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GYRO Overview

Prehistory

The use of gyrokinetic codes at General Atomics began in 1994 with the acquisition of the linear gyrokinetic stability code GSTOTAL. [1] GSTOTAL, the brainchild of Mike Kotschenreuther, was an Eulerian (continuum) initial-value solver with trapped and passing particles (ions and electrons) as well as collisional and electromagnetic physics. GSTOTAL represented an enormous technical advance in simulation capability, comparable in significance to Kotschenreuther's prior invention of the well-known δf method [2] for particle simulation. The time-advance scheme in GSTOTAL was fully implicit, and allowed use of timesteps much larger than those imposed by the electron parallel Courant limit. Historically speaking, GSTOTAL was the first step to a robust, practical nonlinear gyrokinetic code. With the addition of plasma shaping and finite-δB effects, GSTOTAL evolved into the linear stability code GKS. With added plot functionality, GKS has been routinely used for DIII-D experimental studies for over a decade. Combined with results from nonlinear flux-tube gyrofluid simulations, [3] [4] GKS was crucial in the development of GLF23, [5] the transport model of choice at numerous tokamak labs worldwide.

In what follows, we will often draw a distinction between global and flux-tube nonlinear simulations. In this context, flux-tube is synonymous with local. Flux-tube simulations provide the rigorous local (gyroBohm-scaled) limit of global simulations as ρ * vanishes. Here, ρ * = ρs / a is the ratio of gyroradius to system size, with ρs the ion-sound Larmor radius, and a the plasma minor radius.

Nonlinear flux-tube gyrofluid simulations by Beer, Dorland, Hammett, Snyder, Waltz and others provided the key physics discoveries in the mid-1990s. These gyrofluid simulations demonstrated that

  1. nonlinear, self-generated (zonal) flows control the nonlinear saturation of transport;[3][4][6][7][8]
  2. equilibrium ExB shear can quench transport if the shearing rates are comparable to maximum linear growth rates[3][4].

Nonlinear gyrofluid simulation codes were the first to treat trapped electron[8] and electromagnetic [9] turbulence.

The first major contribution from nonlinear gyrokinetic codes was the verification of the importance of Rosenbluth-Hinton [10] residual flows, as demonstrated by the flux-tube PIC code PG3EQ. [11] These flows were shown to give rise to an upshift in the nonlinear threshold for ITG-ae turbulence. Here, ITG-ae means ion-temperature-gradient modes with adiabatic electrons. The nonlinear upshift was later named the Dimits shift after its discoverer. The difficulty in properly treating this residual in gyrofluid models was one motivating factor for a switch from gyrofluid to gyrokinetic simulation. To this end, by building upon the GSTOTAL implicit scheme, Dorland was able to create a working nonlinear gyrokinetic code. This code, GS2, [12] was the first nonlinear gyrokinetic solver to include the crucial nonadiabatic electron dynamics required for trapped electron mode and electromagnetic physics.

In 2000, GS2 became the first nonlinear gyrokinetic code to include trapped and passing electrons at finite beta.

GYRO Design Roadmap

Development of GYRO began in 1999. The primary design target was to include all the physics relevant to simulating microturbulent transport in the core plasma (excluding the H-mode edge pedestal). This meant retaining finite-ρ * effects which would in principle allow

  • stabilization from profile variation (mainly in the driving gradients),
  • nonlocal transport phenomena,
  • deviations from gyroBohm scaling and possibly Bohm scaling.

Note that it is customary to describe the ρ * scaling of diffusivity in comparison to the Bohm scaling with χB = ρscs to the gyroBohm scaling with χgB = ρ * χB, where cs is the ion-sound speed.

The numerical methods for GYRO were patterned after GS2 where possible. In the end, many rather large departures from GS2 were required to meet the GYRO design target and to simultaneously increase computational efficiency. By 2001, GYRO had the ability to operate either globally using Dirichlet (zero-value) radial boundary conditions, or locally using flux-tube (periodic) boundary conditions.

A state-of-the-art implicit-explicit Runge-Kutta (IMEX-RK) integrator [13] was eventually added to ensure stable evolution of the longest wavelength n=0 modes at finite-beta. Independently, a novel poloidal discretization scheme solved the Ampere cancellation problem[14]. This numerical pathology hampered electromagnetic PIC simulation for over a decade. The only PIC code to successfully treat finite-β fluctuations [15] did so only after implementing an analog of the GYRO scheme.

