# Publications by Felix Parra Diaz

## Transport bifurcation in a rotating tokamak plasma

Physical Review Letters **105** (2010)

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 nonzero 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. A bifurcation in the equilibrium gradients is found: for a given input of heat, it is possible, by varying the applied torque, to trigger a transition to significantly higher temperature and flow gradients. © 2010 The American Physical Society.

## Non-physical momentum sources in slab geometry gyrokinetics

PLASMA PHYSICS AND CONTROLLED FUSION **52** (2010) ARTN 085011

## Fulfillment of the kinetic Bohm criterion in a quasineutral particle-in-cell model

Physics of Plasmas **17** (2010)

Quasineutral particle-in-cell models of ions must fulfill the kinetic Bohm criterion, in its inequality form, at the domain boundary in order to match correctly with solutions of the Debye sheaths tied to the walls. The simple, fluid form of the Bohm criterion is shown to be a bad approximation of the exact, kinetic form when the ion velocity distribution function has a significant dispersion and involves different charge numbers. The fulfillment of the Bohm criterion is measured by a weighting algorithm at the boundary, but linear weighting algorithms have difficulties to reproduce the nonlinear behavior around the sheath edge. A surface weighting algorithm with an extended temporal weighting is proposed and shown to behave better than the standard volumetric weighting. Still, this must be supplemented by a forcing algorithm of the kinetic Bohm criterion. This postulates a small potential fall in a supplementary, thin, transition layer. The electron-wall interaction is shown to be of little relevance in the fulfillment of the Bohm criterion. © 2010 American Institute of Physics.

## Transport of momentum in full f gyrokinetics

Physics of Plasmas **17** (2010)

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. © 2010 American Institute of Physics.

## Turbulent transport of toroidal angular momentum in low flow gyrokinetics (vol 52, 045004, 2010)

PLASMA PHYSICS AND CONTROLLED FUSION **52** (2010) ARTN 059801

## Turbulent transport of toroidal angular momentum in low flow gyrokinetics

Plasma Physics and Controlled Fusion **52** (2010)

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. © 2010 IOP Publishing Ltd.

## Comment on On higher order corrections to gyrokinetic Vlasov-Poisson equations in the long wavelength limit [Phys. Plasmas 16, 044506 (2009)]

Physics of Plasmas **16** (2009)

A recent publication [F. I. Parra and P. J. Catto, Plasma Phys. Controlled Fusion 50, 065014 (2008)] warned against the use of the lower order gyrokinetic Poisson equation at long wavelengths because the long wavelength, radial electric field must remain undetermined to the order the equation is obtained. Another reference [W. W. Lee and R. A. Kolesnikov, Phys. Plasmas 16, 044506 (2009)] criticizes these results by arguing that the higher order terms neglected in the most common gyrokinetic Poisson equation are formally smaller than the terms that are retained. This argument is flawed and ignores that the lower order terms, although formally larger, must cancel without determining the long wavelength, radial electric field. The reason for this cancellation is discussed. In addition, the origin of a nonlinear term present in the gyrokinetic Poisson equation [F. I. Parra and P. J. Catto, Plasma Phys. Controlled Fusion 50, 065014 (2008)] is explained. © 2009 American Institute of Physics.

## Vorticity and intrinsic ambipolarity in turbulent tokamaks

Plasma Physics and Controlled Fusion **51** (2009)

Traditional electrostatic gyrokinetic treatments consist of a gyrokinetic Fokker-Planck equation and a gyrokinetic quasineutrality equation. Both of these equations can be found up to second order in a gyroradius over macroscopic length expansion in some simplified cases, but the versions implemented in codes are typically only first order. In axisymmetric configurations such as the tokamak, the accuracy to which the distribution function is calculated is insufficient to determine the neoclassical radial electric field. Moreover, we prove here that turbulence dominated tokamaks are intrinsically ambipolar, as are neoclassical tokamaks. Therefore, traditional gyrokinetic descriptions are unable to correctly calculate the toroidal rotation and hence the axisymmetric radial electric field. We study the vorticity equation, ∇ J = 0, in the gyrokinetic regime, with wavelengths on the order of the ion Larmor radius. We explicitly show that gyrokinetics needs to be calculated at least to third order in the gyroradius expansion if the radial electric field is to be retrieved from quasineutrality. The method employed to study the vorticity equation also suggests a solution to the problem, namely, solving a gyrokinetic vorticity equation instead of the quasineutrality equation. The vorticity equations derived here only obtain the potential within a flux function as required. © 2009 IOP Publishing Ltd.

