Neoclassical Toroidal Plasma Viscosity Research Articles

Threshold density for toroidal rotation reversal bifurcation induced by symmetry breaking

Threshold density for intrinsic toroidal rotation reversal is studied using a newly proposed model [Li et al., Nucl. Fusion 61, 104002 (2021)] based on the neoclassical toroidal plasma viscosity (NTV) effect induced by symmetry breaking due to the existence of internal kink mode. In this model, rotation reversal happens when the NTV steady state flow jumps between “electron root” and “ion root,” which depends on plasma collisionality regimes. To make it convenient for cross machine comparisons or extrapolation to the future device, a scaling of threshold density for rotation reversal is proposed based on the physics of transition between the 1/ν and superbanana plateau regimes, i.e., the normalized collisionality over the particle toroidal precessional frequency rather than over the particle bounce frequency that is frequently used in neoclassical theory. The theoretical scaling of the threshold density on electron temperature and magnetic field strength is well verified by the NTVTOK modeling. Other effects, such as boundary condition, temperature gradient, and temperature ratio, which cannot be included in the theoretical scaling, are also investigated by using the NTVTOK modeling.

Neoclassical toroidal plasma viscosity in bounce-transit and drift resonance regimes in tokamaks

Neoclassical toroidal plasma viscosity in the bounce-transit and drift resonance regimes is calculated using a version of the drift kinetic equation that encompasses the physics of the nonlinear trapping and quasilinear plateau regimes in tokamaks. It is demonstrated that the mirror-force like term controls the transition between these two regimes. When the effective collision frequency is larger than the mirroring or the nonlinear bounce frequency, the quasilinear regime prevails; otherwise, the nonlinear trapping regime reigns. The demonstration is accomplished by using the Eulerian approach and is beyond the grasp of the method of the integration along the unperturbed orbit in solving the drift kinetic equation. The neoclassical toroidal plasma viscosity in the quasilinear plateau regime is calculated. Approximate analytic expressions for the neoclassical toroidal plasma viscosity that include the asymptotic limits of the nonlinear trapping and quasilinear regimes are presented to facilitate thermal and energetic alpha particle transport modeling in tokamaks.

Symmetry breaking driving spontaneous plasma rotation in tokamak fusion devices

Plasma rotation plays a critical role in improving plasma confinement in a magnetically confined fusion device. Spontaneous plasma rotation and its reversal of orientation without external momentum input have been observed in some tokamak fusion devices, while the underlying physics is not well understood. A new mechanism based on neoclassical toroidal plasma viscosity induced by symmetry breaking is proposed and well reproduces both spontaneous toroidal rotation and its reversals in tokamaks by considering a small non-axisymmetric displacement in the plasma center, since internal instabilities are often observed in those experiments. The nonlinear hysteresis process of plasma rotation reversals is also well reproduced in the modeling. The mechanism for driving spontaneous plasma rotation proposed in this paper may be utilized for achieving more economical operation of future tokamak fusion reactors.

Nonlinear trapping in the bounce-transit and drift resonance and neoclassical toroidal plasma viscosity in tokamaks

Bounce-transit and drift resonance is one of the resonances in tokamaks with broken toroidal symmetry. It plays an important role in the wave-particle interactions. The nonlinear consequence of the resonance is the nonlinear particle trapping in the magnetic well, created by the radial drift motion resulting from the perturbed magnetic fields. These nonlinearly trapped particles form superbananas. When the effective collision frequency is less than the bounce frequency of the superbananas, the resonance is resolved by the nonlinear orbits. The transport theory for the superbananas is developed by solving the drift kinetic equation using the Eulerian approach. The neoclassical toroidal plasma viscosity, and the non-axisymmetric transport coefficients have a scaling , which can be significant even for weakly perturbed tokamaks. Here, is the collision frequency, r is the minor radius, is the typical magnitude of the perturbed magnetic field strength, B is the equilibrium magnetic field strength, and U is a function of the magnetic shear parameter, mode numbers, and with being the poloidal gyro-radius. The magnitude of the energy flux can be comparable to that of the axisymmetric tokamaks for energetic alpha particles when ~ and ~ 0.1. Thus, the theory sets a maximum magnitude of the tolerable perturbed magnetic field strength in fusion reactors, when nonlinear trapping is significant.