By 2002, GYRO[14] achieved robust operation with its full physics design targets. This was demonstrated[16] in highly realistic simulations of dimensionally similar Bohm scaled DIII-D L-mode discharges.

Code Capabilities

The code capabilities deemed necessary for realistic core simulations are:

  1. gyrokinetic ions (electrons) for ITG (ETG) physics,
  2. trapped and passing electrons for ITG/TEM physics,
  3. gyrokinetic electron capability for coupled ITG/TEM-ETG simulations,
  4. nonlinear gyrokinetic impurities and energetic particles,
  5. ion-ion and electron-ion pitch-angle collisions,
  6. finite beta fluctuations and associated transport,
  7. shaped plasma geometry,
  8. equilibrium parallel velocity shear for Kelvin-Helmholtz drive,
  9. equilibrium ExB shear (strongly stabilizing in DIII-D),
  10. finite-ρ * effects (profile shear stabilization, nonlocal transport),
  11. input of actual experimental profiles,
  12. diagnostics for particle, momentum, and energy flows, plus heating and current-voltage(dynamo) ,
  13. n=0 driver for neoclassical equilibrium and transport,
  14. adaptive sources to preserve input equilibrium profiles,
  15. predict profiles by feedback profile adjustments to balance simulated transport flows against experimental flows.

The first systematic studies of particle transport and impurity dynamics were made with GYRO as part of (ongoing) thesis work for a UCSD graduate student (item 2 and 4). In particular, studies of temperature-gradient-induced particle pinches, thermal and energetic helium ash transport, differential flows in D-T plasmas, and collisional effects on particle pinches, have been made [17] Also GYRO has bben used to simulate the small ITG/TEM induced tranport of hot fusion alphas. [18] Using the profile feedback algorithm (item 15), simulations using DIII-D L-mode profiles successfully (and slightly) relaxed the input profiles so that simulation power flows matched experimental ones. [19] [20] This capability is crucial for the development of a steady-state gyrokinetic transport code. Such a code is now under development at GA, in collaboration with ORNL CCS.

GYRO is the only gyrokinetic code to simulated all the ion and electron transport channels including momentum trasnport (demonstarting both ExB and coriolis momnetum pinches), [21] turbulent heating, [22] [23] and turbulent dynamos (with little net cirren) in the tokamak current-voltage relation (item12) [24]

Finally, GYRO has recently added the capability to simultaneously treat full electron and ion gyroaverages. Previous ETG adiabatic ion simulations did not always nonlinearly saturate. GYRO (item3) showed that ETG simulations with kinetic ions always saturate. [25] [26] The code efficiency has also been improved in order to treat a very large number of complex toroidal modes. This has allowed GYRO to extend low-k ITG/TEM simulations to much higher k well into the ETG regime. This has allowed GYRO make significant progess on coupled ITG/TEM-ETG turbulence at a reduced ion-to-electron mass ratio. [27] It was found that when low-k ITG/TEM and high-k ETG-ki are strongly driven, there is no need to couple high-k to low-k to get the transport although getting the physical spectrum does require coupling. Furthermore, when coupled to low-k, ETG electron transport wsa found in ETG stable plasmas. In our view, this is the last remaining challenge for core gyrokinetic codes. Gyrokinetic codes for the edge plasma region will need to confront many additional new challenges.

To complement the physics capabilities, a large diagnostic and visualization package (VUGYRO) has been developed and continuously expanded during this period. This feature is by now a critical part of the GYRO project. The next order in ρ * parallel (and drift) nonlinearities (item 17), which are not normally included in gyrokinetic codes, have been added to GYRO for testing. As expected, we find that these small second order in ρ * terms are much too small to dynamically effect the transport flows. They are the physical origin of turbulent heating which GYRO computes diagnostically along with the transport flows in all channels.

GYRO is the flagship gyrokinetic code in the US Fusion program.

Using GYRO for Physics Studies

Utility

From the point of view of utility, the Eulerian codes GS2 and GYRO are set apart from all other codes in the US program in that they have a large (and growing) non-developer users group. A list of GYRO users is here. Realistic simulations of DIII-D, JET, JT60, and NSTX have been done; and simulation work on the UCLA tokamak has also been initiated. There is no published PIC simulation using experimental profiles with real shaped geoemtry and comprehensive physics. R. Bravenec and C. Holland have developed "synthetic diagnostic" tools to analyze GYRO data. GYRO has been the largest source of gyrokinetic data for W. Nevins' (LLNL) analysis code GKV.

