The object of this review is to summarize the achievements of research on the Alcator C-Mod tokamak [Hutchinson et al., Phys. Plasmas 1, 1511 (1994) and Marmar, Fusion Sci. Technol. 51, 261 (2007)] and to place that research in the context of the quest for practical fusion energy. C-Mod is a compact, high-field tokamak, whose unique design and operating parameters have produced a wealth of new and important results since it began operation in 1993, contributing data that extends tests of critical physical models into new parameter ranges and into new regimes. Using only high-power radio frequency (RF) waves for heating and current drive with innovative launching structures, C-Mod operates routinely at reactor level power densities and achieves plasma pressures higher than any other toroidal confinement device. C-Mod spearheaded the development of the vertical-target divertor and has always operated with high-Z metal plasma facing components—approaches subsequently adopted for ITER. C-Mod has made ground-breaking discoveries in divertor physics and plasma-material interactions at reactor-like power and particle fluxes and elucidated the critical role of cross-field transport in divertor operation, edge flows and the tokamak density limit. C-Mod developed the I-mode and the Enhanced Dα H-mode regimes, which have high performance without large edge localized modes and with pedestal transport self-regulated by short-wavelength electromagnetic waves. C-Mod has carried out pioneering studies of intrinsic rotation and demonstrated that self-generated flow shear can be strong enough in some cases to significantly modify transport. C-Mod made the first quantitative link between the pedestal temperature and the H-mode's performance, showing that the observed self-similar temperature profiles were consistent with critical-gradient-length theories and followed up with quantitative tests of nonlinear gyrokinetic models. RF research highlights include direct experimental observation of ion cyclotron range of frequency (ICRF) mode-conversion, ICRF flow drive, demonstration of lower-hybrid current drive at ITER-like densities and fields and, using a set of novel diagnostics, extensive validation of advanced RF codes. Disruption studies on C-Mod provided the first observation of non-axisymmetric halo currents and non-axisymmetric radiation in mitigated disruptions. A summary of important achievements and discoveries are included.

While it is common and correct to frame pure plasma physics phenomena in terms of dimensionless plasma parameters,1,2 practical fusion energy requires prescribed levels of absolute performance. This can be easily understood as a consequence of non-plasma dimensionless parameters, particularly the ratio of plasma temperature to the characteristic energies required for the fusion nuclear reaction (kT/Enuclear) and to the characteristic energies for atomic ionization, recombination, and molecular bonding (kT/Eatomic). The first of these leads directly to the Lawson criterion for the minimum ion temperature in an energy producing fusion plasma. The second is important for edge plasma and plasma-wall interactions and will be discussed in Secs. I A and III. Economic and engineering considerations dictate the optimum level of neutron wall loading in a fusion reactor3 (about 3–4 MW/m2) and consequently to an optimum absolute plasma pressure and density. At the same time, all of the operating limits for a tokamak increase with the magnetic field; the maximum plasma current, which largely determines confinement, and the maximum plasma density are proportional to B,4,5 and the maximum pressure is proportional to B2.6 Thus, absolute performance increases with field, as does robustness against disruptions due to the proximity of operational limits. It is worth noting that the requirement for operation near an optimum density can be problematic for very large low-field fusion reactor designs, since this density range may be above the tokamak density limit.7 Prospective tokamak reactor designs like ARIES-AT assume operation near or above all of these limits8 raising concern about achieving this level of performance and robustness with respect to disruptions. Research at fusion-relevant absolute parameters is required since the plasma and non-plasma physics couple in complicated ways that are well beyond our current abilities to model.

The economic advantage of high fields can be understood by considering the total fusion power from a tokamak device, which is proportional to ( β N / q ) 2 R 3 B 4, where βN is the plasma pressure normalized to the Troyon limit6 and q is the tokamak “safety” factor, the inverse of the rotational transform. Plasma physics sets the upper limit for βN and the lower limit for q. The overall cost for a fusion facility is proportional to the mass of the fusion “core” and thus to the magnetic stored energy R 3 B 2. From these arguments, it is clear that the most cost effective fusion devices would operate with the highest fields that can be safely engineered. On several previous occasions when the U.S. was planning to build its own burning plasma devices, CIT, BPX, and FIRE,9,10 the price to performance argument led to compact high-field designs. Looking forward and considering the substantial costs and extended construction schedule for ITER, which was designed with “well-known” moderate-field superconducting magnet technology, a development path that features higher field seems attractive.

A discussion of the practical limits for the strength of magnetic field in a fusion device is beyond the scope of this paper, but it is worth noting the opportunities presented by recent developments in high temperature superconductors. These materials, YBCO (Yttrium Barium Copper Oxide), for example, have demonstrated significantly higher critical currents at fields above 20 T.11 By operating at elevated temperatures where heat capacities are higher, it should be possible to build magnets with field-demountable joints, allowing much more favorable modes for construction and maintenance. A design concept for a high-field pilot plant has been developed, demonstrating the advantages of this approach.12 A limiting factor, of course, would be the ability to provide the mechanical support for the magnetic stresses produced by high-field magnets, though the design efforts described above suggest that this should be achievable.

Alcator C-Mod is the third in a series of compact high-field tokamaks built and operated on the MIT campus.13,14 Supporting the arguments provided above, these machines have demonstrated high performance at a moderate size and cost—the previous device, Alcator C, being the first controlled fusion experiment to exceed the Lawson product for density times confinement.15 An important early goal of the C-Mod program was to provide a database that is relevant to high-field regimes. This goal encompassed support for the design and operation of ITER, whose toroidal field (TF) of 5.4 T exceeds every other shaped and diverted tokamak in the world except for C-Mod. Table I provides a summary of basic parameters for the device.

TABLE I.

C-Mod physics parameters and symbols used in this manuscript.

Parameter Symbol Range Units/definition
Major radius  0.67 
Minor radius  0.22 
Plasma elongation  κ  1.0–1.9   
Plasma triangularity  δ  0.0–0.85   
Plasma volume  m3 
Toroidal magnetic field  BT  2.4–8.1 
Plasma current  IP  0.24–2.0  MA 
Average plasma density  ne  0.2–8.0  1020/m3 
Central electron temperature  Te  <9  keV 
Central ion temperature  Ti  <6  keV 
Average plasma pressure  <0.18  MPa 
Normalized gyro-radius  ρ*  0.002–0.006  ρi/a 
Normalized pressure  βΝ  <1.8  βT/IP/aBT 
Normalized collisionality  ν*  0.06–1.0  νeiqR/ε3/2vi 
Parameter Symbol Range Units/definition
Major radius  0.67 
Minor radius  0.22 
Plasma elongation  κ  1.0–1.9   
Plasma triangularity  δ  0.0–0.85   
Plasma volume  m3 
Toroidal magnetic field  BT  2.4–8.1 
Plasma current  IP  0.24–2.0  MA 
Average plasma density  ne  0.2–8.0  1020/m3 
Central electron temperature  Te  <9  keV 
Central ion temperature  Ti  <6  keV 
Average plasma pressure  <0.18  MPa 
Normalized gyro-radius  ρ*  0.002–0.006  ρi/a 
Normalized pressure  βΝ  <1.8  βT/IP/aBT 
Normalized collisionality  ν*  0.06–1.0  νeiqR/ε3/2vi 

Operation at high field also allows attainment of uniquely ITER/reactor-relevant physics regimes. Consider, for example, the boundary plasma, where the plasma interacts with the wall, neutral fuel gas, and impurities. The nature of these interactions depends strongly on the plasma temperature normalized to atomic binding energies, which are on the order of a few eV. Thus, survival of plasma-facing components (PCMs) depends on lowering the plasma temperature at the interface to less than 10 eV. Fixing this value as a requirement for safe operation, the remaining boundary plasma parameters depend on the pressure. C-Mod, operating at reactor-like magnetic fields, operates at reactor-like boundary plasma pressures and thus has the same absolute power and particle loads, plasma density, and neutral opacity. As a consequence, a wide range of boundary phenomena can be studied directly on C-Mod, without resort to scaling arguments or excessive dependence on models. Similarly for radio frequency (RF) physics, C-Mod can run with the same cyclotron frequency (same field) and plasma frequency (same plasma density) as ITER and by carrying out experiments with the same RF frequencies can operate with identical wave physics. Figure 1 shows a selection of C-Mod parameters compared to other tokamaks and to the projected parameters for ITER. Data are taken from the International Tokamak Physics Activity (ITPA) H-mode database and also includes the C-Mod I-modes for comparison. These parameter plots are grouped based on the relevant physics—Fig. 1(a) plots ν* vs βN, which are important for core Magneto-Hydro-Dynamic (MHD), Fig. 1(b) shows ωpe (electron plasma frequency) vs ωce (electron cyclotron frequency), which characterize RF physics, and Fig. 1(c) plots the parameters that characterize the boundary plasma challenge as discussed in Sec. III, PB/R vs plasma pressure (where the core pressure stands in as a proxy for the divertor pressure, for which a broad range of data is less available). At the same time, by operating in a unique range of field, input power, and size, C-Mod has made critical contributions to multi-machine databases, which break parameter covariances when combined with larger low-field devices. For example, the inclusion of C-Mod data led to the ITER98 scalings for energy confinement in the H-mode, in which an unconstrained regression yielded a dimensionally correct fit.16 Previous regressions carried out before C-Mod data were available were not dimensionally correct and in fact failed to predict the eventual C-Mod results, pointing out the risks in extrapolating from inadequately conditioned data.17 In a similar vein, C-Mod provided critical data for disruption physics, the L-H threshold, boundary plasmas, H-mode pedestals, and core particle transport used for defining the ITER operational baseline.7,18 Given this background—the ability to operate in relevant regimes, with a good diagnostic set—it was inevitable that C-Mod would make a series of discoveries and address issues important for fusion energy.

FIG. 1.

C-Mod parameters are compared to other tokamaks and those projected for ITER as relevant to physics for (a) core MHD, (b) RF heating and current drive, and (c) boundary plasma physics where PB/R is a proxy for the divertor heat load and the core pressure a proxy for the divertor pressure. Data are mainly from the ITPA H-mode data base and includes C-Mod I-modes for comparison.

FIG. 1.

C-Mod parameters are compared to other tokamaks and those projected for ITER as relevant to physics for (a) core MHD, (b) RF heating and current drive, and (c) boundary plasma physics where PB/R is a proxy for the divertor heat load and the core pressure a proxy for the divertor pressure. Data are mainly from the ITPA H-mode data base and includes C-Mod I-modes for comparison.

Close modal

C-Mod's unique physics capabilities flow directly from its high-field magnet technology.19 The TF magnet consists of 20 6-turn copper coils carrying 225 kA at full field. Each coil is rectangular and composed of 4 straight segments with sliding joints at the corners. The joints are not pinned but rather are free to move under full current, transferring most of the magnetic stress from the coil to an external structure. The magnetic forces, which can reach up to 110 MN, are supported by a cylinder, 0.15 m thick, 4.9 m in diameter together with top and bottom domes, each 0.66 m thick with all three parts forged from high strength 316LN stainless steel and precision machined. The domes are fastened to the cylinder by 96 pretensioned INCONEL 718 drawbars forming a massive pressure vessel. Weighing about 30 tons each, the domes and cylinder were some of the largest stainless steel forgings ever made. A pair of monolithic wedge plates holds the magnet bundles in place and restrains the overturning forces of the magnet. Internal stresses in each bundle are supported by the high-strength copper and reinforced by stainless steel plates that are inserted, with insulation, between each turn. Figure 2 shows a schematic of the machine and the major components mentioned here. A great deal of R&D went into the felt-metal sliding connections that are the key to this design.20 With 120 turns, each made of 4 segments, there are a 480 joints that must slide under full current and full mechanical load with minimal wear, while maintaining very low electrical resistivity. Each joint has 4 felt-metal pads, with a total area of 72 cm2, made of copper wire, sintered onto a copper substrate, silver plated, and coated with colloidal graphite. Spring-plates are hydraulically driven in between the TF joint fingers to provide the required contact pressure. The resulting resistance is below 1.5 μΩ for each joint. The TF magnet is disassembled for inspection roughly every 5000 pulses. The TF and poloidal field (PF) magnets are all cooled to LN2 temperatures to reduce their electrical resistance. Thermal management in C-Mod is challenging, requiring that the vessel and ports be kept at room temperature while the magnets are kept cold. Clearances are small due to the compact size of the device. Table II provides a summary of C-Mod engineering parameters.

FIG. 2.

A schematic of the C-Mod tokamak showing the major components.

FIG. 2.

A schematic of the C-Mod tokamak showing the major components.

Close modal
TABLE II.

C-Mod engineering parameters.19 

Parameter Range Units/definition
Vessel volume  m3 
Vessel toroidal and poloidal resistance  40, 10  μΩ 
Vessel L/R time  20–50  ms 
Effective pumping speed (turbopumps)  500 (D2 l/s 
Effective pumping speed (cryopump)  10 000 (D2 l/s 
Ohmic heating power  1.0–2.7  MW 
ICRF source power  MW 
Lower hybrid source power  MW 
Peak utility power  24  MW 
Peak extracted alternator/flywheel power  250  MV A 
Alternator/flywheel stored energy  GJ 
Toroidal field magnet current  0.225  MA 
Toroidal magnet turns  120   
Forces from toroidal field  110  MN 
Parameter Range Units/definition
Vessel volume  m3 
Vessel toroidal and poloidal resistance  40, 10  μΩ 
Vessel L/R time  20–50  ms 
Effective pumping speed (turbopumps)  500 (D2 l/s 
Effective pumping speed (cryopump)  10 000 (D2 l/s 
Ohmic heating power  1.0–2.7  MW 
ICRF source power  MW 
Lower hybrid source power  MW 
Peak utility power  24  MW 
Peak extracted alternator/flywheel power  250  MV A 
Alternator/flywheel stored energy  GJ 
Toroidal field magnet current  0.225  MA 
Toroidal magnet turns  120   
Forces from toroidal field  110  MN 

Another critical innovation was made in the buss connections, which bring power to several of the PF magnets. To accommodate the high current densities required and dimensional changes during heating and cooling, compliant buss connections were fabricated with electro-forming technology, an additive manufacturing process that produces stress-free high strength joints—compared to standard welding or brazing techniques, which anneal and weaken underlying material. The poloidal field magnets themselves are of more conventional design. The Ohmic Heating (OH) coil is made of 3 segments and is wound directly on the TF central column. The C-Mod OH coils require 30 kA currents to be supplied across magnetic fields above 17 T. A coaxial design allows the inner and outer conductor forces to react against each other to produce a very strong structure. The connection to the OH stack includes electric-discharge-machined 25 μm wide slots acting as springs along with a Belleville stack to provide compliance to the feltmetal contacts. This design has performed extremely well in handling both the extreme electromagnetic forces and the thermal stresses over many thousands of C-Mod shot cycles. The remaining PF coils are supported by the vacuum vessel, which is a structural element of the machine with thickness varying from 1.5 to 5 cm. Power for the magnets is provided by an alternator and flywheel storing 2 GJ of kinetic energy and driven by a 4000 horse-power motor. 250 MV A can be extracted from the alternator during a pulse and is supplemented by 24 MV A from the local electrical utility. Twelve independent power convertors supply current to the machine's magnets. For the first 14 years of its operation, C-Mod plasma control was via a hybrid digital-analog controller provided through a collaboration with the CRRP-EPFL.21 More recently, an all digital real-time control system was implemented using a conventional linux server and I/O cards on a CompactPCI bus.22 Instrumentation and control is handled by ≈30 industrial programmable logic controllers with mimic screens in the control room.23 Pulse coordination, data acquisition, data management, and automated analysis are provided through the MDSplus data system.24 The client-server capabilities of MDSplus allowed C-Mod to demonstrate the first remote operation of a fusion experiment.25 

From the start, plasma facing components (PFCs) in C-Mod were built to withstand the very high heat fluxes and mechanical loads that were anticipated. The design featured a vertical target lower divertor and refractory metals on all surfaces that could come into contact with the plasma. The machine can also run with an upper x-point on a flat target divertor behind which is installed a toroidal cryopump with an effective pumping speed of 10 000 l/s for D2. The choice of high-Z metals was controversial at the time as earlier experience with tungsten limiters on PLT (Princeton Large Torus)26 convinced a generation of fusion scientists that these materials were not practical. However, the C-Mod team believed that graphite and carbon composites would not be acceptable materials in a reactor and that the fusion program needed data that could demonstrate the advantages and overcome the challenges of refractory metals. The 20 years of experience gained on C-Mod in the relevant operational space has been a critical element in decisions made for the ITER first wall design. The C-Mod wall was originally faced with 7000 tiles made of the molybdenum alloy TZM (99.5% Mo, 0.5% Ti, and 0.08% Zr) installed on backing plates made of INCONEL or stainless steel depending on the strength required.27 The large number of relatively small tiles was required to limit the forces due to eddy currents induced in the vessel during disruptions. The metallurgy of the raw material was important for the ability of these tiles to survive the thermal and mechanical shocks that they were subjected to. A belt of tungsten tiles was installed in the highest heat flux areas for several run periods for evaluation of a possible ITER design and to allow measurements of material erosion and migration. Figure 3 is a recent image of the internal hardware, showing the divertor, inner wall tiles, RF launchers, and internal diagnostics. Because of the compact size of the device and port space further limited by the heavy build of the magnets, a large amount of hardware is mounted on the tokamak wall. Over time, the C-Mod team has learned how to design, fabricate, and install hardware that can be subject to significant heat loads and disruption forces.

FIG. 3.

Photo taken inside C-Mod, showing internal components including the divertor and inner wall limiter tiles as well as ICRF antennas and numerous diagnostics.

FIG. 3.

Photo taken inside C-Mod, showing internal components including the divertor and inner wall limiter tiles as well as ICRF antennas and numerous diagnostics.

Close modal

For machine conditioning, the thick, low-resistance vacuum vessel precluded any possibility of pulsed “Taylor” discharge cleaning, which had been the standard procedure on previous Alcator devices. Instead, C-Mod surfaces are prepared for operation via Electron Cyclotron Discharge Cleaning (ECDC) using a 2.5 kW klystron operating at 2.54 GHz.28 The toroidal field is operated near 0.09 T and slowly swept so that the discharge intercepts all of the internal structures. After a period of baking, discharge cleaning and initial operations, the plasma facing surfaces are typically covered with a thin film of boron by running discharge cleaning with deuterated diborane (10% B2D6 in 90% He background). Approximately 100 nm is deposited weekly when operating.29 

While the nature of the C-Mod device allows operation in a wide parameter space, it also drove a research program that was required to address and solve a set of critical scientific and technological challenges imposed by its design. These are prototypical of next-generation devices like ITER or Demo so that research on C-Mod, which was required operationally, is directly and uniquely relevant to meeting future challenges. The necessity of addressing these issues has focused effort on areas that many other research groups could ignore or defer. Among these challenges were

  • Discharge startup with a highly conductive vacuum vessel: The vacuum vessel provides structural support for many of the poloidal field coils and thus was heavily built (1.5–5 cm thick) and with no electrical break. The toroidal resistance of the vessel is 40 μΩ, and its L/R time is 20 ms. The TF support structure, while farther from the coils, has even lower resistance. The result is that at startup, up to 0.5 MA flows in each of these two structures presenting complications for diagnosis and control.13 

  • Very high power outflow: In a compact high-field device running at high absolute pressure, high performance necessarily implies high absolute power and particle loads to the first wall. C-Mod was constructed from the beginning with a divertor design and first wall material that would withstand these loads (by contrast, low-field devices tend to run into β limits at high power well before they attain reactor-like heat fluxes).

  • High-Z metal plasma facing components: The choice of refractory metals meant that solutions to contamination by high-Z impurities needed to be found and required research into the sources and transport of these impurities.