Neoclassical quasilinear theory and universal collision frequency landscape in wave-particle interaction in tokamaks

The neoclassical quasilinear theory is extended to the collisional boundary layer and 1/ν regimes. The theory is for electromagnetic waves with frequencies lower than the bounce frequency of the trapped particles and wavelengths either comparable to or shorter than the width of bananas, but much longer than the gyro-radius. Here, ν is the collision frequency. This is accomplished by solving the banana kinetic equation. The results can be used to model energetic alpha particle and thermal particle losses in the presence of the electromagnetic waves in fusion relevant tokamak plasmas. They can also be employed to quantify transport losses in chaotic magnetic fields; these regimes are not known to exist in the theory of the chaotic magnetic field induced transport. The results of the theory together with those of the theory for neoclassical toroidal plasma viscosity reveal the existence of a universal collision frequency scaling law that governs the physics of the wave-particle interaction. The detailed collision frequency landscape in the theory for neoclassical toroidal viscosity is the universal feature of the wave-particle interaction in non-axisymmetric tori.

Eulerian approach to bounce-transit and drift resonance with magnetic drifts in tokamaks

Bounce-transit and drift resonance can be important to plasma confinement in tokamaks with a broken symmetry. The resonance usually is either treated by integrating along the unperturbed orbits or calculated using an action-angle approach. An Eulerian approach has been developed to take into account the momentum conservation property of the Coulomb collision operator. The difference between the Eulerian approach and other approaches is in the thermodynamic forces of the transport fluxes, and the corresponding toroidal plasma viscosity. The mass and heat flows that are parallel to the equilibrium magnetic field B appear in the thermodynamic forces in the Eulerian approach. However, in the existing Eulerian approach, only the E × B drift is kept in the theory; the magnetic drifts, i.e., ∇B, and curvature drifts are neglected by adopting the large aspect ratio assumption, where E is the electric field and B = |B|. Here, the Eulerian approach is extended to include the magnetic drifts, which is important for energetic alpha particles as well, to calculate neoclassical toroidal plasma viscosity in finite aspect ratio tokamaks. The relation to the nonlinear plasma viscosity in the plateau regime will also be discussed.

Neoclassical toroidal plasma viscosity with effects of finite banana width for finite aspect ratio tokamaks

Theory for neoclassical toroidal plasma viscosity has been developed to model transport phenomena, especially, toroidal plasma rotation for tokamaks with broken symmetry. Theoretical predictions are in agreement with the results of the numerical codes in the large aspect ratio limit. The theory has since been extended to include effects of finite aspect ratio and finite plasma β. Here, β is the ratio of the plasma thermal pressure to the magnetic field pressure. However, there are cases where the radial wavelength of the self-consistent perturbed magnetic field strength B on the perturbed magnetic surface is comparable to the width of the trapped particles, i.e., bananas. To accommodate those cases, the theory for neoclassical toroidal plasma viscosity is further extended here to include the effects of the finite banana width. The extended theory is developed using the orbit averaged drift kinetic equation in the low collisionality regimes. The results of the theory can now be used to model plasma transport, including toroidal plasma rotation, in real finite aspect ratio, and finite plasma β tokamaks with the radial wavelength of the perturbed symmetry breaking magnetic field strength comparable to or longer than the banana width.

