arXiv subjects

Felix I. Parra

Publications and source records attributed to Felix I. Parra.

At least 37 records · Page 2Linked to original sources

arXiv2018

Stellarator impurity flux driven by electric fields tangent to magnetic surfaces

The control of impurity accumulation is one of the main challenges for future stellarator fusion reactors. The standard argument to explain this accumulation relies on the, in principle, large inward pinch in the neoclassical impurity flux caused by the typically negative radial electric field in stellarators. This simplified interpretation was proven to be flawed by Helander et al. [Phys. Rev. Lett. 118, 155002 (2017)], who showed that in a relevant regime (low-collisionality main ions and collisional impurities) the radial electric field does not drive impurity transport. In that reference, the effect of the component of the electric field that is tangent to the magnetic surface was not included. In this Letter, an analytical calculation of the neoclassical radial impurity flux incorporating such effect is given, showing that it can be very strong for highly charged impurities and that, once it is taken into account, the dependence of the impurity flux on the radial electric field reappears. Realistic examples are provided in which the inclusion of the tangential electric field leads to impurity expulsion.

physics.plasm-ph↗

arXiv2018

Solution to a collisionless shallow-angle magnetic presheath with kinetic ions

Using a kinetic model for the ions and adiabatic electrons, we solve a steady state, electron-repelling magnetic presheath in which a uniform magnetic field makes a small angle $α\ll 1$ (in radians) with the wall. The presheath characteristic thickness is the typical ion gyroradius $ρ_{\text{i}}$. The Debye length $λ_{\text{D}}$ and the collisional mean free path of an ion $λ_{\text{mfp}}$ satisfy the ordering $λ_{\text{D}} \ll ρ_{\text{i}} \ll αλ_{\text{mfp}}$, so a quasineutral and collisionless model is used. We assume that the electrostatic potential is a function only of distance from the wall, and it varies over the scale $ρ_{\text{i}}$. Using the expansion in $α\ll 1$, we derive an analytical expression for the ion density that only depends on the ion distribution function at the entrance of the magnetic presheath and the electrostatic potential profile. Importantly, we have added the crucial contribution of the orbits in the region near the wall. By imposing the quasineutrality equation, we derive a condition that the ion distribution function must satisfy at the magnetic presheath entrance --- the kinetic equivalent of the Chodura condition. Using an ion distribution function at the entrance of the magnetic presheath that satisfies the kinetic Chodura condition, we find numerical solutions for the self-consistent electrostatic potential, ion density and flow across the magnetic presheath for several values of $α$. Our numerical results also include the distribution of ion velocities at the Debye sheath entrance. We find that at small values of $α$ there are substantially fewer ions travelling with a large normal component of the velocity into the wall.

physics.plasm-ph↗

arXiv2018

Electrostatic potential variations on stellarator magnetic surfaces in low collisionality regimes

The component of the neoclassical electrostatic potential that is non-constant on the magnetic surface, that we denote by $\tildeφ$, can affect radial transport of highly charged impurities, and this has motivated its inclusion in some modern neoclassical codes. The number of neoclassical simulations in which $\tildeφ$ is calculated is still scarce, partly because they are usually demanding in terms of computational resources, especially at low collisionality. In this paper the size, the scaling with collisionality and with aspect ratio, and the structure of $\tildeφ$ on the magnetic surface are analytically derived in the $1/ν$, $\sqrtν$ and superbanana-plateau regimes of stellarators close to omnigeneity; i. e. stellarators that have been optimized for neoclassical transport. It is found that the largest $\tildeφ$ that the neoclassical equations admit scales linearly with the inverse aspect ratio and with the size of the deviation from omnigeneity. Using a model for a perturbed omnigeneous configuration, the analytical results are verified and illustrated with calculations by the code KNOSOS. The techniques, results and numerical tools employed in this paper can be applied to neoclassical transport problems in tokamaks with broken axisymmetry.

physics.plasm-ph↗

arXiv2017

Optimized up-down asymmetry to drive fast intrinsic rotation in tokamaks

Breaking the up-down symmetry of the tokamak poloidal cross-section can significantly increase the spontaneous rotation due to turbulent momentum transport. In this work, we optimize the shape of flux surfaces with both tilted elongation and tilted triangularity in order to maximize this drive of intrinsic rotation. Nonlinear gyrokinetic simulations demonstrate that adding optimally-tilted triangularity can double the momentum transport of a tilted elliptical shape. This work indicates that tilting the elongation and triangularity in an ITER-like device can reduce the energy transport and drive intrinsic rotation with an Alfvén Mach number on the order of $1\%$. This rotation is four times larger than the rotation expected in ITER and is sufficient to stabilize MHD instabilities. It is shown that this optimal shape can be created using the shaping coils of several experiments.

physics.plasm-ph↗

arXiv2017

Electromagnetic zonal flow residual responses

The collisionless axisymmetric zonal flow residual calculation for a tokamak plasma is generalized to include electromagnetic perturbations. We formulate and solve the complete initial value zonal flow problem by retaining the fully self-consistent axisymmetric spatial perturbations in the electric and magnetic fields. Simple expressions for the electrostatic, shear and compressional magnetic residual responses are derived that provide a fully electromagnetic test of the zonal flow residual in gyrokinetic codes. Unlike the electrostatic potential, the parallel vector potential and the parallel magnetic field perturbations need not relax to flux functions for all possible initial conditions.

