# Invited Talks

Scientists of the Theory Department are frequently invited to give presentations at major physics and computing conferences.

The following presents a selection of the Invited Presentations given by Theory Staff and their collaborators.

### Past

• The International Sherwood Fusion Theory Conference, Annapolis
Gyrokinetic simulation of a fast L-H bifurcation dynamics in a realistic diverted tokamak edge geometry
[#s159: 02 May 2017]
Despite its critical importance in the fusion program and over 30 years of H-mode operation, there has been no fundamental understanding at the kinetic level on how the H-mode bifurcation occurs. We report the first observation of an edge transport barrier formation event in an electrostatic gyrokinetic simulation carried out in a realistic C-Mod like diverted tokamak edge geometry under strong forcing by a high rate of heat deposition. The results show that the synergistic action between two multiscale dynamics, the turbulent Reynolds-stress driven [1] and the neoclassical X-point orbit loss drive [2] sheared ${\bf E}\times{\bf B}$ flows, works together to quench turbulent transport and form a transport barrier just inside the last closed magnetic flux surface. The synergism helps reconcile experimental reports of the key role of turbulent stress in the bifurcation [3] with some other experimental observations that ascribe the bifurcation to X-point orbit loss/neoclassical effects [4,5]. The synergism could also explain other experimental observations that identified a strong correlation between the L-H transition and the orbit loss driven ${\bf E}\times{\bf B}$ shearing rate [6]. The synergism is consistent with the general experimental observation that the L-H bifurcation is more difficult with the $\nabla B$-drift away from the single-null X-point, in which the X-point orbit-loss effect is weaker [2].
[1] P.H. Diamond, S-I Itoh et al., Plasma Phys. Controlled Fusion 47, R35 (2005)
[2] C.S. Chang, Seunghoe Kue & H. Weitzner, Phys. Plasmas 9, 3884 (2002)
[3] G.R. Tynan, M. Xu et al., Nucl. Fusion 53, 073053 (2013)
[4] T. Kobayashi, K. Itoh et al., Phys. Rev. Lett. 111, 035002 (2013)
[5] M. Cavedon, T. Pütterich et al., Nucl. Fusion 57, 014002 (2017)
[6] D.J. Battaglia, C.S. Chang et al., Nucl. Fusion 53, 113032 (2017)
[7] S.M. Kaye, R. Maingi et al., Nucl. Fusion 51, 113109 (2011)
• The International Sherwood Fusion Theory Conference, Annapolis
Parasitic momentum flux in the tokamak core
T. Stoltzfus-Dueck, abstract, slides
[#s160: 01 May 2017]
Tokamak plasmas rotate spontaneously in the absence of applied torque. This so-called 'intrinsic rotation' may be very important for future low-torque devices such as ITER, since rotation can stabilize certain instabilities. In the mid-radius 'gradient region,' which reaches from the sawtooth inversion radius out to the pedestal top, intrinsic rotation profiles are sometimes flat and sometimes hollow. Profiles may even transition suddenly between these two states, an unexplained phenomenon referred to as rotation reversal. Theoretical efforts to identify the origin of the mid-radius rotation shear have focused primarily on quasilinear models, in which the phase relationships of some selected instability result in a nondiffusive momentum flux ("residual stress"). In contrast to these efforts, the present work demonstrates the existence of a robust, fully nonlinear symmetry-breaking momentum flux that follows from the free-energy flow in phase space and does not depend on any assumed linear eigenmode structure. The physical origin is an often-neglected portion of the radial ExB drift, which is shown to drive a symmetry-breaking outward flux of co-current momentum whenever free energy is transferred from the electrostatic potential to ion parallel flows [1]. The resulting rotation peaking is counter-current and scales as temperature over plasma current. As originally demonstrated by Landau [2], this free-energy transfer (thus also the corresponding residual stress) becomes inactive when frequencies are much higher than the ion transit frequency, which may explain the observed relation of density and counter-current rotation peaking in the core. Simple estimates suggest that this mechanism may be consistent with experimental observations, in both hollow and flat rotation regimes.
[1] T. Stoltzfus-Dueck, Phys. Plasmas 24, 030702 (2017)
[2] L. Landau, J. Exp. Theor. Phys. 16, 574 (1946); English translation in J. Phys. (USSR) 10, 25 (1946)
• 11th West Lake International Symposium on Energetic Particle Physics and Microturbulence, Hangzhou
Nonlinear simulations of beam-driven compressional Alfvén eigenmodes in NSTX
E. Belova, abstract, slides
[#s157: 24 Apr 2017]
Results of 3D nonlinear simulations of neutral-beam- driven compressional Alfvén eigenmodes (CAEs) in the National Spherical Torus Experiment (NSTX) are presented. Hybrid MHD- particle simulations for the H-mode NSTX discharge using the HYM code show unstable CAE modes for a range of toroidal mode numbers, $n = 4 - 9$, and frequencies below the ion cyclotron frequency. It is found that the essential feature of CAEs is their coupling to kinetic Alfvén wave (KAW) that occurs on the high-field side at the Alfvén resonance location. High-frequency Alfvén eigenmodes are frequently observed in beam-heated NSTX plasmas, and have been linked to flattening of the electron temperature profiles at high beam power. Coupling between CAE and KAW suggests a new mechanism to explain these observations, namely an energy channeling mechanism, in which beam driven CAEs dissipate their energy at the resonance location, therefore significantly modifying the energy deposition profile. The numerical model in the HYM code allows a full kinetic description of the beam ions, including the cyclotron resonances, but a one fluid MHD description is used to model the thermal plasma; therefore, the radial width of KAW is determined by the beam ion Larmor radius. Nonlinear simulations demonstrate that CAEs can channel the energy of the beam ions from the injection region near the magnetic axis to the location of the resonant mode conversion at the edge of the beam density profile. A set of nonlinear simulations show that the CAE instability saturates due to nonlinear particle trapping, and a large fraction of beam energy can be transferred to several unstable CAEs of relatively large amplitudes and absorbed at the resonant location. Absorption rate shows a strong scaling with the beam power.
• US-Japan Workshop and School on Magnetic Reconnection
General Theory of the Plasmoid Instability
L. Comisso, abstract, slides
[#s137: 21 Mar 2017]
We present the recent formulation of a general theory of the onset and development of the plasmoid instability [1]. We consider the general problem of a reconnecting current sheet that can evolve in time, rather than assuming a fixed Sweet-Parker current sheet. The new theoretical framework has lead to completely new results, which have shown that previously obtained power laws are insufficient to capture the correct properties of the plasmoid instability. The new scaling laws are shown to depend on the initial perturbation amplitude, the characteristic rate of current sheet evolution, and the Lundquist number. The detailed dynamics of the instability is also elucidated, and shown to comprise of a long period of quiescence followed by sudden growth over a short time scale.
[1] L. Comisso, M. Lingam et al., Phys. Plasmas 23, 100702 (2016)
• 58th Annual Meeting of the APS Division of Plasma Physics
Extending geometrical optics: A Lagrangian theory for vector waves
D. Ruiz, abstract, slides
[#s65: 04 Nov 2016]
Even diffraction aside, the commonly known equations of geometrical optics (GO) are not entirely accurate. GO considers wave rays as classical particles, which are completely described by their coordinates and momenta; but rays have another degree of freedom, namely, polarization. As a result, wave rays can behave as particles with spin. A well-known example of polarization dynamics is wave-mode conversion, which can be interpreted as rotation of the (classical) “wave spin”. However, there are other less-known manifestations of the wave spin, such as polarization precession and polarization-driven bending of ray trajectories. This talk presents recent advances in extending and reformulating GO as a first-principle Lagrangian theory, whose effective-gauge Hamiltonian governs both mentioned polarization phenomena simultaneously. Examples and numerical results are presented. When applied to classical waves, the theory correctly predicts the polarization-driven divergence of left- and right- polarized electromagnetic waves in isotropic media, such as dielectrics and non-magnetized plasmas. In the case of particles with spin, the formalism also yields a point-particle Lagrangian model for the Dirac electron, i.e. the relativistic spin-1/2 electron, which includes both the Stern-Gerlach spin potential and the Bargmann-Michel-Telegdi spin precession. Additionally, the same theory contributes, perhaps unexpectedly, to the understanding of ponderomotive effects in both wave and particle dynamics; e.g., the formalism allows to obtain the ponderomotive Hamiltonian for a Dirac electron interacting with an arbitrarily large electromagnetic laser field with spin effects included. [Supported by the NNSA SSAA Program through DOE Research Grant No. DE-NA0002948, by the U.S. DOE through Contract No. DE-AC02-09CH11466, and by the U.S. DOD NDSEG Fellowship through Contract No. 32-CFR-168a.]
• 58th Annual Meeting of the APS Division of Plasma Physics
Radiation effects on the runaway electron avalanche
C. Liu, abstract, slides
[#s67: 04 Nov 2016]
