Neoclassical Toroidal Viscous Torque Research Articles

Ripple-induced neoclassical toroidal viscous torque in Augmented-First Plasma operation phase in ITER

Abstract A systematic calculation is performed on the ripple-induced neoclassical toroidal viscous (NTV) torque for new ITER scenarios designed for the Augmented-First Plasma (A-FP) operation phase with the full tungsten wall, where the plasma-wall gap is varied in view of mitigating the impact of tungsten wall-plasma interactions. The torque calculation includes drift kinetic response of the plasma thermal and energetic particles to the n = 18 (n is the toroidal harmonic number) ripple field. For the plasma scenario with ~45 cm plasma-wall gap at the outboard mid-plane and considering the corrected ripple level of 0.17% by the ferritic steel inserts, the computed net NTV torque acting on the plasma column is in the sub-Nm level. However, with decreasing the plasma-wall gap, the computed net NTV torque can reach a level comparable to that produced by the neutral-beam momentum injection in ITER. Ripple correction by ferritic inserts reduces the net torque by a factor of 3.3 for all the three A-FP scenarios considered. The n ω d = l ω b (with ω d and ω b being the toroidal precession and bounce frequencies of trapped particles, respectively, and l an integer number) type of resonance-enhancement of the NTV torque, due to thermal particles, is found to be weak in ITER despite high-n of 18. The same also holds for the ITER 10 MA steady state scenario from the D-T operation phase, where the aforementioned resonance associated with fusion-born alphas is also included. The ripple-induced NTV torque is well below that produced by the resonant magnetic perturbation applied for controlling the type-I edge-localized mode in ITER.

Alpha particle confinement metrics based on orbit classification in stellarators

We present orbit classification schemes for use as fast metrics for fusion alpha particle losses implemented in the symplectic guiding-centre code SIMPLE. Two variants respectively based on conservation of the parallel adiabatic invariant, and topology of footprints in Poincaré sections are introduced. Like an existing approach based on the Minkowski fractal dimension, those methods estimate whether a guiding-centre orbit is regular and therefore expected to be confined for infinite time in the collisionless case, or chaotic, which might lead to its loss. Compared with the existing approach, the required orbit tracing time for the novel classifiers is shorter by at least an order of magnitude. This enables massive sampling of orbits across the whole phase space to identify regular and chaotic regions for the purpose stellarator optimization. Based on conservation of the perpendicular invariant, we demonstrate how extended regular regions may act as radial barriers for orbits from the chaotic regions on the radially inboard side. We propose to use a quantified version of this property as a new metric for collisionless fusion alpha losses. As pitch-angle scattering becomes only relevant after alphas have already deposited a significant fraction of their energy, such a metric remains useful also for the case with collisions. This is illustrated by comparison with collisional loss computations. Results are presented for applications to two quasi-isodynamic configurations, a quasi-helical configuration and two quasi-axisymmetric configurations. In addition, the Hamiltonian action-angle formalism is used in quasi-axisymmetric configurations to investigate the overlap of drift-orbit resonances leading to chaos. The respective analysis is performed with the NEO-RT code originally developed for investigation of neoclassical toroidal viscous torque in tokamak plasmas.

Account of non-standard orbits in computations of neoclassical toroidal viscous torque in the resonant plateau regime of a tokamak

This article extends theoretical details based on a short paper originally submitted to the 2022 EPS conference in plasma physics [1]. The quasilinear theory of resonant transport regimes in a tokamak is developed for the general case of orbits forming various classes separated in phase space by homoclinic orbits with infinite bounce time. Beyond standard orbits (banana and passing orbits) also all types of non-standard orbits (e.g. “potato” orbits) are taken into account. In case of a weak radial electric field, such orbits are usually present only near the magnetic axis. If the radial electric field cannot be treated as weak, there can be arbitrary many classes, located elsewhere. The present approach covers all such cases and is demonstrated on a specific example of a radial electric field profile. The resulting quasilinear kinetic equation is applicable to compute neoclassical toroidal viscous (NTV) torque in a tokamak with non-axisymmetric magnetic field perturbations. A fully non-local approach to NTV computation has been realized in the upgraded version of the code NEO-RT. Based on a generalization of magnetic flux surfaces to drift surfaces, the notion of a local thermodynamic equilibrium is extended for our purpose. We obtain an expression for the integral toroidal torque within a chosen flux surface and dicuss means to compute such integrals taking singularities in bounce and precession frequencies into account.

