Three-Dimensional Shear-Flow Instability Saturation via Stable Modes

Turbulence in three dimensions ($3$D) supports vortex stretching that has long been known to accomplish energy transfer to small scales. Moreover, net energy transfer from large-scale, forced, unstable flow-gradients to smaller scales is achieved by gradient-flattening instability. Despite such enforcement of energy transfer to small scales, it is shown here not only that the shear-flow-instability-supplied $3$D-fluctuation energy is largely inverse-transferred from the fluctuation to the mean-flow gradient, but that such inverse transfer is more efficient for turbulent fluctuations in $3$D than in two dimensions ($2$D). The transfer is due to linearly stable eigenmodes that are excited nonlinearly. The stable modes, thus, reduce both the nonlinear energy cascade to small scales and the viscous dissipation rate. The vortex-tube stretching is also suppressed. Up-gradient momentum transport by the stable modes counters the instability-driven down-gradient transport, which also is more effective in $3$D than in $2$D ($\mathrm{\approx} 70\% \mathrm{\,\, vs.\,\,}\mathrm{\approx} 50\%$). From unstable modes, these stable modes nonlinearly receive energy via zero-frequency fluctuations that vary only in the direction orthogonal to the plane of $2$D shear flow. The more widely occurring $3$D turbulence is thus inherently different from the commonly studied $2$D turbulence, despite both saturating via stable modes.Comment: To appear in Physics of Fluid

Comparison of local and global gyrokinetic calculations of collisionless zonal flow damping in quasi-symmetric stellarators

The linear collisionless damping of zonal flows is calculated for quasi-symmetric stellarator equilibria in flux-tube, flux-surface, and full-volume geometry. Equilibria are studied from the quasi-helical symmetry configuration of the Helically Symmetric eXperiment (HSX), a broken symmetry configuration of HSX, and the quasi-axial symmetry geometry of the National Compact Stellarator eXperiment (NCSX). Zonal flow oscillations and long-time damping affect the zonal flow evolution, and the zonal flow residual goes to zero for small radial wavenumber. The oscillation frequency and damping rate depend on the bounce-averaged radial particle drift in accordance with theory. While each flux tube on a flux surface is unique, several different flux tubes in HSX or NCSX can reproduce the zonal flow damping from a flux-surface calculation given an adequate parallel extent. The flux-surface or flux-tube calculations can accurately reproduce the full-volume long-time residual for moderate $k_x$, but the oscillation and damping time scales are longer in local representations, particularly for small $k_x$ approaching the system size.Comment: The following article has been accepted by Physics of Plasmas. After it is published, it will be found at https://aip.scitation.org/journal/php. 33 pages, 18 figure

The impact of magnetic fields on momentum transport and saturation of shear-flow instability by stable modes

The Kelvin-Helmholtz (KH) instability of a shear layer with an initially-uniform magnetic field in the direction of flow is studied in the framework of 2D incompressible magnetohydrodynamics with finite resistivity and viscosity using direct numerical simulations. The shear layer evolves freely, with no external forcing, and thus broadens in time as turbulent stresses transport momentum across it. As with KH-unstable flows in hydrodynamics, the instability here features a conjugate stable mode for every unstable mode in the absence of dissipation. Stable modes are shown to transport momentum up its gradient, shrinking the layer width whenever they exceed unstable modes in amplitude. In simulations with weak magnetic fields, the linear instability is minimally affected by the magnetic field, but enhanced small-scale fluctuations relative to the hydrodynamic case are observed. These enhanced fluctuations coincide with increased energy dissipation and faster layer broadening, with these features more pronounced in simulations with stronger fields. These trends result from the magnetic field reducing the effects of stable modes relative to the transfer of energy to small scales. As field strength increases, stable modes become less excited and thus transport less momentum against its gradient. Furthermore, the energy that would otherwise transfer back to the driving shear due to stable modes is instead allowed to cascade to small scales, where it is lost to dissipation. Approximations of the turbulent state in terms of a reduced set of modes are explored. While the Reynolds stress is well-described using just two modes per wavenumber at large scales, the Maxwell stress is not.Comment: 39 pages, 17 figures, preprint forma

Nonlinear mode coupling and energetics of driven magnetized shear-flow turbulence