Productivity

From the point of view of productivity, the fact of the matter is that the PIC code community as a whole has published relatively few well-resolved simulations, and the preponderance of these are based on the Cyclone base case [11]. All results from the GTC code are simply minor variants of the Cyclone case, and all with adiabatic electrons (or ions, in the case of ETG [28] ). The only exception is PG3EQ, for which some non-Cyclone data has been presented [29] for simple ITG-ae turbulence. In comparison, the Eulerian codes have covered a much wider swath in parameter space. In addition to many Cyclone-based scans, GYRO now has as a huge GYRO website transport database of well over 300 well-resolved flux tube simulations based on the GA standard case parameters [5]. Nearly all of these give particle and energy transport coefficients for both electrons and ions. Some also include momentum transport. These include scans in temperature and density gradients (moving from ITG- to TEM-dominated transport), Ti / Te, ExB and parallel velocity shear, [30] safety factor, magnetic shear, MHD alpha parameter, [31] beta, [32] collisionality and impurity fraction [17]. Additional scans are continuously being added to the database. In our view, compiling a database of simulations is a key practical end-product of nonlinear gyrokinetic simulations. This database provides the benchmarks and validation for the GA advanced gyrofluid transport model TGLF [33] [34] (nearly completed development).

Some key physics insights from global simulations

Some of the most novel physics insights from GYRO have followed from operation as a global code with realistic profiles. The first quantification of the transition from Bohm to gyroBohm scaling was achieved using GYRO ITG-ae simulations. [35] Some misunderstanding was subsequently generated by the well-publicized GTC ρ * -scan [36] with highly artificial profiles. The single scan appeared to suggest a universal transition ρ * . GYRO work established that the transition cannot be characterized by a single (universal) ρ * curve; rather, the transition is highly dependent on both the profile shape [35] [37] and the closeness to marginality [16] [35]. GYRO simulations also yielded several examples of nonlocal transport; [38] in particular, turbulence draining from unstable to less unstable (or stable) regions. These properties were confirmed in full-physics simulations of DIII-D L-mode discharges.[16] In this work, for which the discharges were very close to marginality, the transition from gyroBohm to Bohm scaling could be triggered in simulations by the addition of a minor stabilizing effect. That is, turning on ExB shear or collisions was enough to give a Bohm-ratio for diffusivities between DIII-D L-mode discharges. The difference between Bohm and gyroBohm scaling is just outside experimental error bars. GYRO simulations which perfectly projected profiles from dimensionally similar DIII-D discharges verified that the L-modes did indeed have Bohm scaling, but the local experimentally inferred gyroBohm scaling in some H-modes was actually due to experimental profile dissimilarity. [39]

There were also persistent claims from PIC codes that transport is depressed near a qmin-surface where there is a gap in singular surfaces. [40] Global GYRO simulations indicated that transport flows tends to vary monotonically across qmin surfaces [41] (as expected from linear theory and flux-tube gyrofluid simulations[3][4]) due to the appearance of nonresonant modes. These modes are absent in some simplified gyrofluid simulations which at first appeared to confirm the PIC code results.

GYRO simulations have been used to explore the detailed radial structure of nonlinear profile perturbations. We find persistent (i.e., time-averaged) structure tied to rational surfaces[19][32]. To be precise, when electrons are kinetic, there is a strong connection to rational surfaces. With adiabatic electrons, there is persistent time-averaged structure, but it is not so closely connected with these surfaces. These structures, or profile corrugations, are most pronounced for lower-order surfaces q = m/n = 1/1, 2/1, 3/1, and are weaker for successively higher-order surfaces, like q = 3/2, 5/2, 7/2 and so on. The width of these structures is on the order of a few ion gyroradii. The temperature gradient corrugations have the pattern expected of an island, but can exist electrostatically. Also, the turbulent dynamo EMF drives large current density corrugations at low-order rational surfaces, but little net current [24] Most importantly these predicted profile corrugation in the electron temperature gradient have now been observed in q-min = 2 DIII-D discharges, and the attending ExB shear layer is believed to be the trigger for low-power reversed shear ITB formation. [42]

Eulerian codes now have a large and ever-expanding user base, and several years of productive physics application.

