward particle convection [21, 30]. Then, in the absence of the drifts in the model, one has to introduce a strong poloidal variation of the cross-field diffusivities, which now increase significantly in the divertor region [77]. This allows one to reproduce the roll-over of the ion saturation current, Isat, measured by probes in JET, at a lower upstream plasma density, closer to the experimental values, and the in-out asymmetry of the temperature and pressure drop towards the targets by detachment. However, the roll-over of the Isat in the inner and outer divertors occurs at nearly the same value of the

8.4 Physics Results and Model Validation

217

Fig. 8.5 The measured and modeled profiles of Te (top row), ne (middle row), and the specified transport coefficients (D⊥, χi, χe, bottom row, (m2 s1)) at the outer midplane. The temperature measurements are obtained from the Electron Cyclotron Emission (red) and Thomson scattering (green), and the density measurements are from the Integrated Data Analysis (purple, AUG), Li-beam (purple, JET), reflectometry (blue, JET) and Thomson scattering (green). The IDA profile in ASDEX Upgrade is obtained as a combination of the Li-beam and the laser diagnostics. Based on the uncertainties in the radial positioning of the diagnostic data and the separatrix location, as well as the uncertainty range of the measurements, nsep can vary between 0.8–1.6  1019 m3 (#27691) and 1.5–2.8  1019 m3 (#27688) in ASDEX Upgrade, and between 0.7–1.7  1019 m3 in JET. (Reproduced with permission from [76], © IOP Publishing 2017)

upstream density – that is, both divertors start to detach simultaneously, whereas in the experiment, the inner one detaches first. This model does also not reproduce the high density in the far SOL at the inner target (Fig. 8.4a), seen on different machines. Only simultaneous activation of all the drifts and currents, variable cross-field transport and detailed simulation of impurity transport in the multi-fluid code [21] allowed one to see this effect in modeling. However, this, at least, qualitative agreement with the experiment requires specification of rather peculiar profiles of the radial transport coefficients. An example of these profiles used for reproducing the measurements on ASDEX Upgrade and JET [76] is shown in Fig. 8.5. These coefficients vary also in the poloidal direction to reflect the ballooning nature of the radial transport. Profiles of this kind are typical in

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Computational Modeling of the Edge Plasma Transport Phenomena

Fig. 8.6 Comparisons between the modeled and measured target electron temperatures, Te (eV), and ion fluxes, Γk (1024 m2 s1), top and bottom row, respectively. The results are shown for the ASDEX Upgrade discharge #27691 and for the JET discharge #82291 (the so-called low-density reference discharges). For both discharges, the inner divertor comparisons are shown on the lefthand side, and the outer divertor comparisons are on the right-hand side. The Langmuir probe measurements are drawn with black crosses, the simulations with drifts with solid lines (red and blue for the inner and outer divertor, respectively), and the simulations without drifts (but with the currents activated) with dashed green lines. (Reproduced with permission from [76], © IOP Publishing 2017)

the “model validation” studies (see also [15, 21, 75, 78]). Looking at the diffusivities required in the far SOL, one finds that they are often significantly higher than the Bohm diffusivity. For example, the value of ~30 m2/s for χe (Fig. 8.5) corresponds to the Bohm diffusivity at ~900 eV – the value that can hardly be expected in the far SOL. Since the Bohm expression is seen as the upper limit of the diffusivity driven by electrostatic turbulence, see Chap. 7, one can conclude that radial plasma transport in the far SOL is dominated by some processes related to perturbation of the magnetic field or by outward convective transport there [29]. Since this convective flow is formed by filaments (hot plasma formations aligned with the magnetic field and traveling radially through the SOL), it is intermittent [26–28, 75] and its inclusion into the 2D transport codes is not straightforward, see Sect. 8.1.3. It is sometimes emulated as an outward pinch – a convective velocity imposed on the top of diffusion transport in the Eq. (8.1) [21, 30]. In the last decade, a considerable effort has been made to model the radial electric field Er in the plasma edge using 2D transport codes (mostly, the SOLPS family) with drifts and currents. In the code, Er is calculated from the electrostatic potential, whose distribution is found from Eq. (6.52). In particular, the significant difference in the Er values between the L and H modes is reproduced well by the code, thus indicating the neoclassical (drift-related) nature of this field in the H-mode, Fig. 8.7.

8.4 Physics Results and Model Validation

219

Fig. 8.7 Radial electric field in the AUG Ohmic (left) and H-mode (right) shots, calculated with B2SOLPS5.0 code and measured by Doppler reflectometry, compared with the neoclassical values. (Reproduced with permission from [80], © Elsevier 2011)

8.4.2

Modeling the 3D Effects

The 2D models considered above assume the toroidal symmetry of the problem. In a real tokamak experiment, this symmetry is not maintained. The vacuum chamber has discreet ports (NBI, pumping, diagnostics, etc.) and protruding structures, such as limiters or ICRF antennas. The particle source from the gas puff or NBI is toroidally asymmetric. The external magnetic fields applied to the plasma are asymmetric and the plasma itself develops perturbations that are not toroidally symmetric (e.g., ELMs). Therefore, the results of the 2D modeling can only be considered as toroidally average and 3D models are needed to study these 3D effects in more detail. The most published 3D transport codes for the edge plasma modeling are presently EMC3-Eirene [2] and JOREK [79]. EMC3-Eirene is a 3D Monte-Carlo code based on the stationary Braginskii equations for the plasma ions and electrons, coupled to the Monte-Carlo code Eirene that calculates the neutral-related sources in the plasma equations. It is a transport code that runs on a prescribed background magnetic field of nearly arbitrary complexity. JOREK is a finite-element, non-ideal MHD code that solves time-dependent, non-linear MHD and transport equations and calculates the perturbations of the magnetic field self-consistently. Naturally, the increase of the geometrical complexity requires some simplification of the transport models used in these codes. Given the limited amount of the computational resources available, this is unavoidable. The EMC3-Eirene code was originally developed aiming at stellarator applications [52]. The growing interest to its application to tokamak modeling in last decades is related to the perspective of using the resonance magnetic perturbation (RMP) techniques to reduce the size of the ELMs that are seen as one of the key issues for the divertor lifetime in a fusion reactor-tokamak such as ITER [81]. The idea of this method is in introducing toroidally asymmetric, external coils carrying

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Computational Modeling of the Edge Plasma Transport Phenomena

–80

1.0

0.5 –100

Mach Number

Vertical Position [cm]

–90

–110 –120

0.0

–0.5 –130 Flow reversal –140 100 110 120 130 140 Radial Position [cm]

unperturbed configuration

–1.0 150

160

RMP configuration

Flow reversal

Fig. 8.8 Parallel flow pattern in the edge plasma in unperturbed and perturbed magnetic configurations in DIII-D as calculated by EMC3-Eirene. The positive flow is directed from the outer target to the inner one. (Reproduced with permission from [2], © IOP Publishing 2017)

currents in order to disturb the magnetic surfaces at the plasma edge. A number of experiments performed on DIII-D [82], ASDEX Upgrade [83], JET [84], NSTX [85], MAST [86] tokamaks have shown that this scheme allows working in H-mode without Type I ELMs. Effective stochastization of the flux surfaces around the separatrix results in a complex flow pattern that is qualitatively reproduced with EMC3-Eirene, Fig. 8.8. The calculations reproduce also the increase of stochastization by the increase of the current in the RMP coils. However, the simplified transport model together with the prescribed magnetic field in the model (the currents appearing in such a 3D plasma effectively screen the perturbations [87, 88], so the magnetic field depends on the plasma profiles and should be re-calculated consistently) do not allow good quantitative comparison. However, for the problems that do not involve perturbation of the magnetic configuration, the situation is better. For example, a good quantitative comparison of EMC3-Eirene calculations and experimental data can be found in [89]. Here