Runaway electrons are a critical area of research into tokamak disruptions. A thermal quench on ITER can result in avalanche production of a large amount of runaway electrons and a transfer of the plasma current to be carried by runaway electrons. The potential damage caused by the highly energetic electron beam poses a significant challenge for ITER to achieve its mission. It is therefore extremely important to have a quantitative understanding of the avalanche process, including (1) the critical energy for an electron to run away to relativistic energy, and (2) the avalanche growth rate dependence on electric field, which is related to the poloidal flux change required for an e-fold in current. It is found that the radiative energy loss of runaway electrons plays an important role in determining these two quantities. In this talk we discuss three kinds of radiation from runaway electrons, synchrotron radiation, Cerenkov radiation, and electron cyclotron emission (ECE) radiation. Synchrotron radiation, which mainly comes from the cyclotron motion of highly relativistic runaway electrons, dominates the energy loss of runaway electrons in the high-energy regime. The Cerenkov radiation from runaway electrons gives an additional correction to the Coulomb logarithm in the collision operator, which changes the avalanche growth rate. The ECE emission [1] mainly comes from electrons in the energy suprathermal rangee $1.2 <\gamma < 3$, which and gives an important approach to diagnose the runaway electron distribution in momentum and pitch angle. To study the runaway electron dynamics in momentum space including all the radiation and scattering effects, we use a novel tool, the adjoint method [2] to obtain both the runaway probability and the expected slowing-down time. The method is then combined with kinetic simulations to calculate the avalanche threshold and growth rate.
[1] C. Paz-Soldan, R.J. La Haye et al., Nucl. Fusion 56, 056010 (2016)
[2] Chang Liu, Dylan P. Brennan et al., Phys. Plasmas 23, 010702 (2016)
• 58th Annual Meeting of the APS Division of Plasma Physics
Plasmoids formation in a laboratory and large-volume flux closure during simulations of Coaxial Helicity Injection in NSTX-U
F. Ebrahimi, abstract, slides
[#s66: 31 Oct 2016]
In NSTX-U, transient Coaxial Helicity Injection (CHI) is the primary method for current generation without reliance on the solenoid. A CHI discharge is generated by driving current along open field lines (the injector flux) that connect the inner and outer divertor plates on NSTX/NSTX-U, and has generated over 200kA of toroidal current on closed flux surfaces in NSTX. Extrapolation of the concept to larger devices requires an improved understanding of the physics of flux closure and the governing parameters that maximizes the fraction of injected flux that is converted to useful closed flux. Here, through comprehensive resistive MHD NIMROD simulations conducted for the NSTX and NSTX-U geometries, two new major findings will be reported. First, formation of an elongated Sweet-Parker current sheet and a transition to plasmoid instability has for the first time been demonstrated by realistic global simulations [1]. This is the first observation of plasmoid instability in a laboratory device configuration predicted by realistic MHD simulations and then supported by experimental camera images from NSTX. Second, simulations have now, for the first time, been able to show large fraction conversion of injected open flux to closed flux in the NSTX-U geometry [2]. Consistent with the experiment, simulations also show that reconnection could occur at every stage of the helicity injection phase. The influence of 3D effects, and the parameter range that supports these important new findings is now being studied to understand the impact of toroidal magnetic field and the electron temperature, both of which are projected to increase in larger ST devices.
[1] F. Ebrahimi & R. Raman, Phys. Rev. Lett. 114, 205003 (2015)
[2] F. Ebrahimi & R. Raman, Nucl. Fusion 56, 044002 (2016)
• 58th Annual Meeting of the APS Division of Plasma Physics
Understanding and Predicting Profile Structure and Parametric Scaling of Intrinsic Rotation
W. Wang, abstract
[#s68: 31 Oct 2016]
The main focus of this talk is on developing physical understanding and a first-principles-based model for predicting intrinsic rotation profiles in magnetic fusion experiments, including ITER. It is shown for the first time that turbulent fluctuation-driven residual stress (a non-diffusive component of momentum flux) can account for both the shape and magnitude of the observed intrinsic toroidal rotation profile. The orientation and structure of typical residual stress profile is shown to have a complicated dependence on multiple physics parameters including turbulence type, q-profile structure, and collisionality, through which possible rotation profile optimization can be developed. Fluctuation-generated poloidal Reynolds stress, which displays a very similar radial structure, is also shown to significantly modify the poloidal rotation in a way consistent with experimental observations.
• 26th IAEA Fusion Energy Conference
Gyrokinetic projection of the divertor heat-flux width from present tokamaks to ITER
C-S. Chang, abstract, slides
[#s98: 19 Oct 2016]
The edge gyrokinetic code XGC1 shows that the divertor heat flux width $\lambda_q$ in between ELMs of Type-I ELMy H-modes in two representative types of present tokamaks (DIII-D type for conventional aspect ratio and NSTX type for tight aspect ratio) is set mostly by the ion neoclassical orbit spread, which is proportional to $1/I_P$ , while the blobby turbulent spread plays a minor role. This explains the $1/I_P$ scaling of the heat flux width observed in present tokamaks. On the other hand, the XGC1 studies for ITER H-mode like plasmas show that $\lambda_q$ is mostly set by the blobby turbulent spread, with the heat flux width being about 5X wider than that extrapolated from the $1/I_P$ scaling. This result suggests that the achievement of cold divertor plasmas and partial detachment required for power load and W impurity source control may be more readily achieved and be of simpler control issue than predicted on the basis of the $1/I_P$ scaling. A systematic ongoing validation study of the XGC1 results on various existing tokamaks will also be presented, including JET that is the closest existing device to ITER.
• 26th IAEA Fusion Energy Conference
Penetration and amplification of resonant perturbations in 3D ideal-MHD equilibria
S.R. Hudson, abstract, slides
[#s69: 17 Oct 2016]
The nature of ideal-MHD equilibria in three-dimensional geometry is profoundly affected by resonant surfaces, which beget a non-analytic dependence of the equilibrium on the boundary. Furthermore, non-physical currents arise in equilibria with continuously-nested magnetic surfaces and smooth pressure and rotational-transform profiles. We demonstrate that three-dimensional, ideal-MHD equilibria with nested surfaces and $\delta$-function current-densities that produce a discontinuous rotational-transform are well defined and can be computed both perturbatively and using fully-nonlinear equilibrium calculations. The results are of direct practical importance: we predict that resonant magnetic perturbations penetrate past the rational surface (i.e. “shielding” is incomplete, even in purely ideal-MHD) and that the perturbation is amplified by plasma pressure, increasingly so as stability limits are approached.
• Joint Varenna - Lausanne International Workshop
3D MHD equilibria with current sheets, magnetic islands, and chaos in stellarators and tokamaks
J. Loizu, abstract, slides
[#s84: 29 Aug 2016]
Two outstanding questions regarding MHD equilibria in toroidally confined plasmas are: how to reliably compute 3D MHD equilibria (1) in the ideal limit where current sheets are predicted to form at resonant rational surfaces, and (2) in partially relaxed plasmas where flux-surfaces, islands, and chaos coexist? The first question is of fundamental importance for MHD theory and has potential experimental implications. In fact, it has been observed experimentally that under certain conditions self-healing of islands can occur and thus current sheet formation is expected on small scales [1]. Moreover, the understanding of the ideal plasma response to resonant magnetic perturbations (RMPs) in tokamaks is incomplete and still under debate [2]. The second question is clearly vital for a proper description of the magnetic field that is consistent with the established pressure and current profiles. In particular, the $\beta$-limit in stellarators is most likely set by equilibrium degradation rather than stability [3]. In order to address these questions, a theory based on a generalized energy principle, referred to as multi-region, relaxed MHD (MRxMHD) [4], was developed and bridges the gap between Taylor’s relaxation theory and ideal MHD. Using the SPEC code [5], a numerical implementation of MRxMHD, we provide the first numerical proof of the existence of singular current densities and a novel theoretical guideline for the computation of three-dimensional ideal MHD equilibria with current sheets [6]. As an example, we provide new predictions for the ideal response to RMPs in tokamaks, together with an unprecedented verification between linearly and nonlinearly perturbed equilibria [7, 8]. Finally, we use SPEC in stellarator geometry to perform equilibrium calculations for W7-X in experimentally-relevant scenarios, thereby providing new quantitative insights for the effect of pressure and bootstrap current on the generation of magnetic islands and ergodic fields.
[1] Y. Narushima, K.Y. Watanabe et al., Nucl. Fusion 48, 075010 (2008)
[2] A.D. Turnbull, N.M. Ferraro et al., Phys. Plasmas 20, 056114 (2013)
[3] M. Drevlak, D. Monticello & A. Reiman, Nucl. Fusion 45, 731 (2005)
[4] M.J. Hole, S.R. Hudson & R.L. Dewar, Nucl. Fusion 47, 746 (2007)
[5] S.R. Hudson, R.L. Dewar et al., Phys. Plasmas 19, 112502 (2012)
[6] J. Loizu, S.R. Hudson et al., Phys. Plasmas 22, 022501 (2015)
[7] J. Loizu, S.R. Hudson et al., Phys. Plasmas 22, 090704 (2015)
[8] J. Loizu, S.R. Hudson et al., Phys. Plasmas 23, 055703 (2016)
• Quo vadis - Complex plasmas
Electron Emission Effects in Bounded and Dusty Plasma
I.D. Kaganovich, abstract, slides
[#s95: 01 Aug 2016]

