Transport Of Toroidal Angular Momentum Research Articles

Experimental challenges to stiffness as a transport paradigm

Two power scans were carried out in H-mode plasmas in DIII-D; one employed standard co-current neutral beam injection (NBI), while the other used a mixture of co-current and counter-current NBI to scan power while holding the torque to a low fixed value. Analysis of the ion and electron heat transport, ion toroidal angular momentum transport, and thermal deuterium transport from these scans is presented. Invariance of the gradients or gradient scalelengths, as might be expected from stiff transport, was not generally observed. When invariance was seen, it was not accompanied by a strong increase in transport, except in the case of the absolute deuterium ion transport. Conduction in the ion channel is the dominant energy loss mechanism. The variation of the ion heat transport with applied power is similar for the co-injection and fixed torque scans, indicating that E × B shearing is not determining the plasma response to additional power. There is however, a quantitative difference in the transport between the two scans, indicating E × B shearing does play a role in the transport. Comparison of these results with a previous experiment that directly probed stiffness at a single radius leads to the following conclusion: while local stiffness as formally defined may hold, invariance of the gradients or normalized scalelengths does not follow from stiff transport in more practical scaling experiments, such as the power scans discussed here. Possible reasons for the lack of correspondence between the local picture and the global expectations are discussed.

Neoclassical flows in deuterium–helium plasma density pedestals

In tokamak transport barriers, the radial scale of profile variations can be comparable to a typical ion orbit width, which makes the coupling of the distribution function across flux surfaces important in the collisional dynamics. We use the radially global steady-state neoclassical code Perfect (Landreman et al 2014 Plasma Phys. Control. Fusion 56 045005) to calculate poloidal and toroidal flows, and radial fluxes, in the pedestal. In particular, we have studied the changes in these quantities as the plasma composition is changed from a deuterium bulk species with a helium impurity to a helium bulk with a deuterium impurity, under specific profile similarity assumptions. In the presence of sharp profile variations, the poloidally resolved radial fluxes are important for the total fluxes to be divergence-free, which leads to the appearance of poloidal return-flows. These flows exhibit a complex radial–poloidal structure that extends several orbit widths into the core and is sensitive to abrupt radial changes in the ion temperature gradient. We find that a sizable neoclassical toroidal angular momentum transport can arise in the radially global theory, in contrast to the local.

Gyrokinetic Theory and Dynamics of the Tokamak Edge

AbstractThe validity of modern gyrokinetic field theory is assessed for the tokamak edge. The basic structure of the Lagrangian and resulting equations and their conservation laws is reviewed. The conventional microturbulence ordering for expansion is small potential/arbitrary wavelength. The equilibrium ordering for expansion is long wavelength/arbitrary amplitude. The long‐wavelength form of the conventional Lagrangian is derived in detail. The two Lagrangians are shown to match at long wavelength if the E × B Mach number is small enough for its corrections to the gyroaveraging to be neglected. Therefore, the conventional derivation and its Lagrangian can be used at all wavelengths if these conditions are satisfied. Additionally, dynamical compressibility of the magnetic field can be neglected if the plasma beta is small. This allows general use of a shear‐Alfvén Lagrangian for edge turbulence and self consistent equilibrium‐scale phenomena for flows, currents, and heat fluxes for conventional tokamaks without further modification by higher‐order terms. Corrections in polarisation and toroidal angular momentum transport due to these higher‐order terms for global edge turbulence computations are shown to be small. (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Toroidal angular momentum transport with non-axisymmetric magnetic fields

In this study, we calculate the radial transport of the toroidal angular momentum in the presence of non-axisymmetric magnetic fields. It is shown that the radial transport of the toroidal angular momentum, R2∇ζ·V, is proportional to the first order of gyro-radius. This implies that the neoclassical toroidal viscosity caused by asymmetric magnetic fields can change the toroidal rotation significantly.

Radial transport of toroidal angular momentum in tokamaks

The radial flux of toroidal angular momentum is needed to determine tokamak intrinsic rotation profiles. Its computation requires knowledge of the gyrokinetic distribution functions and turbulent electrostatic potential to second-order in ϵ = ρ/L, where ρ is the ion Larmor radius and L is the variation length of the magnetic field. In this article, a complete set of equations to calculate the radial transport of toroidal angular momentum in any tokamak is presented. In particular, the O(ϵ2) equations for the turbulent components of the distribution functions and electrostatic potential are given for the first time without assuming that the poloidal magnetic field over the magnetic field strength is small.

Radial transport with perturbed magnetic field

It is pointed out that the viscosity coefficient describing radial transport of toroidal angular momentum is proportional to the second power of the gyro-radius—like the corresponding coefficients for particle and heat transport—regardless of any geometrical symmetry. The observation is widely appreciated, but worth emphasizing because some literature gives the misleading impression that asymmetry can allow radial moment transport in first-order.