8.4 Physics Results and Model Validation

221

Fig. 8.9 The measured C2+ ion velocity along a field line is shown in blue. Green points are the C2+ parallel ion velocities as extracted from EMC3-EIRENE simulations. (Reproduced with permission from [89], © IAEA 2018)

spreading of the plasma along the flux tubes by a toroidally and poloidally localized gas puff was modeled. This problem involves no self-adjusting of the magnetic field, so comparing the code results with the experiment is easier. The D2 gas was injected locally at the inner mid-plane and the C2+ ion flow velocities were measured from the Doppler shift of the emission lines around the gas injection location. The flow pattern was found to form a 3D structure aligned with the magnetic flux tubes, indicating an acceleration of C2+ ions by friction with the D+ ions spreading along the flux tubes from the gas injection position. The EMC3-Eirene calculations including the carbon impurity in the test fluid approximation reproduce the measured flow velocities reasonably well, Fig. 8.9. A phenomenon in the edge plasma, which is extremely challenging for modeling studies, is the appearance of ELMs. They appear as intense bursts of the energy and particle losses from the core plasma into the SOL in the H-mode [90]. The large (Type I) ELMs carry the energy sufficient to damage the target surfaces in a highpower experiment such as ITER [91]. The experimental data suggest that these bursts have a short duration and a pronounced 3D spatial structure. Several attempts of modeling ELMs in the framework of 2D transport models [92–94] by increasing strongly the cross-field transport coefficients for a short time and following propagation of the heat pulse produced on the non-disturbed magnetic field have mostly phenomenological value. The macro-blob approach described in Sect. 8.1 offers a more accurate description of the radial propagation of a large perturbation of the plasma parameters in the SOL by ELMs [95]. However, 3D perturbations of the magnetic field by ELMs remain outside the scope, so the ELM structure and the pattern of the power deposition on the targets and walls are not resolved. Physically sound modeling of ELMs requires codes that combine 3D MHD and transport models, such as JOREK [96]. This code describes the dynamics of

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Computational Modeling of the Edge Plasma Transport Phenomena

Fig. 8.10 Comparison of the predicted image from non-linear MHD ELM simulation (a) with the visible camera image of an ELM in MAST (b). (Reproduced with permission from [98], © IOP Publishing 2013)

perturbations of the plasma parameters and magnetic field and is capable of following the non-linear stage of development of non-ideal MHD instabilities. The code reproduces well the structure of the plasma perturbations by the ELM crash, Fig. 8.10. More quantitative comparisons involve experiments on ASDEX Upgrade [97], JET [96], MAST [98] and others. They include analysis of the structure of the unstable modes and their interactions that lead to the ELM crash, of mechanisms of the energy and particle loss from the core plasma by ELMs, and of different ways to the ELM control and mitigation.

8.5

Conclusions

Summarizing this chapter, we can say that although the models can reproduce many of the experimental features of the divertor plasmas, their correct application is not a routine procedure and it requires serious consideration for each particular situation in each particular device. None of the models used is complete, even within the restricted range of the time scales of the processes modeled. Given the remaining uncertainty in the description of the plasma transport, one needs to perform a large number of the code runs in order to reveal inter-dependencies between the plasma parameters and to project them onto the experiment. This brings the computational efficiency of the models to the same level of importance as the physical accuracy, so the trade-off between the code efficiency and the completeness of the physical model is the principal issue that determines the success of a modeling study [57].

References

223

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Chapter 9

Physics of Some Edge Plasma Phenomena

Abstract The physics of two phenomena, comprising virtually all processes discussed in the book, so-called Multifaceted Asymmetric Radiation From the Edge (MARFE) and divertor plasma detachment are considered in this chapter. Divertor plasma detachment phenomenon is considered as one of the most important in edge plasma physics, since it allows the solution of the “heat removal” problem, the fundamental technical issue of fusion reactors.

In this chapter, we will discuss the physics of some macroscopic phenomena, which are distinctive for edge plasma. In particular, we will consider (i) MARFE (Multifaceted Asymmetric Radiation From the Edge [1]) and poloidaly symmetric plasma detachment, (ii) self-sustained divertor plasma oscillations and (iii) divertor plasma detachment.

9.1

MARFE and Poloidaly Symmetric Plasma Detachment

Due to high plasma temperature in the core region of magnetic confinement devices, which ensures a high heat conductivity along the magnetic field lines, κk / T5/2, plasma temperature can be considered constant on closed magnetic flux surfaces. However, in the edge region, where temperature and, therefore, plasma heat conduction are relatively low, strongly localized radiation losses can result in an inhomogeneous distribution of both plasma density and temperature over closed magnetic flux surfaces, even for the case where they have no direct contact with the materials of plasma-facing components. Then, obviously, plasma temperature will experience some depression in the region of high radiation losses. It is interesting, however, that the localization of enhanced radiation loss can be related to temperature depression itself. Indeed, recalling Fig. 2.14 we can see that the cooling rate Limp(Te) for low-Z (e.g. carbon) impurity has non-monotonic dependence on electron temperature, and in the temperature range ~ 8–20 eV, Limp(Te) for carbon is increasing with decreasing electron temperature. This feature of Limp(Te) provides positive feedback for the thermal plasma instability, which in the simplest case, can © Springer Nature Switzerland AG 2020 S. Krasheninnikov et al., On the Edge of Magnetic Fusion Devices, Springer Series in Plasma Science and Technology, https://doi.org/10.1007/978-3-030-49594-7_9

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be obtained from the power balance equation where we neglect plasma dynamics and parallel heat conduction: 3n

dT ¼ H  nnimp Limp ðTÞ: dt

ð9:1Þ

Here, for simplicity, we assume equal electron and ion temperatures, n and nimp are plasma and impurity densities, and H is the plasma heating term, which is assumed to be constant. Then, assuming that temperature T ¼ T0 corresponds to e where T e / exp ðγtÞ the steady-state solution of Eq. (9.1) and taking T ¼ T0 þ T, describes a small departure from the equilibrium temperature, from Eq. (9.1) we find the following expression for γ:  nimp dLimp ðTÞ γ¼ : 3 dT T¼T0

ð9:2Þ

Expression (9.2) predicts that the steady-state condition corresponding to dLimp(T)/dT < 0 is unstable and can result in localized temperature drop accompanied by an increase of impurity radiation loss. Such arguments were put forward in [1] to explain experimental observations of toroidally symmetric, rather compact in both poloidal and radial directions, and highly radiative region emerging at the inner side of the torus at high averaged plasma density in Alcator-C tokamak. In Fig. 9.1 one can see that the formation of Fig. 9.1 Time traces from different diagnostics in Alcator-C tokamak. MARFE forms at 120 ms and is accompanied by enhanced: radiation loss at the inner side of the torus, Hα and CIII line radiations. (Reproduced with permission from [1], © IAEA 1984)

9.1 MARFE and Poloidaly Symmetric Plasma Detachment BOLOMETER

10

W.cm–2.sr–1

Fig. 9.2 Brightness profile as seen by the vertical bolometer view before and during MARFE. (Reproduced with permission from [1], © IAEA 1984)

5

0

231

MARFE

–12.0 –8.0 –4.0

0

4.0

8.0

12.0 16.0

HORIZONTAL POSITION (cm)