To comprehensively understand saturation of two-dimensional ($2$D) magnetized Kelvin-Helmholtz-instability-driven turbulence, energy transfer analysis is extended from the traditional interaction between scales to include eigenmode interactions, by using the nonlinear couplings of linear eigenmodes of the ideal instability. While both kinetic and magnetic energies cascade to small scales, a significant fraction of turbulent energy deposited by unstable modes in the fluctuation spectrum is shown to be re-routed to the conjugate-stable modes at the instability scale. They remove energy from the forward cascade at its inception. The remaining cascading energy flux is shown to attenuate exponentially at a small scale, dictated by the large-scale stable modes. Guided by a widely used instability-saturation assumption, a general quasilinear model of instability is tested by retaining all nonlinear interactions except those that couple to the large-scale stable modes. These complex interactions are analytically removed from the magnetohydrodynamic equations using a novel technique. Observations are: an explosive large-scale vortex separation instead of the well-known merger of $2$D, a dramatic enhancement in turbulence level and spectral energy fluxes, and a reduced small-scale dissipation length-scale. These show critical role of the stable modes in instability saturation. Possible reduced-order turbulence models are proposed for fusion and astrophysical plasmas, based on eigenmode-expanded energy transfer analyses.Comment: Selected by the editors of Physics of Plasmas as a Featured article. https://doi.org/10.1063/5.015656

Implications of large scale shifts in tropospheric NOx levels in the remote tropical Pacific

A major observation recorded during NASA's western Pacific Exploratory Mission (PEM-West B) was the large shift in tropical NO levels as a function of geographical location. High-altitude NO levels exceeding 100 pptv were observed during portions of tropical flights 5-8, while values almost never exceeded 20 pptv during tropical flights 9 and 10. The geographical regions encompassing these two flight groupings are here labeled "high" and "low" NOx regimes. A comparison of these two regimes, based on back trajectories and chemical tracers, suggests that air parcels in both were strongly influenced by deep convection. The low NOx regime appears to have been predominantly impacted by marine convection, whereas the high NOx regime shows evidence of having been more influenced by deep convection over a continental land mass. DMSP satellite observations point strongly toward lightning as the major source of NOx in the latter regime. Photochemical ozone formation in the high NOx regime exceeded that for low NOx by factors of 2 to 6, whereas O3 destruction in the low NOx regime exceeded that for high NOx by factors of up to 3. Taking the tropopause height to be 17 km, estimates of the net photochemical effect on the O3 column revealed that the high NOx regime led to a small net production. By contrast, the low NOx regime was shown to destroy O3 at the rate of 3.4% per day. One proposed mechanism for off-setting this projected large deficit would involve the transport of O3 rich midlatitude air into the tropics. Alternatively, it is suggested that O3 within the tropics may be overall near self-sustaining with respect to photochemical activity. This scenario would require that some tropical regions, unsampled at the time of PEM-B, display significant net column O3 production, leading to an overall balanced budget for the "greater" tropical Pacific basin. Details concerning the chemical nature of such regimes are discussed

Progress in unveiling extreme particle acceleration in persistent astrophysical jets

International audienceExtreme blazars emitting teraelectronvolt photons are ideal targets to study particle acceleration processes. The growing number of such sources has been critical for γ-ray cosmology, studying intergalactic magnetic fields and putting constraints on exotic physics

Long-term monitoring of the radio-galaxy M87 in gamma-rays: joint analysis of MAGIC, VERITAS and Fermi-LAT data

M87 was discovered in the very-high-energy band (VHE, E > 100 GeV) with HEGRA in 2003, long before its emission was detected in the high-energy band (HE, E > 100 MeV) with Fermi-LAT in 2009, opening the window to a new family of extragalactic sources with tilted jets. After a series of major VHE flares in 2005, 2008, and 2010, which were detected in multiple bands, the source has been found in a low activity state, interrupted only by comparatively smaller-scale flares. MAGIC and VERITAS, two stereoscopic Cherenkov telescope arrays located at Roque de los Muchachos Observatory (Canary Islands, Spain) and the Fred Lawrence Whipple Observatory (Arizona, US), have monitored M87 continuously and in coordination for more than 10 years. In this work, we present the data for 4 years of MAGIC and VERITAS observations corresponding to 2019, 2020, 2021 and 2022. The resulting light curves are shown in daily and monthly scales where no significant variability is observed. In addition, we show the first joint analysis using combined event data from the two VHE instruments and Fermi-LAT to compute the spectral energy distribution

