Toroidal Angular Momentum Research Articles

Self-consistent poloidal electric field and neoclassical angular momentum flux

A complete expression is obtained for the poloidal variation of the electrostatic potential in the banana regime for large aspect ratio flux surfaces using the method of matched asymptotic expansions. The result exhibits a finite discontinuity at the innermost point of a flux surface instead of a divergence as previously reported. Using this expression in combination with the solution of the linearized drift kinetic equation with a model collision operator, the part of the toroidal angular momentum flux due to the poloidal electric field is calculated. The result is larger than the one in existing works, which neglect the poloidal electric field, by the order of the square root of the aspect ratio.

Angular momentum studies with NBI modulation in JET

In this work we apply the neutral beam modulation technique to a set of JET discharges in order to obtain information on the radial transport of the toroidal angular momentum. This perturbative technique allows one to separate possible pinch terms from the momentum diffusivity, otherwise impossible to decouple with a steady-state momentum balance analysis alone. In this paper a model based on the ion temperature gradient turbulence, which consists of a constant and uniform Prandtl number of the order unity combined with an inward Coriolis pinch term predicted by the gyro-kinetic theory, is validated against data from the JET experiment. The machine size and the good space and time resolution of the charge exchange system make JET particularly suitable for this study. The experimental data are matched with good accuracy, both the steady-state and the transient propagation of the momentum perturbation, choosing a Prandtl number slightly above one, provided the Coriolis pinch is included. Sensitivity studies assess the accuracy range of our predictive modelling.

An intrinsic source of radial electric field and edge flows in tokamaks

We propose a new mechanism for radial electric fields and edge flows in tokamaks that will also serve as an intrinsic momentum source in systems without an up–down symmetry. An essential feature of toroidal plasmas is that charge-dependent ∇B and curvature drifts would lead to a vertical polarization of the discharge if it were not for the Pfirsch–Schlüter currents that neutralize the resulting charge separation. However, in the presence of collisions, there is a residual vertical electric field that drives an E × B flow in the direction of increasing major radius, regardless of the orientation of the fields and currents. This flow is excluded from the hot core and is localized to the more collisional edge plasma. It has many features in common with the edge flows observed in tokamaks such as C-Mod. In an up–down symmetric geometry it carries no net toroidal angular momentum; however, its viscous interaction with asymmetric boundaries leads to a net momentum input to the plasma. Both this momentum input, and the residual vertical electric field, the source of these flows, may play an important role in the ∇B direction-dependence of the power threshold for the L–H transition.

Gyrokinetic particle simulations of toroidal momentum transport

Simulations of toroidal angular momentum transport have been carried out using global toroidal gyrokinetic particle-in-cell code. The significant redistribution of toroidal momentum is observed, driven by the ion temperature gradient turbulence with adiabatic electrons, resulting in a peaked momentum profile in the central region of the radial domain. Cases with rigid and sheared plasma rotation are considered. Diffusive and off-diagonal (pinchlike) fluxes are identified. Toroidal momentum diffusivity is calculated by subtracting pinch contribution from the total momentum flux, and compared to quasilinear estimates. It is found that the ratio of momentum to heat conductivity is smaller than unity even after subtracting pinch contribution when wave-particle resonance energy is larger than thermal energy.

Basic physics of Alfvén instabilities driven by energetic particles in toroidally confined plasmas