MARFE at 120 ms is accompanied by a strong increase of the radiation loss, as well as the Hα and CIII line radiation. In addition, in Fig. 9.2, one sees that the bolometer signal is locally increased in the MARFE region. Predominant formation of MARFE at the inner side of the torus was explained in [1] by ballooning nature of cross-field heat transport from the core, when the heat is largely transported at the outer side of the torus (see Chap. 6 for details) and arrives in the inner side due to parallel heat conduction. However, in [2, 3] it was pointed out that the simple model of plasma thermal instability described by Eq. (9.1) that ignores plasma dynamics, misses an important feature related to the increase of plasma density in the region with reduced temperature. This effect is caused by plasma flow along the magnetic field lines, which is driven by the gradient of plasma pressure. Assuming that plasma flow entrains impurity and stillneglecting electron heat conduction, following [2, 3] we   parallel find γ / 2  dℓn Limp ðTÞ =dTT¼T0 , which describes so-called radiative-condensation instability. Due to plasma “condensation” in a low-temperature region, radiativecondensation instability can develop even for dLimp(T)/dT > 0. This instability plays an important role in many astrophysical and laboratory plasma phenomena (e.g. see [4] and the references therein). Although MARFE is observed at the inner side of the torus, poloidal localization of MARFE can oscillate in time (with the frequency ~ 100 Hz) around the midplane, whereas the localization of stationary MARFE (above or below the midplane) depends on the direction of the toroidal magnetic field [5]. The latter effect is attributed to the impact of drifts [6]. MARFE was observed on many tokamaks (see [5, 7–12] and the references therein) as well on the stellarators LHD [13] and Wendelstein 7-X [14] at plasma density close to density limit [15]. Further studies of the MARFE phenomenon have shown that the physical picture of MARFE formation, which we outlined above and which is based on radiativecondensation instability, associated with impurity radiation is, at least, incomplete. Experimental data from TEXTOR and C-Mod tokamaks demonstrate that Hydrogen radiation loss and plasma recycling within the MARFE region can also play very

232

Physics of Some Edge Plasma Phenomena

(a)

380

98021035; 0.75 sec

5–>2

6–>2

0.1

7–>2

1.0

8–>2

10.0

10–>2 9–>2

400 420 Wavelength (nm)

440

90 95 Wavelength (nm)

4–>1

1021 85

5–>1

1022

980120035; 0.93 sec

(b) 1023

7–>1 6–>1

(W/m2/ster/nm)

100.0

(ph/sec/m3/ster/nm)

Fig. 9.3 Hydrogen radiation spectrum from MARFE in the visible (a) and VUV (b) light. (Reproduced with permission from [17], © American Physical Society 1998)

9

100

important roles [16, 17]. Moreover, C-Mod data show that plasma temperature in the MARFE region falls below 1 eV whereas plasma density reaches ~21015cm3, which stimulates very strong plasma recombination sink, which is close to plasma ionization source in the rest of the tokamak main chamber volume. As a result, the density of neutral Hydrogen in MARFE becomes so high that it traps about 95% of Lyα and more than 50% of Lyβ radiation. The spectrum of neutral Hydrogen radiation from MARFE in C-Mod tokamak, shown in Fig. 9.3, exhibits typical features of recombining plasma. Experimental data from Wendelstein 7-X stellarator also show the presence of plasma recombination in MARFE [14]. Theoretical analysis of an impact of plasma recombination on MARFE [18] shows that even though impurity radiation can be the initial trigger of MARFE formation, plasma recombination can facilitate MARFE development and, actually, determine deeply non-linear evolution of MARFE. Finally, we note that the formation of MARFE is often accompanied by strong fluctuations of MARFE parameters and radiation from the MARFE region (e.g. see [5, 12, 17]). The latter can be associated with the poloidal motion of MARFE or relaxation oscillations, as it was found in numerical solutions of simplified plasma transport and impurity radiation equations describing the nonlinear stage of thermal instabilities and having characteristic frequency ~100 Hz [2, 19, 20]. We note that some of the fluctuations, caused by the thermal force acting on the impurity [19, 20], have the form of self-sustained oscillations. As we already mentioned, MARFE occurs at plasma densities close to the density limit. An increase of plasma density above the MARFE threshold often results in the transition of MARFE to detached plasma characterized by poloidally and toroidally symmetric highly radiative mantle (see Fig. 9.4). In this regime, virtually all plasma heating power is dissipated by the radiation loss from the mantle (see Fig. 9.5) [21]. An excess of radiative power over ohmic heating in Fig. 9.5 is due to the calibration uncertainty.

9.1 MARFE and Poloidaly Symmetric Plasma Detachment

233

Fig. 9.4 Transition from MARFE to detached plasma in the TFTR tokamak. (Reproduced with permission from [8], © Elsevier 1987)

1.5 TFTR Ohmic Detached Plasmas PRAD (MW)

Fig. 9.5 Total radiative power compared to the ohmic heating power in the TFTR tokamak detached plasma. (Reproduced with permission from [21], © Elsevier 1987)

1.0

0.5 I = 600 kA I = 800 kA 0 0

0.5

1.0

OHMIC INPUT POWER (MW)

1.5

234

9

Physics of Some Edge Plasma Phenomena

Fig. 9.6 From the top to bottom: sawtooth oscillations and time variation of loop voltage and the Dα signal from the detached plasma of the FT tokamak. (Reproduced with permission from [22], © Elsevier 1982)

Detached plasma was observed on many tokamaks in ohmic plasmas and with relatively low auxiliary heating (e.g. see [8, 22–24] and the review papers [10, 25]). Similar to the MARFE case, the formation of the radiative mantle can be accompanied by strong fluctuations of plasma parameters. For example, ~100% Dα signal fluctuations at the frequencies ~100 Hz were observed in the FT tokamak [22] (see Fig. 9.6), which shows a strong variation of the plasma recycling process. However, the physics of these fluctuations of plasma recycling is not known. In addition, preliminary spectral measurements in detached plasma in C-Mod indicate that plasma recombination can be the major plasma sink.

9.2

Self-Sustained Divertor Plasma Oscillations

As we have described in the previous section, such macroscopic phenomena as MARFE and detached plasma in limiter discharges are accompanied by the fluctuations of plasma parameters which look as self-sustained oscillations (e.g. see Fig. 9.6). Actually, taking into account complexity and non-linearity of edge plasma processes, in particular, those where strong energy radiation losses play an important role, it is not surprising that edge plasma can exhibit regimes with self-sustained oscillations. Among other tokamak processes resulting in self-sustained oscillations, we can mention sawtooth oscillations and ELMs. However, these oscillations are closely related to MHD phenomena. Somewhat similar to oscillations in MARFE and detached plasma self-sustainedlike oscillations of plasma parameters were also found in the discharges with poloidal divertors (e.g. see [26, 27]). Typical frequencies of these oscillations are about 100 Hz, which is close to the oscillation frequencies observed in MARFE and detached plasmas. Oscillations with similar frequencies were observed in detached

9.2 Self-Sustained Divertor Plasma Oscillations

(1019 m–3)

8

Pulse No: 34154 – n e

7

(MW)

6 div 4 Prad

State Bexp

3 State Aexp

2 0.04

div Pneut

Pneut

0.02 0 10

PFR Pneut

Dαdiv (in)

Dα

5 0 17.2

17.4

17.8 17.6 Time (s)

Dαdiv (out) 18.0

JG99,106/1c/m

(1021 ph m–2 s–1)

(mbar)

Fig. 9.7 Self-sustained oscillations of (from top to bottom) averaged density, divertor radiation, neutral density in inner divertor and private region, and inner/ outer Dα emission observed in the JET tokamak. (Reproduced with permission from [27], © American Physical Society 1999)