Global Linear and Nonlinear Gyrokinetic Simulations of Tearing Modes

To better understand the interaction of global tearing modes and microturbulence in the Madison Symmetric Torus (MST) reversed-field pinch (RFP), the global gyrokinetic code \textsc{Gene} is modified to describe global tearing mode instability via a shifted Maxwellian distribution consistent with experimental equilibria. The implementation of the shifted Maxwellian is tested and benchmarked by comparisons with different codes and models. Good agreement is obtained in code-code and code-theory comparisons. Linear stability of tearing modes of a non-reversed MST discharge is studied. A collisionality scan is performed to the lowest order unstable modes ($n=5$, $n=6$) and shown to behave consistently with theoretical scaling. The nonlinear evolution is simulated, and saturation is found to arise from mode coupling and transfer of energy from the most unstable tearing mode to small-scale stable modes mediated by the $m=2$ tearing mode. The work described herein lays the foundation for nonlinear simulation and analysis of the interaction of tearing modes and gyroradius-scale instabilities in RFP plasmas

Stellarator microinstabilities and turbulence at low magnetic shear

[EN] Gyrokinetic simulations of drift waves in low-magnetic-shear stellarators reveal that simulation domains comprised of multiple turns can be required to properly resolve critical mode structures important in saturation dynamics. Marginally stable eigenmodes important in saturation of ion temperature gradient modes and trapped electron modes in the Helically Symmetric Experiment (HSX) stellarator are observed to have two scales, with the envelope scale determined by the properties of the local magnetic shear and an inner scale determined by the interplay between the local shear and magnetic field-line curvature. Properly resolving these modes removes spurious growth rates that arise for extended modes in zero-magnetic-shear approximations, enabling use of a zero-magnetic-shear technique with smaller simulation domains and attendant cost savings. Analysis of subdominant modes in trapped electron mode (TEM)-driven turbulence reveals that the extended marginally stable modes play an important role in the nonlinear dynamics, and suggests that the properties induced by low magnetic shear may be exploited to provide another route for turbulence saturation.The authors would like to thank F. Jenko for insightful questions that motivated this research and J. Smoniewski and J. H. E. Proll for engaging discussions. This work was supported by US DoE grant nos. DE-FG02-99ER54546, DE-FG02-93ER54222 and DE-FG02-89ER53291. J.E.R. was supported by Agencia Estatal de Investigacion (AEI) under grant TIN2016-75985-P, which includes European Commission ERDF funds. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a US Department of Energy Office of Science User Facility operated under contract no. DE-AC02-05CH11231. This research was performed using the compute resources and assistance of the UW-Madison Center For High Throughput Computing (CHTC) in the Department of Computer Sciences. The CHTC is supported by UW-Madison, the Advanced Computing Initiative, the Wisconsin Alumni Research Foundation, the Wisconsin Institutes for Discovery and the National Science Foundation, and is an active member of the Open Science Grid, which is supported by the National Science Foundation and the US Department of Energy's Office of Science.Faber, BJ.; Pueschel, MJ.; Terry, PW.; Hegna, CC.; Roman, JE. (2018). Stellarator microinstabilities and turbulence at low magnetic shear. Journal of Plasma Physics. 