Photon, electron and ion bombardment of materials leads to the emission of electrons from the materials. This so-called secondary electron emission (SEE phenomenon is a common link between particle-surface interactions in plasmas, particle accelerators, light sources, and space environments. The plasma-surface interaction in the presence of a strong electron emission is omnipresent in numerous plasma applications such as, for example, cathodes, emissive probes, divertor plasma, surface discharges, dusty plasma, plasma thrusters and plasma processing. In a plasma system, electron and ion fluxes to the wall are determined by particles velocity distribution functions and by the sheath potential, which are consistent with the wall properties. Electrons with sufficient energy to overcome the wall sheath potential and ions accelerated by the sheath potential can impact the wall and produce secondary electrons. The secondary electron emission can then reduce the sheath potential, leading to an increased loss of plasma electrons to the wall, increased wall heating, and increased cooling of the bulk plasma.

Although the role of the secondary electron emission in the above processes and applications has been acknowledged, its effects are neither well characterized nor well understood and therefore, cannot be reliably predicted. For example, electron emission significantly changes the space-charge distribution around emissive probes, adding uncertainty to plasma potential measurements. This status quo is in a great part due to a complex synergistic nature of particle-surface interactions, which often involves a coupling between impinging particles and materials properties and surface geometry. This coupling is particularly strong for plasma-surface interactions. In this problem, plasma and materials sciences are not separable – the plasma and surface interact and evolve together. The plasma science challenges are i) to develop an understanding of SEE effects on plasma and plasma effects on SEE, including but not limited to heating and energy relaxation of emitted electrons in the plasma through collisions and collective effects, surface recombination, surface charging, and surface breakdown, ii) to characterize SEE properties and SEE effects directly in plasma rather than in vacuum as it is commonly done, and iii) to develop control of SEE effects. The materials and surfaces sciences challenges are to understand i) how surface evolves from interaction with plasma, ii) how these surface and materials modifications affect the SEE from these materials, and iii) how to control SEE properties of materials. For example, changing surface properties with various coatings or due to wall erosion, trapping of emitted particles in complex surfaces, nanoscale effects all can significantly alter the electron emission properties of plasma facing surfaces.

• 21st International Symposium on Heavy Ion Inertial Fusion
Update on Beam-Plasma Interaction Research at Princeton Plasma Physics Laboratory
I.D. Kaganovich (presented by E.P. Gilson), abstract, slides
[#s78: 19 Jul 2016]
We have performed experimental and theoretical studies of beam neutralization by background plasma. Near-complete space-charge neutralization is required for the transverse compression of high-perveance ion beams for ion-beam-driven warm dense matter experiments and heavy ion fusion. One approach to beam neutralization is to fill the region immediately before the target with sufficiently dense plasma. The plasma provides a charge-neutralizing medium for beam propagation and makes it possible to achieve a high degree of compression beyond the space-charge limit. Experiments were performed on the Princeton Advanced Test Stand to investigate the degree of charge neutralization for different methods of neutralization [1]. A high-perveance 38 keV Ar+ beam was propagated in a plasma produced in a Ferroelectric Plasma Source (FEPS) discharge. By comparing the measured beam radius with the envelope model for space-charge expansion, it was determined that a charge neutralization fraction of 98% is attainable with sufficiently dense FEPS plasma and 83% with neutralization by plasma produced from a background gas. The transverse electrostatic potential of the ion beam is reduced from 15 V before neutralization to 0.3 V, implying that the energy of the neutralizing electrons is below 0.3 eV. Measurements of the time-evolution of beam radius show that near-complete charge neutralization is established 5 μs after the driving pulse is applied to the FEPS and can last for 35 μs. Numerical simulations of effects of the two-stream instability on the propagation of ion beam in background plasma were performed. Development of the two-stream instability between the beam ions and plasma electrons may lead to beam breakup, a slowing down of the beam particles, acceleration of the plasma particles, and transfer of the beam energy to the plasma particles and wave excitations. Making use of the particle-in-cell code LSP, a one-dimensional Vlasov code, the effects of the two-stream instability on beam propagation over a wide range of beam and plasma parameters were simulated. Because of the two-stream instability, the plasma electrons can be accelerated to velocities twice as high as the beam velocity. The resulting return current of the accelerated electrons may completely change the structure of the beam self-magnetic field, thereby changing its effect on the beam from focusing to defocusing. Therefore, previous theories of beam self-electromagnetic fields that did not take into account the effects of the two-stream instability must be significantly modified [2,3]. As a result of the two-stream instability, an ion beam pulse can generate an electron beam propagating ahead of the ion beam pulse and perturb plasma ahead of the ion beam pulse [4]. One simple method to avoid two-stream instability is to use tenuous plasma that can well neutralize ion beam space charge provided enough electrons can be supplied to the beam pulse from the plasma volume or chamber sides [5]. The associated change in plasma parameters affects the two-stream instability and causes its decay. This effect can be observed on the National Drift Compression Experiment-II (NDCX-II) facility by measuring the spot size of the extracted beamlet propagating through several meters of plasma. This research is supported by the U.S. Department of Energy.
[1] Anton D. Stepanov, Eric P. Gilson et al., Phys. Plasmas 23, 043113 (2016)
[2] Erinc Tokluoglu & Igor D. Kaganovich, Phys. Plasmas 22, 040701 (2015)
[3] Edward A. Startsev, Igor Kaganovich & Ronald C. Davidson, Nucl. Instr. Meth. Phys. Res. A 733, 75 (2014)
[4] Kentaro Hara & Igor D. Kaganovich, to be submitted to Phys. Plasmas
[5] William Berdanier, Prabir K. Roy & Igor Kaganovich, Phys. Plasmas 22, 013104 (2015)
• 21st International Symposium on Heavy Ion Inertial Fusion
Special Lecture in Honor of Ron Davidson
I.D. Kaganovich (presented by E.P. Gilson), abstract, slides
[#s79: 18 Jul 2016]
Abstract pending.
• 10th West Lake International Symposium on Magnetic Fusion
12th Asia Pacific Plasma Theory Conference, Hangzhou, China

Kinetic Plasma Turbulence Simulations on Top Supercomputers Worldwide
W. Tang, abstract, slides
[#s99: 09 May 2016]

A major challenge for supercomputing today is to demonstrate how advances in HPC technology translate to accelerated progress in key application domains. This is the focus of an exciting new program in the US called the “National Strategic Computing Initiative (NSCI)” – announced by President Obama as an Executive Order on July 29, 2015 involving all research & development (R&D) programs in the country to “enhance strategic advantage in HPC for security, competitiveness, and discovery.” A strong focus in key application domains is to accelerate progress in advanced codes that model complex physical systems -- especially with respect to reduction in “time-to-solution” as well as “energy to solution.” It is understood that if properly validated against experimental measurements/observational data and verified with mathematical tests and computational benchmarks, these codes can greatly improve much-needed predictive capability in many strategically important areas of interest.