Superthermal energetic particles (EP) often drive shear Alfvén waves unstable in magnetically confined plasmas. These instabilities constitute a fascinating nonlinear system where fluid and kinetic nonlinearities can appear on an equal footing. In addition to basic science, Alfvén instabilities are of practical importance, as the expulsion of energetic particles can damage the walls of a confinement device. Because of rapid dispersion, shear Alfvén waves that are part of the continuous spectrum are rarely destabilized. However, because the index of refraction is periodic in toroidally confined plasmas, gaps appear in the continuous spectrum. At spatial locations where the radial group velocity vanishes, weakly damped discrete modes appear in these gaps. These eigenmodes are of two types. One type is associated with frequency crossings of counterpropagating waves; the toroidal Alfvén eigenmode is a prominent example. The second type is associated with an extremum of the continuous spectrum; the reversed shear Alfvén eigenmode is an example of this type. In addition to these normal modes of the background plasma, when the energetic particle pressure is very large, energetic particle modes that adopt the frequency of the energetic particle population occur. Alfvén instabilities of all three types occur in every toroidal magnetic confinement device with an intense energetic particle population. The energetic particles are most conveniently described by their constants of motion. Resonances occur between the orbital frequencies of the energetic particles and the wave phase velocity. If the wave resonance with the energetic particle population occurs where the gradient with respect to a constant of motion is inverted, the particles transfer energy to the wave, promoting instability. In a tokamak, the spatial gradient drive associated with inversion of the toroidal canonical angular momentum Pζ is most important. Once a mode is driven unstable, a wide variety of nonlinear dynamics is observed, ranging from steady modes that gradually saturate, to bursting behavior reminiscent of relaxation oscillations, to rapid frequency chirping. An analogy to the classic one-dimensional problem of electrostatic plasma waves explains much of this phenomenology. EP transport can be convective, as when the wave scatters the particle across a topological boundary into a loss cone, or diffusive, which occurs when islands overlap in the orbital phase space. Despite a solid qualitative understanding of possible transport mechanisms, quantitative calculations using measured mode amplitudes currently underestimate the observed fast-ion transport. Experimentally, detailed identification of nonlinear mechanisms is in its infancy. Beyond validation of theoretical models, the future of the field lies in the development of control tools. These may exploit EP instabilities for beneficial purposes, such as favorably modifying the current profile, or use modest amounts of power to govern the nonlinear dynamics in order to avoid catastrophic bursts.

Interpretation of edge pedestal rotation measurements in DIII-D

A novel methodology for inferring experimental toroidal angular momentum transfer rates from measured toroidal rotation velocities and other measured quantities has been developed and applied to analyze rotation measurements in the DIII-D [J. Luxon, Nucl. Fusion 42, 6149 (2002)] edge pedestal. The experimentally inferred values have been compared with predictions based on atomic physics processes and on neoclassical toroidal viscosity. The poloidal rotation velocities have been calculated from poloidal momentum balance using neoclassical parallel viscosity and a novel retention of all terms in the poloidal momentum balance, and compared with measured values in the DIII-D edge pedestal.

Gyrokinetic theory and simulation of angular momentum transport

A gyrokinetic theory of turbulent toroidal angular momentum transport as well as modifications to neoclassical poloidal rotation from turbulence is formulated starting from the fundamental six-dimensional kinetic equation. The gyro-Bohm scaled transport is evaluated from toroidal delta-f gyrokinetic simulations using the GYRO code [Candy and Waltz, J. Comput. Phys. 186, 545 (2003)]. The simulations recover two pinch mechanisms in the radial transport of toroidal angular momentum: The slab geometry E×B shear pinch [Dominguez and Staebler, Phys. Fluids B 5, 387 (1993)] and the toroidal geometry “Coriolis” pinch [Peeters, Angioni, and Strintzi, Phys. Rev. Lett. 98, 265003 (2007)]. The pinches allow the steady state null stress (or angular momentum transport flow) condition required to understand intrinsic (or spontaneous) toroidal rotation in heated tokamak without an internal source of torque [Staebler, Kinsey, and Waltz, Bull. Am. Phys. Soc. 46, 221 (2001)]. A predicted turbulent shift in the neoclassical poloidal rotation [Staebler, Phys. Plasmas 11, 1064 (2004)] appears to be small at the finite relative gyroradius (rho-star) of current experiments.