  • Very high input power densities: To attain high-performance regimes, launchers for ion cyclotron range of frequency (ICRF) heating and lower hybrid (LH) current drive needed to operate routinely and reliably at high power densities (∼10 MW/m2).

  • High efficiency, off-axis current drive at higher densities than previously achieved.

  • High plasma performance without Ti > Te, momentum input or core particle sources: The heavy magnet build precludes tangential port space sufficient for high-power neutral beams, thus all auxiliary heating on C-Mod is from RF, which does not directly supply particles or torque to the plasma core and mainly heats electrons. In contrast, on other devices, beams produce high external torque, core fueling, and ion heating, which are all correlated with good confinement.

A fitting preface to a discussion of boundary experiments on C-Mod is a 1983 quote from Peter Stangeby of the University of Toronto “Right now everyone is worried about getting and keeping heat in. Eventually the main problem will be how to handle the heat coming out.” From its inception, the C-Mod team understood that handling power exhaust would be one of its most significant challenges. The operating space for C-Mod is uniquely relevant and reactor prototypical in the following sense. The plasma in contact with material walls is subject to physics scaled to the energy of atomic bonds. Strong interactions with neutrals and impurities through ionization, recombination, and other atomic processes are critical elements for transport of heat, mass, and momentum in this region. Perhaps most importantly, erosion, caused by sputtering processes, drives a requirement to limit ion impact energies below a material-dependent threshold related to the bonding energy in the substrate. These arguments tell us that a reactor must operate with the plasma that is in contact with the wall at a fixed, low temperature. With that temperature at the ∼2–10 eV required, the operating density is given by the plasma pressure and only C-Mod operates at reactor-relevant plasma pressures. Thus, the C-Mod experiments are carried out with the power and particle fluxes, plasma density, neutral density, neutral-neutral collisionality, neutral opacity, and photon opacity similar to what is expected in a reactor. These experiments are not “wind tunnels” with appropriately scaled parameters but are rather discharges with the actual reactor-like values. Experimental results under these conditions are particularly critical as the edge plasma and plasma-material interactions remain far beyond our modeling capabilities. The main difference between C-Mod and a reactor in this region is in the length of the discharges. C-Mod cannot adequately address the set of issues related to machine lifetime and that show themselves only over millions of seconds.

All modern tokamaks are constructed with a toroidal divertor, designed to isolate plasma-wall interactions and to spread heat loads over as broad an area as practical. C-Mod innovated the vertical target divertor, as shown in Fig. 4. The key features of this configuration are a shallow angle between the magnetic field (0.5–1.5°, depending on the plasma equilibrium) and an extended divertor leg.30,31 In this geometry, neutrals arising from recombination at the divertor strike point are directed toward the divertor channel, enhancing reionization and providing a natural baffling. Neutrals created in the divertor are isolated from the main chamber by the divertor plasma itself. One result is better isolation between the divertor and main plasma, leading to a lower density threshold for divertor detachment as discussed below in Sec. III C. The advantages of the vertical target divertor are now widely recognized, and the concept has been adopted for ITER.

FIG. 4.

The C-Mod vertical target divertor features a small incident angle between the magnetic field and the wall, a long divertor leg and natural baffling of neutrals. The separatrix for a typical MHD equilibrium is plotted in red.

FIG. 4.

The C-Mod vertical target divertor features a small incident angle between the magnetic field and the wall, a long divertor leg and natural baffling of neutrals. The separatrix for a typical MHD equilibrium is plotted in red.

Close modal

Also pioneered by C-Mod and adopted by ITER is the use of high-Z metals as a divertor material. C-Mod research has highlighted the advantages and the challenges of these materials and ultimately demonstrated their practicality. Any divertor material needs to withstand steady-state heat loads and to survive transient loads, which cannot be completely eliminated. In a reactor, operating with high availability for extended periods of time, two additional requirements become critical. First, the net erosion rate must be held below 1 sputtered atom for every 106 incident plasma ions.32,33 Second, for safety and limits in supply, the retention of tritium fuel must be kept very low—less than 1 atom of tritium fuel can be retained in the wall for every 107 plasma ions incident.34,35 These requirements effectively rule out low Z materials like carbon even when they can withstand the heat loads. Graphite or carbon composites are a popular choice in current experiments because when introduced as an impurity into the plasma, the power loss from radiation is usually tolerable. High temperature plasmas consisting almost entirely of carbon ions, though useless for fusion, can be sustained. In contrast, the concentration of high-Z impurities must be strictly reduced—for example, concentrations of tungsten in ITER will need to be kept below 2 × 10−5.36 Because refractory metals offered the promise to control erosion and fuel retention, but presented a severe challenge for impurity control, the C-Mod team felt that this was the correct choice—the fusion program would eventually have to step up to this challenge and C-Mod seemed like an ideal platform to begin that research.

Experiments on C-Mod have addressed a large set of operational issues presented by the metal walls. These find no “show stoppers” that would rule out high-Z materials, but do reaffirm previous concerns about impurity sources and point out the need for additional research, particularly at the higher wall temperatures that will be typical of a fusion reactor. Plasma startup is not problematic with metal walls even after disruptions or other deconditioning events. This contrasts to the situation with carbon walls where some form of wall conditioning is typically required to reestablish operations.37 Density control and fueling with metal walls are also straightforward, and recycling is generally high, certainly well above 90% in equilibrium, with the walls adjusting to significant changes in a few shots, i.e., a few seconds of discharge time. In the L-mode, the discharges can be readily gas fueled up to the density limit at currents up to 1 MA ( n ¯ e = 6.5 × 1020/m3). Access to the H-mode is comparably easy, compared to carbon machines—for example, at low q95, Ohmic H-modes are regularly attained.38 The density in the H-modes, normalized to the density limit, is typically 0.5–0.7, a bit below that seen in lower field, neutral beam heated devices. One reason for this is that the very strong gas puffing required for higher densities interferes with ICRF antenna operation,39 though the lack of beam fueling may also be a factor40 along with limitations of fueling and transport through a high-opacity edge and pedestal.41 (The new field-aligned (FA) antenna described in Sec. VI has shown better behavior at very high neutral densities.) Since fusion plasmas have much lower tolerance for high-Z impurities, control of the sources from the wall is critical, especially during ICRF. The first experiments with high power ICRF and bare molybdenum walls found sharply increased molybdenum content, increased core radiation, and difficulty in achieving the high quality H-modes.17,42 It was not clear what parts of the vessel were the principal sources of these impurities. Boronization, as described above, was employed and had the effect of sharply reducing radiation from molybdenum43 and allowing the production of the high quality H-modes.17 Research on impurity challenges in ICRF heated plasmas is described in greater detail in Sec. VI. Operational issues with tungsten plasma facing components are now also under intensive study by the AUG and JET devices.44 

To keep the surface temperature of divertor plates within acceptable limits in a reactor, finite heat conduction dictates that no more than a few mm of material can intervene in front of the cooling channels. Thus, net erosion must be kept on the order of 1 mm over the lifetime of the first wall. One of the key advantages of refractory metals is their potential for lower levels of sputtering when exposed to ions (including impurities) accelerated through the plasma sheath. The energy threshold for sputtering from refractory metals is much higher than for low-Z materials like carbon or beryllium, with exponentially smaller sputtering yields if the edge plasma electron temperature can be held at sufficiently low values. Erosion rates for molybdenum were first determined on C-Mod by analysis of divertor tiles removed between experimental campaigns and measuring the change in depth of a thin chromium marker layer using Rutherford backscattering.45 Net erosion was highest near the outer divertor strikepoint, reaching 150 nm for the 1200 s of discharge time during the campaign, equivalent to removal of 4.5 mm/discharge-year. Gross erosion rates were estimated from physical sputtering yields using measured plasma conditions and were somewhat higher than the measured net erosion—partly attributed to prompt redeposition of sputtered ions. Installation of a toroidally continuous row of bulk tungsten tiles enabled measurement of erosion and migration onto other plasma facing components.46 In this case, the surfaces were analyzed after removal by measuring x-ray emission stimulated by exposure to a 2 MeV proton beam. Analysis of the x-ray spectra allowed determination of the quantity of tungsten on otherwise molybdenum substrates. Figure 5 shows the pattern of deposition found at different poloidal locations. The pattern suggests that scrape-off layer (SOL) flows play an important role in movement of sputtered materials to distant locations. Integration of migrated material yields an estimate for tungsten erosion of 0.014 nm/s or less than a mm per discharge-year—though we must note that the plasma strike-point was not in contact with the row of tungsten divertor tiles at all times during the experiments carried out in this campaign. The values for measured molybdenum and tungsten erosion were, respectively, 10–100 times lower than what has been found for graphite.47 Gross erosion may be a more important measure of acceptable plasma-wall interaction since changes in surface morphology and chemistry associated with redeposition may lead to unacceptable changes in physical properties like thermal conduction. Gross erosion may also increase the amount of dust—a safety issue in a reactor—or allow the build-up of poorly bonded flakes, which would subsequently enter the plasma and cause harmful disruptions.

FIG. 5.

Tungsten redeposition thickness in nm, from a toroidal belt of tiles on the outer divertor (marked “W”). The material deposited can be integrated to estimate the average erosion rate. Reprinted with permission from Barnard et al., J. Nucl. Mater. 415, S301 (2011). Copyright 2011 Elsevier.46 

FIG. 5.

Tungsten redeposition thickness in nm, from a toroidal belt of tiles on the outer divertor (marked “W”). The material deposited can be integrated to estimate the average erosion rate. Reprinted with permission from Barnard et al., J. Nucl. Mater. 415, S301 (2011). Copyright 2011 Elsevier.46 

Close modal

The retention of tritium fuel within the first wall materials is another critical plasma-wall issue for ITER and for future reactors where safety considerations limit tritium inventory to about 1 kg. Using the expected plasma parameters, we find the acceptable limit is less than 1 tritium ion retained for every 107 incident on the plasma wall. A similar number is obtained from economic considerations, given the modest tritium breeding ratios that are expected. The requirement for low fuel retention also drives the interest in high-Z metal walls, since the solubility and reactivity of hydrogen in such metals is much lower than for carbon. Experiments on C-Mod measured retention of D2 gas over a single discharge by “static gas balance,” that is, by looking at the equilibrium pressure attained after running a plasma discharge with all torus pumps valved off compared to a case with the same gas puffing but without a plasma.48 In these experiments, roughly 1% of the incident deuterium ion fluence is retained with no indication that the retention rate is decreasing after 25 s of integrated plasma exposure. The magnitude of retention is significantly larger than what is expected from extrapolation of laboratory results.49 The interpretation of the result is that “traps” are created in the molybdenum substrate by the high incident particle flux.49 The traps are defects in the molecular structure that can hold deuterium atoms, which are otherwise insoluble in the unperturbed matrix. In contrast to single shots, the campaign-integrated retention is about 1000× lower. The difference is apparently due to the occasional disruption, which removes deuterium through transient heating of the tile surfaces. These results point out the importance of conducting experiments at reactor-relevant temperatures, which is with the wall at about 1000 K, where defects in the wall molecular structure are expected to be annealed and retention would be dramatically reduced.

An example of material changes that can be induced by plasma interactions is the growth of tungsten nano-structures (“fuzz”) that has been observed in plasma-wall test stands under suitable conditions.50 The working hypothesis for their formation is that the structures, which consist of small filaments, are extruded by pressure from helium bubbles captured in the metal substrate. An open question was whether the same phenomena would occur on the wall of a confinement experiment or if other plasma-wall processes would destroy the structures before they could grow to significant size. On C-Mod, a careful experiment was performed to raise a tungsten sample to the correct surface temperature, about 2000 K, and expose it to helium plasmas for a sufficient time to match the fluxes and fluences employed on the test stand. Nano-structures, shown in Fig. 6, were created with nearly identical morphology and growth rates (tendril diameter ∼100 nm and growth rate ∼600 nm in 13 s of exposure at temperature).51,52 Helium concentrations in the fuzz layers were measured at 1%–4%, which is well above natural solubility of helium in tungsten, but below the values expected for pressure-driven growth. Erosion rates from sputtering of the tungsten sample were well below the fuzz growth rate; however nearby, molybdenum surfaces operating at lower temperatures were predicted to have faster sputtering than growth. As expected, these surfaces did not show evidence of surface nano-structures. Overall, we conclude that the tokamak environment has little or no impact on tungsten fuzz growth when compared to linear plasma devices. This provides confidence that key growth parameters identified in linear devices can be used to predict surface behavior in future devices. None-the-less, a number of critical questions must still be answered. Largely unknown are the effects of the fuzz on tokamak operations, including wall recycling, fuel retention, erosion, and dust production. Research is also required to clarify the effects on fuzz growth of large Edge Localized Modes (ELMs), impurity seeding, and mixed wall materials.

FIG. 6.

A micrograph of tungsten nanostructures produced by 13 s of helium discharge time on a target operating at about 2000 K. The morphology and growth rate are essentially identical to what is produced in a linear plasma device under similar conditions. Reprinted with permission from Wright et al., Nuclear Fusion 52, 042003 (2012). Copyright 2012 IOP.51 

FIG. 6.

A micrograph of tungsten nanostructures produced by 13 s of helium discharge time on a target operating at about 2000 K. The morphology and growth rate are essentially identical to what is produced in a linear plasma device under similar conditions. Reprinted with permission from Wright et al., Nuclear Fusion 52, 042003 (2012). Copyright 2012 IOP.51 

Close modal

Post-campaign ex-situ measurements usually represent inadequately defined averages over discharge conditions from an entire campaign rather than carefully controlled conditions. A measurement from a single point in time is typically all that is available for an inherently dynamic and complicated process, and the progress is correspondingly difficult and slow. To overcome these limitations, a new diagnostic has been developed and deployed on C-Mod, which is capable of time resolved, in-situ measurements of surface erosion and fuel retention. This diagnostic, AIMS (Accelerator Based in-situ Materials Surveillance), employs a 1 MeV D+ beam that is injected into the torus between shots and steered by the magnetic field produced by running small currents in the TF and PF coils.53 A large selection of wall locations can be accessed by this method and tested between plasma discharges. The beam induces nuclear reactions that allow characterization of the surface composition. Some of the possible reactions and their application to surface analysis are listed in Table III. By preparing tiles with coupons of selected materials, the scope of possible measurements can be further increased, for example, to measure the erosion of high-Z plasma-facing components. A drawing of the AIMS system is shown in Fig. 7.54 Early results have proven the principle of the technique and shown that measurements could be routinely made between shots.55–57 

TABLE III.

A few of the nuclear reactions that can be employed by the AIMS diagnostic.

Probe ion Target Detected particle Surface measurement
D+  Fuel retention 
D+  Li6, Be9, B11  γ  Erosion of surface coating 
D+  C12, N14, O16  γ  Surface impurities 
Probe ion Target Detected particle Surface measurement
D+  Fuel retention 
D+  Li6, Be9, B11  γ  Erosion of surface coating 
D+  C12, N14, O16  γ  Surface impurities 
FIG. 7.

The AIMS diagnostic makes the first time-resolved, in-situ measurements of plasma-wall interactions. It utilizes a 1 MeV deuterium beam, which can be steered between shots by magnetic fields and induce nuclear reactions in the materials of the first wall. Reproduced with permission from Rev. Sci. Instrum. 84, 123503 (2013). Copyright 2013 AIP Publishing LLC.54 

FIG. 7.

The AIMS diagnostic makes the first time-resolved, in-situ measurements of plasma-wall interactions. It utilizes a 1 MeV deuterium beam, which can be steered between shots by magnetic fields and induce nuclear reactions in the materials of the first wall. Reproduced with permission from Rev. Sci. Instrum. 84, 123503 (2013). Copyright 2013 AIP Publishing LLC.54 

Close modal

Meeting the challenges of divertor power handling and erosion require better understanding of the underlying physics, through which improved designs and operating regimes can be achieved. The operating point of the divertor depends in large measure on the balance between parallel and perpendicular transport. Three regimes of parallel transport were identified in C-Mod experiments and are illustrated in Fig. 8, which compares electron pressure and temperature at the midplane to the corresponding profiles measured at the divertor target.58,59 The midplane profiles are measured with fast-scanning Langmuir probes and the divertor profiles with fixed probes that are imbedded in the tiles. At the lowest densities, when the parallel electron mean free path is long compared to the connection length (∼qR), electron temperature and pressure are constant along the field lines. The divertor sheath supports the entire temperature drop from the midplane to the tile surface. In this “sheath limited” regime, the divertor temperature is too high and would lead to unacceptable divertor erosion rates for a reactor. At moderate densities, collisions reduce the parallel thermal conduction and produce a parallel temperature gradient. This results in lower temperatures at the target, about 10 eV, and correspondingly lower erosion rates. The pressure along the field lines is still constant so the density increases near the divertor and supports the required power conduction. At higher densities still the plasma interacts more strongly with neutrals (which increase nonlinearly with plasma density), transferring plasma momentum and energy to them. The momentum transfer causes the plasma pressure to drop, and energy transfer lowers the temperature to the point where volumetric recombination occurs, further reducing the plasma pressure. In this “detached” stage, the temperature at the target drops to about 2 eV and the heat is largely removed from the plasma by radiation and charge exchange, spreading the heat load over a much larger area. From the point of view of erosion and divertor survival, it is highly desirable to operate the divertor in the detached state.34 

FIG. 8.

Three divertor regimes that are produced at increasing density, are identified in this plot of pressure and temperature profiles in the SOL. Reproduced with permission from Phys. Plasmas 2, 2242 (1995). Copyright 1995 AIP Publishing LLC.58 

FIG. 8.

Three divertor regimes that are produced at increasing density, are identified in this plot of pressure and temperature profiles in the SOL. Reproduced with permission from Phys. Plasmas 2, 2242 (1995). Copyright 1995 AIP Publishing LLC.58 

Close modal

The border between the three regimes can be characterized as fractions of the density limit, with the boundaries shifting to higher densities with increased input power. The density and power dependences are partly attributed to the increase in collisionality, consistent with the observations of anomalous cross-field transport discussed in Sec. III E. In typical SOL profiles, such as those shown in Figs. 8 and 9, detachment starts near the strikepoint first and grows outward as the density is raised. Experiments were carried out to explore the role of divertor geometry in the detachment phenomena, comparing the standard vertical target configuration to a flat plate and slot divertor by moving the strike point across the divertor surfaces. Detachment occurred with the vertical target at about half the density of the flat plate with a slight further improvement for the slot divertor.60 These experiments suggest that the main effect is an increase in the interaction between recycled neutrals and the divertor leg for the vertical target. The increase in divertor leg length is apparently a secondary effect. It is worth noting that detachment in C-Mod occurs well below the density limit for all three cases.

FIG. 9.

Typical SOL density profile as a function of global normalized density. Reproduced with permission from Phys. Plasmas 15, 056106 (2008). Copyright 2008 AIP Publishing LLC.91 

FIG. 9.

Typical SOL density profile as a function of global normalized density. Reproduced with permission from Phys. Plasmas 15, 056106 (2008). Copyright 2008 AIP Publishing LLC.91 

Close modal

With the high plasma pressures that were accessible, C-Mod discovered the importance of volume recombination, neutral collisionality, and Lyman α photon opacity on divertor behavior. Modeling of the ITER divertor has confirmed the importance of these parameters.61 At the low temperatures and high densities seen in the detached regime, the plasma can begin to recombine volumetrically, a process that otherwise occurs only on surfaces as recycling. Recombination was confirmed by the distribution of line intensities in the Balmer spectrum, which is markedly different in ionizing and recombining plasmas.62,63 Extensive modeling of the spectra and atomic physics allowed determination of the recombination rate and of the plasma parameters in those regions. Under the conditions that prevailed, the plasma became opaque to Lyα photons,63 with the photon mean free path dropping to about 1 mm, modifying the recombination rate. Also affected by the operation at high densities is the transport of neutrals, with the mean free path for neutrals in C-Mod closer to what is expected in ITER than in any other device. Studies carried out to explore the dynamics and distribution of neutrals showed they are trapped in the divertor by the plasma, providing a natural baffling and building up the neutral pressure in the divertor chamber to levels exceeding 100 mT in some cases.64,65 Recycling impurity gases are preferentially compressed and enriched in the divertor region.66,67 Detachment can be enhanced by injection of impurities, which radiate inside the separatrix and in the divertor, reducing parallel heat exhaust. This effect can be exploited to reduce the divertor heat load, but care must be taken to avoid degrading core performance. The detachment front can be unstable along the field line and move to the x-point where the colder plasma can reduce the H-mode pedestal. Modeling of the divertor region was carried out with the impact of each of these factors assessed.68,69 Even with all of the known effects included, there were important experimental features that could not be modeled. The crucial missing physics may be the spatially dependent, nonlinear cross-field transport that is the subject of Sec. III E.