Modeling plasma toroidal flow profile control via NTV torque with n = 2 3D fields in MAST-U

Toroidal modeling utilizing the MARS-Q code (Liu et al 2013 Phys. Plasmas 20 042503) shows that large (compared to electromagnetic torque) neoclassical toroidal viscous (NTV) torque can be achieved in the plasma core region in reference MAST(-U) L-mode plasmas, by applying the n = 2 (n is the toroidal mode number) 3D fields generated by the magnetic coils used for controlling the edge localized mode and/or by the error field correction coils. Large NTV torque, occurring at relatively slow plasma flow, in combination with strong variation of the torque amplitude versus the coil phasing of the 3D coils, offers a tool to control the plasma toroidal rotation profile in spherical tokamaks. To effectively employ the toroidal NTV torque, the potentially fast initial flow needs to first be damped by 3D fields. MARS-Q quasi-linear initial value simulations demonstrate that this is achievable in MAST-U, within the designed 3D coil current capability for the device.

Non-linear interplay between edge localized infernal mode and plasma flow

Non-linear interaction between the edge localized infernal mode (ELIM) and the plasma toroidal flow is numerically investigated, by solving the initial value problem for the n = 1 ELIM, where n is the toroidal mode number. The linear results show that the ELIM instability is strongly affected by a close-fitting resistive wall. The presence of a resistive wall can fully stabilize an otherwise flow-shear destabilized ELIM. The computed toroidal torques based on linear ELIM eigenfunctions show different radial distributions: neoclassical toroidal viscous torque has a global distribution, whilst the torques associated with the Maxwell and Reynolds stresses largely peak near rational surfaces. The non-linear interplay between the unstable ELIM and the plasma flow tends to push the mode towards a more stable domain, no matter where the instability starts. This may help to understand some of the underlying physics for the experimentally observed edge harmonic oscillations in tokamak H-mode plasmas.

Toroidal modeling of interaction between internal kink mode and plasma flow

Non-linear interaction between the internal kink mode and toroidal plasma rotation is numerically studied using the MARS-Q code [Liu et al., Phys. Plasmas 20, 042503 (2013)]. Simulation results show core plasma flow damping due to various toroidal torques, generated by a weakly stable internal kink mode. The 3-D field perturbation induced torques, including the neoclassical toroidal viscous (NTV) torque, as well as that produced by the Maxwell and Reynolds stresses, act as sink terms in the toroidal momentum balance model. The NTV torque is found to play a dominant role in the flow damping in all cases considered in this study. The modification to the internal kink mode structure is observed during the flow damping. Whilst a steady state can be achieved in the coupled mode-flow evolution with a uniform initial flow, a sheared initial flow affects the linear stability of the mode and consequently changes the non-linear evolution. For cases where the steady state solution is achieved, the saturated plasma flow speed critically depends on the initial flow condition as well as the initial amplitude of the internal kink mode but is less sensitive to the on-axis safety factor q0, as long as the latter stays above 1.

Effect of anisotropic thermal transport on the resistive plasma response to resonant magnetic perturbation field

Plasma response to the resonant magnetic perturbation (RMP) field is numerically investigated by an extended toroidal fluid model, which includes anisotropic thermal transport physics parallel and perpendicular to the total magnetic field. The thermal transport is found to be effective in eliminating the toroidal average curvature induced plasma screening (the so called Glasser-Green-Johnson, GGJ screening) in a slow toroidal flow regime, whilst having minor effect on modifying the conventional plasma screening regimes at faster flow. This physics effect of interaction between thermal transport and GGJ screening is attributed to the modification of the radial structure of the shielding current, which resulted from the plasma response to the applied field. The modification of the plasma response (shielding current, response field, plasma displacement, and the perturbed velocity) also has direct consequence on the toroidal torques produced by RMP. Modelling results show that thermal transport reduces the resonant electromagnetic torque as well as the torque associated with the Reynolds stress, but enhances the neoclassical toroidal viscous torque at slow plasma flow.

