# F. I. Parra

## Publications Authored By F. I. Parra

The bootstrap current and flow velocity of a low-collisionality stellarator plasma are calculated. As far as possible, the analysis is carried out in a uniform way across all low-collisionality regimes in general stellarator geometry, assuming only that the confinement is good enough that the plasma is approximately in local thermodynamic equilibrium. It is found that conventional expressions for the ion flow speed and bootstrap current in the low-collisionality limit are accurate only in the $1/\nu$-collisionality regime and need to be modified in the $\sqrt{\nu}$-regime. Read More

Due to their capability to reduce turbulent transport in magnetized plasmas, understanding the dynamics of zonal flows is an important problem in the fusion programme. Since the pioneering work by Rosenbluth and Hinton in axisymmetric tokamaks, it is known that studying the linear and collisionless relaxation of zonal flow perturbations gives valuable information and physical insight. Recently, the problem has been investigated in stellarators and it has been found that in these devices the relaxation process exhibits a characteristic feature: a damped oscillation. Read More

Introducing up-down asymmetry into the tokamak magnetic equilibria appears to be a feasible method to drive fast intrinsic toroidal rotation in future large devices. In this paper we investigate how the intrinsic momentum transport generated by up-down asymmetric shaping scales with the mode number of the shaping effects. Making use the gyrokinetic tilting symmetry (Ball et al (2016) Plasma Phys. Read More

**Authors:**A Kirk, J Adamek, RJ Akers, S Allan, L Appel, F Arese Lucini, M Barnes, T Barrett, N Ben Ayed, W Boeglin, J Bradley, P K Browning, J Brunner, P Cahyna, M Carr, F Casson, M Cecconello, C Challis, IT Chapman, S Chapman, S Conroy, N Conway, WA Cooper, M Cox, N Crocker, B Crowley, S Cardnell, J Chorley, G Cunningham, A Danilov, D Darrow, R Dendy, D Dickinson, W Dorland, B Dudson, L Easy, S Elmore, M Evans, T Farley, N Fedorczak, A Field, I Fitzgerald, M Fox, S Freethy, L Garzotti, YC Ghim, K Gi, M Gorelenkova, W Gracias, C Gurl, W Guttenfelder, C Ham, D Harting, E Havlickova, N Hawkes, T Hender, S Henderson, J Hillesheim, B Hnat, J Horacek, J Howard, D Howell, D Dunai, G Fishpool, K Gibson, J Harrison, E Highcock, B Huang, M Inomoto, R Imazawa, O Jones, K Kadowaki, S Kaye, D Keeling, M Kocan, L Kogan, M Komm, W Lai, J Leddy, H Leggate, K Imada, I Klimek, J Hollocombe, B Lipschultz, S Lisgo, YQ Liu, B Lloyd, B Lomanowski, V Lukin, G Maddison, J Madsen, J Mailloux, R Martin, G McArdle, I Lupelli, K McClements, B McMillan, A Meakins, H Meyer, C Michael, F Militello, J Milnes, G Motojima, D Muir, G Naylor, A Nielsen, M O'Brien, M O'Mullane, J Olsen, J Omotani, Y Ono, S Pamela, AW Morris, T O'Gorman, L Pangione, F Parra, A Patel, W Peebles, R Perez, S Pinches, L Piron, M Price, M Reinke, P Ricci, F Riva, C Roach, M Romanelli, D Ryan, S Saarelma, A Saveliev, R Scannell, A Schekochihin, S Sharapov, R Sharples, V Shevchenko, K Shinohara, S Silburn, J Simpson, A Stanier, J Storrs, H Summers, Y Takase, P Tamain, H Tanabe, H Tanaka, K Tani, D Taylor, D Thomas, N Thomas-Davies, A Thornton, M Turnyanskiy, M Valovic, R Vann, F Van Wyk, N Walkden, T Watanabe, H Wilson, M Wischmeier, T Yamada, J Young, S Zoletnik, the MAST Team, the EUROfusion MST1 Team

**Category:**Physics - Plasma Physics

New results from MAST are presented that focus on validating models in order to extrapolate to future devices. Measurements during start-up experiments have shown how the bulk ion temperature rise scales with the square of the reconnecting field. During the current ramp up models are not able to correctly predict the current diffusion. Read More

In general, the orbit-averaged radial magnetic drift of trapped particles in stellarators is non-zero due to the three-dimensional nature of the magnetic field. Stellarators in which the orbit-averaged radial magnetic drift vanishes are called omnigeneous, and they exhibit neoclassical transport levels comparable to those of axisymmetric tokamaks. However, the effect of deviations from omnigeneity cannot be neglected in practice. Read More