GYRO from a Computational Science Perspective

GYRO owes it's computational efficiency in part to the strong support from the ORNL Center for Computational Science (CCS). GYRO runs well on a wide range of small clusters to large supercomputers. One can move between platforms seamlessly by setting a single environment variable. GYRO was among the earliest applications ported to the Cray X1 and XT3 at ORNL. The code is modular and the layout is carefully organized. There are few uses of esoteric language features. Initial X1 optimizations to take advantage of multistreaming and vectorization were quite successful for all but the collision operator. A later effort to improve the performance on the collision operator yielded a factor-of-ten improvement on the X1, with an average 10\% improvement on IBM and commodity systems. Recent PERC data is available which analyzes GYRO performance on various HPC systems [43] using the IPM, KOJAK, SvPablo, TAU and PMaC modeling tool suite. Additional GYRO performance data on various systems (including the Cray X1, XD1 and XT3) has been presented by Vetter, [44] Worley [45] and Fahey. [46] GYRO is presently so reliable that it is routinely used by ORNL staff to diagnose system hardware and software issues. For example, chassis interconnect problems on the XD1, filesystem slowdown in the XT3, and memory management issues on the SGI Altix.

Dispelling Some Urban Legends in Gyrokinetics

The Eulerian codes GS2 and GYRO have had to confront a number of urban legends mostly in the form of unpublished/unsubstantiated claims circulating within the Fusion theory community. These seem to originate from researchers having no first-hand experience with either Eulerian schemes or local simulations. Here are our documented counter claims:

The local gyroBohm limit of global codes differs from local codes
This is false. As ρ * vanishes, the transport obtained from a global code reaches a limiting value at a given radial location. This limiting value (i.e., the gyroBohm scaled limit) is identical to the local simulation result. This not only provides the physical meaning of a local simulation, but is an important test of validity for local and global codes alike. GYRO has passed this test repeatedly[37][38][35], but no global PIC code has passed the test with reasonable accuracy.

Full torus simulations are necessary to correctly compute the local transport
This is false. DIII-D full physics simulations with 1/6th of a torus, 1/3rd of a torus, 1/2 of a torus, and a full torus all give the same transport diffusivities to within a few percent[19]. In fact, full torus simulations are generally wasteful of computer resources. Global PIC codes programmed for full torus operation only, could obtain more accurate results by simulating only a fraction of a torus but operating with a higher number of particles per cell to lower the noise level (see 5 below).

GYRO has poorly resolved velocity space
This is false. Published GYRO simulations are always checked for adequate grid convergence by the standard method of grid refinement. GYRO has a particularly efficient velocity-space discretization scheme which suffers no accuracy loss even when the distribution is strongly discontinuous across the trapped-passing boundary. We typically use 128 velocity gridpoints per real-space cell. This is roughly equivalent to 128 particles per cell (PPC) in terms of points where the distribution function is known. We emphasize that this is significantly more than that typically used in PIC simulations (10 PPC in global PIC codes and 32 PPC in local codes). We have verified that no significant fine-scale structure in the distribution is being ignored or coarse-grained. Recent GYRO work [47] demonstrates a detailed steady-state balance between production of fluctuations and (numerical) dissipation, thus resolving the entropy paradox in a manner consistent with the picture developed by Krommes [48] [49] The numerical dissipation is also shown to be so small that it does not affect the observed transport.

The parallel nonlinearity can have a dramatic effect on the transport
This is false for realistic core tokamak parameters. The so-called parallel nonlinearity (a velocity-space nonlinearity which is formally one order smaller in ρ * than other terms in the gyrokinetic equations) is only one of several small terms commonly neglected in the standard operation of gyrokinetic codes. Contrary to previous claims by the GTC PIC code, [50] but in agreement with the UCAN PIC code, [51] GYRO has recently shown [52] that the parallel nonlinearity has no statistically significant effect on the diagnosed transport when ρ * < 0.01. Moreover, the parallel nonlinearity has nothing whatsoever to do with the entropy paradox or with producing steady-states of turbulence. To be clear the parallel nonlinearity (related to the nonlinear Landau damping and to wave-particle trapping) is the physical origin of a small turbulent heating source. GYRO is the first code to diagnostically calculate this heating. [53]

Initial diffusivity overshoots are meaningful and short runs are sufficient This is false. Only time averages over very long, statistically steady states of turbulent transport (well past any initial overshoots) are meaningful. In our experience, GYRO typical runs are substantially longer (we estimate two to five times) than runs from other gyrokinetic codes. Eulerian (continuum) codes can run indefinitely long within computer resource limits. However PIC codes are currently prevented from performing long-time simulations because the mean-square weights grow linearly in time due to discrete particle noise. [49] [54] In fairness we must add that while PIC codes must fight against noise, continuum codes must fight numerical instabilities. While we believe continuums codes have thus far have proved to be more efficient and more successful, the methods are complimentary and both should be pursuded to their best advantage.