Modeling of toroidal torques exerted by internal kink instability in a tokamak plasma

Toroidal modeling efforts are initiated to systematically compute and compare various toroidal torques, exerted by an unstable internal kink in a tokamak plasma, using the MARS-F/K/Q suite of codes. The torques considered here include the resonant electromagnetic torque due to the Maxwell stress (the EM or JXB torque), the neoclassical toroidal viscous (NTV) torque, and the torque associated with the Reynolds stress. Numerical results show that the relative magnitude of the net resonant electromagnetic and the Reynolds stress torques increases with the equilibrium flow speed of the plasma, whilst the net NTV torque follows the opposite trend. The global flow shear sensitively affects the Reynolds stress torque, but not the electromagnetic and the NTV torques. Detailed examinations reveal dominant contributions to the Maxwell and Reynolds stress torques, in terms of the poloidal harmonic numbers of various perturbation fields, as well as their relative toroidal phasing.

Modelling toroidal rotation damping in ITER due to external 3D fields

The linear and quasi-linear plasma response to the n = 3 and n = 4 (n is the toroidal mode number) resonant magnetic perturbation (RMP) fields, produced by the in-vessel edge localized mode control coils, is numerically studied for an ITER 15 MA H-mode baseline scenario. Both single fluid and fluid-kinetic hybrid models are used. The inclusion of drift kinetic effects does not strongly alter the plasma response compared to the fluid approximation for this ITER plasma. The full toroidal drift kinetic model is also used to compute the neoclassical toroidal viscous (NTV) torque, yielding results close to that of an analytic model based on geometric simplifications. The computed NTV torque from low-n RMP fields is generally smaller than the resonant electromagnetic torque for this ITER plasma. The linear response computations show a weak core kink response, contrary to a strong kink response often computed for plasmas from present day tokamak devices. Initial value quasi-linear simulations, including various torque models, show a localized damping of the plasma toroidal flow near the edge, as a result of the applied RMP fields. This localized rotation damping can be weak or strong depending on whether a weakly unstable edge localized peeling mode is present. No qualitative difference is found between the n = 3 and n = 4 RMP field configurations, in both the linear and non-linear response results.

Influence of externally applied magnetic perturbations on neoclassical tearing modes at ASDEX Upgrade

The influence of externally applied magnetic perturbations (MPs) on neoclassical tearing modes (NTM) and the plasma rotation in general is investigated at the ASDEX Upgrade tokamak (AUG). The low n resonant components of the applied field exert local torques and influence the stability of NTMs. The non-resonant components of the error field do not influence MHD modes directly but slow down the plasma rotation globally due to a neoclassical toroidal viscous torque (NTV). Both components slow down the plasma rotation, which in consequence increases the probability for the appearance of locked modes. To investigate the impact of externally applied MPs on already existing modes and the influence on the rotation profile, experimental observations are compared to modelling results. The model used here solves a coupled equation system that includes the Rutherford equation and the equation of motion, taking into account the resonant effects and the resistive wall. It is shown that the NTV torque can be neglected in this modelling. To match the experimental frequency evolution of the mode the MP field strength at the resonant surface has to be increased compared to the vacuum approximation. This leads to an overestimation of the stabilizing effect on the NTMs. The reconstruction of the entire rotation profile via the equation of motion including radial dependencies, confirms that the NTV is negligibly small and that small resonant torques at different resonant surfaces have the same effect as one large one. This modelling suggests that in the experiment resonant torques at different surfaces are acting and slowing down the plasma rotation requiring a smaller torque at the specific resonant surface of the NTM. This additionally removes the overestimated influence on the island stability, whereas the braking of the island's rotation is caused by the sum of all torques. Consequently, to describe the effect of MPs on the evolution of one island, all other islands and the corresponding torques must also be taken into account.