The heat load on the divertor is determined by the physics of the boundary plasma and the geometry of the magnetic field and first wall. While the process is simple to define, critical gaps in our understanding prevent reliable prediction and extrapolation to ITER and to future fusion reactors. C-Mod has carried out important research to help fill these gaps and to make direct measurements of the heat footprint under reactor-like conditions. The measurement of the heat load footprint is challenging on C-Mod for reasons very similar to those facing ITER. It is intrinsically hard to get a good view of the vertical target with an infra-red camera due to its geometry and the highly reflective metal walls have low emissivity. Moreover, the surface emissivity is not constant over time since changes in coatings or surface conditions are routine in the high heat-flux areas under study. To meet these challenges, an innovative set of diagnostics was deployed, summarized in Table IV and shown in Fig. 10.70–74 The diagnostics targeted a region of the outer divertor that was modified to provide a slight radial ramp, ensuring that no tile-to-tile shadowing interfered with accurate measurements. Physics-based calibration strategies allowed redundant cross-comparisons adding to confidence in the results. A measure of success is that the overall energy accounting for each shot—power into the plasma vs power deposited on divertor and limiter surfaces—was balanced within 10% for discharges produced over the 3 years of experiments for which the diagnostics were in place.75 

TABLE IV.

Heat-flux footprint diagnostics.

Diagnostic Measurement Analysis/calibration scheme References
Langmuir probes  Plasma Te, ne  Hat flux compared to surface thermocouples through sheath theory  73 and 74  
Retarding field analyzer  Plasma Ti  Compared to CXRS B5+ ion temperature  76 and 77  
Surface thermocouples  Instantaneous surface temperature and heat flux  Integrated and compared to calorimeters  74   
Calorimeters  Bulk temperature and integrated heat flux  Ice-point compensated  73   
IR camera  Instantaneous surface temperature  Emissivity calibrated by comparison with thermocouples imbedded in viewed tiles  70   
Diagnostic Measurement Analysis/calibration scheme References
Langmuir probes  Plasma Te, ne  Hat flux compared to surface thermocouples through sheath theory  73 and 74  
Retarding field analyzer  Plasma Ti  Compared to CXRS B5+ ion temperature  76 and 77  
Surface thermocouples  Instantaneous surface temperature and heat flux  Integrated and compared to calorimeters  74   
Calorimeters  Bulk temperature and integrated heat flux  Ice-point compensated  73   
IR camera  Instantaneous surface temperature  Emissivity calibrated by comparison with thermocouples imbedded in viewed tiles  70   
FIG. 10.

Divertor heat flux diagnostics. Reproduced with permission from Phys. Plasmas 18, 056104 (2011). Copyright 2011 AIP Publishing LLC.72 

FIG. 10.

Divertor heat flux diagnostics. Reproduced with permission from Phys. Plasmas 18, 056104 (2011). Copyright 2011 AIP Publishing LLC.72 

Close modal

A typical measurement of the heat footprint, mapped to the plasma midplane, is shown in Fig. 11,78 which features the highest peak power and narrowest width of any existing device. Surface temperatures regularly exceed 1300 K. The resulting data from C-Mod challenged empirical scalings that existed at the time.79,80 Contrary to the earlier work, C-Mod found that the dominant scaling was 1/IP (or 1/BP) with no dependence on BT, q95, the connection length or on conducted power.72 Overall, the SOL power density profile at the divertor plate mapped to the pressure profile at the midplane,suggesting that critical gradient physics was responsible for setting the former quantity as well. The heat flux footprint was tied to pedestal conditions, consistent with the picture of the near-SOL and pedestal as a single integrated system. In the L-mode and a variety of H-mode regimes, higher pedestal pressures are associated with narrower heat-flux footprints. The higher pressure pedestals are also associated with better global energy confinement17 reinforcing the inherent challenge of achieving good core performance simultaneous with an acceptable divertor solution. C-Mod heat footprint data contributed to an international database, extending the range in BT, BP, plasma pressure, and heat flux to ITER-like values in multi-machine empirical scaling studies.81 The unique diagnostic set on C-Mod also allowed an accurate determination of the sheath transmission factor that relates plasma properties upstream of the sheath to the heat flux conducted to the underlying material. Theoretical calculations predict a value for this factor ≈7, but experimental measurements of this critical quantity have ranged from 2 to 20 (with the values below 5, physically impossible). Using the measurements from the calibrated surface thermocouples and accounting for the non-zero current flowing through the sheath, a good agreement with theoretical models was found, leading to an excellent match between the measured heat flux profile and the value calculated from probe measurements of the local plasma temperature and density (see Fig. 12).74 

FIG. 11.

The heat flux profile measured with the infra-red camera and calibrated against probes and thermocouples. These profiles show the narrowest width and highest power flux measured on any magnetic confinement experiment. Reproduced with permission from Phys. Plasmas 18, 056104 (2011). Copyright 2011 AIP Publishing LLC.72 

FIG. 11.

The heat flux profile measured with the infra-red camera and calibrated against probes and thermocouples. These profiles show the narrowest width and highest power flux measured on any magnetic confinement experiment. Reproduced with permission from Phys. Plasmas 18, 056104 (2011). Copyright 2011 AIP Publishing LLC.72 

Close modal
FIG. 12.

Heat flux profiles calculated based on plasma measurements compare well to the values taken directly from surface diagnostics. Reproduced with permission from Rev. Sci. Instrum. 83, 033501 (2012). Copyright 2012 AIP Publishing LLC.74 

FIG. 12.

Heat flux profiles calculated based on plasma measurements compare well to the values taken directly from surface diagnostics. Reproduced with permission from Rev. Sci. Instrum. 83, 033501 (2012). Copyright 2012 AIP Publishing LLC.74 

Close modal

Measurement of the divertor heat flux is only half the battle. Given the narrow deposition footprints that are currently predicted for ITER,81 methods to reduce the power load to acceptable engineering limits must be found. One solution is to inject a small level of recycling impurities that would radiate near the plasma edge and spread the heat over a larger surface area. The challenge is to effect this change without greatly reducing the heat flux across the separatrix and thus lowering the pedestal height and the overall plasma performance. Experiments were carried out to find the right types and quantities of impurity gas.82 C-Mod was the first to demonstrate good core performance with Demo-like values of radiated power fraction. Using neon and nitrogen gases, these experiments were able to achieve H98 of 1 with conducted power to the divertor normalized to the loss power (PLOSS = PIN-dWdt) as low as 10% as seen in Fig. 13.83,84 Interestingly, the impurity seeding also improved ICRF coupling.85 That effect is not understood but believed to be caused by changes in the edge plasma profiles or fluctuations.

FIG. 13.

Normalized H-mode confinement, H98 is plotted vs Pdiv/Ploss, the power conducted to the divertor normalized to the net input power. By puffing small amounts of impurities, radiation losses can be increased without degrading confinement—meeting ITER operational requirements. Reprinted with permission from Hughes et al., Nuclear Fusion 51, 083007 (2011). Copyright 2011 IOP.83 

FIG. 13.

Normalized H-mode confinement, H98 is plotted vs Pdiv/Ploss, the power conducted to the divertor normalized to the net input power. By puffing small amounts of impurities, radiation losses can be increased without degrading confinement—meeting ITER operational requirements. Reprinted with permission from Hughes et al., Nuclear Fusion 51, 083007 (2011). Copyright 2011 IOP.83 

Close modal

C-Mod data have contributed to a new view of the nature and importance of cross-field transport in the tokamak boundary. Previously, transport in this region of the plasma had been assumed to be Bohm-like and poloidally symmetric (and often chosen arbitrarily as a free parameter to be adjusted to match models). Observations on C-Mod overturned this view, showing distinctly un-Bohm-like behavior, with no dependence on BT and a strong dependence on collisionality86—particle diffusivity is roughly proportional to ν*2 with profiles held near a critical gradient as explained by marginal stability arguments.87,88 Figure 14 shows a set of SOL profiles for the normalized pressure gradient αMHD, which is proportional to the βP gradient. This characterization of the profiles allows them to be overlain for a wide range in operational parameters. The shape of these critical αMHD profiles is consistent with a dependence on collisionality predicted by several theoretical treatments.89,90 Fig. 15 shows the increase of the normalized pressure gradient with normalized inverse collisionality in the regime of high collisionality88 and can be compared directly, for example, to Fig. 1 from Ref. 90. The models predict a very sharp increase in turbulence and transport when the gradient exceeds some nominal threshold, thus enforcing the marginal stability condition.

FIG. 14.

Plasma profiles in the SOL overlay if they are parameterized by the αMHD parameter (essentially the gradient in βP) supporting the hypothesis that the profiles are set by cross field transport at marginal stability. Reproduced with permission from Phys. Plasmas 18, 056104 (2011). Copyright 2011 AIP Publishing LLC.72 

FIG. 14.

Plasma profiles in the SOL overlay if they are parameterized by the αMHD parameter (essentially the gradient in βP) supporting the hypothesis that the profiles are set by cross field transport at marginal stability. Reproduced with permission from Phys. Plasmas 18, 056104 (2011). Copyright 2011 AIP Publishing LLC.72 

Close modal
FIG. 15.

The normalized pressure gradient (αMHD) in the near-SOL depends on strongly on collisionality.88 Reprinted with permission from LaBombard et al., Nuclear Fusion 45, 1658 (2005). Copyright 2005 IOP. αd is the inverse normalized collisionality as defined in Ref. 90.

FIG. 15.

The normalized pressure gradient (αMHD) in the near-SOL depends on strongly on collisionality.88 Reprinted with permission from LaBombard et al., Nuclear Fusion 45, 1658 (2005). Copyright 2005 IOP. αd is the inverse normalized collisionality as defined in Ref. 90.

Close modal

Turbulence and transport delineate two distinct regions of the boundary plasma. Typical profiles can be seen in Fig. 9.91 In the near-SOL, typically a few mm in C-Mod, the plasma gradients are steep and apparently determined by local marginal stability conditions as described above. Fluctuation statistics in this region are “normal,” that is, with symmetric, Gaussian probability distributions.87 Contrary to earlier expectations, the sharp gradients in the near-SOL profile shapes do not continue indefinitely (or until the plasma encounters a material object). Instead, after a relatively short radial distance, very large, isolated fluctuations are torn from the near-SOL and propagate radially due to un-cancelled particle drifts into the “far-SOL,” creating a region of relatively weak gradients.92 These highly intermittent fluctuations, seen in ultra-high-speed images, Fig. 16,93,94 are often referred to as blobs because of their appearance in poloidal cross-section or as filaments because of their extended structures along the magnetic field lines.95 They cannot be understood from local plasma instabilities in the far-SOL—the gradients are too flat—but can be understood as the byproduct of near-SOL turbulence. Under these conditions, the plasma near the wall is not a vacuum and interactions with physical structures are inevitable. That is, the transport that leads to the flat profiles does not allow isolation of the plasma-wall interactions to the divertor as previously thought.96 In particular, particle exhaust is not exclusively through the divertor leading to the phenomenon of “main chamber recycling,” first recognized on C-Mod. Rather than resulting only from leakage out of the divertor, a significant neutral population is built up in the vessel outer midplane through the interaction of the far-SOL and the wall. This result was most clearly demonstrated by the installation of a novel “divertor bypass flap” system by which the divertor could be opened or closed during a C-Mod discharge.97 With the divertor flaps open, neutral pressures in the divertor would decrease by a factor of two while midplane neutral pressured remained unchanged—that is, the pressure in the main chamber was set by its own dynamics not by leakage from the divertor.98 These experiments also showed that divertor leakage had no effect on the L-H power thresholds or the H-mode confinement, contrary to prevailing ideas at the time. Blob dynamics have been compared to a variety of physical models, which can, at least partially, explain their propagation velocity.99,100 A statistical model has been developed, using measurements from C-Mod, which accurately describe the observed probability distribution function over many decades by characterizing the process with just two numbers—the birth duration and the average waiting time between blobs.101–103 These numbers provide a sensitive metric for testing numerical models of near-SOL turbulence, whose dynamics should produce the same statistical quantities.

FIG. 16.

The far-SOL plasma is composed of large amplitude structures (often called “blobs” or “filaments”) that originate in the near-SOL and propagate poloidally and radially. This image is produced by the gas-puff imaging (GPI) diagnostic.

FIG. 16.

The far-SOL plasma is composed of large amplitude structures (often called “blobs” or “filaments”) that originate in the near-SOL and propagate poloidally and radially. This image is produced by the gas-puff imaging (GPI) diagnostic.

Close modal

1. The tokamak density limit as a consequence of edge turbulence

Observations in C-Mod of anomalous cross-field transport in the plasma boundary also provide a likely mechanism for the tokamak density limit,5,104 which has an empirical scaling nG = IP/πa2. There is a general agreement that the limit is associated with progressive cooling of the plasma edge, leading to a shrinkage of the current profile and MHD instability. Unlike the operational limits on safety factor or pressure, the density limit cannot be understood solely through MHD mechanisms, and despite its observation for more than 40 years, no definitive and self-consistent model for the limit has been developed. One class of models that was prevalent before the C-Mod results explains the edge cooling as a consequence, in one way or another, of impurity radiation. These models are based on the explicit dependence of radiated power on plasma density and typically the dependence of radiation cooling curves on temperature.105,106 However, they fail to explain several important observations. First, the density limit does not depend on input power, nor on impurity content (at least for discharges with ZEFF < 2.5), neither is the limit always associated with very high levels of radiated power. Second, while Marfes and divertor detachment can occur near the limit, often they are triggered harmlessly at substantially lower densities.107 An alternate mechanism, tied instead to changes in plasma transport, was motivated by observed changes in particle confinement near the density limit, the nonlinear increase in gas fueling required as the normalized density, n/nG, increased and the observation that the decrease in density during current ramp-down at the end of a plasma shot is often at the rate required to stay just below the density limit.5 That is, the discharge sheds particles during ramp-down to keep n/nG just below 1.

C-Mod carried out experiments to measure the change in edge temperature along with any changes in fluctuations that accompany the approach to the density limit.87,104 Well before the limit was reached, changes in the time-averaged SOL density profiles were observed, with progressive increases in the far-SOL density and overall flattening of the profiles even with modest increases in the separatrix density as shown in Fig. 9. At the same time, the amplitude, frequency, and velocity of blob production increased.103,108 This picture is supported by fluid models, which predict very strong transport under these conditions.90,109 At still higher densities, the boundary between the near-SOL and far-SOL moved inward, with the region of colder plasma, intermittent fluctuations and blob creation110 eventually crossing the separatrix and intruding onto regions of closed field lines as seen in Figs. 17 and 18. The net cooling mechanism is the exchange between warm plasma convected outward and cold fueling gas entering to replace it. When that boundary reaches roughly to the position of 0.85 normalized flux (a movement of about 3 cm on C-Mod), a density limit disruption is triggered. As the density limit is approached, perpendicular transport of energy is significantly increased and given the low upstream temperatures, the parallel energy transport channel is starved. This contrasts with the situation at lower density where all power is lost via the parallel channel to the divertor. In that case, the upstream temperature is pinned to a narrow range, typically to 60–100 eV, at the boundary between open and closed field lines. At densities close to the limit, perpendicular transport dominates on the open field lines and the temperatures can drop to much lower values. The appearance of Marfes or divertor detachment is then inevitable—if the plasma has not detached at lower densities, it will certainly detach near the limit where virtually no power is available in the parallel channel. While the observations coupled to the predictions of turbulence models make a compelling case for turbulence as the underlying cause of the density limit, work remains to develop a predictive model. What is required is a model that can calculate the change in the equilibrium temperature profile as the density is raised, which will require, at a minimum, a flux-driven solution to equations for turbulence and collisional plasma transport coupled to a neutral transport model.

FIG. 17.

Edge temperature profiles show the progressive edge cooling as the normalized density is increased toward nG.

FIG. 17.

Edge temperature profiles show the progressive edge cooling as the normalized density is increased toward nG.

Close modal
FIG. 18.

Probe measurements show the increase in turbulence amplitude and intermittency that occurs as the normalized density ne/nG is raised. Reproduced with permission from Phys. Plasmas 8, 2107 (2001). Copyright 2001 AIP Publishing LLC.87 

FIG. 18.

Probe measurements show the increase in turbulence amplitude and intermittency that occurs as the normalized density ne/nG is raised. Reproduced with permission from Phys. Plasmas 8, 2107 (2001). Copyright 2001 AIP Publishing LLC.87 

Close modal

2. Poloidally asymmetric transport and sonic SOL flows

An important prediction of turbulence models is that transport would have a significant ballooning structure, that is, the turbulence would be stronger on the low-field side (LFS) of the plasma, which has a bad curvature, compared to the high-field side (HFS) with its good curvature. This prediction was tested on C-Mod using an innovative fast-scanning probe, mounted on the inner wall and driven by the tokamak's strong toroidal field crossed with currents in a small coil in the probe mechanism.97,111 (The design is all the more remarkable in requiring that the probe be normally positioned in a protected location behind the inner-wall limiter, reducing space for the radial build for the entire mechanism to only 1 cm.) Fig. 19 shows the normalized fluctuation induced particle flux profiles from several poloidal locations.112 The flux is computed using the measured potential and density fluctuations, accounting for their phase difference and cross-correlation. The result is clear; there is virtually no turbulent transport on the high-field side of the tokamak as expected for modes driven by pressure and curvature. This is confirmed by observations of the profiles, which for the case of carefully balanced double null plasmas find almost no plasma on the high field side.113 For single null plasmas, this region is populated, but only through parallel flows of plasma lost on the low-field side, as shown in Fig. 20.114 The resulting flows can be measured and are found to be near the sound speed as the plasma expands into a near vacuum.113 The effects of these flows on the H-mode threshold are discussed in Sec. IV A.

FIG. 19.

Normalized turbulent flux profiles from the low- and high-field side of the discharge are compared. There is essentially no turbulent transport on the high-field side of the tokamak, consistent with an important curvature drive for the underlying instabilities. Smick et al., Nuclear Fusion 53, 023001 (2013). Copyright 2013 International Atomic Energy Agency.114 

FIG. 19.

Normalized turbulent flux profiles from the low- and high-field side of the discharge are compared. There is essentially no turbulent transport on the high-field side of the tokamak, consistent with an important curvature drive for the underlying instabilities. Smick et al., Nuclear Fusion 53, 023001 (2013). Copyright 2013 International Atomic Energy Agency.114 

Close modal
FIG. 20.

Schematic showing how asymmetric transport drives sonic flows in the SOL.

FIG. 20.

Schematic showing how asymmetric transport drives sonic flows in the SOL.