235

18.2

regimes of the LHD stellarator [28]. As an example in Fig. 9.7 one can see such selfsustained oscillations found in L-mode discharges on JET tokamak [27], which were explained based on the theory developed in [29, 30]. The main idea of [29, 30], which we also will use for the analysis of tokamak divertor plasma detachment, is based on the properties of the high recycling regime of divertor operation. In these regimes, the neutral ionization mean-free path near the targets becomes very small, smaller than the corresponding width of the SOL. Therefore, the SOL can be considered as an ensemble of weakly interacting magnetic flux tubes filled with plasma and neutrals. The redistribution of plasma ions and neutrals along the magnetic field in the flux tube occurs rather quickly (characteristic time scales of ion and neutral redistribution are determined by the sound speed and inverse ionization frequency respectively, which we will consider being much smaller than the particle exchange time between different flux tubes). As a result, in a steady-state conditions, total pressure (including contributions from both plasma and neutrals) in a flux tube, Ptot ft , can be considered constant (except rather narrow region near the target where neutral friction with the target cannot be ignored) whereas the distributions of plasma and neutral gas parameters along the flux tube are determined by plasma and neutral transport processes and such input parameters as the heat flux propagating through the flux tube to the target, qft, and averaged density of neutrals and ions, Ntot ft , in the flux tube. For simplicity, we ignore, for now, the effects of impurities. We will be looking for the solution for the distributions of plasma and neutral gas parameters along the flux tube as well as for the plasma flux to the target. To elucidate the physics of the variation of plasma and neutral gas parameters, it

236

9

Physics of Some Edge Plasma Phenomena

would be useful to find the dimensionless parameters governing the processes in the flux tube. However, first, we should determine the relevant dimensional parameters. Obviously, they include qft, Ntot ft and the length of the flux tube Lft. In addition, we need to allow for dimensional parameters describing collisional interactions of both the plasma and neutral particles. According to Chap. 2, they are the electron charge, e, the Planck constant, ħ, the electron (or ion) mass m (M), and the speed of light, c. However, it is more convenient to use dimensional parameters having clear physical meaning. Therefore, without any loss of generality, we substitute e, ħ, and c with the Bohr radius, RB, the hydrogen ionization potential, I, and the lifetime of the first excited state of a hydrogen atom, τlt, which is described by the Einstein coefficient from Chap. 2. Thus, we have seven dimensional parameters (qft, Ntot ft , Lft, M (or m), RB, I, and τlt), which completely determine all processes in the flux tube. Strictly speaking, the interactions of both the charged and neutral particles with material surface should provide some additional dimensional parameters (e.g. wall temperature, surface conditions, models describing hydrogen trapping, etc.). However, for simplicity, we assume that the interactions of both the charged and neutral particles with the material surface are described by some dimensionless energy and particle reflection coefficients which depend solely on the parameters of the species impinging onto the surface. As a result, we still have only seven dimensional parameters defining the transport properties of plasma and neutrals within the flux tube and their interactions with the material surface. From these seven dimensional parameters, we can form four dimensionless ones. We choose the dimensionless parameters that have simple physical interpretation: Πq ¼

pffiffiffiffiffiffiffiffi qft 3 tot tot 2 pffiffiffiffiffiffiffiffiffi , Π2 ¼ R2B Ntot I=m: ft Lft , Π3 ¼ RB Nft , Πstep ¼ τlt Nft RB I=M

Ntot ft I

ð9:3Þ

These parameters can be interpreted as follows: the parameter Πq can be considered as the ratio of the available power qft to the power dissipated due to hydrogen recycling; the parameter Π2 can be interpreted as the efficiency of neutral gas trapping (due to neutral ionization) within the domain of interest; Π3 can be viewed as a factor determining the strength of multi-body processes (e.g. three-body recombination or charge screening), and Πstep can be interpreted as a factor controlling the effect of multi-step atomic physic processes (recall Chap. 2). As a result, the distributions of the plasma-neutral gas parameters (being expressed in the corresponding dimensionless form) along the flux tube are functions of the parameters (9.3). For example, the electron temperature distribution Te(x) (where x is the coordinate along the flux tube) can be written as Te(x/Lft)/I ¼ FTe(x/Lft, Πq, Π2, Π3, Πstep), where FTe(x/Lft, Πq, Π2, Π3, Πstep) is some function. We notice that for a given Lft, four dimensionless parameters can be collapsed into only two dimensionless parameters (e.g. Πq and Π3). Therefore, the plasma flux to the target, jd, the plasma temperature at the target, Td, and the pressure Ptot ft can be written as

9.2 Self-Sustained Divertor Plasma Oscillations

237

  Td   Ptot   jd ft p ffiffiffiffiffiffiffiffi ffi Π , Π Π , Π ¼ F ¼ F , , j q 3 T q 3 tot ¼ FP Πq , Π3 , tot I N I Nft I=M ft

ð9:4Þ

where Fj(Πq, Π3), FT(Πq, Π3), and FP(Πq, Π3) are some functions that cannot be determined from dimensionless analysis. However, by adopting some simplifications we can estimate them. To do this, we start with deriving an expression that links the plasma temperature at the divertor tot target Td with Ptot ft and Nft (the latter we consider as the control parameter). From the energy balance equation in the recycling region, allowing for both the flux of the plasma thermal energy to the target and energy dissipation due to hydrogen recycling, we have qft ¼ nd

pffiffiffiffiffiffiffiffiffiffiffiffi Td =MðγTd þ EH ion Þ,

ð9:5Þ

where EH ion is the hydrogen “ionization cost” (recall Chap. 2), which is determined solely by atomic physics represented by the dimensionless parameters Π2, Π3, and Πstep, as well as by the local (in the absence of the radiation trapping) dimensionless electron temperature and density. We assume that the ion temperature is equal to the electron one, γ ~ 58 is the heat transmission coefficient (recall the results from Chap. 4). The first term in the brackets on the right-hand side describes the plasma thermal energy flux to the target, whereas the second one comes from energy dissipation due to hydrogen recycling, taking into account that in the high recycling regime, the plasma flux to the target is virtually equal to the neutral flux from the target. For relatively high Td, the neutral density at the target is lower than the ion density. This follows from the equality of the neutral and ion fluxes from and to the targets and the fact that the plasma flows along the magnetic field lines intercepting the target at a shallow angle. Therefore, the total momentum flux (pressure) is attributed to the plasma contribution, so we have Ptot ft ¼ 2nd Td (e.g. see [31]). And from Eq. (9.5) we find [29, 30]: Ptot ft

2qft ¼ γTd þ EH ion

rffiffiffiffiffi Td : M

ð9:6Þ

One can see from Eq. (9.6) that the function Ptot ft ðTd Þ is non-monotonic. At large Td, Ptot ft ðTd Þ increases with decreasing Td, then reaches a maximum at Td ¼ T  EH ion =γ: 



Ptot ft max

sffiffiffiffiffiffiffiffiffiffi M , ¼ qft γEH ion

and then Ptot ft ðTd Þ decreases with increasing Td.