Nonlinear gyrokinetic theory of toroidal momentum pinch

The turbulent convective flux of the toroidal angular momentum density is derived using the nonlinear toroidal gyrokinetic equation which conserves phase space density and energy [T. S. Hahm, Phys. Fluids, 31, 2670 (1988)]. A novel pinch mechanism is identified which originates from the symmetry breaking due to the magnetic field curvature. A net parallel momentum transfer from the waves to the ion guiding centers is possible when the fluctuation intensity varies on the flux surface, resulting in imperfect cancellation of the curvature drift contribution to the parallel acceleration. This mechanism is inherently a toroidal effect, and complements the k‖ symmetry breaking mechanism due to the mean E×B shear [O. Gurcan et al., Phys. Plasmas 14, 042306 (2007)] which exists in a simpler geometry. In the absence of ion thermal effects, this pinch velocity of the angular momentum density can also be understood as a manifestation of a tendency to homogenize the profile of “magnetically weighted angular momentum density,” nmiR2ω‖∕B2. This part of the pinch flux is mode-independent (whether it is trapped electron mode or ion temperature gradient mode driven), and radially inward for fluctuations peaked at the low-B-field side, with a pinch velocity typically, VAngTEP∼−2χϕ∕R0. Ion thermal effects introduce an additional radial pinch flux from the coupling with the curvature and grad-B drifts. This curvature driven thermal pinch can be inward or outward, depending on the mode-propagation direction. Explicit formulas in general toroidal geometry are presented.

Collisional bulk ion transport and poloidal rotation driven by neutral beam injection

Neutral beam injection (NBI) is known to significantly affect radial transport in a tokamak plasma. Furthermore, recent observations have shown poloidal velocities, in the presence of NBI, significantly in excess of the standard neoclassical value. Motivated by this, the additional collisional radial bulk ion fluxes of particles, heat and toroidal angular momentum, and the poloidal velocity, driven by fast ions from NBI have been evaluated for a low-collisionality, pure plasma, with strong toroidal rotation and arbitrary aspect ratio. Higher order velocity space structure of the fast ion distribution function can be significant, whilst the effects of toroidal acceleration caused by strong NBI dominate at large aspect ratio. The driven poloidal velocity depends strongly on system parameters, becoming larger at higher beam density and lower beam energy.

Intrinsic rotation in DIII-D

In the absence of any auxiliary torque input, the DIII-D plasma consists of nonzero toroidal angular momentum, in other words, it rotates. This effect is commonly observed in tokamaks, being referred to as intrinsic rotation. Measurements of intrinsic rotation profiles have been made in DIII-D [J. Luxon, Nucl. Fusion 42, 614 (2002)] H-mode discharges, with both Ohmic heating (OH) and electron cyclotron heating (ECH) in which there is no auxiliary torque. Recently, the H-mode data set has been extended with the newly configured DIII-D simultaneous co- and counter-directed neutral beam injection (NBI) capability resulting in control of the local torque deposition, where co and counter refer to the direction relative to the toroidal plasma current. Understanding intrinsic rotation is important for projection toward burning plasma performance where any NBI torque will be relatively small. The toroidal velocity is recognizably important regarding issues of stability and confinement. In DIII-D ECH H-modes the rotation profile is hollow, co-directed at large minor radius and depressed, or actually counter-directed, nearer the magnetic axis. This profile varies with the ECH power deposition profile to some extent. In contrast, OH H-modes have a relatively flat co-directed rotation profile. There is a scaling of the DIII-D intrinsic toroidal velocity with W∕Ip, as seen in intrinsic rotation in Alcator C-Mod [J. Rice, Nucl. Fusion 39, 1175 (1999)], where W is the total plasma thermal energy and Ip is the magnitude of the toroidal plasma current. This common scaling resulted in a dimensionless similarity experiment between DIII-D and Alcator C-Mod on intrinsic rotation, obtaining a single spatial point match in the toroidal velocity normalized to the ion thermal velocity. The balanced NBI capability in DIII-D is a useful tool to push scaling studies to higher values of the plasma normalized energy, notwithstanding the details of torque deposition for co-NBI versus counter-NBI. There are theories which address intrinsic rotation, both extensions of neoclassical theory and related to turbulent transport. At this time, the comparisons with theory are qualitative.