84(5). https://doi.org/10.1017/S0022377818001022S845Connor, J. W., & Hastie, R. J. (2004). Microstability in tokamaks with low magnetic shear. Plasma Physics and Controlled Fusion, 46(10), 1501-1535. doi:10.1088/0741-3335/46/10/001Terry, P. W., Faber, B. J., Hegna, C. C., Mirnov, V. V., Pueschel, M. J., & Whelan, G. G. (2018). Saturation scalings of toroidal ion temperature gradient turbulence. Physics of Plasmas, 25(1), 012308. doi:10.1063/1.5007062Hernandez, V., Roman, J. E., & Vidal, V. (2005). SLEPc. ACM Transactions on Mathematical Software, 31(3), 351-362. doi:10.1145/1089014.1089019Friedman, B., Carter, T. A., Umansky, M. V., Schaffner, D., & Joseph, I. (2013). Nonlinear instability in simulations of Large Plasma Device turbulence. Physics of Plasmas, 20(5), 055704. doi:10.1063/1.4805084Eiermann, M., & Ernst, O. G. (2006). A Restarted Krylov Subspace Method for the Evaluation of Matrix Functions. SIAM Journal on Numerical Analysis, 44(6), 2481-2504. doi:10.1137/050633846Connor, J. W., Hastie, R. J., & Taylor, J. B. (1978). Shear, Periodicity, and Plasma Ballooning Modes. Physical Review Letters, 40(6), 396-399. doi:10.1103/physrevlett.40.396Xanthopoulos, P., & Jenko, F. (2007). Gyrokinetic analysis of linear microinstabilities for the stellarator Wendelstein 7-X. Physics of Plasmas, 14(4), 042501. doi:10.1063/1.2714328Hegna, C. C., & Hudson, S. R. (2001). Loss of Second-Ballooning Stability in Three-Dimensional Equilibria. Physical Review Letters, 87(3). doi:10.1103/physrevlett.87.035001Hatch, D. R., Terry, P. W., Jenko, F., Merz, F., Pueschel, M. J., Nevins, W. M., & Wang, E. (2011). Role of subdominant stable modes in plasma microturbulence. Physics of Plasmas, 18(5), 055706. doi:10.1063/1.3563536Faber, B. J., Pueschel, M. J., Proll, J. H. E., Xanthopoulos, P., Terry, P. W., Hegna, C. C., … Talmadge, J. N. (2015). Gyrokinetic studies of trapped electron mode turbulence in the Helically Symmetric eXperiment stellarator. Physics of Plasmas, 22(7), 072305. doi:10.1063/1.4926510Sugama, H., & Watanabe, T.-H. (2006). Collisionless damping of zonal flows in helical systems. Physics of Plasmas, 13(1), 012501. doi:10.1063/1.2149311Hegna, C. C., Terry, P. W., & Faber, B. J. (2018). Theory of ITG turbulent saturation in stellarators: Identifying mechanisms to reduce turbulent transport. Physics of Plasmas, 25(2), 022511. doi:10.1063/1.5018198Zocco, A., Xanthopoulos, P., Doerk, H., Connor, J. W., & Helander, P. (2018). Threshold for the destabilisation of the ion-temperature-gradient mode in magnetically confined toroidal plasmas. Journal of Plasma Physics, 84(1). doi:10.1017/s0022377817000988Merz, F. 2008 Gyrokinetic simulation of multimode plasma turbulence. PhD thesis.Dorland, W., Jenko, F., Kotschenreuther, M., & Rogers, B. N. (2000). Electron Temperature Gradient Turbulence. Physical Review Letters, 85(26), 5579-5582. doi:10.1103/physrevlett.85.5579Xanthopoulos, P., Cooper, W. A., Jenko, F., Turkin, Y., Runov, A., & Geiger, J. (2009). A geometry interface for gyrokinetic microturbulence investigations in toroidal configurations. Physics of Plasmas, 16(8), 082303. doi:10.1063/1.3187907Dinklage, A., Beidler, C. D., Helander, P., Fuchert, G., Maaßberg, H., … Zhang, D. (2018). Magnetic configuration effects on the Wendelstein 7-X stellarator. Nature Physics, 14(8), 855-860. doi:10.1038/s41567-018-0141-9Hatch, D. R., Kotschenreuther, M., Mahajan, S., Valanju, P., Jenko, F., Told, D., … Saarelma, S. (2016). Microtearing turbulence limiting the JET-ILW pedestal. Nuclear Fusion, 56(10), 104003. doi:10.1088/0029-5515/56/10/104003Hatch, D. R., Terry, P. W., Jenko, F., Merz, F., & Nevins, W. M. (2011). Saturation