Close modal

3. Impact of cross-field transport on boundary physics

Experimental results from C-Mod have highlighted the centrality of turbulent transport to a wide range of boundary plasma phenomena. These results challenged the conventional view that anomalous cross-field transport was a secondary effect that could be represented in a simplified parametric form in plasmas that were understood mainly through the lens of collisional transport and atomic physics. Particle exhaust was found to have an important perpendicular component, wherein the plasma-wall interactions could not be isolated to the divertor. The dynamics and thresholds for divertor regimes were found to be sensitive functions of perpendicular transport, which not only competed with parallel processes but also determined the plasma-neutral interactions through the nonlinear increase in fueling required as the normalized density was increased. The same physics led to the tokamak density limit, which should be understood fundamentally as a transport phenomenon in which edge cooling is driven by collisionality-dependent turbulence. The poloidal asymmetry of turbulent transport, which is the result of curvature driven instabilities, causes sonic flows in the SOL. (We will see in Sec. IV that these are likely responsible for important variation in the L-H threshold as well.) The width of the heat-load footprint, at least in the attached state, can also be understood as a manifestation of turbulent transport since the pressure profile at the target maps to the transport-determined midplane pressure. Overall, the conclusion must be that any prediction of plasma boundary and plasma-wall prediction requires deeper understanding of cross-field transport.

The improvement in energy confinement provided by the H-modes is required for ITER baseline operations as well as most tokamak-based reactor designs. Edge transport barriers raise the temperature at the boundary of the plasma, increasing the core gradients through profile stiffness as described in Sec. V A. C-Mod has carried out research in all three key areas of edge barrier physics: access conditions for barrier formation; profile structures in the barrier region and relaxation mechanisms that saturate the profile at equilibrium. The emphasis has been on regimes compatible with high core performance and acceptable divertor physics—that is, on regimes featuring complete suppression of large ELMs. To support these studies, profile diagnostics with resolution close to 1 mm were designed and deployed to measure electron and ion temperature, electron density, and plasma rotation.115–117 C-Mod is also equipped with a set of fluctuation diagnostics including Langmuir probes, magnetic probes, gas-puff imaging (GPI), correlation reflectometry, phase-contrast imaging (PCI), and polarimetry with similar spatial resolution and sensitivity to the short wavelength modes that dominate the edge.94,118–128

Prediction of transport bifurcations, though challenging due to the complexity of the physics, is critical for extrapolation into burning plasma regimes. Without a computable, first-principles model, prediction of the threshold has been based on empirical fits to global operating parameters. At the time that C-Mod was under construction, existing empirical scaling laws for the L-H transition predicted power thresholds that ranged from 100 kW to 10 MW. Given the expected Ohmic and auxiliary power available, the breadth of this range implied that C-Mod might be “always in H-mode” or “never in H-mode.” The wide range arose because of significant correlations in the existing data where machine size, plasma current, and input power all increased together and magnetic field had only a limited variation. Thus, the covariance of the regressors was such that multivariate fits had difficulty separating the effects of the different parameters. When experiments began, C-Mod quickly found a power threshold on the order of 1–2 MW.38 The inclusion of C-Mod data into multi-machine databases improved their overall condition, modified the empirical fit, and led to ostensibly more reliable predictions.129 It is worth noting however that the empirical scalings do not yet capture all of the important dependences seen in the data. A crucial question related to the threshold is the minimum power requirement for full-performance H-modes—driving a need for data that supports a prediction for ITER, where the available power is not far above the empirical scaling. C-Mod experiments showed that the H-factor increases moderately, but linearly with power conducted through the pedestal and that H98 ≈ 1 could be achieved with PCONDUCTED/PTHRESHOLD of about 1 as seen in Fig. 2 of Ref. 83.

As part of its critical contributions to the ITPA databases in support of ITER, C-Mod has carried out a series of dedicated experiments aimed at elucidating the role of parameters not included in the threshold scaling studies and supporting development of first-principles models through characterization of the transition in terms of local physics values. An important observation was the so-called “low-density limit” for the L-H transition.130 Originally characterized as a density threshold,131 carefully controlled studies in C-Mod with otherwise fixed conditions found that the dependence of the power threshold on density, which was roughly linear for the multi-machine power-law regressions, had instead a parabolic shape, with a distinct minimum power point and stronger than linear upturns at both lower and higher densities,132 as seen in Fig. 21. C-Mod was the first device to test directly the empirical scaling of the optimum density nth,opt with magnetic field, confirming that nth,opt ∼ BT,133 a result recently confirmed by experiments on JET.134 The transition between the low and high density branches is consistent with the transition between the sheath-limited and conduction-limited divertor regimes, as considered by Fundamenski et al.,135 although further work is required to understand this connection. Similar results were reported from other experiments, suggesting that the multi-machine fits were capturing only the average behavior of an inherently more complicated dependence. The implications for extrapolation to ITER are still unclear, but it is certain that a future machine cannot count on achieving H-mode at arbitrarily low power by simply lowering the L-mode target density. Neither can a burning plasma device assume that fusion power, increasing as the ion density squared for fixed temperature, will increase as fast as the threshold—that is, the plasma may not be guaranteed to stay in the H-mode during densification as previously assumed. The impact of divertor geometry was also studied on C-Mod where a significant drop, by as much as 50%, in the power threshold was found for a slot divertor when compared to the standard vertical target.136 This reduction is best correlated to the extended low-field side connection length along the divertor arm. In the low density branch, the power threshold is found to be largely insensitive to divertor configuration.

FIG. 21.

The L-H power threshold vs plasma density has a distinct minimum and rises faster than linearly on either side. A strong dependence of the threshold on divertor topology is found on the high density side. Greenwald et al., Nuclear Fusion 53, 14, 104004 (2013). Copyright 2013 International Atomic Energy Agency.197 

FIG. 21.

The L-H power threshold vs plasma density has a distinct minimum and rises faster than linearly on either side. A strong dependence of the threshold on divertor topology is found on the high density side. Greenwald et al., Nuclear Fusion 53, 14, 104004 (2013). Copyright 2013 International Atomic Energy Agency.197 

Close modal

C-Mod carried out some of the first studies on local edge plasma conditions at the transition, finding a critical Te (or ∇Te) at the threshold137,138 that is independent of density, as seen in Fig. 22. These data were used to test emerging theoretical models.139,140 Below the minimum threshold density, the transition may be better characterized as a critical pressure.132 The local threshold is seen to increase roughly linearly with magnetic field, consistent with global scaling. Overall, the results suggest that some of the parametric dependence seen in the scaling laws arises from transition physics (for example, the BT dependence) and some from the nature of L-mode turbulent transport (for example, the density dependence). Studies of hysteresis in the transition dynamics showed stable and unstable operating regions on the bifurcation curve.141,142 The threshold L-mode profiles are roughly consistent with a model that had derived a collisionality-dependent critical pressure gradient for the transition.90 Studies of edge turbulence with GPI have shown nonlinear turbulent kinetic energy transfer from the background drift-wave turbulence into sheared quasi-static flows.143 As suggested by earlier work144,145 these results found that this energy transfer rate equals the local drift-wave growth rate just before the L-H transition. The work on C-Mod showed for the first time that the large H-mode edge profile gradients develop after the transient zonal flow generation and turbulence suppression phenomena—clearly demonstrating the temporal sequence of events that leads to the H-mode regime.

FIG. 22.

The L-H transition was found to have a sharp threshold at a fixed temperature independent of density.137 Reprinted with permission from Hubbard et al., in Proceedings of the 16th International Atomic Energy Agency Conference on Fusion Energy (IAEA, Montreal, 1996), Vol. 1, p. 875. Copyright 1996 IAEA.

FIG. 22.

The L-H transition was found to have a sharp threshold at a fixed temperature independent of density.137 Reprinted with permission from Hubbard et al., in Proceedings of the 16th International Atomic Energy Agency Conference on Fusion Energy (IAEA, Montreal, 1996), Vol. 1, p. 875. Copyright 1996 IAEA.

Close modal

The topology-dependent flows seen in the C-Mod L-modes, described in Sec. III E 2 and shown in Fig. 23, have contributed toward an explanation to a longstanding mystery—that is, the effect of the ion drift direction on the H-mode threshold. Starting with the earliest work on ASDEX, all tokamaks have seen a substantially higher power threshold when the ion ∇B drift direction is away from a single-null x-point (usually called the unfavorable drift direction) when compared to otherwise identical conditions with the ion drift toward the x-point (the favorable drift direction).137,146,147 The difference in the power thresholds, which can be a factor of 2–3, has had no satisfactory explanation. The SOL flows, we have described, are driven by poloidally asymmetric turbulent transport and are always in the co-current direction when in the favorable drift condition and in the counter-current direction for the unfavorable case. This is true for all combinations of toroidal field direction, plasma current direction, and x-point location.113 The flows in the SOL are mirrored by intrinsic flows measured in the core.148 This may be a result of momentum transported from the boundary into the core as described below in Sec. V B. Figure 24 shows the behavior of these flows as a function of magnetic geometry and demonstrates the strong correlation between the geometry, flows, and L-H threshold. In this plot, the x-axis variable named SSEP is the distance between the primary and secondary separatrices mapped to the midplane. Positive values of SSEP correspond to upper single-null geometries and negative values correspond to lower single-null. In this case, the ion drifts are down; thus, negative SSEP is the favorable drift direction. One can see that the flows and threshold are sensitive to geometry on the scale of a few mm—which is a scale length characteristic of the SOL. Since prevalent theories and experimental evidence for the L-H transition points toward flow shear suppression,149 it seems plausible that this change in equilibrium flow direction results in the different power threshold observed. Confirmation will only come, however, with a comprehensive, validated, first-principles model for boundary transport and the L-H bifurcation.

FIG. 23.

The direction of SOL flows driven by poloidally asymmetric radial transport depends only on the direction of the ∇B drift relative to the direction to the x-point. For all combinations of toroidal field, plasma current or X-point direction. Reprinted with permission from LaBombard et al., Nuclear Fusion 44, 1047 (2004). Copyright 2004 IOP.112 

FIG. 23.

The direction of SOL flows driven by poloidally asymmetric radial transport depends only on the direction of the ∇B drift relative to the direction to the x-point. For all combinations of toroidal field, plasma current or X-point direction. Reprinted with permission from LaBombard et al., Nuclear Fusion 44, 1047 (2004). Copyright 2004 IOP.112 

Close modal
FIG. 24.

The L-H power threshold is well-correlated with the topology dependent flows seen in the plasma edge and core. The independent axis, SSEP is the distance between the primary and secondary separatrices, mapped to the midplane. Reproduced with permission from Phys. Plasmas 12, 056111 (2005). Copyright 2005 AIP Publishing LLC.113 

FIG. 24.

The L-H power threshold is well-correlated with the topology dependent flows seen in the plasma edge and core. The independent axis, SSEP is the distance between the primary and secondary separatrices, mapped to the midplane. Reproduced with permission from Phys. Plasmas 12, 056111 (2005). Copyright 2005 AIP Publishing LLC.113 

Close modal

The EDA (Enhanced D-Alpha described in Sec. IV C 1) was the first type of stationary H-mode seen on C-Mod and is, by far, the most prevalent H-mode regime. ELMy H-modes, common on most devices, are also routinely achieved. The ELMy form of H-mode was first seen in dimensionless scaling experiments where C-Mod was run with a shape similar to the JFT-2M tokamak.150 These discharges have good plasma performance, with H98 ≈ 1 and are stationary, with particle and impurity transport apparently controlled by the periodic ELMs and residual fluctuations seen between ELMs. The power threshold for transition to ELMy and EDA discharges are similar. A key ingredient in producing this type of discharge in C-Mod is to place the strike-point deep in the divertor slot. The recycling patterns of this geometry combined with the particle transport intrinsic to the ELMy regime, provide density control and allow operation at plasma densities lower than the more common EDA regime. This leads to lower collisionality and thus to higher edge bootstrap current. In addition the reduced shaping of the discharge in these cases lowers their stability to peeling-ballooning modes. While sufficient to bring the discharge to a stationary state, the ELMs are always “small” in the sense that the reduction in particle and energy inventory is well under 1% per ELM. Stability calculations with ELITE are consistent with operation near the high-n or ballooning side of the peeling-ballooning stability diagram.151 This combination of conditions exist on C-Mod in a restricted window in shaping (δU < 0.3, δL > 0.7, and κ < 1.6).

Data from the ELMy discharges were compared successfully to the EPED model,152 substantially extending the tested data range for magnetic field and pedestal pressure, approaching the values predicted for the ITER pedestal (see Fig. 25).153,154 The EPED model predicts pedestal height and width through the simultaneous solution to linear stability models for MHD peeling-ballooning and kinetic-ballooning modes (KBMs). A good agreement was found, demonstrating weak βP dependence of the pedestal width, consistent with the KBM arguments. Ideal infinite-n ballooning mode calculations as a proxy for the KBM also show marginal stability to KBM. Recent electromagnetic signatures observed between ELMs and described below are possible evidence for KBM pedestal-regulating activity.155 Separately, ELM precursors have also been documented along with multiple “secondary” filaments following the primary ELM filament ejection.156 

FIG. 25.

The measured profile structure is consistent with the EPED model, extending the world database to within a factor of three of what is expected on ITER. Hughes et al., Nuclear Fusion 53, 8, 043016 (2013). Copyright 2013 International Atomic Energy Agency.154 

FIG. 25.

The measured profile structure is consistent with the EPED model, extending the world database to within a factor of three of what is expected on ITER. Hughes et al., Nuclear Fusion 53, 8, 043016 (2013). Copyright 2013 International Atomic Energy Agency.154 

Close modal

While the H-mode provides a good energy confinement needed for burning plasma devices like ITER and fusion reactors, it brings with it several unfavorable characteristics. The particle transport barrier can be too good, with the potential to accumulate high-Z impurities and the concomitant high levels of radiated power. Even more daunting is the prospect of large ELMs, resulting from an overly steep pressure gradient. In the absence of other mechanisms, large ELMs relax this gradient leading to periodic exhausts of power that cannot be tolerated in large-scale devices.157,158 Methods of external control that increase the frequency of ELMs and thus decrease their impact are being explored,159,160 but their applicability and reliability in a reactor environment is uncertain. Thus, there is an unmet need for intrinsic operating regimes with a good energy confinement but with either very small ELMs or with none at all. Two such ELM-suppressed regimes have been discovered and studied on C-Mod. These are the EDA or the Enhanced D-Alpha H-mode and the “Improved” or I-mode, which are obtained at a high and low collisionality, respectively.

1. EDA H-mode

The EDA regime is the standard H-mode on C-Mod, seen early in its operation upon the first application of high-power ICRF in a well-conditioned machine.17,161 Compared to ELM-free operation, EDA tends to be favored at higher collisionality (or higher density) and higher safety factor (q95).162 It is also found that EDA is achievable in hydrogen at lower q95 than it is in deuterium. A dependence on shaping has also been seen but this is complicated and not fully understood.162,163 The EDA regime is not specific to ICRF heating as it is obtained in Ohmic H-modes when similar access conditions are met. Energy confinement in EDA can be variable, but H89, the energy confinement time normalized to the ITER89 L-mode scaling164 in the range 1.6–2.0 was readily obtained.17 C-Mod EDA data were part of the international collection used to develop the ITER98 H-mode scaling laws.165,166 The salient feature of the EDA regime (and the reason for its name) is the high levels of recycling light. Compared to an ELM-free H-mode, where the level of radiation from neutral deuterium (or hydrogen) drops sharply at the L-H transition and remains very low, in EDA, this signal returns quickly to or exceeds L-mode like values. The implication is that edge particle transport is much higher in EDA. Impurity transport is even more strongly affected. While impurities can accumulate in an ELM-free discharge, they pump out readily in EDA as seen in Fig. 26 as a sudden change in the time derivative of radiated power. The result is that impurity radiation and electron density are under control, allowing a stationary state to be achieved. The main features of the EDA can be seen in Fig. 27, which compares traces from similar 1 MA, 5.4T EDA, and ELM-free discharges. Notable is the stationary particle and energy content, the lower levels of impurity radiation and the difference in deuterium Balmer-α brightness. The change in particle transport is associated with and attributed to a plasma fluctuation not seen in ELM-free discharges. The transport caused by this mode is apparently sufficient to hold the pressure gradient below the MHD stability threshold and avoid any large ELMs.154,167 More detail on the characteristics and effects of this fluctuation, termed the “quasi-coherent mode” or QCM, are given in Sec. IV D. In some EDA discharges, typically with βN > 1.2, very small, energetically insignificant ELMs are also observed.

FIG. 26.

A sudden transition from the ELMfree to the EDA H-mode is accompanied by a change in impurity confinement and the appearance of the QCM. Greenwald et al., Fusion Sci. Tech. 51, 266 (2007). Copyright 2007 American Nuclear Society.185 

FIG. 26.

A sudden transition from the ELMfree to the EDA H-mode is accompanied by a change in impurity confinement and the appearance of the QCM. Greenwald et al., Fusion Sci. Tech. 51, 266 (2007). Copyright 2007 American Nuclear Society.185 

Close modal
FIG. 27.

A comparison of waveforms between the ELM-free and the EDA H-modes shows the essential stationary character of the EDA and the drop in particle and impurity transport. Greenwald et al., Fusion Sci. Tech. 51, 266 (2007). Copyright 2007 American Nuclear Society.185 

FIG. 27.

A comparison of waveforms between the ELM-free and the EDA H-modes shows the essential stationary character of the EDA and the drop in particle and impurity transport. Greenwald et al., Fusion Sci. Tech. 51, 266 (2007). Copyright 2007 American Nuclear Society.185 

Close modal

The pedestal in the EDA H-mode is narrow in C-Mod, typically 2–4 mm,168 and spans roughly the same fraction of the normalized poloidal flux as pedestals in ELMy discharges.154 The pressure at the top of the pedestal scales with IP2, with the dependence equally partitioned between density and temperature. The dependence on other parameters such as plasma density, toroidal field, or shaping is weaker—suggesting that, as in the SOL, βP or αMHD is the controlling parameter. To further investigate the importance of plasma physics vs atomic physics (i.e., neutral penetration) in determining the pedestal structure, a series of dimensionless identity experiments was carried out in collaboration with DIII-D. The experiments matched all geometric and plasma dimensionless parameters at the top of the pedestal. The result was a good match across the entire pedestal—suggesting that plasma physics alone can account for the structure of the density and temperature profiles.169 Later similarity studies conducted in ELMy H-modes showed evidence of a mismatch in density pedestals, with the DIII-D pedestal being wider in flux space.170 These matches required DIII-D to operate at its lowest feasible H-mode densities, in a regime known to show a dependence of the pedestal width on neutral penetration171 that is not seen on C-Mod, suggesting that the neutral penetration range can have an effect on sufficiently transparent pedestals.172 However, C-Mod has the highest neutral opacity of any operating tokamak and should be more prototypical of ITER/Reactor conditions.

2. I-mode

By operating at high power under conditions where the L-H threshold is elevated, C-Mod has explored a new and even more promising regime—the I-mode (short for Improved Mode).116,173,174 The I-mode combines the H-mode like energy confinement (H98 ≈ 1) with the L-mode particle and impurity confinement, and is generally ELM-free. The change in global confinement and the drop in core thermal diffusivity are mirrored by a drop in core fluctuations, with δne/ne decreasing by 30% and δTe/Te by at least 70%.175 (The I-mode regime described here must be clearly distinguished from “I-phase” a regime of fast dithering between the L and H-modes reported on some machines at powers near the L-H threshold.176) Most commonly, the I-mode is accessed by running with the ion ∇B drift in the direction unfavorable for the H-mode operation, though it has also been observed in the standard configuration. The window in input power is higher for the “unfavorable” field direction, and allows powers up to about 2× the L-I threshold before an I-H transition is encountered. The I-mode was probably first obtained in some of the earliest ICRF heating experiments on C-Mod in 1996, where improved energy confinement and higher pedestal temperatures were seen for “reversed field L-modes.” They were categorized as L-modes due to the lack of density rise that accompanies the H-mode.17 Similar plasmas were called improved L-mode on ASDEX-Upgrade.177 Limitations on diagnostic coverage and the lack of an accurate H-mode scaling law at the time prevented clear recognition of this phenomena as a new and distinct confinement regime. The higher local Te threshold and multi-step dynamics with unfavorable drift were later studied in more detail.178 These early observations suggests that the I-mode is not an exotic and elusive regime, but is rather a standard behavior under suitable conditions.