We believe research on both local and global PIC codes should continue as a means of validating the more efficient Eulerian codes. However the US PIC community must join their European and Japanese colleagues in solving or mitigating the PIC noise problem. Certainly no future PIC simulation can be trusted without demonstrating the Nevins-Hammett[54] noise diagnostic test.

References

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  2. ↑ M. Kotschenreuther. Invited talk 9I4. Bull. Am. Phys. Soc. 33 (1998) 2107.
  3. ↑ 3.0 3.1 3.2 3.3 R.E. Waltz, G.R. Kerbel, and J. Milovich, Toroidal gyro-landau fluid model turbulence simulations in a nonlinear ballooning mode representation with radial modes. Phys. Plasmas 1 (1994) 2229.
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  5. ↑ 5.0 5.1 R.E. Waltz, G.M. Staebler, W. Dorland, G.W. Hammett, M. Kotschenreuther, and J.A Konings, A gyro-landau fluid transport model. Phys. Plasmas 4 (1997) 2482.
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  15. ↑ Y. Chen, S.E. Parker, B.I. Cohen, A.M. Dimits, W.M. Nevins, D. Schumaker, V.K. Decyk, and J.N. Leboeuf, Simulations of turbulent transport with kinetic electrons and electromagnetic effects. Nucl. Fusion 43 (2003) 1121.
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  18. ↑ C. Estrada-Mila, J. Candy, and R.E. Waltz, Turbulent Transport of Alpha Particles and Helium Ash in Reactor Plasmas. Phys. Plasmas 13 (2006) 0112303.
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  23. ↑ R.E. Waltz and G. M. Staebler, Gyrokinetic Theory and Simulation of Turbulent Energy Exchange, submitted to Phys. Plasmas.
  24. ↑ 24.0 24.1 F.L. Hinton, R.E. Waltz, and J. Candy, Effect of electromagnetic turbulence in the neoclassical Ohm's law. Phys. Plasmas 11 (2004) 2433.
  25. ↑ J. Candy, R.E. Waltz, M. Fahey, and C. Holland, The Effect of Ion-Scale Dynamics on Electron-Temperature-Gradient Turbulence. Plas. Phys. Control. Fusion 49 (2007) 1209.
  26. ↑ J. Candy and R.E. Waltz, Coupled ITG/TEM-ETG Gyrokinetic Simulation,” Proc. 21st IAEA Fusion Energy Conf., Chengdu, China, 2006, Paper TH/2-1.
  27. ↑ R.E. Waltz, J. Candy, and M. Fahey, Coupled Ion Temperature Gradient and Trapped Electron Mode to Electron Temperature Gradient Mode Gyrokinetic Simulations. APS06 Issue Phys. Plasmas 14 (2007) 0056116.
  28. ↑ Z. Lin, L. Chen, and F. Zonca, Role of nonlinear toroidal coupling in electron temperature gradient turbulence. Phys. Plasmas 12 (2005) 056125.
  29. ↑ A.M. Dimits, B.I. Cohen, W.M Nevins, and D.E Shumaker, Parameter dependences of ion thermal transport due to toroidal ITG turbulence. Nucl. Fusion 40 (2001) 1725.
  30. ↑ J.E. Kinsey, R.E. Waltz, and J. Candy, Nonlinear gyrokinetic turbulence simulations of ExB shear quenching of transport. Phys. Plasmas 12 (2005) 062302.
  31. ↑ J.E. Kinsey, R.E. Waltz, and J. Candy, The effect of safety factor and magnetic shear on turbulent transport in nonlinear gyrokinetic simulations. Phys. Plasmas 13 (2006) 022305.
  32. ↑ 32.0 32.1 J. Candy, Beta scaling of transport in microturbulence simulations. Phys. Plasmas 12 (2005) 072307.