The flow shear associated with the differential toroidal rotation of tokamak plasmas breaks an underlying symmetry of the turbulent fluctuations imposed by the up-down symmetry of the magnetic equilibrium. Using experimental Beam-Emission-Spectroscopy (BES) measurements and gyrokinetic simulations, this symmetry breaking in ion-scale turbulence in MAST is shown to manifest itself as a tilt of the spatial correlation function and a finite skew in the distribution of the fluctuating density field. The tilt is a statistical expression of the "shearing" of the turbulent structures by the mean flow. Read More

Measurements of the relaxation of a zonal electrostatic potential perturbation in a non-axisymmetric magnetically confined plasma are presented. A sudden perturbation of the plasma equilibrium is induced by the injection of a cryogenic hydrogen pellet in the TJ-II stellarator, which is observed to be followed by a damped oscillation in the electrostatic potential. The waveform of the relaxation is consistent with theoretical calculations of zonal potential relaxation in a non-axisymmetric magnetic geometry. Read More

We develop a gyrokinetic treatment for ions in the magnetic presheath, close to the plasma-wall boundary. We focus on magnetic presheaths with a small magnetic field to wall angle, $\alpha \ll 1$ (in radians). Characteristic lengths perpendicular to the wall in such a magnetic presheath scale with the typical ion Larmor orbit size, $\rho_{\text{i}}$. Read More

Achieving impurity and helium ash control is a crucial issue in the path towards fusion-grade magnetic confinement devices, and this is particularly the case of helical reactors, whose low-collisionality ion-root operation scenarios usually display a negative radial electric field which is expected to cause inwards impurity pinch. In these work we discuss, based on experimental measurements and standard predictions of neoclassical theory, how plasmas of very low ion collisionality, similar to those observed in the impurity hole of the Large Helical Device, can be an exception to this general rule, and how a negative radial electric field can coexist with an outward impurity flux. This interpretation is supported by comparison with documented discharges available in the International Stellarator-Heliotron Profile Database, and it can be extrapolated to show that achievement of high ion temperature in the core of helical devices is not fundamentally incompatible with low core impurity content. Read More

In electrostatic simulations of MAST plasma at electron-gyroradius scales, using the local flux-tube gyrokinetic code GS2 with adiabatic ions, we find that the long-time saturated electron heat flux (the level most relevant to energy transport) decreases as the electron collisionality decreases. At early simulation times, the heat flux "quasi-saturates" without any strong dependence on collisionality, and with the turbulence dominated by streamer-like radially elongated structures. However, the zonal fluctuation component continues to grow slowly until much later times, eventually leading to a new saturated state dominated by zonal modes and with the heat flux proportional to the collision rate, in approximate agreement with the experimentally observed collisionality scaling of the energy confinement in MAST. Read More

Tokamaks with up-down asymmetric poloidal cross-sections spontaneously rotate due to turbulent transport of momentum. In this work, we investigate the effect of the Shafranov shift on this intrinsic rotation, primarily by analyzing tokamaks with tilted elliptical flux surfaces. By expanding the Grad-Shafranov equation in the large aspect ratio limit we calculate the magnitude and direction of the Shafranov shift in tilted elliptical tokamaks. Read More

We review in a tutorial fashion some of the causes of impurity density variations along field lines and radial impurity transport in the moment approach framework. An explicit and compact form of the parallel inertia force valid for arbitrary toroidal geometry and magnetic coordinates is derived and shown to be non-negligible for typical TJ-II plasma conditions. In the second part of the article, we apply the fluid model including main ion-impurity friction and inertia to observations of asymmetric emissivity patterns in neutral beam heated plasmas of the TJ-II stellarator. Read More

Breaking the up-down symmetry of tokamaks removes a constraint limiting intrinsic momentum transport, and hence toroidal rotation, to be small. Using gyrokinetic theory, we study the effect of different up-down asymmetric flux surface shapes on the turbulent transport of momentum. This is done by perturbatively expanding the gyrokinetic equation in large flux surface shaping mode number. Read More

A poloidal tilting symmetry of the local nonlinear $\delta f$ gyrokinetic model is demonstrated analytically and verified numerically. This symmetry shows that poloidally rotating all the flux surface shaping effects with large poloidal mode number by a single tilt angle has an exponentially small effect on the transport properties of a tokamak. This is shown using a generalization of the Miller local equilibrium model to specify an arbitrary flux surface geometry. Read More