ð9:7Þ

238

9

Physics of Some Edge Plasma Phenomena

Note that the ratio of the left- to the right-hand sides of Eq. (9.5) for Td ¼ T and tot Ptot ft ¼ Pft max gives us virtually parameter Π1, which, as we discussed before, sets the limit, imposed by the available power, on the rate of plasma recycling. To find the relation between Ntot ft and Td, we take into account the plasma outside the narrow, Lft, recycling region, where there is no ionization source and the plasma flow is stagnant. Therefore, outside the recycling region, the only available mechanism to provide the energy flux qft is heat conduction. So we have qft ¼ κe(T)dT/dℓ, where κe(T) / T5/2 is the electron heat conductivity and ℓ is the coordinate along the magnetic field line (we take ℓ ¼ 0 at the target). Then we find  2=7 7=2 the following expression for the electron temperature: TðℓÞ ¼ Td þ qft ℓ=b κ , where b κ ¼ ð2=7Þκe ðTÞT5=2 ¼ const: As a result, assuming T(Lft)  Td, using the expression for the plasma density nðℓÞ ¼ Ptot ft =TðℓÞ and taking into account Eq. (9.6), we find the following expression for the upstream plasma density nup  n(Lft):  nup ðTd Þ ¼

Ptot ft ðTd Þ

b κ qft Lft

2=7 :

ð9:8Þ

Note that the expression virtually identical to Eq. (9.7), (9.8) was used in [32] as the justification for the SOL plasma density limit. Similarly to the derivation of Eq. (9.8) we find the dependence Ntot ft ðTd Þ: Ntot ft ðTd Þ

¼

L1 ft

ZLft

 2=7 b 7 tot κ nðℓÞdℓ Pft ðTd Þ , 5 qft Lft

ð9:9Þ

0

and finally, we have Ntot ft ðTd Þ ¼

2qft γTd þ EH ion

rffiffiffiffiffi  2=7 b Td 7 κ : M 5 qft Lft

ð9:10Þ

Examining the expression (9.10) one finds that in accordance with our dimensionless analysis it can be re-written in terms of the dimensionless parameters (9.3) and the functions (9.5). As we see from Eqs. (9.6), (9.8), and (9.9), the functions nup(Td), Ntot ft ðTd Þ, and tot Pft ðTd Þ have a similar dependence on Td, which seems to suggest that like Ptot ft , both  tot  tot nup(Td) and Nft ðTd Þ have some maximum values, nup ðTd Þ / Nft max / ðqft Þ5=7 . However, more detailed analysis and numerical simulations [29] show that the model we consider here is too crude to describe properly the recycling region for  small Td. In practice, the dependence Td Ntot for some cases can be described by an ft N-shaped curve shown schematically in Fig. 9.8. Moreover, as it often happens,  tot  some part of the N-shaped curve T N is unstable (see Fig. 9.8), and there is a d ft    tot  tot dependence around N bifurcation of Td Ntot N [29, 30, 33]. With ft ft ft max

9.2 Self-Sustained Divertor Plasma Oscillations

239

   tot    Fig. 9.8 (a) Schematic dependences of Td Ntot Nft . The unstable part of Td Ntot and Pneut d ft  tot  ft neut dependence is shown in red. (b) Pd Nft dependence found from one-dimensional numerical modeling employing fluid plasma and Monte-Carlo neutral descriptions. The horizontal blue line corresponds to the ambient neutral pressure Pneut d,amb

increasing Ntot ft , there is an accumulation of neutrals  tot  in the vicinity of the target. tot N Therefore the neutral pressure at the target, Pneut d ft , increases with increasing Nft  tot but it also exhibits the bifurcation at Ntot ft Nft max due to the transition from one  tot  stable branch of Td Nft to the other one, see Fig. 9.8. However, such a bifurcated solution exists only in the simplified one-dimensional model of the plasma within an isolated flux tube. When we allow for the effects of particle exchange between different flux tubes, we may see self-sustained oscillations of plasma parameters. The physics of these oscillations can be illustrated with a very simple example. Assume that some flux tube is surrounded by plasma having the neutral pressure in , such that it corresponds to the gap between the vicinity of the divertor target, Pneut  tot  d,amb the stable branches of Pneut N in the flux tube (see the blue line in Fig. 9.8b). In d ft this case, due to the neutral flow between the flux tube under consideration and the surrounding plasma (which can exceed the similar plasma flow caused by anomalous cross-field transport [30]), no steady-state equilibrium between the plasma within the flux tube and the ambiance becomes possible. As a result, self-sustained oscillations, corresponding to the limiting cycle indicated in Fig. 9.8 by arrows, develop. In our simplified description of plasma parameters within the isolated magnetic flux tube, we ignored, for simplicity, the impact of impurity. However, the impurity radiation loss in the SOL plasma is ubiquitous. To incorporate the impurity radiation loss in the framework of plasma behavior within an isolated magnetic flux tube, we notice that the main energy losses caused by most of the impurities occur at plasma temperature higher than the temperature in the hydrogen recycling region (exceptions can be light elements having low ionization potential such as lithium). Impurity radiation loss per se does not alter the plasma pressure, but just decreases the power available to sustain hydrogen recycling (we assume here that the fraction of impurity in overall plasma particle balance is small). As a result, the

240

9 1.3e+07

Physics of Some Edge Plasma Phenomena

peak power (total)

1.2e+07 1.1e+07 1e+07 0e+06 8e+06 7e+06 6e+06 0.18 0.185 0.19 0.195

0.2

0.205 0.21 0.215 0.22 0.225 Time

0.2

Fig. 9.9 Impurity (neon) driven self-sustained oscillations of the heat load on the outer divertor target found in 2D modeling of ITER. (Reproduced with permission from [33], © IOP Publishing 2019)

power flux reaching the hydrogen recycling region is qrecycl ¼ qft  qimp, where qft is the power flux propagating in the flux tube farther upstream and qimp accounts for the reduction of this power due to impurity radiation. Then, we find that the Eqs. (9.5), (9.6) and (9.7) still hold with the substitution of qrecycl instead of qft. If the impurity radiation region is localized at relatively small,  Lft, distance from the divertor target, then the expressions (9.8) and (9.9) also hold. As a result, we find the following expression for Ntot ft ðTd Þ:   rffiffiffiffiffi  2=7 2 qft  qimp b Td 7 κ tot Nft ðTd Þ ¼ : M 5 qft Lft γTd þ EH ion

ð9:11Þ

  Thus, we conclude that Td Ntot ft can have an N-shape indicating bifurcation with impurity radiation loss. However, now the value of Ntot corresponding to the ft bifurcation depends on qimp and not only the neutral hydrogen pressure but also the neutral impurity pressure can bifurcate. Then, the impurity exchange between the flux tubeand ambiance will result in a along the Ntot change of qimp and a shift of the N-shaped curve Td Ntot ft ft axis. A mismatch of the ambient neutral impurity pressure and the neutral impurity pressure corresponding to the stable branches will also result in self-sustained oscillations of plasma parameters. An example of such impurity-driven oscillations is shown in Fig. 9.9. Note that careful numerical analysis performed in [34] confirmed that selfsustained oscillations driven by impurity radiation, which are observed in numerical simulations of ITER divertor plasmas, are not related to computational issues.

