# Publications by Felix Parra Diaz

## Semianalytical calculation of the zonal-flow oscillation frequency in stellarators

Plasma Physics and Controlled Fusion IOP Publishing **59** (2017) 065005-

Due to their capability to reduce turbulent transport in magnetized plasmas, understanding the dynamics of zonal flows is an important problem in the fusion program. Since the pioneering work by Rosenbluth and Hinton in axisymmetric tokamaks, it is known that studying the linear and collisionless relaxation of zonal flow perturbation s 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. The frequency of this oscillation might be a relevant parameter in the regulation of turbulent transport, and therefore its efficient and accurate calculation is important. Although an analytical expression can be derived for the frequency, its numerical evaluation is not simple and has not been exploited systematically so far. Here, a numerical method for its evaluation is considered, and the results are compared with those obtained by calculating the frequency from gyrokinetic simulations. This 'semianalytical' approach for the determination of the zonal-flow frequency is revealed to be accurate and faster than the one based on gyrokinetic simulations.

## On the effect of neoclassical flows on intrinsic momentum in ASDEX Upgrade Ohmic L-mode plasmas

NUCLEAR FUSION **57** (2017) ARTN 046008

## Stellarator bootstrap current and plasma flow velocity at low collisionality

Journal of Plasma Physics Cambridge University Press **83** (2017) 1-25

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. The correction due to finite collisionality is also discussed and is found to scale as $\nu^{2/5}$.

## The effect of tangential drifts on neoclassical transport in stellarators close to omnigeneity

Plasma Physics and Controlled Fusion IOP Publishing **59** (2017) 055014

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, and it is more deleterious at small collisionalities. For sufficiently low collision frequencies (below the values that define the 1/ν regime), the components of the drifts tangential to the flux surface become relevant. This article focuses on the study of such collisionality regimes in stellarators close to omnigeneity when the gradient of the non-omnigeneous perturbation is small. First, it is proven that closeness to omnigeneity is required to actually preserve radial locality in the drift-kinetic equation for collisionalities below the 1/ν regime. Then, using the derived radially local equation, it is shown that neoclassical transport is determined by two layers located at different regions of phase space. One of the layers corresponds to the so-called √ν regime and the other to the so-called superbanana-plateau regime. The importance of the superbanana-plateau layer for the calculation of the tangential electric field is emphasized, as well as the relevance of the latter for neoclassical transport in the collisionality regimes considered in this paper. In particular, the role of the tangential electric field is essential for the emergence of a new subregime of superbanana-plateau transport when the radial electric field is small. A formula for the ion energy flux that includes the √ν regime and the superbanana-plateau regime is given. The energy flux scales with the square of the size of the deviation from omnigeneity. Finally, it is explained why below a certain collisionality value the formulation presented in this article ceases to be valid.

## The effect of lower hybrid waves on JET plasma rotation

NUCLEAR FUSION **57** (2017) ARTN 034002

## Symmetry breaking in MAST plasma turbulence due to toroidal flow shear

Plasma Physics and Controlled Fusion Institute of Physics **59** (2016) 034002-

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. The skewness of the distribution is related to the emergence of long-lived density structures in sheared, near-marginal plasma turbulence. The extent to which these effects are pronounced is argued (with the aid of the simulations) to depend on the distance from the nonlinear stability threshold. Away from the threshold, the symmetry is effectively restored.