Computational advances in magnetic fusion energy research have produced global particle-in-cell (PIC) simulations of turbulent kinetic dynamics for which computer run-time and problem size scale very well with the number of processors on massively parallel many-core supercomputers. For example, the GTC-Princeton (GTC-P) code, which has been developed with a “co-design” focus, has demonstrated the effective usage of the full power of current leadership class computational platforms worldwide at the petascale and beyond to produce efficient nonlinear PIC simulations that have advanced progress in understanding the complex nature of plasma turbulence and confinement in fusion systems for the largest problem sizes. Instead of the familiar Fortran-90 language, this is a modern code written in C and deploying OpenMP/MPI, CUDA, and OpenACC programming strategies with a strong focus on performance optimization of key operational functions within particle-in-cell codes in general. This has produced significant advances in scalability, performance, and portability on path-to-exascale supercomputing systems worldwide. Results have provided strong encouragement for being able to include increasingly realistic dynamics in extreme-scale computing campaigns with the goal of enabling predictive simulations characterized by unprecedented physics resolution/realism needed to help accelerate progress in delivering fusion energy.

• 10th West Lake International Symposium on Magnetic Fusion
12th Asia Pacific Plasma Theory Conference, Hangzhou, China

Computation of Multi-Region, Relaxed MHD Equilibria
S.R. Hudson, abstract, slides
[#s100: 09 May 2016]

We describe the construction of stepped-pressure equilibria as extrema of a multi-region, relaxed magnetohydrodynamic (MHD) energy functional that combines elements of ideal MHD and Taylor relaxation, and which we call MRxMHD. The model is compatible with Hamiltonian chaos theory and allows the three-dimensional MHD equilibrium problem to be formulated in a well-posed manner suitable for computation, and numerical solutions are constructed using the stepped-pressure equilibrium code, SPEC. Highlights of recent calculations will be presented and discussed, including: that the self-organized single-helical-axis (SHAx) and double-axis (DAx) states in reversed field pinch experiments can be reproduced; that MRxMHD can recover ideal MHD; and the SPEC code is used to compute (for the first time) the singular current densities predicted in ideal MHD equilibria in three-dimensional geometry and a new class of solution to the ideal MHD equilibrium equation will be presented. Some ongoing developments of MRxMHD and SPEC will be discussed, including: vacuum verification calculations of W7-X equilibria; free-boundary, non-up-down symmetric DIIID calculations, and including a non-trivial flow into the energy principle.