9.3 Divertor Plasma Detachment

9.3

241

Divertor Plasma Detachment

In the first subsection, we already discussed detached plasma regimes. However, that was toroidally and poloidaly symmetric detachment of plasma situated on “closed” magnetic flux surfaces, where plasma interaction with material surfaces (e.g. limiters, main chamber wall) is driven largely by cross-field transport. Here we consider detachment of plasma situated in the divertor volume on “open” magnetic field lines intersecting the divertor targets. In this case, plasma interaction with the divertor target material is mainly driven by plasma transport along the magnetic field lines. There are different ways to define the “depth” of divertor plasma detachment. Here we will call the divertor plasma detached when there is a rollover of the ion flux to the divertor target, similar to that shown in Fig. 1.5a. Note that experimental data show that similarly to poloidaly symmetric plasma detachment, divertor plasma detachment does also occur at plasma densities close to the density limit (e.g. see [25]). Although some signatures of divertor plasma detachment were observed a long time ago (e.g. see Fig. 8 in [35]), intensive study of such regimes became one of the focal points of the magnetic fusion research only since 1990 (see the corresponding references in [25]). The interest to the detached divertor regime was driven by the need to reduce the power loading on the divertor targets in future tokamak-reactors, including ITER, to a tolerable level. The only way to do this is to re-radiate a significant fraction of the power, generated in the reactor, with impurity. However, the concept of strong impurity radiation from the core plasma has two issues. First, due to peculiarities of impurity cross-field transport in the core, strongly radiating, high-Z impurities (recall Fig. 2.14) have a tendency of accumulation in the very core plasma, cooling it down and reducing the rate of fusion reactions. The radiation from low-Z impurities that are less prone to accumulation in the core plasma is rather weak, so for a sizeable effect, the concentration of the low-Z impurity must be high, with the corresponding dilution of the fusion D-T plasma and reduction of its performance. Secondly, a strong reduction of the heat flux from the core to the edge can prevent the transition to the improved confinement regime (H-mode), which may be needed for self-sustained burning of the fusion plasma. Therefore, it is widely accepted that it is necessary to increase power dissipation by impurity radiation in the divertor region as much as possible. There are two possible ways to do this: (i) to increase the plasma/impurity density in the divertor region (recall that the density of the impurity radiation loss is proportional to the product of the electron and impurity densities) and (ii) to increase the divertor volume by implementing so-called advanced divertor geometry (see Fig. 1.6). It is very likely that in practice both ways will be combined. However, an increase of the divertor plasma density will likely result in the increase of the plasma flux to the target and to the increase of the so-called “irreducible” power flux to the target, associated with the deposition of the internal energy of electron-ion pair – the ionization potential. Estimates made for ITER have

242

Physics of Some Edge Plasma Phenomena

4 Heat Flux (MW/m2)

Fig. 9.10 Reduction of the power loading on the outer divertor target in DIII-D after the transition to the detached divertor regime. (Reproduced with permission from [36], © IAEA 1999)

9

IRTV Divertor Heat Flux

3

Before Gas Puffing After Detachment

2 1 0 1.0

1.1

1.2

1.3

1.4 1.5 Radius (m)

1.6

1.7

1.8

shown that such “irreducible” power flux can exceed the tolerable power loading. As a result, in addition to the dissipation of plasma thermal power by impurity, it is also necessary to reduce the plasma particle flux to the target to a tolerable level. It seems that the detached divertor regime can meet both of these criteria (see Figs. 1.6 and 9.10). Shortly after initial experimental results on divertor plasma detachment became available, two main theoretical models, claiming their explanations, were put forward. The first one [37, 38] was relying on elastic (including charge-exchange) collisions of the plasma ions with the neutral gas in the divertor volume, which can switch plasma transport along the magnetic field lines from the fast, “ballistic” regime to the slow, “diffusive” one. As a result, “diffusive” plasma transport would cause a large plasma pressure drop between the upstream SOL region and the vicinity of the target (the so-called plasma “momentum removal”) similar to that shown in [39]. This, according to [38], can explain the reduction of the plasma flux to the target with increasing neutral gas density in the divertor and decreasing plasma temperature when the elastic ion-neutral collisions prevail over the electron impact ionization of neutrals. This model seems to be supported by experimental data from linear divertor simulators [40–42]. However, we will see later that the data from linear divertor simulators, in this case, cannot be translated directly to the situation in a tokamak divertor. The second model [43, 44] was based on energy and particle balance, including both the impurity radiation and the hydrogen “ionization” cost, as well as on the plasma recombination effect. In this model, the ion-neutral collisions per se do not result in the reduction of the plasma flux to the target. Nonetheless, they play an important role in the dissipation of the plasma momentum (via effective neutral viscosity) and thermal energy (via neutral heat conduction) at low temperatures when both the impurity and hydrogen radiation losses become inefficient. Note that over the years, different models of the reduction of the plasma heat flux to the material surfaces, including both the ion-neutral collisions and plasma recombination were considered [45–48]. 2D numerical simulations of edge plasma transport performed with both UEDGE and SOLPS codes have shown that in agreement with [43, 44], the ion-neutral collisions alone cannot cause the reduction of the plasma flux to divertor targets [49–52].

9.3 Divertor Plasma Detachment

243

a

b 15 Γ , (1023 s–1) w

8

Γw, (1023 s–1)

6 10

Γw

4

Γrec Γion

5 2 0 0

5

10

15 ˆ , (1020) N 3D

20

0

0

5

10

15 ˆ , (1020) N 3D

b 3D . Red lines: QSOL ¼ 8 MW, Qimp ¼ 0 w/o recombination Fig. 9.11 (a) Γw as the function of N (dashed); QSOL ¼ 8 MW, Qimp ¼ 0 with recombination (solid). Green lines: QSOL ¼ 8 MW, Qimp ¼ 4 MW w/o recombination (dashed); QSOL ¼ 8 MW, Qimp ¼ 4 MW with recombination (solid). Blue lines: QSOL ¼ 4 MW, Qimp ¼ 0 w/o recombination (dashed); QSOL ¼ 4 MW, Qimp ¼ 0 with recombination (solid). (b) Dependences of Γw, plasma ionization source, Γion, and recombib 3D for QSOL ¼ 4 MW, Qimp ¼ 0 with recombination turned on. (Reproduced nation sink, Γrec, on N with permission from [51], © Cambridge University Press 2017)

As an example, in Fig. 9.11a the total plasma flux to the divertor targets and main chamber wall, Γw, is shown as the function of the total number of hydrogenic b 3D, for different particles (including atoms and ions) in the computational domain, N b 3D power input, QSOL, from the core plasma and the impurity radiation loss, Qimp. N tot can be considered a natural extension of the parameter Nft (which we used for the analysis of the SOL plasma parameters within a magnetic flux tube) to 2D modeling b 3D is of edge plasma. The approach of the edge plasma simulations with fixed N called the “closed box” model, which, as demonstrated in [53], is a very good approximation for the high recycling regime where the plasma flux to the target exceeds by orders of magnitude both the puffing and pumping rates. b 3D , plasma temperature in the vicinity of the targets falls Note that at large N below 1 eV and the neutral density becomes comparable to the plasma one, which, according to [38], is supposed to result in a reduction of the plasma flux. However, b 3D and then from Fig. 9.11a we see that Γw initially increases with increasing N saturates unless plasma recombination is turned on. In addition, we notice that the b 3D is proportional to QSOL  Qimp. From Fig. 9.11b, saturation level of Γw at large N b 3D the plasma ionization source, Γion, it follows that for a given QSOL, at large N saturates, whereas the plasma recombination sink, Γrec, becomes almost equal to the ionization source, which causes the reduction of Γw. All these observations have a simple physical explanation based on the model developed in [43, 44]. Following this model, we consider the energy and particle balance equations in the SOL for the high recycling conditions (ignoring hydrogen puffing and pumping):