## Gyrokinetic treatment of a grazing angle magnetic field

Plasma Physics and Controlled Fusion Institute of Physics **59** (2017) 025015-

>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, α ⟪ 1. Characteristic lengths perpendicular to the wall in such a magnetic presheath scale with the typical ion Larmor orbit size, pi. The smallest scale length associated with variations parallel to the wall is taken to be across the magnetic field, and ordered l = ρi/δ, where δ ⟪ 1 is assumed. The scale lengths along the magnetic field line are assumed so long that variations associated with this direction are neglected. These orderings are consistent with what we expect close to the divertor target of a tokamak. We allow for a strong electric field E in the direction normal to the electron repelling wall, with strong variation in the same direction. The large change of the electric field over an ion Larmor radius distorts the orbit so that it is not circular. We solve for the lowest order orbits by identifying coordinates, which consist of constants of integration, an adiabatic invariant and a gyrophase, associated with periodic ion motion in the system with α = δ = 0. By using these new coordinates as variables in the limit α ~ δ ⟪ 1, we obtain a generalized ion gyrokinetic equation. We find another quantity that is conserved to first order and use this to simplify the gyrokinetic equation, solving it in the case of a collisionless magnetic presheath. Assuming a Boltzmann response for the electrons, a form of the quasineutrality equation that exploits the change of variables is derived. The gyrokinetic and quasineutrality equations give the ion distribution function and electrostatic potential in the magnetic presheath if the entrance boundary condition is specified.

## Turbulent momentum transport due to the beating between different tokamak flux surface shaping effects

Plasma Physics and Controlled Fusion IOP Publishing **59** (2017) 024007

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. Control. Fusion 58 045023), we study the effect of envelopes created by the beating of different high-order shaping effects. This reveals that the presence of an envelope can change the scaling of the momentum flux from exponentially small in the limit of large shaping mode number to just polynomially small. This enhancement of the momentum transport requires the envelope to be both up–down asymmetric and have a spatial scale on the order of the minor radius.

## Implementation of multiple species collision operator in gyrokinetic code GS2

44th EPS Conference on Plasma Physics, EPS 2017 (2017)

## Toroidal rotation reversals in JET plasmas

44th EPS Conference on Plasma Physics, EPS 2017 (2017)

© 2017 IOP Publishing Ltd. Recent experiments at JET studied the effect of density on the rotation of Ohmic divertor plasmas. As the density increased, two core rotation reversals were observed, showing two regimes of peaked co-current rotation. The experiment was done with hydrogen and deuterium plasmas, critical densities for reversal appear to be independent on isotope type.

## Moderation of neoclassical impurity accumulation in high temperature plasmas of helical devices

Nuclear Fusion IOP Publishing **57** (2016) 016016-

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 this 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 (Yoshinuma et al and The LHD Experimental Group 2009 Nucl. Fusion 49 062002, Ida et al and The LHD Experimental Group 2009 Phys. Plasmas 16 056111 and Yokoyama et al and LHD Experimental Group 2002 Nucl. Fusion 42 143), 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.

## Effect of the Shafranov shift and the gradient of β on intrinsic momentum transport in up-down asymmetric tokamaks

Plasma Physics and Controlled Fusion Institute of Physics **58** (2016) 125015-

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. The results show that, while the Shafranov shift becomes updown asymmetric and depends strongly on the tilt angle of the flux surfaces, it is insensitive to the shape of the current and pressure profiles (when the geometry, total plasma current, and average pressure gradient are kept fixed). Next, local nonlinear gyrokinetic simulations of these MHD equilibria are performed with GS2, which reveal that the Shafranov shift can significantly enhance the momentum transport. However, to be consistent, the effect of β′ (i.e. the radial gradient of β) on the magnetic equilibrium was also included, which was found to significantly reduce momentum transport. Including these two competing effects broadens the rotation profile, but leaves the on-axis value of the rotation roughly unchanged. Consequently, the shape of the β profile has a significant effect on the rotation profile of an up-down asymmetric tokamak.

## Effect of the Shafranov shift and the gradient of β on intrinsic momentum transport in up-down asymmetric tokamaks

Plasma Physics and Controlled Fusion IOP Publishing **58** (2016) 125015-

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. The results show that, while the Shafranov shift becomes up–down asymmetric and depends strongly on the tilt angle of the flux surfaces, it is insensitive to the shape of the current and pressure profiles (when the geometry, total plasma current, and average pressure gradient are kept fixed). Next, local nonlinear gyrokinetic simulations of these MHD equilibria are performed with GS2, which reveal that the Shafranov shift can significantly enhance the momentum transport. However, to be consistent, the effect of ${{\beta}^{\prime}}$ (i.e. the radial gradient of β) on the magnetic equilibrium was also included, which was found to significantly reduce momentum transport. Including these two competing effects broadens the rotation profile, but leaves the on-axis value of the rotation roughly unchanged. Consequently, the shape of the β profile has a significant effect on the rotation profile of an up–down asymmetric tokamak.