• 2016 International Sherwood Fusion Theory Conference
Adjoint method and runaway electron dynamics in momentum space
C. Liu, abstract, slides
[#s71: 06 Apr 2016]
Runaway electron physics is an important aspect of the post-thermal collapse in disruptions, and is a critical area for current research. Theoretical and experimental studies have shown that various kinds of kinetic effects, including the drag force, the pitch angle scattering, and the synchrotron and bremsstrahlung radiation can change the runaway electron distribution in momentum space. In this study, we use a novel tool, the adjoint method [1], to study the runaway electron momentum space structure. The adjoint method includes all the aforementioned kinetic effects, and overcomes some of the limitations of previous methods such as the test-particle and Monte-Carlo methods. Using the adjoint method, one can obtain results like the runaway probability function and the expected slowing-down time. Theses results are consistent with previous studies, including the increase of the critical electric field for runaway electron growth due to radiation effects [2] and the runaway electron hysteresis [3]. In addition, we use the adjoint method to study the role of large angle scattering in the runaway electron population decay when the electric field is close to but less than the critical value (the marginal case). For this case, we develop a new collision operator for runaway electrons that includes both small and large angle scattering self-consistently. In contrast with the common belief that small angle scattering is much more important than large angle scattering in weakly coupled plasmas, we find that for the marginal case large-angle scattering plays an important role and cannot be ignored. Kinetic simulations with the new collision operator show an upward shift of the critical electric field value compared to previous results. These results can help us better understand the runaway electron momentum space structure, and give insights into quiescent runaway electron (QRE) experiments and runaway electron mitigation in disruptions. This research is supported by the US Department of Energy.
[1] Chang Liu, Dylan P. Brennan et al., Phys. Plasmas 23, 010702 (2016)
[2] A. Stahl, E. Hirvijoki et al., Phys. Rev. Lett. 114, 115002 (2015)
[3] Pavel Aleynikov & Boris N. Breizman, Phys. Rev. Lett. 114, 155001 (2015)
• 2016 International Sherwood Fusion Theory Conference
First realistic characterizations of chirping instabilities in tokamaks
Vinícius Duarte, abstract, slides
[#s73: 06 Apr 2016]
In tokamak plasmas, the dynamics of phase-space structures with their associated frequency chirping is a topic of major interest in connection with mechanisms for fast ion losses. The onset of phase-space holes and clumps which produce chirping phenomena, has been theoretically shown to be related to the emergence of an explosive solution of an integro-differential, nonlocal cubic equation (IDNC) [1,2] that governs the early mode amplitude evolution in the nonlinear regime near marginal stability. We have extended the analysis of the IDNC model to quantitatively account for multiple resonance surfaces of a given mode in the presence of drag and diffusion (due to collisions and microturbulence) operators. Then a more realistic criterion is found, that takes into account the details of the mode structure and the variation of transport coefficients in phase space, to determine whether steady state solutions can or cannot exist. Stable steady state solutions indicate that chirping oscillations do not arise, while the lack of steady solutions due to the predominance of drag, is indicative that a frequency chirping response is likely in a plasma. Waves measured in experiments have been analyzed using NOVA and NOVA-K codes, with which we can realistically account for the mode structure and varying resonance responses spread over phase space. In the experiments presently analyzed, compatibility has been found between the theoretical predictions for whether chirping should or should not arise and the experimental observation or lack of observation of toroidicity-induced Alfvén eigenmodes in NSTX, DIII-D and TFTR. We have found that stochastic diffusion due to wave microturbulence is the dominant energetic particle transport mechanism in many plasma experiments, and its strength is the key as to whether chirping solutions are likely to arise.
[1] H.L. Berk, B.N. Breizman & M. Pekker, Phys. Rev. Lett. 76, 1256 (1996)
[2] M.K. Lilley, B.N. Breizman & S.E. Sharapov, Phys. Rev. Lett. 102, 195003 (2009)
• 2016 International Sherwood Fusion Theory Conference
Penetration and amplification of resonant perturbations in 3D ideal-MHD equilibria
S.R. Hudson, abstract, slides
[#s70: 04 Apr 2016]
Understanding 3D ideal-MHD equilibria, as described by the ideal force-balance equation, $\nabla p = {\bf j} \times {\bf B}$, is fundamentally important for understanding both tokamaks & stellarators. Edge-localized modes are believed to be ideal, peeling-ballooning modes; and a ‘hot-topic’ of current research is to suppress these modes using resonant magnetic perturbations (RMPs). However, the nature of ideal-MHD equilibria in 3D geometry is profoundly affected by resonant surfaces, which beget a non-analytic dependence on the boundary. And, in order to preserve quasi-neutrality, non-physical infinite currents arise in equilibria with continuously-nested magnetic surfaces and smooth pressure & transform profiles. These difficulties are not fundamental flaws in ideal-MHD, which remains perhaps the most successful, relevant yet simplest model of plasma dynamics. It is just that, until recently, self-consistent tractable solutions to the ideal-MHD equilibrium equation for arbitrary 3D geometry had not been discovered. Recently, for the first time, we computed the $1/x$ and $\delta$-function current-densities, and we realized that self-consistent solutions demand locally-infinite shear at the resonant surfaces. We introduced a new class of solutions that admit additional delta-function current-densities that produce a discontinuity in the rotational-transform that removes the singularities. Our equilibrium solutions can be computed both perturbatively and using fully-nonlinear equilibrium calculations (with the SPEC code), and we present precise verification calculations. Most importantly, our solutions yield predictions that are in sharp contrast to previous predictions, with direct implications for understanding the penetration of RMPs: in ideal-MHD, a resonant perturbation penetrates past the rational surface and into the core of the plasma; and the perturbation is magnified by pressure inside the resonant surface, increasingly so as stability limits are approached.
• 57th Annual Meeting of the APS Division of Plasma Physics
A new class of three-dimensional ideal-MHD equilibria with current sheets
J. Loizu, abstract, slides
[#s82: 20 Nov 2015]
Ideal MHD predicts singular current densities in 3D equilibria with nested flux surfaces: a pressure-driven $1/x$ current density that arises around resonant rational surfaces, and a Dirac $\delta$-function current that develops at those surfaces. Only recently have these currents been computed numerically [1], and we provide details of the calculation. We show that locally-infinite shear (i.e. discontinuous rotational-transform) at the resonant surfaces ensures well-defined solutions. Singularities in the current density are allowed in ideal-MHD, but the current passing through any surface must be finite for any physically-acceptable equilibrium model. While the integral of the $\delta$-current density is finite, the $1/x$ current diverges over certain surfaces. This led to the conclusion that pressure gradients cannot exist in the vicinity of rational surfaces and thus that the possible pressure profiles are either fractal [2] or discontinuous [3]. In this talk, we present a new class of 3D, globally-ideal, MHD equilibria with (i) continuously nested surfaces, (ii) arbitrary, continuous and smooth pressure profiles, (iii) arbitrary, 3D boundaries, (iv) without unphysical currents, and which are (v) analytic functions of the boundary [4]. Examples of such equilibria, computed with the SPEC code [5], are shown and verified against generalized solutions to Newcomb equation, showing excellent convergence. The results imply that a resonant magnetic perturbation can penetrate all the way into the centre of a tokamak without being shielded at the resonant surface, even within ideal MHD.
[1] J. Loizu, S.R. Hudson et al., Phys. Plasmas 22, 022501 (2015)
[2] Harold Grad, Phys. Fluids 10, 137 (1967)
[3] Oscar Bruno & Peter Laurence, Commun. Pure Appl. Math. 49, 717 (1996)
[4] J. Loizu, S.R. Hudson et al., Phys. Plasmas 22, 090704 (2015)
[5] S.R. Hudson, R.L. Dewar et al., Phys. Plasmas 19, 112502 (2012)
• 57th Annual Meeting of the APS Division of Plasma Physics
Acceleration of plasma electrons by intense nonrelativistic ion beams propagating in background plasma due to two-stream instability
I.D. Kaganovich, abstract, slides
[#s75: 19 Nov 2015]