of Gyrokinetic Turbulence through Damped Eigenmodes. Physical Review Letters, 106(11). doi:10.1103/physrevlett.106.115003Proll, J. H. E., Xanthopoulos, P., & Helander, P. (2013). Collisionless microinstabilities in stellarators. II. Numerical simulations. Physics of Plasmas, 20(12), 122506. doi:10.1063/1.4846835Whelan, G. G., Pueschel, M. J., & Terry, P. W. (2018). Nonlinear Electromagnetic Stabilization of Plasma Microturbulence. Physical Review Letters, 120(17). doi:10.1103/physrevlett.120.175002Friedman, B., & Carter, T. A. (2014). Linear Technique to Understand Non-Normal Turbulence Applied to a Magnetized Plasma. Physical Review Letters, 113(2). doi:10.1103/physrevlett.113.025003High mode number stability of an axisymmetric toroidal plasma. (1979). Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 365(1720), 1-17. doi:10.1098/rspa.1979.0001Boozer, A. H. (1998). What is a stellarator? Physics of Plasmas, 5(5), 1647-1655. doi:10.1063/1.872833Boozer, A. H. (1981). Plasma equilibrium with rational magnetic surfaces. Physics of Fluids, 24(11), 1999. doi:10.1063/1.863297Dewar, R. L. (1983). Ballooning mode spectrum in general toroidal systems. Physics of Fluids, 26(10), 3038. doi:10.1063/1.864028Anderson, F. S. B., Almagri, A. F., Anderson, D. T., Matthews, P. G., Talmadge, J. N., & Shohet, J. L. (1995). The Helically Symmetric Experiment, (HSX) Goals, Design and Status. Fusion Technology, 27(3T), 273-277. doi:10.13182/fst95-a11947086Bhattacharjee, A. (1983). Drift waves in a straight stellarator. Physics of Fluids, 26(4), 880. doi:10.1063/1.864229Baumgaertel, J. A., Belli, E. A., Dorland, W., Guttenfelder, W., Hammett, G. W., Mikkelsen, D. R., … Xanthopoulos, P. (2011). Simulating gyrokinetic microinstabilities in stellarator geometry with GS2. Physics of Plasmas, 18(12), 122301. doi:10.1063/1.3662064Blackford, L. S., Choi, J., Cleary, A., D’Azevedo, E., Demmel, J., Dhillon, I., … Whaley, R. C. (1997). ScaLAPACK Users’ Guide. doi:10.1137/1.9780898719642Martin, M. F., Landreman, M., Xanthopoulos, P., Mandell, N. R., & Dorland, W. (2018). The parallel boundary condition for turbulence simulations in low magnetic shear devices. Plasma Physics and Controlled Fusion, 60(9), 095008. doi:10.1088/1361-6587/aad38aPlunk, G. G., Xanthopoulos, P., & Helander, P. (2017). Distinct Turbulence Saturation Regimes in Stellarators. Physical Review Letters, 118(10). doi:10.1103/physrevlett.118.105002Candy, J., Waltz, R. E., & Rosenbluth, M. N. (2004). Smoothness of turbulent transport across a minimum-q surface. Physics of Plasmas, 11(5), 1879-1890. doi:10.1063/1.1689967Nagaoka, K., Takahashi, H., Murakami, S., Nakano, H., Takeiri, Y., Tsuchiya, H., … Komori, A. (2015). Integrated discharge scenario for high-temperature helical plasma in LHD. Nuclear Fusion, 55(11), 113020. doi:10.1088/0029-5515/55/11/113020Canik, J. M., Anderson, D. T., Anderson, F. S. B., Likin, K. M., Talmadge, J. N., & Zhai, K. (2007). Experimental Demonstration of Improved Neoclassical Transport with Quasihelical Symmetry. Physical Review Letters, 98(8). doi:10.1103/physrevlett.98.085002Dimits, A. M., Williams, T. J., Byers, J. A., & Cohen, B. I. (1996). Scalings of Ion-Temperature-Gradient-Driven Anomalous Transport in Tokamaks. Physical Review Letters, 77(1), 71-74. doi:10.1103/physrevlett.77.71Beer, M. A., Cowley, S. C., & Hammett, G. W. (1995). Field‐aligned coordinates for nonlinear simulations of tokamak turbulence. Physics of Plasmas, 2(7), 2687-2700. doi:10.1063/1.871232Gates, D. A., Anderson, D., Anderson, S., Zarnstorff, M., Spong, D. A., Weitzner, H., … Glasser, A. H. (2018). Stellarator Research