Dependence of intrinsic rotation reversals on collisionality in MAST

Tokamak plasmas rotate even without external injection of momentum. A Doppler backscattering system installed at MAST has allowed this intrinsic rotation to be studied in ohmic L-mode and H-mode plasmas, including the first observation of intrinsic rotation reversals in a spherical tokamak. Experimental results are compared to a novel 1D model, which captures the collisionality dependence of the radial transport of toroidal angular momentum due to the effect of neoclassical flows on turbulent fluctuations. The model is able to accurately reproduce the change in sign of core toroidal rotation, using experimental density and temperature profiles from shots with rotation reversals as inputs and no free parameters fit to experimental data.

A variational principle and its implications for radial electric field and rotation in tokamaks

A variational principle that minimizes an energy functional of the total electric field and the kinetic energy of the plasma flows with respect to the radial electric field is presented. Minimization is performed order by order in a small-gyroradius expansion, retaining up to second-order electric fields and third-order flows in the small parameter δ = ρi/L. The electric fields and flows obtained variationally are interpreted in the context of a global momentum conservation integral, while comparing them with neoclassical results based on the radial transport of toroidal angular momentum.

Turbulent momentum transport in core tokamak plasmas and penetration of scrape-off layer flows

The turbulent transport of toroidal angular momentum in the core of a tokamak plasma is investigated in global, full-f gyrokinetic simulations, performed with the GYSELA code in the flux-driven regime. During the initial turbulent phase, a front of positive Reynolds stress propagates radially, generating intrinsic toroidal rotation from a vanishing initial profile. This is also accompanied by a propagating front of turbulent heat flux. In the statistical steady-state regime, turbulent transport exhibits large-scale avalanche-like events which are found to transport both heat and momentum, and similar statistical properties are obtained for both fluxes. The impact of scrape-off layer flows is also investigated by modifying the boundary conditions in the simulations. The observed impact is radially localized for L-mode like poloidal profiles of parallel velocity at the edge, while a constant velocity at the edge can modify the core toroidal rotation profile in a large fraction of the radial domain.

Long-wavelength limit of gyrokinetics in a turbulent tokamak and its intrinsic ambipolarity

Recently, the electrostatic gyrokinetic Hamiltonian and change of coordinates have been computed to order ϵ2 in general magnetic geometry. Here ϵ is the gyrokinetic expansion parameter, the gyroradius over the macroscopic scale length. Starting from these results, the long-wavelength limit of the gyrokinetic Fokker–Planck and quasineutrality equations is taken for tokamak geometry. Employing the set of equations derived in the present paper, it is possible to calculate the long-wavelength components of the distribution functions and of the poloidal electric field to order ϵ2. These higher order pieces contain both neoclassical and turbulent contributions, and constitute one of the necessary ingredients (the other is given by the short-wavelength components up to second order) that will eventually enter a complete model for the radial transport of toroidal angular momentum in a tokamak in the low flow ordering. Finally, we provide an explicit and detailed proof that the system consisting of second-order gyrokinetic Fokker–Planck and quasineutrality equations leaves the long-wavelength radial electric field undetermined; that is, the turbulent tokamak is intrinsically ambipolar.

Gyrokinetic simulations with external resonant magnetic perturbations: Island torque and nonambipolar transport with plasma rotation

Static external resonant magnetic field perturbations (RMPs) have been added to the gyrokinetic code GYRO [J. Candy and R. E. Waltz, J. Comp. Phys. 186, 545 (2003)]. This allows nonlinear gyrokinetic simulations of the nonambipolar radial current flow jr, and the corresponding j→×B→ plasma torque (density) R[jrBp/c], induced by magnetic islands that break the toroidal symmetry of a tokamak. This extends the previous GYRO formulation for the transport of toroidal angular momentum (TAM) [R. E. Waltz, G. M. Staebler, J. Candy, and F. L. Hinton, Phys. Plasmas 14, 122507 (2007); errata 16, 079902 (2009)]. The focus is on electrostatic full torus radial slice simulations of externally induced q=m/n=6/3 islands with widths 5% of the minor radius or about 20 ion gyroradii. Up to moderately strong E×B rotation, the island torque scales with the radial electric field at the resonant surface Er, the island width w, and the intensity I of the high-n micro-turbulence, as ErwI. The radial current inside the island is carried (entirely in the n=3 component) and almost entirely by the ion E×B flux, since the electron E×B and magnetic flutter particle fluxes are cancelled. The net island torque is null at zero Er rather than at zero toroidal rotation. This means that while the expected magnetic braking of the toroidal plasma rotation occurs at strong co- and counter-current rotation, at null toroidal rotation, there is a small co-directed magnetic acceleration up to the small diamagnetic (ion pressure gradient driven) co-rotation corresponding to the zero Er and null torque. This could be called the residual stress from an externally induced island. At zero Er, the only effect is the expected partial flattening of the electron temperature gradient within the island. Finite-beta GYRO simulations demonstrate almost complete RMP field screening and n=3 mode unlocking at strong Er.