Observation of Plasma Toroidal-Momentum Dissipation by Neoclassical Toroidal Viscosity

Dissipation of plasma toroidal angular momentum is observed in the National Spherical Torus Experiment due to applied nonaxisymmetric magnetic fields and their plasma-induced increase by resonant field amplification and resistive wall mode destabilization. The measured decrease of the plasma toroidal angular momentum profile is compared to calculations of nonresonant drag torque based on the theory of neoclassical toroidal viscosity. Quantitative agreement between experiment and theory is found when the effect of toroidally trapped particles is included.

Effect of the dynamic ergodic divertor in the TEXTOR tokamak on MHD stability, plasma rotation and transport

With the dynamic ergodic divertor in TEXTOR fundamental effects of the coupling of external magnetic field perturbations to the confined plasma have been studied. The non-linear coupling between external (m/n = 12/4) and internal modes (m/n = 3/1) has been investigated. The critical perturbation field (m/n = 3/1) for the excitation of an m/n = 2/1 tearing mode depends not only on the magnitude but also on the direction of the toroidal angular momentum input by neutral beam injection (NBI). Below the excitation threshold of this mode a toroidal spin-up of the plasma has been observed, which only depends on the strength of the perturbation field. It is independent of both the rotation direction of the external perturbation field and the toroidal angular momentum supplied by the NBI.

A semi-analytical method to evaluate the dielectric response of a tokamak plasma accounting for drift orbit effects

A semi-analytical method is proposed to evaluate the dielectric response of a plasma to electromagnetic waves in the ion cyclotron domain of frequencies in a D-shaped but axisymmetric toroidal geometry. The actual drift orbit of the particles is accounted for. The method hinges on subdividing the orbit into elementary segments in which the integrations can be performed analytically or by tabulation, and it relies on the local book-keeping of the relation between the toroidal angular momentum and the poloidal flux function. Depending on which variables are chosen, the method allows computation of elementary building blocks for either the wave or the Fokker–Planck equation, but the accent is mainly on the latter. Two types of tangent resonance are distinguished.

Evaluation of the neoclassical radial electric field in a collisional tokamak

The neoclassical electric field in a tokamak is determined by the conservation of toroidal angular momentum. In the steady state in the absence of momentum sources and sinks it is explicitly evaluated by the condition that radial flux of toroidal angular momentum vanishes. For a collisional or Pfirsch-Schlüter short mean-free path ordering with subsonic plasma flows we find that there are two limiting cases of interest. The first is the simpler case of a strongly up-down asymmetric tokamak for which the lowest order gyroviscosity does not vanish and must be balanced by the leading order collisional viscosity in order to determine the radial electric field. The second case is the more complicated case of an up-down symmetric tokamak for which the gyroviscosity must be evaluated to higher order and again balanced by the lowest order collisional viscosity to determine the radial electric field. In general, both the lowest and the next order contributions from the gyroviscosity must be retained.

The influence of gas fuelling location on H-mode access in the MAST spherical tokamak

The observation that high-field side (HFS) gas puff refuelling facilitates access to the improved confinement (H-mode) regime on the COMPASS-D and MAST tokamaks prompted a theoretical investigation of the role of the neutral gas dynamics in controlling the edge plasma rotation and radial E-field, Er. Within the framework of neo-classical theory, higher edge plasma flow, and hence Er, are predicted when fuelling from the HFS—rather than from the more usual low-field side (LFS)—provided neutral viscosity dominates the transport of toroidal angular momentum. Here, these predictions are compared with experiments on MAST, where the influence of the gas-puff location on the edge Er profile is measured spectroscopically. An increase in Er is indeed observed with HFS refuelling in the region where the edge transport barrier forms, provided the neutral density at the LFS is sufficiently low so as not to damp the toroidal flow.

Toroidal rotation in neutral beam heated discharges in DIII-D

It is known that the toroidal angular momentum and the ion thermal energy are correlated in tokamak discharges heated by neutral beam injection. Here, data from ten years of measurements on DIII-D are considered, for representative discharges from all types and all conditions. The ratio of simple replacement times for momentum and energy is found to order this correlation indicating that these times are approximately equal, across the minor radius. Representative discharges of several types are discussed in more detail, as well as transport analysis results for the momentum and thermal ion diffusivities.