Neoclassical toroidal plasma viscosity in the vicinity of the magnetic axis in tokamaks with broken symmetry

Theory for neoclassical toroidal plasma viscosity in the low collisionality regime is extended to the vicinity of the magnetic axis in tokamaks with broken symmetry. The toroidal viscosity is induced by particles drifting off the perturbed magnetic surface under the influence of the symmetry breaking magnetic field. In the region away from the magnetic axis, the drift orbit dynamics is governed by the bounce averaged drift kinetic equation in the low collisionality regimes. In the vicinity of the magnetic axis, it is the drift kinetic equation, averaged over the trapped particle orbits, i.e., potato orbits, that governs the drift dynamics. The orbit averaged drift kinetic equation is derived when collision frequency is low enough for trapped particles to complete their potato trajectories. The resultant equation is solved in the 1/ν regime to obtain transport fluxes and, thus, toroidal plasma viscosity through flux-force relation. Here, ν is the collision frequency. The viscosity does not vanish on the magnetic axis, and has the same scalings as that in the region away from magnetic axis, except that the fraction of bananas is replaced by the fraction of potatoes. It also has a weak radial dependence. Modeling of plasma flow velocity V for the case where the magnetic surfaces are broken is also discussed.

Finite toroidal flow generated by unstable tearing mode in a toroidal plasma

The neoclassical toroidal plasma viscosity torque and electromagnetic torque, generated by tearing mode (TM) in a toroidal plasma, are numerically investigated using the MARS-Q code [Liu et al., Phys. Plasmas 20, 042503 (2013)]. It is found that an initially unstable tearing mode can intrinsically drive a toroidal plasma flow resulting in a steady state solution, in the absence of the external momentum input and external magnetic field perturbation. The saturated flow is in the order of 0.5%ωA at the q=2 rational surface in the considered case, with q and ωA being the safety factor and the Alfven frequency at the magnetic axis, respectively. The generation of the toroidal flow is robust, being insensitive to the given amplitude of the perturbation at initial state. On the other hand, the flow amplitude increases with increasing the plasma resistivity. Furthermore, the initially unstable tearing mode is fully stabilized by non-linear interaction with the self-generated toroidal flow.

Transport theory for energetic alpha particles and tolerable magnitude of error fields in tokamaks with broken symmetry

A transport theory for energetic fusion born alpha particles in tokamaks with broken symmetry has been developed. The theory is a generalization of the theory for neoclassical toroidal plasma viscosity for thermal particles in tokamaks. It is shown that the radial energy transport rate can be comparable to the slowing down rate for energetic alpha particles when the ratio of the typical magnitude of the perturbed magnetic field strength to that of the equilibrium magnetic field strength is of the order of 10−4 or larger. This imposes a constraint on the magnitude of the error fields in thermonuclear fusion reactors. The implications on stellarators as potential fusion reactors are also discussed.

Intrinsic plasma rotation determined by neoclassical toroidal plasma viscosity in tokamaks

Intrinsic toroidal plasma rotation due to the neoclassical toroidal plasma viscosity (NTV) effect induced by a three-dimensional helical magnetic field ripple in tokamaks is investigated in this paper. The intrinsic rotation is determined self-consistently by searching for the roots of the ambipolarity constraint, after evaluation of the particle fluxes from the numerical modelling. In the low-collisionality case, there are three roots, in which two are stable roots. One corresponds to the ‘ion root’ in the counter-current direction, and the other stable one corresponds to the ‘electron root’ in the co-current direction, near which the electron flux is dominant. Both of the two stable roots scale like the diamagnetic frequency. In the high-collisionality case, there is only one ‘ion’ root. The application of this modelling for International Thermonuclear Experimental Reactor (ITER) cases is discussed. In a large range of plasma radii, there are three roots. The NTV torque drives plasma rotation in ITER towards one of the stable roots, depending on the initial condition. The amplitudes of the electron roots near the pedestal in both baseline and steady-state scenarios are much larger than that of the ion roots. The amplitudes of the NTV torque density and the electron roots near the pedestal increase with increasing height of the temperature pedestal in the ITER baseline scenario.