Up-down symmetry of the turbulent transport of toroidal angular momentum in tokamaks

Two symmetries of the local nonlinear δf gyrokinetic system of equations in tokamaks in the high flow regime are presented. The turbulent transport of toroidal angular momentum changes sign under an up-down reflection of the tokamak and a sign change of both the rotation and the rotation shear. Thus, the turbulent transport of toroidal angular momentum must vanish for up-down symmetric tokamaks in the absence of both rotation and rotation shear. This has important implications for the modeling of spontaneous rotation.

Transport of momentum in full f gyrokinetics

Full f electrostatic gyrokinetic formulations employ two gyrokinetic equations, one for ions and the other for electrons, and quasineutrality to obtain the ion and electron distribution functions and the electrostatic potential. We demonstrate with several examples that the long wavelength radial electric field obtained with full f approaches is extremely sensitive to errors in the ion and electron density since small deviations in density give rise to large, nonphysical deviations in the conservation of toroidal angular momentum. For typical tokamak values, a relative error of 10−7 in the ion or electron densities is enough to obtain the incorrect toroidal rotation. Based on the insights gained with the examples considered, three simple tests to check transport of toroidal angular momentum in full f simulations are proposed.

Turbulent transport of toroidal angular momentum in low flow gyrokinetics

We derive a self-consistent equation for the turbulent transport of toroidal angular momentum in tokamaks in the low flow ordering that only requires solving gyrokinetic Fokker–Planck and quasineutrality equations correct to second order in an expansion on the gyroradius over scale length. We also show that according to our orderings the long wavelength toroidal rotation and the long wavelength radial electric field satisfy the neoclassical relation that gives the toroidal rotation as a function of the radial electric field and the radial gradients of pressure and temperature. Thus, the radial electric field can be solved for once the toroidal rotation is calculated from the transport of toroidal angular momentum. Unfortunately, even though this methodology only requires a gyrokinetic model correct to second order in gyroradius over scale length, current gyrokinetic simulations are only valid to first order. To overcome this difficulty, we exploit the smallish ratio Bp/B, where B is the total magnetic field and Bp is its poloidal component. When Bp/B is small, the usual first order gyrokinetic equation provides solutions that are accurate enough to employ for our expression for the transport of toroidal angular momentum. We show that current δf and full f simulations only need small corrections to achieve this accuracy. Full f simulations, however, are still unable to determine the long wavelength, radial electric field from the quasineutrality equation.

Effects of electron dynamics in toroidal momentum transport driven by ion temperature gradient turbulence

Several issues in the toroidal angular momentum transport by the ion temperature gradient (ITG) turbulence are addressed in this work: the system size effect in the momentum transport, the symmetry breaking mechanism for the generation of the spontaneous rotation and the effect of the trapped electron dynamics. We find that the magnitude of the momentum flux scales with the system size according to the gyroBohm scaling, with no significant size effect on the radial structure of the perturbed toroidal angular momentum. The symmetry breaking due to the shear of the radial electric field is found to be a mechanism for generating the residual momentum flux. However, it is small compared with the momentum pinch term in the case of a finite background rotation. The trapped electrons in the ITG turbulence increase the intensity and modify the spectral properties of the electrostatic fluctuations, leading to the increase in the toroidal momentum pinch.

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.

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.

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.

Spontaneous poloidal spin-up of tokamaks and the transition to the H mode.

The radial transport of toroidal angular momentum and circulation in a tokamak resulting from diffusion that is poloidally asymmetric is shown to produce an instability of the poloidal rotation. This instability, due to Stringer, sets in where the local particle-confinement time is smaller than the damping time of poloidal flow and leads to poloidal velocity shar. The nonlinear interplay between the poloidal spin-up and turulence-driven anoamlous transport is shown to lead to bifurcated equilibria of the type observed in the L-to-H-mode transition in tokamaks.

Neoclassical ion transport in rotating axisymmetric plasmas

The neoclassical theory of ion transport in rotating axisymmetric plasmas is formulated. The flow speed is allowed to be of the order of the ion thermal speed. It is shown that the ion distribution function becomes Maxwellian, with temperature uniform on a magnetic surface, and the poloidal flow decays, in a few transit or collision times, in general. A drift kinetic equation is derived which is a simple generalization of the drift kinetic equation for nonrotating plasmas. The radial gradient of the toroidal angular velocity appears as a driving term like the temperature gradient. Both gradients drive the transport of toroidal angular momentum and energy, in general; Onsager relations for the two-by-two transport matrix are derived. The off-diagonal transport coefficients are shown to be zero if the magnetic field has up–down symmetry. A simple expression for the enhancement of the ion thermal conductivity in the banana regime, caused by rotation, is derived. The neoclassical viscosity is shown not to be enhanced by rotation in the banana regime, and to be small in the collisionality parameter in the collisional regime, assuming up–down symmetry. In the collisional regime, the thermal conductivity is shown to be suppressed by the effects of rotation.