## Parallel impurity dynamics in the TJ-II stellarator

Plasma Physics and Controlled Fusion IOP Science **58** (2016) 074009

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. The model is able to explain qualitatively several features of the radiation asymmetry, both in stationary and transient conditions, based on the calculated in-surface variations of the impurity density.

## Sensitivity of detachment extent to magnetic configuration and external parameters

Nuclear Fusion IOP Publishing **56** (2016) 056007

Divertor detachment may be essential to reduce heat loads to magnetic fusion tokamak reactor divertor surfaces. Yet in experiments it is difficult to control the extent of the detached, low pressure, plasma region. At maximum extent the front edge of the detached region reaches the x-point and can lead to degradation of core plasma properties. We define the `detachment window' in a given position control variable C (for example, the upstream plasma density) as the range in C within which the front location can be stably held at any position from the target to the x-point; increased detachment window corresponds to better control. We extend a 1D analytic model[1] to determine the detachment window for the following control variables: the upstream plasma density, the impurity concentration and the power entering the scrape-off layer (SOL). We find that variations in magnetic configuration can have strong effects; Increasing the ratio of the total magnetic field at the x-point to that at the target, Bx/Bt , (total flux expansion, as in the Super-X divertor configuration) strongly increases the detachment window for all control variables studied, thus strongly improving detachment front control and the capability of the divertor plasma to passively accommodate transients while still staying detached. Increasing flux tube length and thus volume in the divertor, through poloidal flux expansion (as in the snowflake or x-divertor configurations) or length of the divertor, also increases the detachment window, but less than the total ux expansion does. Thesensitivity of the detachment front location, zh, to each control variable, C, defined as δzh/δC , depends on the magnetic configuration. The size of the radiating volume and the total divertor radiation increase α (Bx/Bt)^2 and α Bx/Bt , respectively, but not by increasing divertor poloidal flux expansion or field line length. We believe this model is applicable more generally to any thermal fronts in flux tubes with varying magnetic field, and similar sources and sinks, such as detachment fronts in stellarator divertors and solar prominences in coronal loops.

## Scaling of up-down asymmetric turbulent momentum flux with poloidal shaping mode number in tokamaks

PLASMA PHYSICS AND CONTROLLED FUSION **58** (2016) ARTN 055016

## Poloidal tilting symmetry of high order tokamak flux surface shaping in gyrokinetics

PLASMA PHYSICS AND CONTROLLED FUSION **58** (2016) ARTN 045023

## Residual zonal flows in tokamaks and stellarators at arbitrary wavelengths

PLASMA PHYSICS AND CONTROLLED FUSION **58** (2016) ARTN 045018

## Scaling of up-down asymmetric turbulent momentum flux with poloidal shaping mode number in tokamaks

Plasma Physics and Controlled Fusion IOP Publishing **58** (2016) 055016

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. It is found that the momentum flux generated by shaping that lacks mirror symmetry (which is necessarily up-down asymmetric) has a power law scaling with the shaping mode number. However, the momentum flux generated by mirror symmetric flux surface shaping (even if it is up-down asymmetric) decays exponentially with large shaping mode number. These scalings are consistent with nonlinear local gyrokinetic simulations and indicate that low mode number shaping effects (e.g. elongation, triangularity) are optimal for creating rotation. Additionally it suggests that breaking the mirror symmetry of flux surfaces may generate significantly more toroidal rotation

## Poloidal tilting symmetry of high order tokamak flux surface shaping in gyrokinetics

Plasma Physics and Controlled Fusion IOP Publishing **58** (2016) 045023-

A poloidal tilting symmetry of the local nonlinear δf 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. With this geometry specification we find that, when performing an expansion in large flux surface shaping mode number, the governing equations of gyrokinetics are symmetric in the poloidal tilt of the high order shaping effects. This allows us to take the fluxes from a single configuration and calculate the fluxes in any configuration that can be produced by tilting the large mode number shaping effects. This creates a distinction between tokamaks with mirror symmetric flux surfaces and tokamaks without mirror symmetry, which is expected to have important consequences for generating toroidal rotation using updown asymmetry.