## Limitations, insights and improvements to gyrokinetics

Nuclear Fusion **49** (2009)

We first consider gyrokinetic quasineutrality limitations when evaluating the axisymmetric radial electric field in a non-turbulent tokamak by an improved examination of intrinsic ambipolarity. We next prove that the background ions in a pedestal of poloidal ion gyroradius scale must be Maxwellian and nearly isothermal in Pfirsch-Schlüter and banana regime tokamak plasmas, and then consider zonal flow behaviour in a pedestal. Finally, we focus on a simplifying procedure for our transport time scale hybrid gyrokinetic-fluid treatment that removes the limitations of gyrokinetic quasineutrality and remains valid in the pedestal. © 2009 IAEA, Vienna.

## Gyrokinetic equivalence

Plasma Physics and Controlled Fusion **51** (2009)

We compare two different derivations of the gyrokinetic equation: the Hamiltonian approach in Dubin D H E et al (1983 Phys. Fluids 26 3524) and the recursive methodology in Parra F I and Catto P J (2008 Plasma Phys. Control. Fusion 50 065014). We prove that both approaches yield the same result at least to second order in a Larmor radius over macroscopic length expansion. There are subtle differences in the definitions of some of the functions that need to be taken into account to prove the equivalence. © 2009 IOP Publishing Ltd.

## Gyrokinetic limitations and improvements

35th EPS Conference on Plasma Physics 2008, EPS 2008 - Europhysics Conference Abstracts **32** (2008) 1418-1421

For a tokamak, we consider gyrokinetic quasineutrality limitations when evaluating the axisymmetric radial electric field; a gyrokinetic entropy production restriction on the ion temperature pedestal; and a hybrid gyrokinetic-fluid treatment valid on transport time scales.

## Electrostatic turbulence in tokamaks on transport time scales

Plasma Physics and Controlled Fusion **50** (2008)

Simulating electrostatic turbulence in tokamaks on transport time scales requires retaining and evolving a complete turbulence modified neoclassical transport description, including all the axisymmetric neoclassical and zonal flow radial electric field effects, as well as the turbulent transport normally associated with drift instabilities. Neoclassical electric field effects are particularly difficult to retain since they require evaluating the ion distribution function to higher order in gyroradius over background scale length than standard gyrokinetic treatments. To avoid extending gyrokinetics an alternate hybrid gyrokinetic-fluid treatment is formulated that employs moments of the full Fokker-Planck kinetic equation to remove the need for a higher order gyrokinetic distribution function. The resulting hybrid description is able to model all electrostatic turbulence effects with wavelengths much longer than an electron Larmor radius such as the ion temperature gradient (ITG) and trapped electron modes (TEM). © 2008 IOP Publishing Ltd.

## Limitations of gyrokinetics on transport time scales

Plasma Physics and Controlled Fusion **50** (2008)

We present a new recursive procedure to find a full f electrostatic gyrokinetic equation correct to first order in an expansion of gyroradius over magnetic field characteristic length. The procedure provides new insights into the limitations of the gyrokinetic quasineutrality equation. We find that the ion distribution function must be known at least to second order in gyroradius over characteristic length to calculate the long wavelength components of the electrostatic potential self-consistently. Moreover, using the example of a steady-state -pinch, we prove that the quasineutrality equation fails to provide the axisymmetric piece of the potential even with a distribution function correct to second order. We also show that second order accuracy is enough if a more convenient moment equation is used instead of the quasineutrality equation. These results indicate that the gyrokinetic quasineutrality equation is not the most effective procedure to find the electrostatic potential if the long wavelength components are to be retained in the analysis. © 2008 IOP Publishing Ltd.

## Improvements on particle accuracy in a Hall thruster hybrid code

Collection of Technical Papers - AIAA/ASME/SAE/ASEE 42nd Joint Propulsion Conference **7** (2006) 5076-5084

Time-centering algorithms to increment the temporal accuracy of the state of macroparticles in the hybrid code HPHall-2 are presented. As a result, one orderof-magnitude increment in temporal precision is achieved. Errors in the conservation of mass flow along the chamber are reduced by one order-of-magnitude too. In a second part, the algorithm controlling the population of particles per cell is improved with the aim of reducing both the statistical oscillations and the computational cost.