In this paper we study the effects of the two-stream instability on the propagation of intense nonrelativistic ion and electron beams in background plasma. Development of the two-stream instability between the beam ions and plasma electrons leads to beam breakup, a slowing down of the beam particles, acceleration of the plasma particles, and transfer of the beam energy to the plasma particles and wave excitations. Making use of the particle-in-cell codes EDIPIC and LSP, and analytic theory we have simulated the effects of the two-stream instability on beam propagation over a wide range of beam and plasma parameters. Because of the two-stream instability, the plasma electrons can be accelerated to velocities as high as twice the beam velocity. The resulting return current of the accelerated electrons may completely change the structure of the beam self-magnetic field, thereby changing its effect on the beam from focusing to de-focusing. Therefore, previous theories of beam self-electromagnetic fields that did not take into account the effects of the two-stream instability must be significantly modified. This effect can be observed on the National Drift Compression Experiment-II (NDCX-II) facility by measuring the spot size of the extracted beamlet propagating through several meters of plasma. Particle-in-cell, fluid simulations, and analytical theory also reveal the rich complexity of beam-plasma interaction phenomena: intermittency and multiple regimes of the two-stream instability in dc discharges; band structure of the growth rate of the two-stream instability of an electron beam propagating in a bounded plasma and repeated acceleration of electrons in a finite system. In collaboration with E. Tokluoglu, D. Sydorenko, E. A. Startsev, J. Carlsson, and R. C. Davidson. Research supported by the U.S. Department of Energy.
• 57th Annual Meeting of the APS Division of Plasma Physics
Free-Boundary 3D Equilibria and Resistive Wall Instabilities with Extended-MHD
N. Ferraro, abstract, slides
[#s76: 19 Nov 2015]
The interaction of the plasma with external currents, either imposed or induced, is a critical element of a wide range of important tokamak phenomena, including resistive wall mode (RWM) stability and feedback control, island penetration and locking, and disruptions. A model of these currents may be included within the domain of extended-MHD codes in a way that preserves the self-consistency, scalability, and implicitness of their numerical methods. Such a model of the resistive wall and non-axisymmetric coils is demonstrated using the M3D-C1 code for a variety of applications, including RWMs, perturbed non-axisymmetric equilibria, and a vertical displacement event (VDE) disruption. The calculated free-boundary equilibria, which include Spitzer resistivity, rotation, and two-fluid effects, are compared to external magnetic and internal thermal measurements for several DIII-D discharges. In calculations of the perturbed equilibria in ELM suppressed discharges, the tearing response at the top of the pedestal is found to correlate with the onset of ELM suppression. Nonlinear VDE calculations, initialized using a vertically unstable DIII-D equilibrium, resolve in both space and time the currents induced in the wall and on the plasma surface, and also the currents flowing between the plasma and the wall. The relative magnitude of these contributions and the total impulse to the wall depend on the resistive wall time, although the maximum axisymmetric force on the wall over the course of the VDE is found to be essentially independent of the wall conductivity. This research was supported by US DOE contracts DE-FG02-95ER54309, DE-FC02-04ER54698 and DE-AC52-07NA27344.
• 57th Annual Meeting of the APS Division of Plasma Physics
Magnetic self-organization in Tokamaks
S.C. Jardin, abstract, slides
[#s74: 17 Nov 2015]
We report here on a nonlinear mechanism that forms and maintains a self-organized stationary (sawtooth free) state in tokamaks. This process was discovered by way of extensive long-time simulations using the M3D-C1 3D extended MHD code in which new physics diagnostics have been added. It is well known that most high-performance modes of tokamak operation undergo “sawtooth” cycles, in which the peaking of the toroidal current density triggers a periodic core instability which redistributes the current density. However, certain modes of operation are known, such as the “hybrid” mode in DIII-D, ASDEX-U, JT-60U and JET, and the long-lived modes in NSTX and MAST, which do not experience this cycle of instability. Empirically, it is observed that these modes maintain a non-axisymmetric equilibrium which somehow limits the peaking of the toroidal current density. The physical mechanism responsible for this has not previously been understood, but is often referred to as “flux-pumping”, in which poloidal flux is redistributed in order to maintain $q_0 > 1$. In this talk, we show that in long-time simulations of inductively driven plasmas, a steady-state magnetic equilibrium may be obtained in which the condition $q_0 > 1$ is maintained by a dynamo driven by a stationary marginal core interchange mode. This interchange mode, unstable because of the pressure gradient in the ultra-low shear region in the center region, causes a $(1,1)$ perturbation in both the electrostatic potential and the magnetic field, which nonlinearly cause a $(0,0)$ component in the loop voltage that acts to sustain the configuration. This hybrid mode may be a preferred mode of operation for ITER. We present parameter scans that indicate when this sawtooth-free operation can be expected.
• 2015 International Sherwood Fusion Theory Conference
Computation of singular currents at rational surfaces in non-axisymmetric MHD equilibria
J. Loizu, abstract, slides
[#s81: 17 Mar 2015]
Ideal MHD predicts the existence of singular current densities forming at rational surfaces in three-dimensional equilibria with nested flux surfaces. These current singularities consist of a Pfirsch-Schlüter, $1/x$ current that arises around rational surfaces as a result of finite pressure gradient and a Dirac $\delta$-function current that develops at rational surfaces and presumably prevents the formation of islands that would otherwise develop in a non-ideal plasma. These currents play a crucial role in describing (1) the plasma response to non-axisymmetric boundary perturbations, (2) the equilibrium and stability of non-axisymmetric, toroidally confined plasmas, and (3) the triggering of reconnection phenomena such as tokamak sawteeth. While analytical formulations have been developed to describe such currents in simplified geometries, a numerical proof of their existence has been hampered by the assumption of smooth functions made in conventional MHD equilibrium models such as VMEC. Recently, a theory based on a generalized energy principle, referred to as multi-region, relaxed MHD (MRxMHD), was developed and incorporates the possibility of non-smooth solutions to the MHD equilibrium problem. Using SPEC [1], a nonlinear implementation of MRxMHD, we provide the first numerical proof of their mutual existence and a novel theoretical guideline for the numerical computation of three-dimensional ideal MHD equilibria with singular currents [2]. Examples of such kind of equilibria are shown for both slab and cylindrical geometries, and the numerical results are thoroughly verified against analytical predictions from linearly perturbed ideal MHD equilibria. Based on these results, we present the hypothesis that non-axisymmetric ideal MHD equilibria only exist for discontinuous rotational transform profiles [3].
[1] S.R. Hudson, R.L. Dewar et al., Phys. Plasmas 19, 112502 (2012)
[2] J. Loizu, S.R. Hudson et al., Phys. Plasmas 22, 022501 (2015)
[3] J. Loizu, S.R. Hudson et al., Phys. Plasmas 22, 090704 (2015)
• 56th Annual Meeting of the APS Division of Plasma Physics
X-Point-Position-Dependent Intrinsic Rotation in the Edge of TCV
T. Stoltzfus-Dueck, abstract, slides
[#s97: 28 Oct 2014]
A simple transport-based theoretical model predicts that intrinsic toroidal rotation in the tokamak edge should depend strongly on $R_X$, the major-radial position of the X-point, including a sign change to counter-current rotation for adequately outboard X-point. To test the prediction, an $R_X$ scan was conducted in Ohmic L-mode shots on TCV, in both USN and LSN configurations. The strong linear dependence on $R_X$ was experimentally observed, with quantitative magnitude corresponding to a realistic value for the theory’s corresponding input parameter. Although peaked rotation profiles complicate the comparison of absolute rotation values, the data is consistent with the predicted sign change. The core rotation profile shifted fairly rigidly with the edge rotation value, maintaining a relatively constant core rotation gradient. Core rotation reversals, triggered accidentally in a few shots, had little effect on the edge rotation velocity. Edge rotation was modestly more counter-current in USN than LSN discharges.
• Joint 19th ISHW and 16th IEA-RFP workshop
Minimally constrained model of self-organised helical states in RFX
G.R. Dennis, abstract, slides
[#s83: 17 Sep 2013]
We show that the self-organized single-helical-axis (SHAx) and double-helical-axis (DAx) states [1, 2] in reversed field pinches can be reproduced in a minimally constrained equilibrium model using only five parameters [3] (see Figure 1). This is a significant reduction on previous representations of the SHAx which have required an infinite number of constraints [4]. The DAx state, which has a non-trivial topology, has not been previously reproduced using an equilibrium model that preserves this topological structure. We show that both states are a consequence of transport barrier formation in the plasma core, in agreement with experimental results.