Neoclassical toroidal viscosity in perturbed equilibria with general tokamak geometry

This paper presents a calculation of neoclassical toroidal viscous torque independent of large-aspect-ratio expansions across kinetic regimes. The Perturbed Equilibrium Nonambipolar Transport (PENT) code was developed for this purpose, and is compared to previous combined regime models as well as regime specific limits and a drift kinetic δf guiding center code. It is shown that retaining general expressions, without circular large-aspect-ratio or other orbit approximations, can be important at experimentally relevant aspect ratio and shaping. The superbanana plateau, a kinetic resonance effect recently recognized for its relevance to ITER, is recovered by the PENT calculations and shown to require highly accurate treatment of geometric effects.

Toroidal modeling of penetration of the resonant magnetic perturbation field

A toroidal, quasi-linear model is proposed to study the penetration dynamics of the resonant magnetic perturbation (RMP) field into the plasma. The model couples the linear, fluid plasma response to a toroidal momentum balance equation, which includes torques induced by both fluid electromagnetic force and by (kinetic) neoclassical toroidal viscous (NTV) force. The numerical results for a test toroidal equilibrium quantify the effects of various physical parameters on the field penetration and on the plasma rotation braking. The neoclassical toroidal viscous torque plays a dominant role in certain region of the plasma, for the RMP penetration problem considered in this work.

Toroidal modeling of interaction between resistive wall mode and plasma flow

The non-linear interplay between the resistive wall mode (RWM) and the toroidal plasma flow is numerically investigated in a full toroidal geometry, by simultaneously solving the initial value problems for the n = 1 RWM and the n = 0 toroidal force balance equation. Here, n is the toroidal mode number. The neoclassical toroidal viscous torque is identified as the major momentum sink that brakes the toroidal plasma flow during the non-linear evolution of the RWM. This holds for a mode that is initially either unstable or stable. For an initially stable RWM, the braking of the flow, and hence the eventual growth of the mode, depends critically on the initial perturbation amplitude.

Toroidal modeling of plasma response and resonant magnetic perturbation field penetration

The penetration dynamics of the resonant magnetic perturbation (RMP) field is simulated in the full toroidal geometry, under realistic plasma conditions in MAST experiments. The physics associated with several aspects of the RMP penetration—the plasma response and rotational screening, the resonant and non-resonant torques and the toroidal momentum balance—are highlighted. In particular, the plasma response is found to significantly amplify the non-resonant component of the RMP field for some of the MAST plasmas. A fast rotating plasma, in response to static external magnetic fields, experiences a more distributed electromagnetic torque due to the resonance with continuum waves in the plasma. At fast plasma flow (such as for the MAST plasma), the electromagnetic torque is normally dominant over the neoclassical toroidal viscous (NTV) torque. However, at sufficiently slow plasma flow, the NTV torque can play a significant role in the toroidal momentum balance, thanks to the precession drift resonance enhanced, so-called superbanana plateau regime.

Neoclassical toroidal viscosity and error-field penetration in tokamaks

A model for field error penetration is developed that includes nonresonant as well as the usual resonant field error effects. The nonresonant components cause a neoclassical toroidal viscous torque that tries to keep the plasma rotating at a rate comparable to the ion diamagnetic frequency. The new theory is used to examine resonant error-field penetration threshold scaling in ohmic tokamak plasmas. Compared to previous theoretical results, the plasma is found to be less susceptible to error-field penetration and locking, by a factor that depends on the nonresonant error-field amplitude.

Effect of Neoclassical Toroidal Viscosity on Error-Field Penetration Thresholds in Tokamak Plasmas

A model for field-error penetration is developed that includes nonresonant as well as the usual resonant field-error effects. The nonresonant components cause a neoclassical toroidal viscous torque that keeps the plasma rotating at a rate comparable to the ion diamagnetic frequency. The new theory is used to examine resonant error-field penetration threshold scaling in Ohmic tokamak plasmas. Compared to previous theoretical results, we find the plasma is less susceptible to error-field penetration and locking, by a factor that depends on the nonresonant error-field amplitude.