physics.plasm-ph↗

arXiv2017

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

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. 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 preserve radial locality in the drift-kinetic equation for collisionalities below the $1/ν$ regime. Then, it is shown that neoclassical transport is determined by two layers in phase space. One of the layers corresponds to the $\sqrtν$ regime and the other to the 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 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 $\sqrtν$ 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.

physics.plasm-ph↗

arXiv2017

Gyrokinetic treatment of a grazing angle magnetic field

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, $α\ll 1$ (in radians). Characteristic lengths perpendicular to the wall in such a magnetic presheath scale with the typical ion Larmor orbit size, $ρ_{\text{i}}$. The smallest scale length associated with variations parallel to the wall is taken to be across the magnetic field, and ordered $l = ρ_{\text{i}} / δ$, where $ δ\ll 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 component of the electric field $\vec{E}$ 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 $α\sim δ\ll 1$, 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.

physics.plasm-ph↗

arXiv2016

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

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.

physics.plasm-ph↗

arXiv2016

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

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 $β'$ (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.

physics.plasm-ph↗

arXiv2016

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

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.

physics.plasm-ph↗

arXiv2016

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

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 up-down asymmetry.

physics.plasm-ph↗

arXiv2015

Equivalence of two different approaches to global $δf$ gyrokinetic simulations

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 $δf$ gyrokinetic equations to an order higher in $l_\bot/L$ than the usual flux tube formulations.

physics.plasm-ph↗

arXiv2015

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 $O(ε^2)$ 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.

physics.plasm-ph↗

arXiv2015

Intrinsic rotation in tokamaks: theory

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. Two different turbulence regimes are considered: turbulence with characteristic perpendicular lengths of the order of the ion gyroradius, $ρ_i$, and turbulence with characteristic lengths of the order of the poloidal gyroradius, $(B/B_p) ρ_i$. Intrinsic rotation driven by gyroradius scale turbulence is mainly due to the effect of neoclassical corrections and of finite orbit widths on turbulent momentum transport, whereas for the intrinsic rotation driven by poloidal gyroradius scale turbulence, the slow variation of turbulence characteristics in the radial and poloidal directions and the turbulent particle acceleration can be become as important as the neoclassical and finite orbit width effects. The magnetic drift is shown to be indispensable for the intrinsic rotation driven by the slow variation of turbulence characteristics and the turbulent particle acceleration. The equations are written in a form conducive to implementation in a flux tube code, and the effect of the radial variation of the turbulence is included in a novel way that does not require a global gyrokinetic formalism.

physics.plasm-ph↗

arXiv2015

Less constrained omnigeneous stellarators

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. In the pioneering and influential article [Cary~J~R and Shasharina~S~G 1997 {\it Phys. Plasmas} {\bf 4} 3323], the conditions that the magnetic field of a stellarator must satisfy to be omnigeneous are derived. However, reference [Cary~J~R and Shasharina~S~G 1997 {\it Phys. Plasmas} {\bf 4} 3323] only considered omnigeneous stellarators in which all the minima of the magnetic field strength on a flux surface must have the same value. The same is assumed for the maxima. We show that omnigenenous magnetic fields can have local minima and maxima with different values. Thus, the parameter space in which omnigeneous stellarators are possible is larger than previously expected. The analysis presented in this article is only valid for orbits with vanishing radial width, and in principle it is not applicable to energetic particles. However, one would expect that improving neoclassical confinement would improve energetic particle confinement.

physics.plasm-ph↗

arXiv2015

Radial penetration of flux surface shaping in tokamaks

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. Then, a technique to separate the effects of magnetic pressure and tension in the Grad-Shafranov equation is presented and used to calculate radial profiles of strong elongation for nearly constant current profiles. Lastly, it is shown that more hollow toroidal current profiles are significantly better at conveying shaping from the edge to the core.

physics.plasm-ph↗

arXiv2014

Intrinsic momentum transport in up-down asymmetric tokamaks

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 $δf$ gyrokinetic code that self-consistently calculates momentum transport, to permit up-down asymmetric configurations. Tokamaks with tilted elliptical poloidal cross-sections were simulated to determine nonlinear momentum transport. The results, which are consistent with experiment in magnitude, suggest that a toroidal velocity gradient, $(\partial u_{ζi} / \partial ρ) / v_{th i}$, of 5% of the temperature gradient, $(\partial T_{i} / \partial ρ) / T_{i}$, is sustainable. Here $v_{th i}$ is the ion thermal speed, $u_{ζi}$ is the ion toroidal mean flow, $ρ$ is the minor radial coordinate normalized to the tokamak minor radius, and $T_{i}$ is the ion temperature. Since other intrinsic momentum transport mechanisms scale poorly to larger machines, these results indicate that up-down asymmetry is the most feasible method to generate the current experimentally-measured rotation levels in reactor-sized devices.

physics.plasm-ph↗

arXiv2014

Flow damping in stellarators close to quasisymmetry

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. The answer to this question depends strongly on the size of the spatial gradients of the deviation from quasisymmetry and on the collisionality regime. Recently, formal criteria for closeness to quasisymmetry have been derived in a variety of situations. In particular, the case of deviations with large gradients was solved in the $1/ν$ regime. Denoting by $α$ a parameter that gives the size of the deviation from quasisymmetry, it was proven that particle fluxes do not scale with $α^{3/2}$, as typically claimed, but with $α$. It was also shown that ripple wells are not necessarily the main cause of transport. This paper reviews those works and presents a new result in another collisionality regime, in which particles trapped in ripple wells are collisional and the rest are collisionless.

physics.plasm-ph↗