Figure 1: Comparison of the ideal MHD representation of the SHAx state in RFX-mod and the minimal model (MRXMHD) of this state presented in this work. Figures (a)–(d) show the (poloidal) magnetic flux contours of the ideal MHD plasma equilibrium at equally spaced toroidal angles covering one period of the helical solution. Figures (e)–(h) show Poincaré plots of the minimal model at the same toroidal locations as (a)–(d). The thick black lines mark the location of the transport barrier separating the two plasma volumes.
[1] D.F. Escande, P. Martin et al., Phys. Rev. Lett. 85, 1662 (2000)
[2] P. Martin, L. Marrelli et al., Nucl. Fusion 43, 1855 (2003)
[3] G.R. Dennis, S.R. Hudson et al., Phys. Rev. Lett. 111, 055003 (2013)
[4] D. Terranova, D. Bonfiglio et al., Plasma Phys. Controlled Fusion 52, 124023 (2010)
• 2013 International Sherwood Fusion Theory Conference
A minimally constrained model of self-organised helical states in reversed-field pinches
G.R. Dennis, abstract, slides
[#s96: 17 Apr 2013]
We show that the self-organized single-helical-axis (SHAx) and double-helical-axis (DAx) states [1, 2] in reversed field pinches can be reproduced in a minimally constrained equilibrium model using only five parameters [3] (see Figure 1). This is a significant reduction on previous representations of the SHAx which have required an infinite number of constraints [4]. The DAx state, which has a non-trivial topology, has not been previously reproduced using an equilibrium model that preserves this topological structure. We show that both states are a consequence of transport barrier formation in the plasma core, in agreement with experimental results.