Numerical validation of the refined formula of neoclassical toroidal plasma viscosity in tokamaks

Neoclassical toroidal plasma viscosity (NTV) theory in collisionless regimes in tokamaks has been well developed in the past. The NTV evaluated from the connected formula developed by Shaing et al (2010 Nucl. Fusion 50 025020) was in good agreement with the numerical results in most cases. The boundary condition in the superbanana plateau regime has been found to be important in the numerical modelling when the resonant pitch is close to the boundaries of the pitch angle space, but it was not included in the original connected formula. This generates a big discrepancy between the numerical results and those evaluated from the smoothly connected formula as the resonant pitch is close to 0 or 1. Recently, the connected formula was refined. In this paper, we present the method of how to apply this refinement for practical NTV modelling and demonstrate the improvement of this refined formula by comparing the results evaluated from it with the numerical ones. Some techniques are developed to accurately model the NTV with the refined formula. The accuracy of the results modelled from the refined formula is strongly improved over the previous formula as the resonant pitch is close to 0 or 1 and they agree very well with the numerical ones.

Theory for neoclassical toroidal plasma viscosity in tokamaks

Error fields and magnetohydrodynamic modes break toroidal symmetry in tokamaks. The broken symmetry enhances the toroidal plasma viscosity, which results in a steady-state toroidal plasma flow. A theory for neoclassical toroidal plasma viscosity in the low-collisionality regimes is developed. It extends stellarator transport theory to include multiple modes and to allow for |m − nq| ∼ 1. Here, m is the poloidal mode number, n is the toroidal mode number and q is the safety factor. The bounce averaged drift kinetic equation is solved in several asymptotic limits to obtain transport fluxes. These fluxes depend non-linearly on the radial electric field except for those in the 1/ν regime. Here, ν is the collision frequency. The theory is refined to include the effects of the superbanana plateau resonance at the phase space boundary and the finite ∇B drift on the collisional boundary layer fluxes. Analytical expressions that connect all asymptotic limits are constructed and are in good agreement with the numerical results. The flux–force relations that relate transport fluxes to forces are used to illustrate the roles of transport fluxes in the momentum equation. It is shown that the ambipolar state is reached when the momentum equation is relaxed. It is also shown that the origin of the momentum for plasma flow generated without momentum sources is the local unbalance of particles' momenta and is diamagnetic in nature regardless of the details of the theory.

Non-resonant magnetic braking on JET and TEXTOR

The non-resonant magnetic braking effect induced by a non-axisymmetric magnetic perturbation is investigated on JET and TEXTOR. The collisionality dependence of the torque induced by the n = 1, where n is the toroidal mode number, magnetic perturbation generated by the error field correction coils on JET is observed. The observed torque is located mainly in the plasma core (normalized radius ρ < 0.4) and increases with decreasing collisionality. The neoclassical toroidal plasma viscosity (NTV) torque in the collisionless regime is modelled using the numerical solution of the bounce-averaged drift kinetic equation. The calculated collisionality dependence of the NTV torque is in good agreement with the experimental observation on JET. The reason for this collisionality dependence is that the torque in the plasma core on JET mainly comes from the flux of the trapped electrons, which are still mainly in the 1/ν regime. The strongest NTV torque on JET is also located near the plasma core. The magnitude of the NTV torque strongly depends on the plasma response, which is also discussed in this paper. There is no obvious braking effect with n = 2 magnetic perturbation generated by the dynamic ergodic divertor on TEXTOR, which is consistent with the NTV modelling.

Superbanana plateau resonance in the vicinity of the phase space boundary in tokamaks

A theory for the superbanana plateau resonance that occurs at the phase space boundary is presented to refine the comprehensive theory for neoclassical toroidal plasma viscosity in tokamaks. The results of the theory reproduce those of the standard superbanana plateau resonance theory when the resonance occurs away from the boundary. It shows that the strength of the superbanana plateau resonance weakens in the vicinity of the phase space boundary. It also indicates that it is important to know the resonance pitch angle parameter when comparing the results of the superbanana plateau resonance with those of the experiments. The theory is used to refine the kernel for the superbanana plateau resonance in the approximate expression for the neoclassical toroidal plasma viscosity.