Figure 28 shows time traces from a typical I-mode and illustrates the salient characteristics.179 While the edge temperature and stored energy increase markedly as they would in an H-mode, there is no change observed in the plasma density at the L-I transition or thereafter. This difference is seen dramatically in Fig. 29, which compares the edge profiles between the L-mode, the I-mode, and the H-mode. The I-mode temperature profile is distinctly H-mode like, while the density profile retains the L-mode shape and values.154 Just as in the H-mode, an Er well is observed in the I-mode,116,180 though shallower than in the H-mode. The pedestal pressure gradient is lower in the I-mode than in the H-mode and stability analysis finds it stable to peeling-ballooning.151,154 The L-mode like density profile is probably responsible for the lack of ELMs due to its impact on the pressure gradient and a drop in the density gradient drive for bootstrap current. In contrast to the EDA regime, where high collisionality is responsible for reducing the bootstrap current, I-modes are amongst the lowest collisionality, improved confinement regimes in C-Mod, with ν* at the top of the pedestal as low as 0.1. Weak, energetically insignificant ELMs are observed in some I-modes, arising from pedestals far from the peeling-ballooning boundary and are often triggered by sawteeth. Impurity transport in the I-mode is essentially at the L-mode levels as seen in Fig. 30, which plots the energy and impurity confinement time measured from calcium impurities injected via laser blow-off (LBO)181 for the three regimes. As a result, the I-mode performance is considerably less sensitive to wall conditioning (boronization) than the H-mode's and more easily compatible with high Z PFCs and impurity seeding than the H-modes.

FIG. 28.

The transition from the L to I-mode is shown, demonstrating the increase in energy confinement without a change in particle transport or the appearance of ELMs.179 Reprinted with permission from Hubbard et al., in Proceedings of the 24th IAEA Fusion Energy Conference (IAEA, San Diego, 2012), see http://www-naweb.iaea.org/napc/physics/FEC/FEC2012/papers/171_EX13.pdf. Copyright 2012 IAEA.

FIG. 28.

The transition from the L to I-mode is shown, demonstrating the increase in energy confinement without a change in particle transport or the appearance of ELMs.179 Reprinted with permission from Hubbard et al., in Proceedings of the 24th IAEA Fusion Energy Conference (IAEA, San Diego, 2012), see http://www-naweb.iaea.org/napc/physics/FEC/FEC2012/papers/171_EX13.pdf. Copyright 2012 IAEA.

Close modal
FIG. 29.

The profiles of electron temperature and density are compared for the L-mode, H-mode, and I-mode.

FIG. 29.

The profiles of electron temperature and density are compared for the L-mode, H-mode, and I-mode.

Close modal
FIG. 30.

The confinement times of calcium impurities, injected via the laser blow-off technique, are plotted vs normalized energy confinement and confirm the L-mode-like particle transport for the I-mode plasmas.

FIG. 30.

The confinement times of calcium impurities, injected via the laser blow-off technique, are plotted vs normalized energy confinement and confirm the L-mode-like particle transport for the I-mode plasmas.

Close modal

Overall, the I-mode has the advantages of the H-mode without its drawbacks. The I-mode is an ELM-suppressed, high-temperature, low collisionality regime without impurity accumulation. The density can be controlled by gas puffing, and the density profiles are mildly peaked as in the L-modes or the low-collisionality H-modes.182 Strong fueling also seems to help avoid the I-H transition. So far, limits to the I-mode performance have been set by the power available on C-Mod (4–5 MW). Based on current results, it might be possible to operate at Q = 10 if an I-mode could be achieved in ITER,179 though much more information is needed on density, field, power, and size dependence for the I-mode access and on the confinement properties of the regime across a larger range of machines. The divertor heat footprint in the I-mode is somewhat wider than that in the H-mode, and more equal power sharing is seen between the inner and outer strike point—both favorable characteristics for divertor power handling.78 And aside from the intrinsic interest in the I-mode as an attractive reactor regime, the decoupling of energy and particle barriers should also illuminate the physics of edge barriers.

Short-wavelength electromagnetic fluctuations apparently play an essential role in regulating pedestal profiles in all edge barrier regimes observed on C-Mod. When sufficiently strong, these fluctuations can maintain the pressure gradient below the threshold for peeling-ballooning and effectively suppressing ELMs. These observations suggest the possibility of external control by launching waves that stimulate or destabilize this class of plasma fluctuation.

1. The QCM in EDA H-modes

In EDA H-modes, the increase of particle and impurity transport over ELM-free discharges is due to very large amplitude, narrow-band fluctuations observed by every diagnostic capable of detecting short-wavelength fluctuations in the plasma edge, including reflectometry,119,183 phase-contrast imaging,121 Langmuir probes,184 magnetic loops,120 gas-puff imaging,110 and polarimetry.127 The QCM frequency is typically in the range of f ∼ 50–150 kHz and field aligned (k·B = 0) with an outer-midplane poloidal wavenumber, kθ ∼ 1.5 cm−1. As suggested by the name, the QC frequency spread is typically small, δf/f ∼ 0.05–0.15. The evolution of the autopower spectrum of this mode in typical EDA H-modes is shown in Fig. 31. It may be notable that broadband fluctuations in the same ω and k range are prevalent in L-mode discharges and believed to be a key component in boundary plasma transport. Multi-field measurements of the mode have been made recently using “mirror probe” electronics,184 showing mode amplitudes in plasma density, δne/ne ∼ 0.3; electron temperature δTe/Te ∼ 0.45, plasma potential δϕ/Te ∼ 0.45, and magnetic field, δB ∼ 0.4 mT, δJ ∼ 25 A/cm2. Measurements from an active antenna (described in Sec. IV D 4) find that the mode has weak damping or growth rates with γ/ω on the order of 5%–10%. This suggests weak nonlinearities in the mode dynamics and may explain the narrow frequency width even as the mode grows to such large amplitudes. The mode can be precisely located in Ohmic EDA H-modes, where power levels are low enough to make probe measurements across the pedestal, and is found to span the separatrix with a full-width at half maximum of 2–3 mm. That places it in a region of positive Er, that is, with the E × B flow in the ion diamagnetic direction. These measurements allow the calculation of the wave propagation in the plasma frame, which is found to be unambiguous in the electron diamagnetic direction. As seen in Fig. 31, the mode often chirps to lower frequency as the EDA H-mode develops—likely a result of the change in the Doppler shift as the equilibrium electric field well deepens. The connection between the mode and enhanced particle transport can be seen macroscopically—as the near-SOL particle diffusivity is proportional to the mode amplitude185—or can be computed microscopically from the fluctuation amplitude and phase relations. The plasma potential fluctuations are found to lag the density fluctuations by 16 deg and the temperature fluctuations by 7 deg, consistent with the identification of the mode as a drift wave. The Te response is not simple Boltzmann, perhaps not surprising given the electromagnetic character of the wave. With these observations, we would describe the mode as an electromagnetic drift-wave, driven by pressure gradients and curvature. Fluid simulations90,186 find similar modes and suggest that the mode is probably modified by the magnetic shear near the plasma x-point.

FIG. 31.

The density fluctuation spectra are shown for a discharge with three H-mode periods. The first is ELM-free followed by two EDA intervals with the presence of a strong QCM. Greenwald et al., Fusion Sci. Tech. 51, 266 (2007). Copyright 2007 American Nuclear Society.185 

FIG. 31.

The density fluctuation spectra are shown for a discharge with three H-mode periods. The first is ELM-free followed by two EDA intervals with the presence of a strong QCM. Greenwald et al., Fusion Sci. Tech. 51, 266 (2007). Copyright 2007 American Nuclear Society.185 

Close modal

2. The weakly-coherent mode (WCM) in the I-mode

The fluctuations apparently responsible for regulating the pedestal in the I-mode are at somewhat higher frequency than the QCM, typically in the range of 200–300 kHz and considerably broader, δf/f ∼ 20%–50%, but with a similar wavelength, kθ ∼ 1.5 cm−1. The WCM fluctuations can readily be seen in diagnostics looking at electron density, electron temperature, and magnetic field. The fluctuation amplitudes are smaller than for the QCM with δne/ne on the order of 10% and δTe/Te in the range of 1%–2%.187 The appearance of the WCM is accompanied by a drop in lower frequency, broad band fluctuations as seen in Fig. 32. The effective particle diffusivity, DEFF ∝ Γ/∇n, is found to be proportional to the amplitude of the WCM,188 supporting its role in the I-mode particle transport. By contrast, the amplitude of fluctuations below 150 kHz are strongly correlated with energy diffusivity,174 further suggesting that turbulence in this range is responsible for energy transport in the L-mode target plasmas. Geodesic Acoustic Modes (GAMs), a fluctuating form of zonal flows have been observed in the I-mode and persist throughout the regime. These play a critical role in the development of the WCM as demonstrated by bispectral analysis that shows power transfer between the WCM and the GAM.189 This observation also suggests that the GAM is responsible for depleting power from the lower frequency turbulence and thus in the suppression of energy transport in the I-mode. The I-mode is the only regime in C-Mod with coexisting strong mean and fluctuating shear flow. These observations may help us understand how the transport bifurcation occurs in two distinct steps—L to I and then I to H, along with the difference in transport characteristics in each of the regimes.

FIG. 32.

Density fluctuations are shown a discharge transitions from the L- to I- to H-mode. The I-mode is typically accompanied by the appearance of a WCM at frequencies above 200 kHz and the reduction in lower frequency fluctuations. Reproduced with permission from Phys. Plasmas 18, 056115 (2011). Copyright 2011 AIP Publishing LLC.152 

FIG. 32.

Density fluctuations are shown a discharge transitions from the L- to I- to H-mode. The I-mode is typically accompanied by the appearance of a WCM at frequencies above 200 kHz and the reduction in lower frequency fluctuations. Reproduced with permission from Phys. Plasmas 18, 056115 (2011). Copyright 2011 AIP Publishing LLC.152 

Close modal

3. Fluctuations that regulate transport in ELMy discharges

Short-wavelength electromagnetic fluctuations also play a role in ELMy discharges, though in these cases the effects are not large enough to prevent the larger-scale MHD instabilities from arising. The conventional picture is that the pedestal pressure profiles come into equilibrium rather quickly between the ELMs, which are caused by unstable current profiles that take longer to evolve.190,191 (Recent results suggest some modification of this picture, suggesting relatively rapid pedestal current evolution in the ELM cycle,192 and detailed profile analysis has shown that more subtle evolution is possible, with pressure gradients saturating early in the ELM recovery phase, followed by a slower increase of both the pedestal height and the width that eventually results in a peeling-ballooning instability.170) A long-standing question concerns the transport processes that dominate the pressure profile evolution between the ELMs. As discussed above, the successful EPED pedestal model is based on the hypothesis that KBMs control the pressure profile during the build-up to an ELM.152 On C-Mod, Te drops after each ELM then quickly recovers. As the temperature recovers, pedestal localized fluctuations are observed to build up, as seen in Fig. 33, with kθρs ∼ 0.04, that is, with wavenumbers somewhat lower than for the QCM and WCM discussed above.155 The mode amplitude scales with electron β (Te and Ti are equilibrated) consistent with expectations for the KBM. Immediately after each ELM, this mode disappears. Stability analysis with the GS2 gyrokinetic code193 finds a mode at kθρs ∼ 0.03 with KBM characteristics and work on mode identification is ongoing.

FIG. 33.

Magnetic fluctuations are shown to grow rapidly as the plasma temperature recovers between the ELMs. These fluctuations may be evidence for kinetic ballooning, a key element in the EPED model for pedestal structure. Hughes et al., Nuclear Fusion 53, 8, 043016 (2013). Copyright 2013 International Atomic Energy Agency.154 

FIG. 33.

Magnetic fluctuations are shown to grow rapidly as the plasma temperature recovers between the ELMs. These fluctuations may be evidence for kinetic ballooning, a key element in the EPED model for pedestal structure. Hughes et al., Nuclear Fusion 53, 8, 043016 (2013). Copyright 2013 International Atomic Energy Agency.154 

Close modal

4. External control of edge transport

These results open the possibility that the pedestal transport, the overall plasma performance, and the presence of ELMs might be controlled through an external means. Early success was achieved using microwaves in the LH range of frequencies. In these experiments, a modest level of LH power at 4.6 GHz is applied to the high density EDA H-mode discharges (ne > 2 × 1020/m3) where the waves have little or no accessibility to the core plasma.194 A large fraction of the launched power appears promptly on the outer divertor target, supporting modeling that shows that the waves propagate only in the plasma edge. The experimental result can be seen in Fig. 34(a), which shows an increase in pedestal temperature when the LH was applied to an ICRF heated H-mode. In the highest density cases, the overall stored energy increase can be as much as 30%, which would require almost 3 MW of additional heating if the confinement were fixed and followed the H98 scaling. In the experiment, this was accomplished with only 0.6 MW of the LH power. At the same time, we have observed the level of edge fluctuations on flux tubes, which pass in front of the LH launcher, to drop by almost an order of magnitude as seen in Fig. 34(b). The mechanism by which the LH decreases energy transport is under investigation.

FIG. 34.

(a) The increase in the temperature pedestal after the application of a small increment in RF power in the lower-hybrid range of frequencies applied to plasmas strongly heated by ICRF. (b) The increase in temperature is accompanied by a drop of almost an order of magnitude in edge fluctuations.

FIG. 34.

(a) The increase in the temperature pedestal after the application of a small increment in RF power in the lower-hybrid range of frequencies applied to plasmas strongly heated by ICRF. (b) The increase in temperature is accompanied by a drop of almost an order of magnitude in edge fluctuations.

Close modal

Another approach tries to more directly mimic the intrinsic plasma behavior by driving the QCM- or WCM-like fluctuations in the plasma with an external antenna.195 This so-called “shoe-lace” is essentially an active MHD antenna for short wavelength electromagnetic modes. The antenna is named after the geometry of the antenna winding, which can drive currents in the plasma edge in the relevant k range (see Fig. 35). An innovative matching network allows consistent coupling across a wide range of frequencies, 50–300 kHz.196 With the existing RF sources, the antenna currents can reach about 80 A and could be increased further without excess antenna heating. Because of the rapid fall-off in these short wavelength perturbations, the launching structure incorporates protection tiles and is designed to operate safely with no more than 1 cm clearance between the windings and the plasma edge. The antenna can be operated in a passive mode as a sensitive receiver, but the primary mode is active, where waves launched by the antenna are observed with the array of C-Mod edge fluctuation diagnostics. The antenna frequency can be swept to look for resonances with the plasma or phase locked to existing perturbations. In the H-mode, when a plasma pressure pedestal is present, the antenna will drive both the magnetic and density fluctuations. The density response is absent in the L-mode, suggesting that the antenna strongly interacts with modes that are driven by the pressure gradient. From the plasma response, a transfer function is computed, peaking when the drive frequency equals the QC frequency, as seen in Fig. 36. The density perturbation approaches the intrinsic mode amplitude when the drive is on resonance. A response is seen in the H-mode plasmas even if the QC mode is absent, indicating a damped rather than growing instability in those cases. From the transfer function calculation, it is determined that the plasma mode is propagating in the electron diamagnetic direction and is weakly damped or growing with γ/ω ∼ 0.05–0.10. This observation helps account for the high level of coherency. The narrow spectrum is consistent with the lack of strong nonlinear damping, limiting the spread in k and ω space. Future experiments will attempt use this actuator as a tool to modify pedestal transport.

FIG. 35.

A 3D rendering of the “shoelace” antenna, which can drive short wavelength magnetic perturbations in the plasma edge.

FIG. 35.

A 3D rendering of the “shoelace” antenna, which can drive short wavelength magnetic perturbations in the plasma edge.

Close modal
FIG. 36.

The magnetic perturbation applied by the shoe-lace antenna drives a strong plasma density response when the drive frequency is at or near resonance with the naturally occurring QCM. Reproduced with permission from Phys. Plasmas 21, 056111 (2014). Copyright 2014 AIP Publishing LLC.113 

FIG. 36.

The magnetic perturbation applied by the shoe-lace antenna drives a strong plasma density response when the drive frequency is at or near resonance with the naturally occurring QCM. Reproduced with permission from Phys. Plasmas 21, 056111 (2014). Copyright 2014 AIP Publishing LLC.113 

Close modal

Core transport studies in C-Mod generally feature strong electron heating, equilibrated electrons and ions, no external torque, and no core particle sources. The exclusive use of RF for heating provides a particularly good platform for studies of intrinsic rotation and particle transport. Several of the dimensioned quantities, BT, ne stand well apart from other experiments, but discharges with substantial overlap in dimensionless plasma parameters are also obtainable.185,198 C-Mod provided important contributions to the H-mode confinement database. Operating at higher current and input power than other small devices, these data broke important covariances between size, current and power, the most important scaling parameters, and led to the ITER98y scaling laws.165,166 It is worth noting that the C-Mod data, used in this database, were obtained in the ELM-suppressed regimes, with dominant electron heating, low torque, Te ∼ Ti, and in a device with metal walls. All of these are ITER-typical and different from conditions behind most of the data in the confinement database. Recent results from AUG and JET find a drop in energy confinement under similar conditions,44 suggesting that ITER98 may overestimate the results that will be obtained on ITER.

Early H-mode studies noted the simultaneous increase in core energy confinement and the formation of an edge transport barrier;131 however, the first quantitative studies of the correlation between the pedestal and core transport were carried out in C-Mod.17 These studies found a linear relationship between the height of the temperature pedestal and the normalized confinement time as shown in Fig. 37, unifying the C-Mod database across confinement regimes. It was found that the correlation was due to the self-similarity of temperature profiles. Figure 38 shows temperature profiles for a collection of 100 randomly chosen shots and times, at a wide variety of plasma density, heating power, impurity content, and regime (OH, L, and H).185 The temperature is plotted on a log scale, demonstrating constancy of the logarithmic gradient 1/LT = ∇T/T over almost an order of magnitude variation in temperature magnitude. This result is consistent with transport theory that predicts a drift-wave stability threshold dependent on R/LT and a strong turbulence and transport for normalized gradients that exceed the threshold.199 These theories also predict a dependence of the critical gradient length on magnetic shear; thus, the shots in Fig. 38 were selected at the same magnetic field and plasma current. Nonlinear gyrokinetic simulations found, in fact, quantitative agreement between the experimental temperature gradient and the gradient computed to match the experimental heat flux.200 These results also help to explain the insensitivity of the L-mode confinement to impurity radiation. It was observed that the L-mode confinement followed the empirical scaling even when virtually all power was lost through radiation before reaching the plasma edge, as seen in Fig. 39.17 Apparently even the greatly reduced levels of heat conduction seen for the high radiated power were sufficient to sustain the plasma near the marginal stability point. In contrast, the H-modes are sensitive to the radiated power fraction through the degradation of the pedestal. Taken together, this work has suggested that the flux-gradient response is a more useful model than one that characterized heat transport in terms of the thermal diffusivity. The implications for burning plasmas, like ITER, are that fusion power will be strongly linked to the pedestal temperature. It is also worth noting that these observations combined with those for the pedestal and SOL, described in Secs. III and IV, suggest that most of the plasma is organized by marginal stability conditions.

FIG. 37.

The energy confinement time, normalized to the ITER89 L-mode scaling law, is plotted vs the pedestal temperature, unifying data over a wide range of parameters and confinement regimes. Reprinted with permission from Greenwald et al., Nuclear Fusion 37, 793 (1997). Copyright 1997 IOP.17 

FIG. 37.