## A two-dimensional hybrid model of the Hall thruster discharge

Journal of Applied Physics **100** (2006)

Particle-in-cell methods are used for ions and neutrals. Probabilistic methods are implemented for ionization, charge-exchange collisions, gas injection, and particle-wall interaction. A diffusive macroscopic model is proposed for the strongly magnetized electron population. Cross-field electron transport includes wall collisionality and Bohm-type diffusion, the last one dominating in most of the discharge. Plasma quasineutrality applies except for space-charge sheaths, which are modeled taking into consideration secondary-electron-emission and space-charge saturation. Specific weighting algorithms are developed in order to fulfil the Bohm condition on the ion flow at the boundaries of the quasineutral domain. The consequence is the full development of the radial plasma structure and correct values for ion losses at lateral walls. The model gains in insight and physical consistency over a previous version, but thrust efficiency is lower than in experiments, indicating that further model refinement of some phenomena is necessary. © 2006 American Institute of Physics.

## A model of the two-stage Hall thruster discharge

Journal of Applied Physics **98** (2005)

The effect of a third, active electrode placed inside the ceramic chamber of a Hall thruster is analyzed. Both electron-collecting and electron-emitting modes are considered. Significant efficiency enhancement with respect to the single-stage operation can be obtained for a good electron-emitting electrode, placed in an intermediate location of the acceleration region, and for an anode-to-electrode (inner-stage) potential significantly larger than the ionization potential. Optimum values of the electrode location and voltage are determined. The performance improvement is due to a reduction of the energy losses to the chamber walls. This is the consequence of lower Joule heating and thus lower electron temperature in the outer stage. When the ionization process is efficient already in the single-stage operation, (i) two-stage operation does not affect practically the propellant and voltage utilizations and (ii) thrust efficiency decreases when the intermediate electrode works as an electron collector. © 2005 American Institute of Physics.

## Partial trapping of secondary-electron emission in a Hall thruster plasma

Physics of Plasmas **12** (2005) 1-7

Secondary-electron emission at the ceramic walls of a Hall thruster modifies the potential jump of the wall Debye sheaths and thus the electron energy losses to the wall. Because of the low plasma collisionality the two counterstreaming beams of secondary electrons are not expected to be totally trapped within the bulk of the discharge. In order to analyze the effects of partial trapping of secondary electrons on the presheathsheath radial structure, a macroscopic model is formulated. The plasma response depends on the secondary electron emission yield and the trapped fraction of secondary electrons. The sheath potential and wall energy losses are determined mainly by the net current of secondary electrons in the sheaths. For any practical value of the secondary emission yield, the zero-trapping solution is very similar to the zero secondary emission case. Space charge saturation of the sheaths is unattainable for weak trapping. In all cases, secondary electrons have a weak effect on the presheath solution and the ion flux recombined at the walls. © 2005 American Institute of Physics.

## Improvement of the plasma-wall model on a fluid-PIC code of a hall thruster

European Space Agency, (Special Publication) ESA SP (2004) 707-714

Two issues are discussed. First, a new sheath model that takes into account charge-saturation is implemented in HPHall. Second, the transition between the quasineutral solution and the sheaths at the lateral walls is found to be treated deficiently in the original code. The use of finer meshes yields better solutions but do not solve the problem completely. Hall thrusters; particle-in-cell codes; sheaths.

## Study of a Hall thruster discharge with an intermediate electrode

39th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit (2003)

An axial model for a two-stage discharge with an electron-emissive electrode is further examined. Scaling laws are derived and help to understand two-stage physics. Efficiency gains are obtained when the second-stage is placed in the upstream part of the acceleration region and the first and two stage voltages are comparable. A parametric study to determine the best position and voltage of the intermediate electrode is carried out. © 2003 by The Authors.

## Model of radial plasma-wall interactions in a Hall thruster

38th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit (2002)

Recent achievements on a sheath/presheath/sheath model of the radial interaction of the plasma discharge with the dielectric walls of a Hall thruster are presented. Trapping and thermalization of the secondary electron emission (SEE) depend on the plasma collisionality. The main new contribution is the sheath structure for partial and zero trapping of the SEE. The SEE yield necessary for charge saturation of the sheath increases as collisionality decreases, and the charge-saturation limit is beyond reach for low collisionality and real materials. Other contributions are the analysis of the ion temperature and of asymmetric profiles in non-planar channel geometries. © 2002 by the authors.