  33. ↑ G.M. Staebler, J.E. Kinsey, and R.E. Waltz, Gyro-Landau fluid equations for trapped and passing particles, Phys. Plasmas 12 (2005) 102508.
  34. ↑ G.M. Staebler, J.E. Kinsey, and R.E. Waltz, A Theory Based Transport Model with Comprehensive Physics, APS06 Issue Phys. Plasmas 14 (2007) 0055909.
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  39. ↑ R.E. Waltz, J. Candy, and C.C. Petty, Projected Profile Similarity in Bohm and GyroBohm Scaled DIII-D L- and H-Modes, Phys. Plasmas 13(2006) 072304 .
  40. ↑ Y. Kishimoto, J.Y. Kim, W. Horton, T. Tajima, M.J. LeBrun, S.A. Dettrick, J.Q. Li, and S. Shirai, Discontinuity model for internal transport formation in reversed magnetic shear plasmas. Nucl. Fusion 40 (2000) 667.
  41. ↑ J. Candy, R.E. Waltz, and M.N. Rosenbluth, Smooothness of turbulent transport across a minimum-q surface. Phys. Plasmas 11 (2004) 1879.
  42. ↑ R.E. Waltz, M.E. Austin, K.H. Burrell, and J. Candy, Gyrokinetic Simulations of Off-Axis Minimum-q Profile Corrugations, Phys. Plasmas 13 (2006) 052301 .
  43. ↑ P. Worley, J. Candy, L. Carrington, K. Huck, T. Kaiser, G. Mahinthakumar, A. Maloney, S. Moore, D. Reed, P. Roth, H. Shan, S. Shende, A. Snavely, S. Sreepathi, F. Wolf, and Y. Zhang, Performance analysis of gyro: A tool evaluation. In Proceedings of the 2005 SciDAC Conference, San Francisco, CA, June 26-30, 2005.
  44. ↑ J.S. Vetter, S. Alam, T.H. Dunigan Jr., M.R. Fahey, P.C. Roth, and P. Worley, Early evaluation of the cray XT3 at ORNL. In Proceedings of the 47th Cray User Group Conference, Knoxville, TN, May 16-19, 2005.
  45. ↑ P. Worley, S. Alam, T.H. Dunigan Jr., M.R. Fahey, and J.S. Vetter, Comparative analysis of interprocess communication on the X1, XD1, and XT3. In Proceedings of the 47th Cray User Group Conference, Knoxville, TN, May 16-19, 2005.
  46. ↑ M.R. Fahey and J. Candy, GYRO: A 5-D gyrokinetic-Maxwell solver. In ACM/IEEE Proceedings of SC2004: High Performance Computing, Networking and Storage (Pittsburgh, PA), 2004.
  47. ↑ J. Candy and R.E. Waltz, Velocity-space resolution, entropy production and upwind dissipation in Eulerian gyrokinetic simulations. Phys. Plasmas 13 (2006) 074501.
  48. ↑ J.A. Krommes and G. Hu, The role of dissipation in the theory and simulations of homogeneous plasma turbulence, and resolution of the entropy paradox. Phys. Plasmas 1 (1994) 321.
  49. ↑ 49.0 49.1 J.A. Krommes, Thermostatted δf. Phys. Plasmas 6 (1999) 1477.
  50. ↑ W.W. Lee, Invited talk FI1B3. Bull. Am. Phys. Soc. 49 (2004) 135.
  51. ↑ J.C. Kniep, J.N. Leboeuf, and V.K. Decyk, Gyrokinetic particle-in-cell calculations of ion temperature gradient driven turbulence with parallel nonlinearity and strong flow corrections. Comput. Phys. Commun. 164 (2004) 98.
  52. ↑ J. Candy, R.E. Waltz, S.E. Parker, and Y. Chen, Relevance of the Parallel Nonlinearity in Gyrokinetic Simulations of Tokamak Plasmas. Phys. Plasmas 13 (2006) 074501 .
  53. ↑ F.L. Hinton and R.E. Waltz, Gyrokinetic turbulent heating. Phys. Plasmas 13 (2005) 102301.
  54. ↑ 54.0 54.1 W.M. Nevins, G.W. Hammett, A.M. Dimits, W. Dorland, and D.E. Shumaker, Discrete particle noise in particle-in-cell simulations of plasma microturbulence. Phys. Plasmas (2005).
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