Intrinsic toroidal rotation in a tokamak can be driven by turbulent momentum transport due to neoclassical flow effects breaking a symmetry of turbulence. In this paper we categorize the contributions due to neoclassical effects to the turbulent momentum transport, and evaluate each contribution using gyrokinetic simulations. We find that the relative importance of each contribution changes with collisionality. Read More

In the linear collisionless limit, a zonal potential perturbation in a toroidal plasma relaxes, in general, to a non-zero residual value. Expressions for the residual value in tokamak and stellarator geometries, and for arbitrary wavelengths, are derived. These expressions involve averages over the lowest order particle trajectories, that typically cannot be evaluated analytically. Read More

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 $\epsilon = \rho/L$, where $\rho$ 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. Read More

A stellarator is said to be omnigeneous if all particles have vanishing average radial drifts. In omnigeneous stellarators, particles are perfectly confined in the absence of turbulence and collisions, whereas in non-omnigeneous configurations, particle can drift large radial distances. One of the consequences of omnigeneity is that the unfavorable inverse scaling with collisionality of the stellarator neoclassical fluxes disappears. Read More

A set of flux tube gyrokinetic equations that includes the effect of the spatial variation of the density, temperature and rotation gradients on the turbulence is derived. This new set of equations uses periodic boundary conditions. In the limit $l_\bot/L \ll 1$, where $l_\bot$ is the characteristic perpendicular length of turbulent structures and $L$ is the characteristic size of the device, this new set of flux tube gyrokinetic equations is shown to be equivalent to the traditional global $\delta f$ gyrokinetic equations to an order higher in $l_\bot/L$ than the usual flux tube formulations. Read More

Using analytic calculations, the effects of the edge flux surface shape and the toroidal current profile on the penetration of flux surface shaping are investigated in a tokamak. It is shown that the penetration of shaping is determined by the poloidal variation of the poloidal magnetic field on the surface. This fact is used to investigate how different flux surface shapes penetrate from the edge. Read More

The difference between the guiding center phase-space Lagrangians derived in [J.W. Burby, J. Read More

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. Read More

Quasisymmetric stellarators are a type of optimized stellarators for which flows are undamped to lowest order in an expansion in the normalized Larmor radius. However, perfect quasisymmetry is impossible. Since large flows may be desirable as a means to reduce turbulent transport, it is important to know when a stellarator can be considered to be sufficiently close to quasisymmetry. Read More

Self-consistent equations for intrinsic rotation in tokamaks with small poloidal magnetic field $B_p$ compared to the total magnetic field $B$ are derived. The model gives the momentum redistribution due to turbulence, collisional transport and energy injection. Intrinsic rotation is determined by the balance between the momentum redistribution and the turbulent diffusion and convection. Read More

Plasma flow is damped in stellarators because they are not intrinsically ambipolar, unlike tokamaks, in which the flux-surface averaged radial electric current vanishes for any value of the radial electric field. Only quasisymmetric stellarators are intrinsically ambipolar, but exact quasisymmetry is impossible to achieve in non-axisymmetric toroidal configurations. By calculating the violation of intrinsic ambipolarity due to deviations from quasisymmetry, one can derive criteria to assess when a stellarator can be considered quasisymmetric in practice, i. Read More

A generic non-symmetric magnetic field does not confine magnetized charged particles for long times due to secular magnetic drifts. Stellarator magnetic fields should be omnigeneous (that is, designed such that the secular drifts vanish), but perfect omnigeneity is technically impossible. There always are small deviations from omnigeneity that necessarily have large gradients. Read More

Recent work demonstrated that breaking the up-down symmetry of tokamak flux surfaces removes a constraint that limits intrinsic momentum transport, and hence toroidal rotation, to be small. We show, through MHD analysis, that ellipticity is most effective at introducing up-down asymmetry throughout the plasma. We detail an extension to GS2, a local $\delta f$ gyrokinetic code that self-consistently calculates momentum transport, to permit up-down asymmetric configurations. Read More

Conventional radially-local neoclassical calculations become inadequate if the radial gradient scale lengths of the H-mode pedestal become as small as the poloidal ion gyroradius. Here, we describe a radially global $\delta f$ continuum code that generalizes neoclassical calculations to allow stronger gradients. As with conventional neoclassical calculations, the formulation is time-independent and requires only the solution of a single sparse linear system. Read More