Three‐dimensional Magnetohydrodynamic Simulations of an Accretion Disk with Star‐Disk Boundary Layer

We present global 3D MHD simulations of geometrically thin but unstratified accretion disks in which a near Keplerian disk rotates between two bounding regions with initial rotation profiles that are stable to the MRI. The inner region models the boundary layer between the disk and an assumed more slowly rotating central, non magnetic star. We investigate the dynamical evolution of this system in response to initial vertical and toroidal fields imposed in a variety of domains contained within the near Keplerian disk. Cases with both non zero and zero net magnetic flux are considered and sustained dynamo activity found in runs for up to fifty orbital periods at the outer boundary of the near Keplerian disk. Simulations starting from fields with small radial scale and with zero net flux lead to the lowest levels of turbulence and smoothest variation of disk mean state variables. For our computational set up, average values of the Shakura & Sunyaev (1973) $\alpha$ parameter in the Keplerian disk are typically $0.004\pm 0.002.$ Magnetic field eventually always diffuses into the boundary layer resulting in the build up of toroidal field inward angular momentum transport and the accretion of disk material. The mean radial velocity, while exhibiting large temporal fluctuations is always subsonic. Simulations starting with net toroidal flux may yield an average $\alpha \sim 0.04.$ While being characterized by one order of magnitude larger average $\alpha$, simulations starting from vertical fields with large radial scale and net flux may lead to the formation of persistent non-homogeneous, non-axisymmetric magnetically dominated regions of very low density.

Minimum energy state of tokamak plasmas with an internal transport barrier

The minimum energy state under the constraint of the total toroidal angular momentum is examined. The result is applied to actual tokamak plasmas with an internal transport barrier (ITB). The study shows that tokamak plasmas with ITB are in the minimum energy state under the constraint of the total toroidal angular momentum and that they are likely to be self-organized.

Comparision of neoclassical rotation theory with experiment under a variety of conditions in DIII-D

A neoclassical theory of gyroviscous radial momentum transport and poloidal and toroidal rotation has been compared with experiment in DIII-D [Luxon, Anderson, Batty et al., Plasma Physics and Controlled Nuclear Fusion Research 1986 (IAEA, Vienna, 1987), Vol. 1, p. 159] discharges in different confinement regimes, with a range of neutral beam powers and with co- and counter-injection, and with various types of dominant impurity species present. Calculated central toroidal rotation velocities and momentum confinement times agreed with experiment over a wide range of these conditions, with one notable exception in which a drift correction may be needed to reduce the gyroviscous toroidal force. Radial distributions of toroidal and poloidal rotation velocities and radial electric field, calculated using the radial distribution of toroidal angular momentum input density, agreed with measured distributions for the one time in an L-mode discharge that was examined in detail.

Fast particle finite orbit width and Larmor radius effects on low-n toroidicity induced Alfvén eigenmode excitation

The effects of finite drift orbit width (FOW) and Larmor radius (FLR) of fast particles on the stability of low-n toroidicity-induced Alfvén eigenmodes (TAE) are studied. The formulation is based on the solution of the low frequency gyrokinetic equation (ω≪ωc, where ωc is particle cyclotron frequency). A quadratic form has been derived in terms of invariant variables; energy E, magnetic moment μ, and toroidal angular momentum Pφ. The growth rate of the TAE is computed perturbatively using numerical averaging over the fast particle drift orbit. This new computational capability improves the NOVA-K code [G. Y. Fu, C. Z. Cheng, and K. L. Wong, Phys. Fluids B 5, 4040 (1994)] which included FOW effects in the growth rate calculation based on small radial orbit width approximation. The new NOVA-K version has been benchmarked for different regimes of particle TAE excitation. It is shown that both FOW and FLR effects are typically stabilizing; the TAE growth rate can be reduced by as much as a factor of 2 for tokamak fusion test reactor supershots [D. J. Grove and D. M. Meade, Nucl. Fusion 25, 1167 (1985)]. However, FOW may be destabilizing for the global modes, which are localized at the plasma edge.