Modelling of the neoclassical toroidal plasma viscosity torque in tokamaks

Neoclassical toroidal plasma viscosity (NTV) torque induced by non-axisymmetric magnetic perturbation in the collisionless regimes in tokamaks is modelled by solving the bounce-averaged drift kinetic equation numerically. The detailed comparison between the numerical and the analytic solutions of NTV is discussed in this paper. In different asymptotic limits of the collisionless regimes, the numerical solutions are in good agreement with the analytic results. The numerical results are different from the analytic results calculated from the smoothly connected formula in the transit regimes. The pitch angle scattering is especially important in the regime. The final difference between the numerical and the analytic results can be up to a factor of 2 near the transition between the non-resonant and resonant regimes. This reveals the importance of the boundary condition of the pitch angle space. The sign of the electric field is found to be important in the calculation of the NTV torque. It shows that the effect of the resonant particles makes the NTV torque more important for the lower collisionality and lower rotation cases, which are the International Thermonuclear Experimental Reactor relevant conditions. It also shows that the electron NTV torque is important in the low collisionality case. This numerical method can be applied for modelling the NTV torque in different collisionality regimes and their transitions in tokamaks without additional approximations.

Theory for island induced neoclassical toroidal plasma viscosity in tokamaks

Error fields and resistive magnetohydrodynamic modes are ubiquitous in real tokamaks. They break the toroidal symmetry in |B| in tokamaks. Here, B is the magnetic field. There are two mechanisms that break the symmetry on the perturbed magnetic surface: one is the perturbed field itself and the other results from the distortion of the magnetic surface due to the perturbed field. The broken toroidal symmetry leads to enhanced neoclassical toroidal plasma viscosity and consequently the rate of the toroidal flow damping. The neoclassical toroidal plasma viscosity also results in a steady-state toroidal plasma flow. In addition, the neoclassical toroidal plasma viscosity in the vicinity of the magnetic islands provides a mechanism to determine the island rotation frequency, which is an important quantity for the island stability. Here, the theory for neoclassical toroidal plasma viscosity in the vicinity of the magnetic island is extended to include the effects of the collisional boundary layer that lead to scaling in the transport fluxes, where ν is the collision frequency.

Flux–force relation for non-axisymmetric tori in general flux coordinates and neoclassical toroidal plasma viscosity

Flux–force relation, a fundamental relation that relates transport fluxes to forces, for non-axisymmetric tori in general magnetic flux coordinates that are not Hamada coordinates, is derived. The derivation is based on kinetic theory instead of fluid theory. It is shown that pressure force also contributes to the relation in non-Hamada coordinates in general to make the relation compatible with kinetic theory and to make it coordinates invariant. The results are applied to the theory for the neoclassical toroidal viscosity in tokamaks that have error fields or resistive magnetohydrodynamic (MHD) modes.

Effects of finite gradient B drift on collisional boundary layer analysis for neoclassical toroidal plasma viscosity in tokamaks

The effects of finite gradient B drift are included in the collisional boundary layer analysis to improve the accuracy of the neoclassical toroidal plasma viscosity in tokamaks that have error fields or magnetohydrodynamic activities present. Depending on the sign of the electric charge of the species and that of the radial electric field, the effects of finite gradient B drift can either reduce, if the E × B drift is in the same direction of the gradient B drift, or enhance, if these two drifts are in the opposite direction, the magnitude of the neoclassical toroidal plasma viscosity. Here, E is the electric field and B is the magnetic field. However, because the gradient B drift depends on the effective pitch angle, the net effects have to be properly weighted by integrating over the particle energy.

Neoclassical Toroidal Plasma Viscosity Torque in Collisionless Regimes in Tokamaks

Bumpiness in a magnetic field enhances the magnitude of the plasma viscosity and increases the rate of the plasma flow damping. A general solution of the neoclassical toroidal plasma viscosity (NTV) torque induced by nonaxisymmetric magnetic perturbation (NAMP) in the collisionless regimes in tokamaks is obtained in this Letter. The plasma angular momentum can be strongly changed, when there is a small deviation of the toroidal symmetry caused by a NAMP of the order of 0.1% of the toroidal field strength.