244

9

Physics of Some Edge Plasma Phenomena

QSOL ¼ Qimp þ QH þ QCX þ γTw Γw ,

ð9:12Þ

Γion ¼ Γw þ Γrec ,

ð9:13Þ

where QH is the power loss associated with hydrogen ionization; QCX describes the power delivered to the plasma-facing components by neutrals via the neutral-ion energy exchange (in dense divertor plasma, this energy loss is related to neutral heat conductivity). The last term on the right-hand side of Eq. (9.12) describes the transfer of the plasma thermal energy to the wall, and Tw is the average plasma temperature at the wall. By using the hydrogen ionization cost we have QH ¼ EH ion Γion. Since at high plasma density, both neutral heat and particle transport have diffusive nature, we have the estimate QCX ¼ ζκ/DTionΓion, where Tion is the temperature in the neutral ionization region and ζκ/D 2.5 [54] is the ratio of the neutral hydrogen heat and particle diffusivities. For the detached divertor regime, Tion 3  5 eV and does not vary strongly since for lower temperature, the ionization rate constant drops sharply (see Fig. 2.5). For small Tw the last term in Eq. (9.12) can be ignored and from Eq. (9.12) and (9.13) we find Γw ¼

QSOL  Qimp  Γrec  Γmax ion  Γrec , Eeff ion

ð9:14Þ

H where Eeff ion ¼ Eion þ ζκ=D Tion is the effective ionization cost of neutral hydrogen accounting also for the energy loss associated with neutral heat conduction. We notice that the Γmax ion is limited by the power available for neutral ionization and this limit corresponds to the saturation level of Γw in Fig. 9.11 for the case of no recombination. In agreement with the data shown in this figure, from Eq. (9.14) we have Γmax ion / QSOL  Qimp . Thus, from Eq. (9.14) it follows that for low Tw, the reduction of Γw is only possible either by increasing the impurity radiation or by the plasma recombination processes (see Chap. 2) or by both. Available experimental data fully support the idea that the impurity radiation loss and plasma recombination are the main parameters determining the plasma flux to the targets at low Tw. Depending on the plasma conditions, either one can play the dominant role. Clear signatures of volumetric plasma recombination were observed with spectroscopic diagnostics on many tokamaks [55, 57–62]. For example, in Fig. 9.12 one can see the intensities of the Balmer series lines, which are typical for recombining plasmas (see also Fig. 9.3), from the C-Mod and NSTX tokamak divertors. Careful analysis of the plasma ionization source and the volumetric recombination sink in the C-Mod tokamak, performed in [63], has shown that Γrec can exceed 80% of Γion. We notice that both electron-ion and molecular activated recombination (MAR), recall Chap. 2, can contribute to the volumetric plasma particle loss. Recent experimental data from the TCV tokamak show that in the detached divertor regime, the

9.3 Divertor Plasma Detachment

245

a

b Intensity (uW/cm2/nm) 106

x 1000

view

CD B-X band

2

Brightness (mW/cm /ster/nm)

1000

H7

105 H8

100

H9

BD band

104

H10 H11

H12 H13

103

10 T = 1.0–1.1 eV

120324

360

370

380

390

400

410

420

3700

3800

Wavelength (nm)

3900

4000

Wavelength (A)

– ne(1020 m–3)

0 4

4

0 4

4

Γion Γrec Γw

2 0

2

980116038

2

ICRF

980213021

Γ (1022/s)

6

Γ (1022/s)

Fig. 9.13 Impact of the ion-cyclotron (ICRF) heating and nitrogen radiation loss on ionization source Γion and plasma recombination sink Γrec in C-Mod tokamak. (Reproduced with permission from [63], © AIP Publishing 1999)

– ne(1020 m–3)

Fig. 9.12 Intensities of the Balmer series lines in (a, reproduced with permission from [55], © AIP Publishing 1998) detached recombining divertor plasma of C-Mod and (b, reproduced with permission from [56], AIP Publishing 2007) attached (upper curve) and detached recombining (lower curve) divertor plasma of NSTX tokamaks

N2 gas 2 H-mode 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Time (s)

contribution of MAR to the overall volumetric plasma recombination can reach ~40% [64]. Experimental data from both tokamaks [63, 65–68] and stellarators [69, 70] show that in cold divertor plasma, the impurity radiation, in accordance with Eq. (9.14), can also reduce the plasma flux to the targets. An example demonstrating the impact of both plasma recombination and impurity radiation on the plasma flux to the divertor targets is shown in Fig. 9.13. As we see in the upper panel, just before the ICRF heating is on, plasma recombination is very close to the ionization source, which, however, is almost doubled when additional power for neutral ionization becomes available due to ICRF heating. On

246

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Physics of Some Edge Plasma Phenomena

the lower panel, one sees that nitrogen puffing causes a strong reduction of plasma ionization source and, correspondingly reduction of the plasma flux to the targets, whereas the recombination sink remains small. However, what to do with experimental data from linear divertor simulators [40– 42], which seem to show that neutrals play an important role in the reduction of plasma flux to the target? We notice that in these experiments, the plasma was produced by the source situated in a separate chamber and only some portion of the generated plasma was flowing through the orifice into the working chamber. Therefore, the flux to the end target was significantly impacted by the neutral density in the working chamber. In addition, cross-field plasma transport in these experiments was relatively large and, for example, in [40, 41] it was concluded that cross-field plasma transport is the main reason for the reduction of the plasma flux to the end target. As we see, the conditions of the plasma flow to the target in linear divertor simulators are very different from the tokamak ones, where all generated plasma particles are supposed either to flow to the plasma-facing components or to recombine volumetrically. Therefore, even though the experimental data obtained in linear divertor simulators on such issues as atomic physics and material erosion (see Chap. 3) appear to be relevant for the edge plasma conditions in fusion devices (e.g. see [71– 73] and the references therein), the results on plasma detachment cannot be transferred directly to the tokamak experiments. Even though the simple physical picture, boiled down to Eq. (9.14), for the plasma flux to the target in cold divertor plasma allows explaining the key experimental observations, it only describes the integral plasma flux to the plasma-facing components. However, from Fig. 1.5b one sees that the divertor plasma detachment process does not happen uniformly over the entire divertor target. Instead, it starts from some particular flux tubes. Therefore, we need to find some local conditions for the onset of detachment, which we define as the beginning of the rollover of the specific plasma flux jd. Following [74] we will use the same concept of the “closed box” for some particular magnetic flux tube, which we used for the analysis of the self-sustained tokamak divertor plasma oscillations. We notice that such an approach might be not directly suitable for the onset of divertor detachment in stellarators having a complex divertor magnetic geometry. In the previous subsection, we found that for high recycling conditions, the plasma in the flux tube is sustained largely by the plasma recycling processes in the divertor region. However, the total pressure Ptot ft within the flux tube, which can be supported by recycling, is limited by the power needed for hydrogen ionization and is determined by Eq. (9.7), which can be expressed as follows Ptot ft

e
Qrecycl, which

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Fig. 9.20 Neutrals bypassing impurity radiation region can face large power available for neutral ionization

can greatly complicate the rollover of Γw. Experimental data also demonstrate that the so-called “closed”, well baffled, divertors have a lower threshold of the upstream plasma density for detachment (e.g. see [88]).

9.4

Conclusions

As we have discussed, the edge plasma exhibits different nonlinear phenomena resulting in the formation of macroscopic, strongly radiative and recombining structures (e.g. MARFE, detached divertor), nonlinear oscillations and bifurcations. In many cases, atomic physics effects, including impurity and hydrogen radiation, drive some of these phenomena. Others are associated with the interplay of atomic physics effects and cross-field plasma transport (including drifts and anomalous transport) as well as with plasma interactions with the materials of plasma-facing components. Some of these phenomena (e.g. divertor plasma detachment) can be affected by magnetic configuration and geometry of plasma-facing components. Whereas some of these phenomena can be very beneficial for the performance of future fusion reactors (e.g. divertor plasma detachment can drastically reduce power and particle loading on divertor targets), others (e.g. MARFE formation close to separatrix) can result in the degradation of core plasma confinement and the reduction of overall reactor performance. Even though large progress was made in the understanding of the physics of these nonlinear phenomena (e.g. divertor plasma detachment), still much more work is needed for better assessment of edge plasma behavior in future reactors.