Opportunities: A Report of the National Stellarator Coordinating Committee. Journal of Fusion Energy, 37(1), 51-94. doi:10.1007/s10894-018-0152-7Helander, P. (2014). Theory of plasma confinement in non-axisymmetric magnetic fields. Reports on Progress in Physics, 77(8), 087001. doi:10.1088/0034-4885/77/8/087001Peeters, A. G., Camenen, Y., Casson, F. J., Hornsby, W. A., Snodin, A. P., Strintzi, D., & Szepesi, G. (2009). The nonlinear gyro-kinetic flux tube code GKW. Computer Physics Communications, 180(12), 2650-2672. doi:10.1016/j.cpc.2009.07.001Jenko, F., Dorland, W., Kotschenreuther, M., & Rogers, B. N. (2000). Electron temperature gradient driven turbulence. Physics of Plasmas, 7(5), 1904-1910. doi:10.1063/1.874014Fulton, D. P., Lin, Z., Holod, I., & Xiao, Y. (2014). Microturbulence in DIII-D tokamak pedestal. I. Electrostatic instabilities. Physics of Plasmas, 21(4), 042110. doi:10.1063/1.4871387Bañón Navarro, A., Morel, P., Albrecht-Marc, M., Carati, D., Merz, F., Görler, T., & Jenko, F. (2011). Free energy balance in gyrokinetic turbulence. Physics of Plasmas, 18(9), 092303. doi:10.1063/1.3632077Xanthopoulos, P., Merz, F., Görler, T., & Jenko, F. (2007). Nonlinear Gyrokinetic Simulations of Ion-Temperature-Gradient Turbulence for the Optimized Wendelstein 7-X Stellarator. Physical Review Letters, 99(3). doi:10.1103/physrevlett.99.035002Goldhirsch, L., Orszag, S. A., & Maulik, B. K. (1987). An efficient method for computing leading eigenvalues and eigenvectors of large asymmetric matrices. Journal of Scientific Computing, 2(1), 33-58. doi:10.1007/bf01061511Plunk, G. G., Helander, P., Xanthopoulos, P., & Connor, J. W. (2014). Collisionless microinstabilities in stellarators. III. The ion-temperature-gradient mode. Physics of Plasmas, 21(3), 032112. doi:10.1063/1.4868412Nadeem, M., Rafiq, T., & Persson, M. (2001). Local magnetic shear and drift waves in stellarators. Physics of Plasmas, 8(10), 4375-4385. doi:10.1063/1.1396842Candy, J., & Waltz, R. E. (2003). An Eulerian gyrokinetic-Maxwell solver. Journal of Computational Physics, 186(2), 545-581. doi:10.1016/s0021-9991(03)00079-2Al-Mohy, A. H., & Higham, N. J. (2010). A New Scaling and Squaring Algorithm for the Matrix Exponential. SIAM Journal on Matrix Analysis and Applications, 31(3), 970-989. doi:10.1137/09074721xTerry, P. W., Baver, D. A., & Gupta, S. (2006). Role of stable eigenmodes in saturated local plasma turbulence. Physics of Plasmas, 13(2), 022307. doi:10.1063/1.2168453Higham, N. J. (2008). Functions of Matrices. doi:10.1137/1.9780898717778Cuthbert, P., & Dewar, R. L. (2000). Anderson-localized ballooning modes in general toroidal plasmas. Physics of Plasmas, 7(6), 2302-2305. doi:10.1063/1.874064Waltz, R. E., & Boozer, A. H. (1993). Local shear in general magnetic stellarator geometry. Physics of Fluids B: Plasma Physics, 5(7), 2201-2205. doi:10.1063/1.860754Pueschel, M. J., Faber, B. J., Citrin, J., Hegna, C. C., Terry, P. W., & Hatch, D. R. (2016). Stellarator Turbulence: Subdominant Eigenmodes and Quasilinear Modeling. Physical Review Letters, 116(8). doi:10.1103/physrevlett.116.085001Pearlstein, L. D., & Berk, H. L. (1969). Universal Eigenmode in a Strongly Sheared Magnetic Field. Physical Review Letters, 23(5), 220-222. doi:10.1103/physrevlett.23.220Meerbergen, K., & Sadkane, M. (1999). Using Krylov approximations to the matrix exponential operator in Davidson’s method. Applied Numerical Mathematics, 31(3), 331-351. doi:10.1016/s0168-9274(98)00134-2Chen, Y., Parker, S. E., Wan, W., & Bravenec, R. (2013). Benchmarking gyrokinetic simulations in a toroidal flux-tube. Physics of Plasmas, 20(9), 092511. doi:10.1063/1.482198