The energy confinement time, normalized to the ITER89 L-mode scaling law, is plotted vs the pedestal temperature, unifying data over a wide range of parameters and confinement regimes. Reprinted with permission from Greenwald et al., Nuclear Fusion 37, 793 (1997). Copyright 1997 IOP.17 

Close modal
FIG. 38.

Profile self-similarity is demonstrated. Te profiles are plotted, on a semi-log scale for a random selections of shots and time including the OH, L-mode, and H-modes at a wide range in density and input power. Greenwald et al., Fusion Sci. Tech. 51, 266 (2007). Copyright 2007 American Nuclear Society.185 

FIG. 38.

Profile self-similarity is demonstrated. Te profiles are plotted, on a semi-log scale for a random selections of shots and time including the OH, L-mode, and H-modes at a wide range in density and input power. Greenwald et al., Fusion Sci. Tech. 51, 266 (2007). Copyright 2007 American Nuclear Society.185 

Close modal
FIG. 39.

Normalized energy confinement for the L-mode can be maintained, even at very low levels of conducted power. In contrast, the H-mode confinement deteriorates at a high radiated power because of the decrease in pedestal temperature. Reprinted with permission from Greenwald et al., Nuclear Fusion 37, 793 (1997). Copyright 1997 IOP.17 

FIG. 39.

Normalized energy confinement for the L-mode can be maintained, even at very low levels of conducted power. In contrast, the H-mode confinement deteriorates at a high radiated power because of the decrease in pedestal temperature. Reprinted with permission from Greenwald et al., Nuclear Fusion 37, 793 (1997). Copyright 1997 IOP.17 

Close modal

The nonlinear flux-gradient response is mirrored by observations of core fluctuations and transient heat transport. For example, transport in the H-mode is reduced compared to the L-mode, not only at the edge but also in the core. A matched pair of the L and H-modes with the same IP, BT, and PRF have the same temperature gradient scale length and the same heat flux but with the plasma temperature, temperature gradient, and plasma density significantly higher in the H-mode (from which one would calculate a factor of 2 reduction in thermal diffusivity).17 The appropriate normalization for heat flux is the gyro-Bohm power ∝ nT3/2 and is thus substantially higher for the H-mode parameters, implying that the normalized heat flux is lower for the H-mode case than for the L-mode. In the experiments, fluctuations, n ̃ e / n e , T ̃ e / T e , in the core of L-mode are found to be higher, as expected from these arguments.201 Similar observations are seen when comparing the core of the L and I-mode plasmas.175 The rapid propagation of temperature perturbations, for example, from sawtooth oscillations is consistent with this picture.202 The perturbations respond to the steep slope of flux vs gradient that exists at the discharge operating point. By comparison, the thermal diffusivity is simply proportional to the ratio of flux to gradient and is thus much lower than the local slope and does not predict the fast evolution of the profile that is observed.

Enabled by a high-resolution X-ray imaging crystal spectrometer (XICS),203 capable of measuring plasma rotation profiles without injecting momentum (as with beam based diagnostics), C-Mod has pioneered studies of self-generated equilibrium flows.204,205 Strong co-current rotation, with toroidal velocities up to 130 km/s (about 0.3 times the sound speed), has been observed and is strongest in enhanced confinement plasmas, i.e., the H- and I-mode. Under otherwise similar condition, the rotation state is independent of whether the plasma is heated with ICRF power or Ohmically, suggesting that the underlying mechanism is independent of the specific heating method.206,207 As seen in Fig. 40, the core rotation velocity scales in proportion to the global plasma energy (or pressure) divided by the plasma current, that is, generally increasing with input power, but is significantly higher in the H-mode than in the L-mode for the same power.205,208,209 Subsequent analysis of a multi-machine database found that the rotation could be characterized as a toroidal Mach number vs βN.209,210 By following the evolution of the profiles, it was shown that the intrinsic rotation originates at the plasma edge and propagates into the core.211,212 Core rotation in the H-mode is strongly coupled to the pedestal temperature gradient for both the H-modes and I-modes, as seen in Fig. 41. This dependence is consistent with the model that this rotation is driven by residual stress, that is, the part of momentum flux which is not proportional to the flow velocity or its gradient.213 The E × B shearing rate of intrinsic rotation is apparently large enough to effect transport through well-known turbulence stabilization mechanisms214,215 and is thought to play a role in the formation of internal transport barriers (ITB) in C-Mod216,217 as described in Sec. V D.

FIG. 40.

Intrinsic rotation scales with stored energy divided by the plasma current and is independent of the heating method. Reprinted with permission from Rice et al., Nuclear Fusion 41, 277 (2001). Copyright 2001 IOP.251 

FIG. 40.

Intrinsic rotation scales with stored energy divided by the plasma current and is independent of the heating method. Reprinted with permission from Rice et al., Nuclear Fusion 41, 277 (2001). Copyright 2001 IOP.251 

Close modal
FIG. 41.

Intrinsic toroidal rotation is proportional to the pedestal temperature gradient for both H-modes (green) and I-modes (red), consistent with the model that this rotation is driven by residual stress. Rice et al., Phys. Rev. Lett. 106, 265001 (2011). Copyright 2011 American Physical Society.213 

FIG. 41.

Intrinsic toroidal rotation is proportional to the pedestal temperature gradient for both H-modes (green) and I-modes (red), consistent with the model that this rotation is driven by residual stress. Rice et al., Phys. Rev. Lett. 106, 265001 (2011). Copyright 2011 American Physical Society.213 

Close modal

In Ohmic plasmas, the intrinsic rotation has a complicated dependence on collisionality, plasma current, and geometry.218 A substantial counter-current rotation, up to −60 km/s, has been observed in some discharges with unfavorable ∇B drift. As noted in Sec. IV, this observation is connected to the H-mode power threshold.148 Core rotation reversals, abrupt changes of the toroidal rotation direction, have been observed to occur at a q-dependent, critical value of the collisionality.219 The reversals, seen in Fig. 42, can be induced with slight changes in the electron density, plasma current, or toroidal magnetic field, and are often accompanied by abrupt changes in turbulence characteristics220,221 and also, unexpectedly, are associated with the transition from the linear Ohmic confinement (LOC) to the saturated Ohmic confinement (SOC) regimes.222,223 This confinement transition is generally attributed to a transition from an electron-transport to ion-dominated turbulent transport.224 The SOC seems in most respects to be identical to the ITG (ion thermal gradient) dominated L-mode while in the LOC, an electron transport channel opens up as the density is lowered and causes energy confinement to drop precipitously.

FIG. 42.

Reversal of intrinsic rotation occurs dramatically at a q-dependent, critical density. The transition from LOC-SOC confinement and from non-local to local transient transport occur at the same density. Reprinted with permission from Rice et al., Nuclear Fusion 53, 033004 (2013). Copyright 2013 IOP.225 

FIG. 42.

Reversal of intrinsic rotation occurs dramatically at a q-dependent, critical density. The transition from LOC-SOC confinement and from non-local to local transient transport occur at the same density. Reprinted with permission from Rice et al., Nuclear Fusion 53, 033004 (2013). Copyright 2013 IOP.225 

Close modal

Other seemingly unrelated changes occur at the same critical collisionality. The non-local heat transport, core toroidal rotation reversals, energy confinement saturation, and up/down impurity density asymmetry are correlated with each other experimentally. That is, at low densities in the linear Ohmic confinement regime, with collisionality ν* ≤ 0.35 (evaluated inside of the q = 3/2 surface), heat transport exhibits non-local behavior, core toroidal rotation is directed co-current, edge impurity density profiles are up/down symmetric, and a turbulent feature in line-integrated core density fluctuations with kθ up to 15 cm−1 (kθρs ∼ 1) is present. At high density/collisionality with saturated ohmic confinement, the electron thermal transport is diffusive, the core rotation is in the counter-current direction, the edge impurity density profiles are up/down asymmetric, and the high kθ turbulent feature is absent. The rotation reversal stagnation point (just inside of the q = 3/2 surface) coincides with the non-local electron temperature profile inversion radius.225,226 Rotation “reversals” have also been observed in discharges with the LH current drive (LHCD). For target plasmas with high plasma current, the intrinsic rotation experiences a change in the counter-current direction,227,228 while for low plasma current targets, the rotation increment is in the co-current direction.229,230 This reversal of the change in rotation with LHCD has been traced to the current density, through the q profile. At low collisionality, ICRF can also cause core rotation to decrease markedly and even reverse direction.231 

Particle transport studies on C-Mod began with modulated gas puff experiments.232 These experiments followed the response of the electron density profiles to periodic gas puffs using a singular-value decomposition analysis of interferometer chords. From the response, the profiles of transport coefficients, D (particle diffusivity) and V (convection velocity), were obtained. For the typical OH and L-mode plasmas, these two quantities increased monotonically with minor radius, reaching values on the order of 0.2 m2/s and 1.5 m/s, respectively, at mid-radius. Particle transport was typically slower than energy transport with D/χi ∼ 0.2–1. For Ohmic LOC plasmas, both D and V decreased with density from 0.3 m2/s and 3 m/s at ne = 7 × 1019/m2 to 0.07 m2/s and 0.3 m/s at ne = 1.3 × 1020/m2 corresponding to stronger peaking at low density.

Turning to the H-mode, results from AUG and JET40 showed moderate peaking at low collisionality, but could not distinguish between dependence on collisionality and n/nG. (In these studies, a slightly modified form of collisionality, νEFF, is used.) This was critical for ITER since it uniquely would run simultaneously at a very low collisionality and high n/nG. Depending on which of these normalizations for density was correct, this could imply either peaked or flat density profiles and thus rather different fusion yields and stability properties. Experiments were carried out on C-Mod to break this covariance. Moreover, by operating with ICRF only, it could verify the role of Neutral Beam Injection (NBI) fueling found in the earlier work. These experiments also featured strong electron heating, Ti = Te, and a very weak neutral penetration—all ITER-like characteristics. In various parameter scans, the density peaking factors ne(0)/〈ne〉 were clearly higher at low collisionality.182 Figure 43 compares the C-Mod data with results from AUG and JET. It is clear that the overlay is better when the peaking is plotted vs νEFF than with n/nG. The C-Mod results, without a core particle source, demonstrate that the main effect is via transport rather than fueling locations and strongly support the notion that ITER will operate with mildly peaked density profiles; ne(0)/〈ne〉 may be up to 1.5. Gyrokinetic modeling was carried out for these discharges, by adjusting density profiles to match a zero particle flux condition, which is required for equilibrium.233 The dependence on collisionality was recovered in these simulations, with shorter wavelength fluctuations (kθρs > 0.5) responsible for much of the difference in particle transport. The key to the pinch seems to be a reduction in the ITG drive, which may not be applicable in ITER. Overall, this work is consistent with the recent models of particle transport that depend on the interplay of ITG and Trapped Electron Mode (TEM) drift wave turbulence.234 

FIG. 43.

Density peaking ratios in C-Mod is overlaid on data from AUG and JET showing that the appropriate scaling is collisionality (a) rather than n/nG (b) thus implying a moderate level of peaking for ITER baseline discharges. Reprinted with permission from Greenwald et al., Nuclear Fusion 47, L26 (2007). Copyright 2007 IOP.182 

FIG. 43.

Density peaking ratios in C-Mod is overlaid on data from AUG and JET showing that the appropriate scaling is collisionality (a) rather than n/nG (b) thus implying a moderate level of peaking for ITER baseline discharges. Reprinted with permission from Greenwald et al., Nuclear Fusion 47, L26 (2007). Copyright 2007 IOP.182 

Close modal

Early studies of impurity particle transport used a ruby LBO system to inject trace amounts of non-intrinsic, non-recycling impurities, observing various impurity charge states with a wide range of spectroscopic diagnostics.207,235,236 The LBO system effectively provides a delta-function impurity source in space and time. The subsequent evolution of spectral line brightness is then analyzed to obtain impurity transport properties. Impurity confinement times, τZ, in the L-mode were on the same order as the energy confinement time, that is, 0.020–0.030 s. In EDA H-modes, τZ is on the order of 0.1–0.2 s (see Fig. 30) somewhat longer than τE. In ELM-free H-modes, impurities tend to accumulate, with a confinement time long compared to the discharge length. Using the MIST impurity transport code,237 impurity diffusion, DZ, and convection, VZ coefficients consistent with the evolution of spectral brightness were obtained. In the core of both the L-mode and EDA H-modes, the transport coefficients are well above the levels predicted by neoclassical theory. However, in the vicinity of the H-mode transport barrier, DZ and VZ are significantly smaller, approaching neoclassical levels. Studies of soft x-ray emission from the pedestals of H-modes found a strong inward convection of impurities in the pedestal.207,238,239 This pinch velocity was larger for ELM-free H-modes and led to extremely sharp profiles of impurity density in the pedestal, consistent with neoclassical predictions. These early studies also investigated poloidal asymmetries in impurity transport. More recently, new insights on the poloidal variation of plasma parameters in the pedestal region of C-Mod have been obtained with Charge eXchange Recombination Spectroscopy (CXRS) measurements from both the HFS and LFS midplane. This reveals large (>6×) in-out impurity density asymmetries in the H-mode and nearly symmetric impurity density profiles between the HFS and LFS pedestals in the I-mode.240 Furthermore, the HFS and LFS measurements in the I- and H-mode show that potential and impurity temperature cannot both be flux functions in the pedestal.180 These results are currently being investigated with numerical models and in particular support the idea that two-dimensional transport effects need to be retained in impurity modeling of the pedestal region.

A newer LBO system,181 employing a multi-pulse YAG laser, was paired with the XICS diagnostic to measure, for the first time, the full profile evolution of a particular impurity charge state, in this case Ca+17, following injection. Transport coefficients were derived by using the STRAHL code,241 which simulates impurity transport and atomic physics, fitted with a synthetic diagnostic to replicate the XICS and VUV (vacuum ultraviolet) measurements.242,243 These calculations were performed inside of an iteration loop that varied the DZ and VZ profiles and minimized the difference between the synthetic measurements and those obtained on the experiment. Uncertainties in the transport coefficients were calculated from the sensitivity of the calculation to input parameters (mainly, Te and ne) and the spectroscopic measurement uncertainties. This approach was a significant improvement on the “guess and test” method typically applied to this problem, and the realistic error estimates allowed for meaningful comparisons with theoretical models for the first time. With the temperature dependence of the charge state density and emission under observation, good estimates of transport were obtained for r/a < 0.6. Inside of r/a of 0.3, little turbulent transport was calculated, but instead impurity transport seemed to be governed by the MHD activity of the sawtooth instability. In the L-modes, in the region dominated by turbulence, DZ profiles were well above neoclassical levels and far from constant, increasing sharply from the inversion radius and reaching values on the order of 5–6 m2/s by r/a = 0.6 (see Fig. 44). Similar profile shapes were calculated for VZ.244 Values of the peaking factor, RVZ/DZ were on the order of 3, similar to ne profile for the L-modes. These data were then compared in detail to nonlinear multi-channel gyrokinetic simulations, the results of which are discussed below in Sec. V E 2.

FIG. 44.

Profiles of impurity transport coefficients, Dz (a) Vz (b), are obtained from impurity injection experiments. These are compared to gyrokinetic simulations, which can simultaneously match the ion energy (c) and impurity particle transport within experimental uncertainties. Electron energy transport is under-predicted in these simulations (d). Reproduced with permission from Phys. Plasmas 19, 056110 (2012). Copyright 2012 AIP Publishing LLC.268 

FIG. 44.

Profiles of impurity transport coefficients, Dz (a) Vz (b), are obtained from impurity injection experiments. These are compared to gyrokinetic simulations, which can simultaneously match the ion energy (c) and impurity particle transport within experimental uncertainties. Electron energy transport is under-predicted in these simulations (d). Reproduced with permission from Phys. Plasmas 19, 056110 (2012). Copyright 2012 AIP Publishing LLC.268 

Close modal

ITBs are important tools to raise overall performance and in particular to achieve the high values of βP necessary for high bootstrap current in steady-state regimes. Most research in this area has used strong NBI that drives rotation and high levels of E × B shear. The core particle source from the beams can also contribute to peaking density, which reduces ITG drive as does the higher ratio of Ti/Te that is typically found in the beam heated plasmas. The traditional recipe for ITB formation often includes modification of the current profile and thus the magnetic shear by strong heating during the current ramp-up.245,246 On C-Mod, as in reactors, strong NBI heating is not available, and current relaxation is relatively fast compared to the discharge time. This prompts a search for actuators that will extrapolate into the reactor regime.247 

In C-Mod, the ITBs have been produced in several ways:248 

  1. In OH and ICRF heated plasmas with deuterium or lithium pellet injection, the core fueling creates peaked density profiles and suppresses transport, likely by stabilizing ITG turbulence, which is sensitive to the gradient scale length ratio η ≡ Ln/LT.15,249 The decrease in turbulent transport helps maintain the peaked density profile and sustain the regime.

  2. At many H-L transitions, a transient enhancement in central ion temperature and neutron production is seen. At the transition, the loss of H-mode particle confinement causes the edge density to drop quickly, but the core density remains at H-mode levels.250 A possible explanation is that the transient increase in density gradient that follows from the drop in edge density, suppresses ITG turbulence via the same mechanism at work in pellet fueled discharges. This regime is transient and destroyed after a few sawteeth periods.

  3. Internal transport barriers also arise spontaneously in Ohmic H-modes.250,251 If the mode can be maintained, the density profile slowly peaks, the sawteeth period lengthens, and the sawteeth often cease entirely. (A modification in the q profile is also seen in pellet fueled discharges and attributed to peaking of light impurities.252) This regime can last up to 10 energy confinement times, often until current ramp-down. The mechanism is not understood.

  4. The principal tool for creating ITBs in C-Mod is off-axis ICRF heating.202,250,251,253–255 These barriers form in sawtoothing H-mode discharges (that is, with monotonic q profiles and qmin < 1) without beam-driven rotation, without a core particle source and with Ti = Te. This regime can be made stationary by application of modest on-axis heating. The remainder of this section will describe this regime.

In this last and most common ITB scenario, the ICRF resonance must be moved well off the magnetic axis, accomplished by changing the magnetic field or the ICRF frequency or both. Figure 45 shows the critical dependence of barrier formation on resonance location via a magnetic field scan for ICRF frequency fixed at 70 MHz. The most obvious sign of ITB formation is strong density peaking, which develops slowly—on a time scale consistent with the Ware pinch.256 Starting with flat, H-mode like densities, in the range of 2.5–4 × 1020 m−3, the central density will peak to values above 6 × 1020 m−3. The profiles outside of the barrier foot remain at their H-mode level and shape. The temperature peaking inside the barrier is modest, but the overall pressure peaking is pronounced with analysis showing strongly suppressed thermal diffusivity inside the barrier foot. Thermal diffusivity can drop to ion-neoclassical levels inside a fully developed barrier.255 The barrier foot location is itself a function of the safety factor, with a location at r/a= 0.5 at q95 = 3 moving in to r/a = 0.25 at q95 = 7.

FIG. 45.

ITB formation, as indicated by changes in the density profile and intrinsic rotation, depends critically on the ICRF resonance location as seen in this scan of BT. Reprinted with permission from Fiore et al., Fusion Sci. Tech. 51, 303 (2007). Copyright 2007 American Nuclear Society.248 

FIG. 45.