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Chapter 10

Conclusions and Outlook

Abstract Some overarching conclusions on the current status and the suggestions for the directions of further development in the edge plasma physics are presented.

Over the whole book, we were emphasizing the complexity and multifaceted nature of the processes and phenomena in the edge plasma. To unravel the physics of these processes and phenomena one needs to involve: (i) Classical and anomalous (often highly intermittent) transport of multispecies plasma including impurities that are either released due to erosion of the plasma-facing components or introduced into the edge plasma deliberately to radiate the high power coming from the core (see Chaps. 6 and 7); (ii) Different issues of atomic physics, including ionization, recombination and radiation processes involving different species, which often occur in optically opaque media (see Chap. 2); (iii) Sheath physics and plasma interactions with the materials of the plasma-facing components, including material erosion and deposition, saturation with hydrogenic species and impurities, which can result in modification of the surface morphology and material properties, as well as an impact of the eroded material, including dust particles, on edge plasma performance (see Chaps. 3, 4 and 5). All these processes have strongly nonlinear behavior and in many cases exhibit strong synergistic effects. Therefore, it is very difficult to build simplified analytic or semi-analytic models that can properly describe the edge plasma phenomena. In most cases, the outcome of such models needs to be verified with more sophisticated numerical simulations (e.g. see Chap. 9). However, because of the complexity of the edge plasma and a large span of spatiotemporal scales which needs to be covered, the numerical codes are capable of treating only some “patches” of the necessary spatiotemporal domain. In addition, they often do not work smoothly even though some significant compromises in the physics embedded in these codes are usually made. For example, edge plasma turbulence and edge plasma transport are usually described with different models and even though the relative turbulent fluctuations of the edge plasma parameters in many cases exceed 100%, edge plasma transport is described with “laminar” models where the impact of these fluctuations is ignored (see Chap. 8). Also, most of the codes (e.g. the edge plasma transport codes such as SOLPS) are based on a © Springer Nature Switzerland AG 2020 S. Krasheninnikov et al., On the Edge of Magnetic Fusion Devices, Springer Series in Plasma Science and Technology, https://doi.org/10.1007/978-3-030-49594-7_10

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combination of the discrete (e.g. complex 3D3V neutral transport Monte-Carlo codes treating a vast body of atomic physics) and continuous (e.g. plasma transport based on the fluid approximation) parts, which complicates convergence and forces implementation of different algorithms for their numerical solution (see Chap. 8). Various experimental tools are used in the edge plasma studies, including different Langmuir probes, Thomson scattering, visible and UV spectroscopy, postmortem analysis of the plasma-facing components (including the nuclear reaction analysis) and in situ surface temperature measurements with infrared radiation, etc. However, the spatiotemporal resolution of all these experimental techniques is rather limited and the raw data obtained in the experiments do often require postprocessing (e.g. conversion of the spectroscopic data obtained by integration of the signal along the viewing chords into the distributions over the magnetic flux coordinates). Nonetheless, in spite of all these issues with both the experimental and theoretical/computational studies, substantial progress has been made in the last two decades in our understanding of the physics of the edge plasma phenomena. In particular, (i) The main ingredients governing the divertor plasma detachment process have been identified and confirmed by both numerical simulations and experimental data. (ii) The impact of the sheared plasma flow on suppression of anomalous cross-field plasma transport has been clearly shown theoretically and confirmed by both the results of numerical simulations and the experimental data. Moreover, it was demonstrated that such a flow can be generated by the plasma turbulence itself. (iii) It was shown that anomalous plasma cross-field transport can not always be described by diffusion type equations and in many cases, it can have more complex and nonlocal nature. In particular, it can be associated with large intermittent bursts of the plasma particle and energy fluxes. (iv) Dust particles, under some conditions, can play an important role in plasma contamination with impurity. (v) Erosion and re-deposition of the materials of the plasma-facing components can result in the formation of rather thick layers of the co-deposited material. Such loose co-deposits can cause the formation of hot spots and emission of dust particles and impurity atoms/molecules, severely limiting the operational window of fusion devices. In addition, they can retain a large amount of hazardous tritium. (vi) Both the fluid-based and gyro-kinetics-based codes capable of describing edge plasma turbulence have been developed. They are more and more frequently used for the understanding of anomalous transport of the edge plasma and in many cases produce the experiment-relevant results. However, there are still many things that can be improved. Here we outline only the most important gaps in our understanding of the edge plasma phenomena. Obviously, this list reflects just the view of the authors and it is pretty much possible that other researches working in the field of the physics of edge plasma in magnetic confinement devices would alter it. We think that one of the most important showstoppers on the way to a better understanding of the edge plasma phenomena is related to our poor comprehension of edge plasma turbulence. Edge plasma turbulence has an overarching impact on the processes in the edge plasma. It can govern the heat loading on and erosion of the

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plasma-facing components including the divertor targets and main chamber wall. These issues are crucial for both the feasibility of fusion reactor design and the reactor lifetime. We also notice that cross-field plasma transport in the SOL and divertor regions plays an important role in divertor plasma detachment which seems to be the only plausible solution for the mitigation of large divertor heat load in the future reactors. In particular, today it is not clear if the plasma turbulence observed in the SOL is driven by the SOL plasma itself or originates inside the separatrix and then spreads into the SOL. It is also not clear, through what mechanisms the divertor plasma conditions (e.g. the attached or detached states) affect edge plasma turbulence, as seems to be observed in experiments. Another issue, which is closely related to the study of edge plasma turbulence, is how to incorporate it into the codes simulating the edge plasma. It is clear that without the use of comprehensive numerical simulations, no progress in edge plasma studies is possible. However, today, edge plasma transport and turbulence are mostly simulated with, correspondingly, 2D and 3D codes (the limitations associated with such an approach were discussed in Chap. 8). Taking into account the multifaceted nature of the physics of the edge plasma and strong synergy between the different processes in it, we believe that both edge plasma turbulence and transport should be described simultaneously by 3D codes. This is a very challenging task combining different ranges of the timescales, which requires incorporation into the turbulence codes of both the physics of multispecies plasma and the atomic physics effects. Obviously, the development and real application of such codes require close collaboration of physicists and computer scientists as well as a new generation of computers. A further issue in the edge plasma physics is related to the interactions of the plasma and neutral particles with the materials of the plasma-facing components. In the past, such interactions were described by some particle and energy reflection coefficients for the impinging species and sputtering coefficients for the surface material. However, nowadays it is well recognized that these interactions are more complex. They involve not only reflection of the impinging species and erosion and re-deposition of the wall material, but also the modification of subsurface layers of the wall material due to complex phenomena associated, in particular, with the penetration of the impinging particles into the lattice of the wall material (see Chap. 3). As a result, the wall response to the impact of the charged and neutral particles of the edge plasma becomes very complex and nonlinear. Since all these processes determine such crucial parameters of a magnetic fusion reactor as the lifetime of the plasma-facing components and tritium retention, the topic of the plasma-material interactions becomes one of the top priorities in the edge plasma physics. We notice that the neutron damage of the lattice of the plasma-facing materials, inevitable in fusion reactors, brings additional complication to this issue. Because of the synergy among edge plasma transport, plasma recycling, impurity radiation and transport, plasma-material interactions, physics of strongly modified subsurface layers and wall material erosion, etc., it seems that it is inevitable that more and more integrated models and codes of different sophistication, describing the edge plasma in fusion devices, will be developed in the future. As a matter of fact, such a development is already underway.