Gyrokinetic simulations have greatly improved our theoretical understanding of turbulent transport in fusion devices. Most gyrokinetic models in use are delta-f simulations in which the slowly varying radial profiles of density and temperature are assumed to be constant for turbulence saturation times, and only the turbulent electromagnetic fluctuations are calculated. New massive simulations are being built to self-consistently determine the radial profiles of density and temperature. Read More

The induced electric field in a tokamak drives a parallel electron current flow. In an inhomogeneous, finite beta plasma, when this electron flow is comparable to the ion thermal speed, the Alfven mode wave solutions of the electromagnetic gyrokinetic equation can become nearly purely growing kink modes. Using the new "low-flow" version of the gyrokinetic code GS2 developed for momentum transport studies [Barnes et al 2013 Phys. Read More

A local magnetic equilibrium solution is sought around the magnetic axis in order to identify the key parameters defining the magnetic-surface's up-down asymmetry in the core of tokamak plasmas. The asymmetry is found to be determined essentially by the ratio of the toroidal current density flowing on axis to the fraction of the external field's odd perturbation that manages to propagate from the plasma boundary into the core. The predictions are tested and illustrated first with an analytical Solovev equilibrium and then using experimentally relevant numerical equilibria. Read More

Rotation is favorable for confinement, but a stellarator can rotate at high speeds if and only if it is sufficiently close to quasisymmetry. This article investigates how close it needs to be. For a magnetic field $\mathbf{B} = \mathbf{B}_0 + \alpha \mathbf{B}_1$, where $\mathbf{B}_0$ is quasisymmetric, $\alpha\mathbf{B}_1$ is a deviation from quasisymmetry, and $\alpha\ll 1$, the stellarator can rotate at high velocities if $\alpha < \epsilon^{1/2}$, with $\epsilon$ the ion Larmor radius over the characteristic variation length of $\mathbf{B}_0$. Read More

The effect of small deviations from a Maxwellian equilibrium on turbulent momentum transport in tokamak plasmas is considered. These non-Maxwellian features, arising from diamagnetic effects, introduce a strong dependence of the radial flux of co-current toroidal angular momentum on collisionality: As the plasma goes from nearly collisionless to weakly collisional, the flux reverses direction from radially inward to outward. This indicates a collisionality-dependent transition from peaked to hollow rotation profiles, consistent with experimental observations of intrinsic rotation. Read More

The ion toroidal rotation in a tokamak consists of an $E\times B$ flow due to the radial electric field and a diamagnetic flow due to the radial pressure gradient. The turbulent pinch of toroidal angular momentum due to the Coriolis force studied in previous work is only applicable to the $E\times B$ flow. In this Letter, the momentum pinch for the rotation generated by the radial pressure gradient is calculated and is compared with the Coriolis pinch. Read More

Using a two-dimensional fluid description, we investigate the nonlinear radial-longitudinal dynamics of intense beams in storage rings and cyclotrons. With a multiscale analysis separating the time scale associated with the betatron motion and the slower time scale associated with space-charge effects, we show that the longitudinal-radial vortex motion can be understood in the frame moving with the charged beam as the nonlinear advection of the beam by the $\mathbf{E}\times\mathbf{B}$ velocity field, where $\mathbf{E}$ is the electric field due to the space charge and $\mathbf{B}$ is the external magnetic field. This interpretation provides simple explanations for the stability of round beams and for the development of spiral halos in elongated beams. Read More

Beam Emission Spectroscopy (BES) measurements of ion-scale density fluctuations in the MAST tokamak are used to show that the turbulence correlation time, the drift time associated with ion temperature or density gradients, the particle (ion) streaming time along the magnetic field and the magnetic drift time are consistently comparable, suggesting a "critically balanced" turbulence determined by the local equilibrium. The resulting scalings of the poloidal and radial correlation lengths are derived and tested. The nonlinear time inferred from the density fluctuations is longer than the other times; its ratio to the correlation time scales as $\nu_{*i}^{-0. Read More

Scaling laws for the transport and heating of trace heavy ions in low-frequency, magnetized plasma turbulence are derived and compared with direct numerical simulations. The predicted dependences of turbulent fluxes and heating on ion charge and mass number are found to agree with numerical results for both stationary and differentially rotating plasmas. Heavy ion momentum transport is found to increase with mass, and heavy ions are found to be preferentially heated, implying a mass-dependent ion temperature for very weakly collisional plasmas and for partially-ionized heavy ions in strongly rotating plasmas. Read More

The injection of lower hybrid waves for current drive into a tokamak affects the profile of intrinsic rotation. In this article, the momentum deposition by the lower hybrid wave on the electrons is studied. Due to the increase in the poloidal momentum of the wave as it propagates into the tokamak, the parallel momentum of the wave increases considerably. Read More