ITB formation, as indicated by changes in the density profile and intrinsic rotation, depends critically on the ICRF resonance location as seen in this scan of BT. Reprinted with permission from Fiore et al., Fusion Sci. Tech. 51, 303 (2007). Copyright 2007 American Nuclear Society.248 

Close modal

These C-Mod experiments provided the first evidence that intrinsic equilibrium toroidal rotation can generate sufficient E × B shearing to influence the formation of an internal transport barrier.216,217,257 Creation of the ITB seems to require two essential elements. First, the off-axis heating reduces the temperature profile gradient, and thus the drive for ITG instabilities.256,258 However, by itself, this mechanism is not strong enough to suppress the instability and account for barrier formation. The second ingredient, E × B stabilization, arises because of changes in the rotation profile that occur when the RF heating resonance is positioned off-axis. The rotation decreases in the center of the plasma while remaining unchanged in the outer part of the discharge, forming a well in the inner half-radius.216,217,251 The result is a radial toroidal rotation profile with strong E × B shear (>1.5 × 105 rad/s) in the region where the ITB foot is observed. Linear and nonlinear gyrokinetic analyses indicate that this spontaneous shearing rate is comparable to the linear ion temperature gradient (ITG) growth rate at this location and that the shearing rate is sufficient to reduce the turbulent particle and energy transport. Figure 46 shows the linear growth rate and E × B shearing rate at the barrier foot location and demonstrates the clear difference in these quantities for discharges with off-axis compared to central heating.

FIG. 46.

The E × B shearing rate from intrinsic rotation can reach the ITG growth rate for discharges with off-axis ICRF that transition to an ITB. In similar H-mode discharges, with on-axis heating, the E × B rate is far below the growth rate. Reproduced with permission from Phys. Plasmas 19, 056113 (2012). Copyright 2012 AIP Publishing LLC.217 

FIG. 46.

The E × B shearing rate from intrinsic rotation can reach the ITG growth rate for discharges with off-axis ICRF that transition to an ITB. In similar H-mode discharges, with on-axis heating, the E × B rate is far below the growth rate. Reproduced with permission from Phys. Plasmas 19, 056113 (2012). Copyright 2012 AIP Publishing LLC.217 

Close modal

A dramatic illustration of barrier physics is the propagation of heat-pulses from sawteeth. The propagation is fast in core, then slow through barrier, then fast again outside the barrier.202 This observation suggests that the turbulence drive at the barrier itself is well below the nonlinear critical gradient, i.e., well below marginality with turbulent transport entirely suppressed. Away from the barrier foot, both inside the barrier region and outside, the plasma is apparently above the marginal stability point (χincremental ≫ χpower balance). Overall, the picture is that in ITB discharges, turbulent transport is strongly reduced in the plasma core, entirely suppressed in the narrow barrier region and unchanged outside the barrier region.

The reduction in turbulent transport in ITB discharges can lead to density and impurity accumulation, leading to excess radiation, a sharp reduction in conducted power, and loss of the barrier. However, by adding a small level of on-axis heating, the peaking and impurity accumulation can be controlled.253,255,256,259,260 The mechanism seems to be through the stimulation of TEM instabilities driven by the steep density gradient.256 The growth rate of transport due to density gradient driven turbulence in this regime increases strongly with Te, which responds to the heating. The simulation work behind this interpretation is discussed in Sec. V E 2.

Over the time period covered in this review of C-Mod research, there has been a dramatic change in the role of computer simulations in turbulence studies. New theory, better computational algorithms, and faster computers have allowed the development of models sufficiently rich in physics to be reasonably compared to experimental measurements. At the same time, improvements in profile and fluctuation diagnostics have widened the scope of those comparisons. Broadly based in the national and international fusion programs, the long-term aim of this work is to develop computationally tractable models that can produce predictions of plasma behavior sufficiently reliable to be used for design of future machines. It is worth noting that transport prediction for the ITER design was based almost entirely on empirical scaling. Before we can make the step from empirical to physics-based predictive models, they will need to be rigorously and systematically tested against experimental observation. Fusion plasma research has begun adopting code verification, validation, and uncertainty quantification methods that were developed originally for computational fluid dynamics.261–263 This should be understood as an extension of the scientific method into a research domain where advanced simulations are required to compute the implications of theory. The work has tended to focus on turbulent transport because (1) nonlinear behavior is critical to the predictions but requires difficult computations and (2) a good physical model is available. Anomalous transport in the plasma core is thought to be due to electrostatic drift-wave turbulence and well described by gyrokinetic theory, obeying the ordering required for the validity of that theory.

1. Simulations of ion and electron energy transport

The first nonlinear simulations of C-Mod discharges were motivated by discrepancies in the predictions of two widely used quasi-linear models. Though well-tested on data from other devices, the IFS-PPPL199 and Multi-mode264 models systematically and significantly under-predicted the core temperature gradients that were observed on C-Mod. This result suggested that the codes were not correctly calculating the nonlinear upshift265 in the critical temperature gradient. Nonlinear simulations using the GS2 code200,266 studied the parametric dependence of the upshift and found that proper treatment of zonal flow growth and damping required a calculation with kinetic (rather than adiabatic) electron dynamics and realistic collisionality. With these features, the model was found to be consistent with experimental fluxes and gradients, within uncertainties.

Further simulations of ion energy transport267–269 using the GYRO code270 found agreement with experiments over a wider range of discharges, even under conditions similar to those where a so-called “transport shortfall” has been reported in DIII-D L-mode plasmas. In those cases, the ion and electron heat fluxes and turbulence levels in DIII-D were significantly under-predicted in the outer part of the plasma core by nonlinear gyrokinetic calculations using the GEM and GYRO codes.271,272 In contrast, the simulations of C-Mod experiments predicted ion energy transport consistent with experimental data at the corresponding radial locations. Further, in a power scan, the electron energy transport was correctly predicted at higher powers, leaving a discrepancy only at lower powers and only in the electron channel.269 A comparison of the predicted and measured heat fluxes for these discharges is shown in Fig. 47.

FIG. 47.

Experimental ion and electron heat fluxes are shown for low-power (blue— (a), (b)) and high-power (red—(c), (d)) L-modes. The ion transport matches ion-scale gyrokinetic simulations at all radii, not showing the “shortfall” reported on DIII-D. At high powers, TEM instabilities in the simulation are excited and can explain electron heat transport. A discrepancy remains in the low-power case. Reproduced with permission from Phys. Plasmas 20, 032510 (2013). Copyright 2013 AIP Publishing LLC.269 

FIG. 47.

Experimental ion and electron heat fluxes are shown for low-power (blue— (a), (b)) and high-power (red—(c), (d)) L-modes. The ion transport matches ion-scale gyrokinetic simulations at all radii, not showing the “shortfall” reported on DIII-D. At high powers, TEM instabilities in the simulation are excited and can explain electron heat transport. A discrepancy remains in the low-power case. Reproduced with permission from Phys. Plasmas 20, 032510 (2013). Copyright 2013 AIP Publishing LLC.269 

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More generally, understanding anomalous electron thermal transport has proven to be a daunting and unmet challenge.273 Recent multi-scale simulations suggest that ETG (Electron Thermal Gradient) turbulence plays an important role in electron thermal transport in C-Mod.274 These simulations, using the GYRO code included both electron and ion gyrokinetic dynamics, using the actual ion-electron mass ratio and realistic plasma profiles—that is, they were near marginal stability.275 The interplay between fluctuations at different scales was crucial in these simulations, with short wavelength fluctuations downshifted from the peak of their linear growth rate. An important feature of ETG turbulence is radially extended structures, called streamers, which have been seen in previous simulations of electron-scale turbulence.193 Without the streamers, the radial scale of fluctuations would be far too small to drive transport at the levels seen in experiments. It had been believed that strong, long wavelength turbulence would destroy the streamers reducing the predicted transport rates.273 However, crucially in the multi-scale simulations reported here, the streamers can coexist with ion-scale eddies (as seen in Fig. 48, which plots the potential fluctuations from the simulation) and the levels of both the ion and electron energy transport predicted are close to the measured values.

FIG. 48.

Multi-scale gyrokinetic simulations, which include both electron and ion-scale dynamics show that fine scale ETG streamers can coexist with larger ITG structures and produce electron heat flux consistent with experiments.

FIG. 48.

Multi-scale gyrokinetic simulations, which include both electron and ion-scale dynamics show that fine scale ETG streamers can coexist with larger ITG structures and produce electron heat flux consistent with experiments.

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Another approach has been to address the electron transport in the least complicated case possible, with a series of experiments and modeling activities focused on low-density Ohmic plasmas. In this regime, all input power is into electrons through the resistive dissipation of plasma current and coupling to ions is weak. Energy confinement, in this Alcator or LOC regime, is proportional to plasma density—thus, we must conclude that electron thermal transport increases substantially at low densities. Studies of these discharges included GYRO simulations and a synthetic diagnostic for the PCI diagnostic, which was capable of measuring density fluctuations with wave numbers up to 55 cm−1.276 Overall, the intensity of density fluctuations increases with density and a higher frequency, higher k feature (kθ up to 15 cm−1, kθρs ∼ 1) is present in the LOC regime but not the SOC.223,225,277 In the SOC regime, the simulated ion and electron thermal diffusivities agree with experiments after varying the ion temperature gradient within experimental uncertainty. The absolute fluctuation intensity agrees with the simulation within experimental error (±60%). However, in LOC, the model substantially over-predicts the ion transport and under-predicts the electron transport. This work has since been extended to include the role of ion dilution on the ITG drive278 which reduces the ion density and the computed ion energy transport. This effect had previously been reported in low-density discharges.279,280 This approach does not explain the discrepancy in the electron channel.

In the first measurements of long wavelength (kyρs < 0.3) electron temperature fluctuations in Alcator C-Mod made with a new correlation electron cyclotron emission diagnostic281 electron temperature fluctuations decrease significantly (∼40%) crossing from LOC to SOC, consistent with a change from TEM turbulence domination to ion temperature gradient (ITG) turbulence as the density and collisionality is increased.221 Linear stability analysis shows that TEMs are dominant for long wavelength turbulence in the LOC regime and the ITG modes are dominant in the SOC regime at the radial location (ρ ∼ 0.8) where the changes in electron temperature fluctuations are measured. In contrast, deeper in the core (ρ < 0.8), linear stability analysis indicates that ITG modes remain dominant across the LOC/SOC transition. This radial variation suggests that the robust global changes in confinement of energy and momentum occurring across the LOC/SOC transition are correlated to local changes in the dominant turbulent mode near the edge, which coincides with very minor changes in collisionality locally in that edge region.

2. Simulations of particle transport

Gyrokinetic simulations were used in studies of particle transport in ITB discharges. With off-axis heating only, core turbulence and transport are greatly reduced and the Ware pinch is sufficient to account for the rate of density peaking. The density peaks sufficiently destabilize TEMs (before radiation leads to a back-transition).256,260 A synthetic PCI diagnostic was developed to compare with GS2 simulations, resulting in the first direct comparison between measured fluctuation spectra and gyrokinetic simulations and finding a good agreement with the spectrum and the increase in amplitude of measured fluctuations.260 The particle and energy fluxes also match transport analysis within uncertainties. Later simulations found, for the first time, a strong nonlinear upshift (illustrated in Fig. 49(a)) in the critical gradient for the density gradient driven TEMs, analogous to the effect of the Dimits shift on the temperature gradient and similarly associated with turbulent energy transfer into zonal flows.282 The predicted upshift is much weaker at low collisionalities and thus is sensitive to temperature and can be modified by heating. In fact, as noted above, modest levels of on-axis heating were sufficient to control the peaking of the ITB density profiles and produce a steady state. Experimentally, the density gradient is found to be limited by the predicted nonlinear gradient, well above the linear calculation, and can be reduced with an increase in heating as seen in Fig. 49(b). Figure 50 shows the results of modulated heating experiments that helped localize the induced turbulence to inside the transport barrier and to demonstrate the response of the turbulence to localized heating consistent with the theory.

FIG. 49.

A nonlinear upshift in density-gradient driven TEM was discovered in simulations of C-Mod ITB discharges. The upshift increases at higher collisionality providing a mechanism for transport control within the barrier.

FIG. 49.

A nonlinear upshift in density-gradient driven TEM was discovered in simulations of C-Mod ITB discharges. The upshift increases at higher collisionality providing a mechanism for transport control within the barrier.

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FIG. 50.

Modulated on-axis heating in ITB discharges allows measured fluctuations to be localized within the barrier and supports the theory of barrier control via density gradient driven TEMs.

FIG. 50.

Modulated on-axis heating in ITB discharges allows measured fluctuations to be localized within the barrier and supports the theory of barrier control via density gradient driven TEMs.

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Simultaneous comparison of impurity particle transport and energy transport were carried out between GYRO simulations and LBO experiments.268 In these L-mode experiments, it was possible, for the first time, to match the ion energy transport and the profiles of both the impurity diffusion, DZ, and inward convection, VZ, as shown in Fig. 44. An extensive set of sensitivity studies were carried out for both the linear and nonlinear calculations in order to propagate the uncertainties from experimental profiles, which are inputs for the code, into the nonlinear turbulence calculation.243 These studies also looked at the roughly linear dependence of impurity confinement time on plasma current, finding a corresponding decrease in both DZ, and VZ. The simulations were able to match the trend and the values of these transport coefficients as shown in Fig. 51. The effect is apparently from the change in magnetic shear that accompanies the change in edge safety factor. The discharges are dominated by ITG turbulence but TEMs are beginning to go unstable at lower values of IP. In these simulations, it was not possible to simultaneously match the electron energy transport, likely due to the lack of high-k dynamics in the simulations and that is currently being addressed in multi-scale work as discussed in Sec. V E 1.

FIG. 51.

Impurity transport coefficients, Dz, Vz, from an Ip are compared to gyrokinetic simulations which match the values and trends found in the experiments. Reproduced with permission from Phys. Plasmas 19, 056110 (2012). Copyright 2012 AIP Publishing LLC.268 

FIG. 51.

Impurity transport coefficients, Dz, Vz, from an Ip are compared to gyrokinetic simulations which match the values and trends found in the experiments. Reproduced with permission from Phys. Plasmas 19, 056110 (2012). Copyright 2012 AIP Publishing LLC.268 

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3. Momentum transport

Models for momentum transport, in the low flow regime, are only now emerging but it has been possible to test some general ideas about the origins of intrinsic rotation. These ideas focus on the role of “residual stress,” Πr, that is, the portion of the angular momentum flux that is not proportional to the velocity or to its gradient.283,284 The divergence of the residual stress constitutes the intrinsic torque. In the enhanced confinement regimes, Πr is a function of the temperature gradient. Πr depends upon the underlying turbulence, and can change sign if the turbulence mode propagation changes direction. During some rotation reversals, induced through collisionality changes, the dominant drift-wave turbulence regime is close to the boundary between the two dominant long-wavelength drift waves the ITG-TEM. Πr is also a function of the q profile, which is in qualitative agreement with the rotation changes observed in the LHCD plasmas.

More recent experiments studying rotation reversal in the ICRF heated plasmas suggest that the situation is more complex.231,285 In this experiment, designed for validating gyrokinetic models of the energy and particle transport, a base-case steady, sawtoothing L-mode plasma with 1.2 MW of on-axis RF heating was established. When the density was raised by 20%, it was found that the measured rotation profiles changed from peaked to hollow in shape while the electron density and impurity profiles remain peaked. Ion and electron heat fluxes in the two plasmas were the same. Direct quantitative comparisons with GYRO were carried out, and a good agreement with experimental ion heat flux, impurity particle transport, and trends in the fluctuation level ratio, n ̃ e / n e , T ̃ e / T e , was found though the electron heat flux was under-predicted.269 However, the observed changes in momentum transport (rotation profiles changing from peaked to hollow) did not correlate with changes in particle transport, and also did not correlate with changes in linear mode dominance, i.e., ITG vs TEM. These new results suggest that the drive for momentum transport differs from drives for heat and particle transport, possibly entering the gyrokinetic model formulation at a higher order.286 

From the start of C-Mod operation, ICRF was the principal auxiliary heating tool and underlies the entire research program. The need for routine operation of these systems at high power density (routinely up to 10 MW/m2) in efficient heating scenarios motivated the development of robust and reliable engineering solutions and drove research into related physics and technology.287,288 Using a set of innovative diagnostics, studies of wave coupling, propagation, absorption and mode conversion physics contributed to validation of emerging full-wave RF models for the first time. Engineering challenges had to be faced and solved by employing advanced design and analysis codes backed up by two decades of field testing. And while a tremendous amount has been learned about RF physics in the process, the importance of “everyday” use as a driver for technology development and a metric for performance cannot be overstated. The similarity of the C-Mod plasma density, magnetic field, and RF frequencies compared to ITER, as discussed in Sec. I, argues for the strong and immediate relevance of the results produced.

Using RF sources originally procured for the Fusion Materials Irradiation Test Facility, the C-Mod facility has available 8 MW of source power; 4 MW fixed at 80 MHz, and 4 MW tunable from 40 to 80 MHz. Power coupled into the plasma has been as high as 6 MW. The transmission network is carefully engineered to maximize the transmitted power, voltage handling, and impedance matching to the ICRF antennas.289,290 A set of fast ferrite tuners has been deployed to improve the tolerance of the matching to changes in the plasma loading,291,292 particularly in response to L-H transitions and ELMs. Five different antennas have been built and tested in the machine, beginning with a simple monopole design and advancing through a pair of 2-strap dipole antennas, to a 4-strap, toroidally-aligned (TA) design, and finally to a 4-strap field-aligned version.293–295 The design of the in-vessel RF feeds has also evolved based on modeling and testing aimed at reducing power limits imposed by high-voltage breakdown. Several types of protection circuitry have been implemented to prevent damage to the antennas, feeds, transmission line, and RF tubes. Three of these antennas can be seen in Fig. 3. The most common ICRF scenario employed has been hydrogen minority in deuterium majority plasmas, D(H), at 5.4 T, which puts the resonance on axis and provides highly efficient heating typically with 80%–90% of the coupled power absorbed in the core plasma.293 Also tested were He3 minority heating, D(He3) at 7.9 T, a variety of mode conversion scenarios and 2nd harmonic heating of H minority at 2.6 T.294,296–301

In ICRF heating, power is transmitted from the antenna through the plasma to an absorbing region as a compressional-Alfven wave (also called the fast magnetosonic wave). Absorption can be via cyclotron damping on minority ions or through electron Landau damping of the incoming fast wave or short-wavelength, mode converted waves. In a typical D(H) minority heating case, modeling suggests that 70% of absorbed power is coupled to a fast minority ion tail, 20% to majority ions, via second harmonic deuterium cyclotron damping, and 10% of power directly to electrons via Landau damping. Heating efficiency is optimum with a few percent minority concentration. Under these conditions, a strong minority ion tail develops. Since most of the minority ion tail slows down on electrons, overall heating power to electrons is about twice the power to ions. At higher minority fractions, the fast wave will mode-convert near the ion-ion hybrid resonance layer. With either regime, C-Mod is a dominantly electron heated device, though over much of its operating range, the ions and electrons are strongly coupled.

While the basic physics mechanism for ICRF heating is well established,302,303 calculations that can model the full-wave propagation and damping, with proper treatment of kinetic wave-particle interactions and realistic geometry have become available only relatively recently. New algorithms written for efficient parallel computation were required, especially to model the shorter wavelength mode-conversion phenomena.304,305 Confidence in these models must be earned through comparison with experiments, carefully testing the predictions of each of their constituent elements. The computation of wave propagation was tested for the first time in minority and mode-conversion regimes by direct measurements of plasma density perturbed by the wave fields in the plasma using the PCI diagnostic and comparing to the output of the TORIC306 and AORSA307 codes fitted with a matching synthetic diagnostic. In general, these experiments found agreement between the predictions and experiments, and featured the experimental discovery of an RF wave that had been predicted theoretically many years earlier308 but never reported in experiments or codes. This ion cyclotron wave (ICW) originates in the mode conversion process, propagates back toward the low-field situated antenna, and has a wavelength intermediate between the launched fast wave and the more familiar IBW (Ion Bernstein Wave). All three types of ICRF waves, the fast wave, the IBW, and the ICW can be seen in Fig. 52.309 Comparisons of the predicted and measured wave intensity, measured with PCI, are shown in Fig. 53(a).

FIG. 52.