Recently, the electrostatic gyrokinetic Hamiltonian and change of coordinates have been computed to order $\epsilon^2$ in general magnetic geometry. Here $\epsilon$ 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. Read More

Sheared toroidal flows can cause bifurcations to zero-turbulent-transport states in tokamak plasmas. The maximum temperature gradients that can be reached are limited by subcritical turbulence driven by the parallel velocity gradient. Here it is shown that q/\epsilon (magnetic field pitch/inverse aspect ratio) is a critical control parameter for sheared tokamak turbulence. Read More

The generation of intrinsic rotation by turbulence and neoclassical effects in tokamaks is considered. To obtain the complex dependences observed in experiments, it is necessary to have a model of the radial flux of momentum that redistributes the momentum within the tokamak in the absence of a preexisting velocity. When the lowest order gyrokinetic formulation is used, a symmetry of the model precludes this possibility, making small effects in the gyroradius over scale length expansion necessary. Read More

Using theoretical arguments, a simple scaling law for the size of the intrinsic rotation observed in tokamaks in the absence of momentum injection is found: the velocity generated in the core of a tokamak must be proportional to the ion temperature difference in the core divided by the plasma current, independent of the size of the device. The constant of proportionality is of the order of $10\,\mathrm{km \cdot s^{-1} \cdot MA \cdot keV^{-1}}$. When the intrinsic rotation profile is hollow, i. Read More

First-principles numerical simulations are used to describe a transport bifurcation in a differentially rotating tokamak plasma. Such a bifurcation is more probable in a region of zero magnetic shear than one of finite magnetic shear because in the former case the component of the sheared toroidal flow that is perpendicular to the magnetic field has the strongest suppressing effect on the turbulence. In the zero-magnetic-shear regime, there are no growing linear eigenmodes at any finite value of flow shear. Read More

Scaling laws for ion temperature gradient driven turbulence in magnetized toroidal plasmas are derived and compared with direct numerical simulations. Predicted dependences of turbulence fluctuation amplitudes, spatial scales, and resulting heat fluxes on temperature gradient and magnetic field line pitch are found to agree with numerical results in both the driving and inertial ranges. Evidence is provided to support the critical balance conjecture that parallel streaming and nonlinear perpendicular decorrelation times are comparable at all spatial scales, leading to a scaling relationship between parallel and perpendicular spatial scales. Read More

A low flow, $\delta f$ gyrokinetic formulation to obtain the intrinsic rotation profiles is presented. The momentum conservation equation in the low flow ordering contains new terms, neglected in previous first principles formulations, that may explain the intrinsic rotation observed in tokamaks in the absence of external sources of momentum. The intrinsic rotation profile depends on the density and temperature profiles and on the up-down asymmetry. Read More

Two symmetries of the local nonlinear delta-f 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. Read More

The effect of momentum injection on the temperature gradient in tokamak plasmas is studied. A plausible scenario for transitions to reduced transport regimes is proposed. The transition happens when there is sufficient momentum input so that the velocity shear can suppress or reduce the turbulence. Read More

Gyrokinetic theory is based on an asymptotic expansion in the small parameter $\epsilon$, defined as the ratio of the gyroradius and the characteristic length of variation of the magnetic field. In this article, this ordering is strictly implemented to compute the electrostatic gyrokinetic phase-space Lagrangian in general magnetic geometry to order $\epsilon^2$. In particular, a new expression for the complete second-order gyrokinetic Hamiltonian is provided, showing that in a rigorous treatment of gyrokinetic theory magnetic geometry and turbulence cannot be dealt with independently. Read More

The effect of flow shear on turbulent transport in tokamaks is studied numerically in the experimentally relevant limit of zero magnetic shear. It is found that the plasma is linearly stable for all non-zero flow shear values, but that subcritical turbulence can be sustained nonlinearly at a wide range of temperature gradients. Flow shear increases the nonlinear temperature gradient threshold for turbulence but also increases the sensitivity of the heat flux to changes in the temperature gradient, except over a small range near the threshold where the sensitivity is decreased. Read More

Nonlinear gyrokinetic simulations have been conducted to investigate turbulent transport in tokamak plasmas with rotational shear. At sufficiently large flow shears, linear instabilities are suppressed, but transiently growing modes drive subcritical turbulence whose amplitude increases with flow shear. This leads to a local minimum in the heat flux, indicating an optimal E x B shear value for plasma confinement. Read More