TORIC simulations of ICRF propagation show the incoming fast wave, the anticipated, mode-converted, forward-propagating IBW and the re-discovered, backward propagating ICW. Lin et al., Plasma Physics and Controlled Fusion 47, 1207 (2005). Copyright 2005 International Atomic Energy Agency.309 

FIG. 52.

TORIC simulations of ICRF propagation show the incoming fast wave, the anticipated, mode-converted, forward-propagating IBW and the re-discovered, backward propagating ICW. Lin et al., Plasma Physics and Controlled Fusion 47, 1207 (2005). Copyright 2005 International Atomic Energy Agency.309 

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FIG. 53.

(a) Mode-converted RF waves measured with phase-contrast imaging are compared to full-wave simulations. Lin et al., Plasma Physics and Controlled Fusion 47, 1207 (2005). Copyright 2005 International Atomic Energy Agency.309 (b) The energy spectra of non-thermal ions are compared to AORSA simulations of minority heating. These simulations show agreement in the equilibrium distribution function as well as its dependence on plasma current and input power. Reproduced with permission from in Proceedings of the 19th Topical Conference on Radio Frequency Power in Plasmas (2011), Vol. 1406. Copyright 2011 AIP Publishing LLC.341 (c) Experimental measurements of local heat deposition are compared to TORIC simulations of mode conversion heating. Lin et al., Plasma Physics and Controlled Fusion 45, 1013 (2003). Copyright 2003 International Atomic Energy Agency.299 

FIG. 53.

(a) Mode-converted RF waves measured with phase-contrast imaging are compared to full-wave simulations. Lin et al., Plasma Physics and Controlled Fusion 47, 1207 (2005). Copyright 2005 International Atomic Energy Agency.309 (b) The energy spectra of non-thermal ions are compared to AORSA simulations of minority heating. These simulations show agreement in the equilibrium distribution function as well as its dependence on plasma current and input power. Reproduced with permission from in Proceedings of the 19th Topical Conference on Radio Frequency Power in Plasmas (2011), Vol. 1406. Copyright 2011 AIP Publishing LLC.341 (c) Experimental measurements of local heat deposition are compared to TORIC simulations of mode conversion heating. Lin et al., Plasma Physics and Controlled Fusion 45, 1013 (2003). Copyright 2003 International Atomic Energy Agency.299 

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A second set of predictions tested on C-Mod involves the fast ion distribution created in minority heating. Using a novel, multi-chord, compact neutral particle analyzer (CNPA)310 to look at neutrals created by passive and active charge exchange reactions, proton energy spectra were obtained. These spectra were compared to simulations using the full-wave codes TORIC and AORSA coupled to the Fokker-Planck solvers FPPRF311 and CQL3D312 fitted with synthetic diagnostics to match the CNPA viewing geometry and sensitivity.313 The measurements showed that the superthermal ions were peaked off-axis due to incomplete wave focusing and preferential heating of trapped ions. This observation was consistent with measurements of local electron heating from observations of sawtooth reheat rates. The codes could reproduce the experimental features in steady-state with reasonable agreement, as seen in Fig. 53(b), and also reproduced the observed dependence of the fast proton spectra with IP and PICRF.314 However, the codes failed to predict the transient evolution of the spectra, finding a significantly slower build-up and decay when the RF was pulsed. This disagreement might be related to the finite banana width and gyro-orbit size of the ion tail or to non-diffusive effects of the RF on the distribution function.315 Results from the ORBIT-RF316 and DC317 codes suggest that the wave-particle interactions modify the distribution function, causing it to evolve faster than expected from collisional processes alone. A computational approach was developed that uses the ICRF wave fields from the AORSA solver in DC. The DC code directly integrates the Lorentz force equation for ions using the full-wave fields and reconstructs an RF operator from a statistical ensemble of RF particle kicks in the wave field in order to capture non-diffusive effects. This RF operator is then used in CQL3D to evolve the non-thermal particle distribution. Preliminary application of this technique has greatly improved the agreement between the measured and simulated formation times of the energetic tail in C-Mod minority heating experiments.315 

In the mode conversion regime, direct, localized heating of the electrons near the mode conversion layer is expected. This prediction was tested using modulated RF power and a break in slope analysis of the electron temperature profiles.298,299,301,318 The simulations required proper treatment of electron Landau damping for short wavelength modes.319 The predicted and measured profiles of mode conversion electron heating in a D(H) plasma can be seen in Fig. 53(c).299 The predicted position and localization (Δ(r/a) ∼ 0.2) of the heating layer in D(He3) plasmas was confirmed in the simulations as well as the dependence of heating efficiency on the He3 content for fractions, nHe3/ne, below 0.20. At higher fractions, the code initially under-predicted the measured heating, likely due to a lack of resolution in the poloidal mode expansion of the RF fields. This disagreement was addressed in later versions of TORIC with improved numerical algorithms and parallel execution that allowed much higher poloidal resolution.304 

Plasma rotation has been shown to be beneficial in stabilizing MHD modes320 and improving confinement in experiments215 with a strong external torque applied by the neutral beam heating systems. However, reactor scale devices like ITER or Demo will need to run with a low or zero applied torque, and it is not clear yet whether intrinsic rotation will be large enough to realize all of the desired effects. The prospect of modifying plasma transport directly through E × B stabilization has motivated studies of plasma flow driven by RF waves.321 ICRF codes have predicted that such flows could be driven by IBW; however, uncertainties in the physics of the plasma response to RF and in plasma momentum transport have led to corresponding uncertainty in the predictions.322,323 On C-Mod, flow drive has been demonstrated for the first time in the mode conversion regime. In these experiments, plasma rotation was measured at levels well above those expected from the response of intrinsic rotation to the added heating.324 Simulations with the TORIC code showed that the mechanism is through IC wave interaction with He3 ions, while the shorter wavelength IBW only caused electron heating.325 Figure 54 shows a comparison of two discharges with the same RF power, one with D(H) minority heating and the second in the D(He3) mode conversion regime. Although the stored energy increase is somewhat larger for the minority heating case, the toroidal rotation is much greater for mode conversion. Driven rotation can exceed 100 km/s, which is on the order of 20% the sound speed. The E × B shearing rate of these flows approached the linear growth rate for ITG drift waves, the level at which strong effects on plasma transport are expected. So far, the limitation on driven rotation is connected with the stimulation of neoclassical tearing modes (NTM) due to low collisionality at the very high electron temperatures produced. A series of parameter scans allowed the derivation of an empirical scaling law for the driven rotation ΔV(km/s) ≈ 10PRF(MW)1.1Te0(keV)1.3IP(MA)0.4ne0(1020 m−3)−0.9.326 Notable is the linear dependence on power per particle.

FIG. 54.

ICRF flow drive is demonstrated in this comparison of minority heating (blue) with only intrinsic rotation (which is proportional to plasma energy) and mode-conversion heating (red). With equal power in each regime, significantly higher flow is produced in the mode conversion case. Reproduced with permission from Phys. Plasmas 16, 056102 (2009). Copyright 2009 AIP Publishing LLC.316 

FIG. 54.

ICRF flow drive is demonstrated in this comparison of minority heating (blue) with only intrinsic rotation (which is proportional to plasma energy) and mode-conversion heating (red). With equal power in each regime, significantly higher flow is produced in the mode conversion case. Reproduced with permission from Phys. Plasmas 16, 056102 (2009). Copyright 2009 AIP Publishing LLC.316 

Close modal

The importance of controlling impurity sources in plasmas with strong ICRF heating has long been recognized.327 This issue becomes particularly important when the antenna and other PFCs are made from reactor-compatible materials,namely, high-Z metals. Early experiments on C-Mod confirmed these concerns, with performance degraded by impurity radiation in the H-mode discharges with untreated molybdenum PFCs.17,328,329 Contributing to this effect is the improved impurity confinement in the H-mode and the sensitivity of the H-mode confinement to power conducted across the separatrix. High-Z impurities, which have their peak radiation at temperatures that prevail well into the plasma core, must be minimized for full performance. For example, tolerance to tungsten in a reactor would be no more than 10 parts per million. The molybdenum source rate was found to be proportional to RF power and originated mainly from the antenna protection tiles, with sources from the wall and divertor significantly less important.43,330 As a palliative measure, the walls of the vacuum vessel were boronized,331 that is, covered by a thin layer of boron by discharge cleaning with deuterated diborane gas.19 The boron layers on the order of 100 nm were sufficient to restore the H-mode confinement for several days of operation This approach is satisfactory for a short pulsed experiment, but is not extrapolatable to a steady-state reactor. To make further progress, it was necessary to understand the mechanism by which impurities were generated and transported into the plasma and to develop techniques to mitigate this problem.

Research into the impurity source has centered on the role of the RF sheath, an increase in the plasma potential on field lines that contact material surfaces and pass near the antenna or other locations with large wave energy density. The sheath is produced by rectification of the RF waves due to the difference between electron and ion mobility.332,333 The resulting potential accelerates ions, increasing their sputtering yield when they impact a material surface. An indirect evidence for an RF specific mechanism is from boron film erosion rates, estimated to be in the range of 15–20 nm/s, consistent with the eroding species having an energy far above the thermal background.330 Direct experimental evidence for the sheath mechanism was obtained with Langmuir probes, operated in a variety of modes, which measured plasma potentials of over 100 V, high enough to cause significant sputtering of molybdenum by D+ ions.334 The measured potentials increased with RF power and were lower when the walls were well boronized. These potentials, seen in Fig. 55, have a threshold dependent on the density in front of the antenna, consistent with theoretical predictions.335 Rectified potentials were observed336 and modeled337 on surfaces not magnetically connected to the antenna. These RF-induced potentials were also inferred from the modification of the fluctuation phase velocity as measured with gas-puff imaging.126,338 Interpreting the change in phase velocity profile as a change in the E × B flow yields an estimate for Er in excess of 10 kV/m, consistent with the sheath potential measurement. In addition, the GPI measurements suggest that the modified potential structure could be influencing particle transport through the generation of large convective cells.

FIG. 55.

Significant RF sheath potentials are measured with an amplitude proportional to input power. The acceleration of ions through this sheath and onto the first wall is believed to contribute to the increased impurity content of ICRF heating plasmas.

FIG. 55.

Significant RF sheath potentials are measured with an amplitude proportional to input power. The acceleration of ions through this sheath and onto the first wall is believed to contribute to the increased impurity content of ICRF heating plasmas.

Close modal

In order to reduce the magnitude of the RF sheath and control the level of impurities, an innovative antenna concept—the FA antenna—was developed.339,340 The idea is to minimize the component of the RF electric field that is parallel to the tokamak magnetic field. Modeling showed that by symmetrizing the antenna and surrounding structures, E would be reduced, circumventing the RF sheath mechanism. The resulting antenna geometry is shown schematically in Fig. 56, with the antenna box, straps, and Faraday screens all aligned with the total magnetic field. This is a much more challenging engineering task than simply aligning the Faraday screens, as it requires design and fabrication of a helical structure that fits snuggly against the wall of the axisymmetric toroidal vacuum vessel. Initial results have been promising, with the molybdenum source from the antenna lower by an order of magnitude, as seen in Fig. 57, along with an overall reduction in radiated power.340 The electrical properties of the antenna are excellent. Power densities up to 9 MW/m2 have been achieved; greater load tolerance and a very low RF power deposited in the antenna itself (0.4%), which is below the requirement for the ITER antenna (0.625%). While successful in its ultimate aim—addressing the impurity issue—the results differed from expectations in important details. The observed changes in the induced potential and its dependence on antenna phasing do not match theory or models. In fact, the measured RF sheath induced by the FA antenna is similar to what is produced by the conventional TA antenna. Recent work has begun to explain the discrepancy. For example, the original model had suggested that monopole phasing would lead to an improvement with respect to impurity sources compared to a dipole, but the measured impurity response was worse. Measurements made of waves in plasma with PCI found poor wave coupling and penetration for monopole phasing—a difference apparently due to modification of the RF spectrum by structures surrounding the antenna straps. The overall conclusion is that the FA approach has a great promise for solving the impurity problem associated with ICRF, but much work remains to be done.

FIG. 56.

The geometry of the traditional TA ICRF antenna is compared to the new FA design. A sample field line, which passes directly in front of the FA antenna is shown in purple.

FIG. 56.

The geometry of the traditional TA ICRF antenna is compared to the new FA design. A sample field line, which passes directly in front of the FA antenna is shown in purple.

Close modal
FIG. 57.

The molybdenum source rate during ICRF heating drops by almost an order of magnitude in FA design. Reproduced with permission from Phys. Plasmas 20, 056117 (2013). Copyright 2013 AIP Publishing LLC.295 

FIG. 57.

The molybdenum source rate during ICRF heating drops by almost an order of magnitude in FA design. Reproduced with permission from Phys. Plasmas 20, 056117 (2013). Copyright 2013 AIP Publishing LLC.295 

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A future tokamak reactor will need efficient off-axis, non-inductive current drive to allow steady-state operation, even with substantial bootstrap current. Further, the driver technology must be viable in steady state and in the reactor's nuclear environment. LHCD is among the very few options available and has been well demonstrated at low to moderate densities.342 Recent C-Mod experiments have extended these studies to reactor-relevant fields, density, RF, and magnetic geometry343 and allowed tests of emerging LHCD models.344 The principal question being addressed is whether results from low density can be extended to the higher densities required for reactor level performance, particularly to the values of βP required to provide sufficient bootstrap current for a steady-state scenario. Experiments to date have launched up to 1 MW of RF power at 4.6 GHz through a phased-array launcher. The launcher has 16 toroidal by 4 poloidal elements employing a novel design based on a four-way splitter, with one waveguide feed for each vertical column of antenna array. The design was developed with the aid of an advanced finite element code to model the RF fields and to account for the electromagnetic, thermal, and structural interactions.345–347 The column-to-column phase delay can be adjusted electronically to launch waves with the high directivity required for current drive experiments. Experiments were typically run with the parallel refractive index, n in the range from 1.6 to 2.2, which should interact strongly with and accelerate electrons from a distribution whose initial temperature is on the order of 5 keV. Studies of LH coupling elucidated the role of the ponderomotive effect and E × B drifts through experiments and modeling. The LH waves can reduce the plasma density in front of the launcher by this mechanism.346–348 ICRF waves from nearby antennas can have a similar effect, lowering the density and increasing reflections.349 

At moderate densities, up to 0.6 × 1020/m3, the LH system on C-Mod can drive 100% of the plasma current (0.5 MA) for multiple L/R times,347,350,351 as seen in Fig. 58(a). With the plasma density, magnetic field, and RF frequency in these experiments, approximately what is projected for ITER “steady-state” scenarios, C-Mod provides a directly relevant test-bed for studies of current profile control, stability, and transport. Global current drive efficiency, η = n20RILH/PLH, is on the order of 0.25 (A/m2W), consistent with previous experiments, theoretical expectations, and the values assumed for ITER steady-state scenarios. The population of nonthermal electrons predicted by LH theory was measured with a multi-chord hard x-ray diagnostic and found to build up and decay at a rate consistent with coupled ray-tracing/Fokker-Planck models in response to pulses of LH power.352,353 The same measurements showed that perpendicular transport of the fast electrons during the slowing down time was small compared to the device size, indicating that fast electrons stay near the flux surfaces on which they are generated. Measurements with a Motional Stark Effect (MSE) diagnostic, used to constrain a magnetic equilibrium reconstruction, have shown that the current can be driven well off-axis.354–356 The current profile can be modified sufficiently to create a sawtooth-free reversed shear regime.357 In these plasmas, the change in q profile is accompanied by development of an electron energy transport barrier, leading to a sharp rise in the core temperature, as seen in Fig. 58(b). The barrier formation is likely attributable to the stabilization of drift wave turbulence in response to the change in magnetic shear. This regime is often curtailed by development of n = 2, m = 1 MHD activity.

FIG. 58.

(a) Efficient LHCD can produce fully non-inductive discharges at densities above 0.5 × 1020. Reproduced with permission from Phys. Plasmas 19, 062505 (2012). Copyright 2012 AIP Publishing LLC.359 (b) Off axis-current drive can modify the magnetic shear and allow an electron ITB to form. Shiraiwa et al., Nuclear Fusion 53, 113028 (2013). Copyright 2013 International Atomic Energy Agency.357 

FIG. 58.

(a) Efficient LHCD can produce fully non-inductive discharges at densities above 0.5 × 1020. Reproduced with permission from Phys. Plasmas 19, 062505 (2012). Copyright 2012 AIP Publishing LLC.359 (b) Off axis-current drive can modify the magnetic shear and allow an electron ITB to form. Shiraiwa et al., Nuclear Fusion 53, 113028 (2013). Copyright 2013 International Atomic Energy Agency.357 

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At high densities, n ¯ e > 1020/m3, but below the limit for wave accessibility, LHCD efficiency drops faster than expected.358,359 Figure 59 compares experimental measurements of the hard x-rays produced by fast electrons and the prediction of a ray tracing calculation that neglects propagation and absorption processes in the plasma edge. The changes in the measured ionization rates and profiles in the SOL suggest that wave-plasma interactions in the edge are significant. These effects have been studied with ray tracing (GENRAY360) and full-wave (LHEAF,345,361 TORLH362) RF simulations coupled to the Fokker-Planck models CQL3D and VERD.346 LHEAF is a new code, developed by the C-Mod group, which uses finite element methods to compute wave coupling and propagation. The model computes the RF fields as they propagate in the launcher, through the plasma edge and into the plasma, allowing better modeling of the interactions in the edge plasma. Several mechanisms have been identified so far—all connected to low single-pass absorption—including spectral broadening due to full-wave effects and plasma density fluctuations, nonlinear interactions such as parametric decay instabilities (PDIs), collisional damping, and loss of fast electrons in the plasma SOL.358 Visible spectroscopy and imaging reveal local limiter heating and enhanced erosion in areas with magnetic field-line mapping to the LH antenna horns. Although direct scattering by edge fluctuations was found to modify the LH wave spectrum, it was not found to limit wave penetration.363 Figure 60 is the predicted wave amplitude from a full-wave calculation for a high-density plasma, showing that a large fraction of the wave energy propagates in the plasma edge and SOL at high densities. The high wave amplitude, particularly in high-field SOL region of the plasma, can drive PDI, resulting in a loss of current drive by dramatically upshifting the n of the daughter waves. Fig. 61 shows evidence of PDI in measurements of the RF frequency spectra taken at the high-field side midplane with broadening of the drive frequency and strong sidebands separated from the pump wave by multiples of the local ion gyro-frequency.364–366 Modeling suggests that weak absorption enhances the wave amplitude in the regions where local conditions allow the PDI to grow—consistent with a model by Takase,367 though the quantitative role of PDI in reducing LHCD efficiency at a high density is not certain. Based on the modeling, running with higher single-pass damping (as in ITER) could mitigate all of the identified mechanisms and lead to a higher current drive efficiency. Experiments at higher plasma temperature, which increases the damping and reduces edge effects, do in fact demonstrate the LHCD recovery of LH driven electrons at densities near the accessibility limit (see Fig. 62). Based on these results, a new LH launcher is being designed and modeled. This launcher will be located off the midplane where improved single pass absorption can be achieved.357,368 The design reduces reflected power via a toroidal bi-junction while retaining the control of the n spectrum. Velocity space synergy with the midplane launcher is predicted to maximize the driven current at ITER relevant densities.

FIG. 59.

At high densities, but below wave accessibility limits, driven current—indicated here by the decrease in hard x-rays—drops well below the expectations of a simple ray tracing model. Reproduced with permission from Phys. Plasmas 19, 062505 (2012). Copyright 2012 AIP Publishing LLC.359 

FIG. 59.

At high densities, but below wave accessibility limits, driven current—indicated here by the decrease in hard x-rays—drops well below the expectations of a simple ray tracing model. Reproduced with permission from Phys. Plasmas 19, 062505 (2012). Copyright 2012 AIP Publishing LLC.359 

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FIG. 60.