Figure 1: Comparison of the ideal MHD representation of the SHAx state in RFX-mod and the minimal model (MRXMHD) of this state presented in this work. Figures (a)–(d) show the (poloidal) magnetic flux contours of the ideal MHD plasma equilibrium at equally spaced toroidal angles covering one period of the helical solution. Figures (e)–(h) show Poincaré plots of the minimal model at the same toroidal locations as (a)–(d). The thick black lines mark the location of the transport barrier separating the two plasma volumes.
[1] D.F. Escande, P. Martin et al., Phys. Rev. Lett. 85, 1662 (2000)
[2] P. Martin, L. Marrelli et al., Nucl. Fusion 43, 1855 (2003)
[3] G.R. Dennis, S.R. Hudson et al., Phys. Rev. Lett. 111, 055003 (2013)
[4] D. Terranova, D. Bonfiglio et al., Plasma Phys. Controlled Fusion 52, 124023 (2010)
• 38th EPS Conference on Plasma Physics
Partially-relaxed, partially-constrained MHD equilibria
S.R. Hudson, abstract, slides
[#s87: 01 Jul 2011]
The commonly used equation of ideal force balance, $\nabla p = {\bf j} \times {\bf B}$, is pathological when the magnetic field, ${\bf B}$, is chaotic. Any continuous, non-trivial pressure that satisfies ${\bf B}\cdot\nabla p = 0$ with a chaotic field will have an infinity of discontinuities in the pressure gradient. The perpendicular current ${\bf j}_\perp = {\bf B} \times \nabla p/B^2$ is either zero or discontinuous, and $\nabla \cdot {\bf j}_\perp$ is zero or not defined. This pathological structure causes problems for the so-called “Spitzer” iterative approach, which is fundamentally ill-posed as it depends on inverting magnetic differential equations, e.g. ${\bf B} \cdot \nabla (j_\parallel/B) = −\nabla \cdot {\bf j}_\perp$, and such equations have a dense set of singularities. We suggest instead a well-posed equilibrium construction based on an extension of Taylor relaxation: a weakly-resistive plasma will relax to minimize the plasma energy subject to the constraint of conserved helicity. To obtain non-trivial pressure profiles we add additional topological constraints on a selection of KAM surfaces on which the constraints of ideal MHD are imposed. Consider a plasma region comprised of a set of $N$ nested annular regions which are separated by a discrete set of toroidal interfaces, ${\cal I}_l$. In each volume, ${\cal V}_l$, bounded by the ${\cal I}_{l-1}$ and ${\cal I}_l$ interfaces, the plasma energy, $U_l$, the global-helicity, $H_l$, and the “mass”, $M_l$, are given by the integrals: \begin{eqnarray} U_l \equiv \int_{{\cal V}_l} \left( \frac{p}{\gamma-1} + \frac{B^2}{2} \right)dv, \;\; H_l \equiv \int_{{\cal V}_l} {\bf A}\cdot{\bf B}\;dv, \;\; M_l \equiv \int_{{\cal V}_l} p^{1/\gamma}\;dv, \end{eqnarray} where ${\bf A}$ is the vector potential, ${\bf B} = \nabla \times {\bf A}$. The equilibrium states that we seek minimize the total plasma energy, subject to the constraints of conserved helicity and mass in each annulus. Arbitrary variations in both the magnetic field in each annulus and the geometry of the interfaces are allowed, except that we assume the magnetic field remains tangential to the interfaces which act as “ideal barriers” and coincide with pressure gradients. The Euler-Lagrange equations show that in each annulus the magnetic field satisfies $\nabla \times {\bf B} = \mu_l {\bf B}$, and across each interface the total pressure is continuous, $[[p + B^2/2]] = 0$. We have implemented this model in a code, the Stepped Pressure Equilibrium Code (SPEC), which uses a mixed Fourier, finite-element representation for the vector potential. Quintic polynomial basis functions give rapid convergence in the radial discretization, and the freedom in the poloidal angle is exploited to minimize a “spectral-width”. For given interface geometries the Beltrami fields in each annulus are constructed in parallel, and a Newton method (with quadratic-convergence) is implemented to adjust the interface geometry to satisfy force-balance. Convergence studies and three-dimensional equilibrium solutions with non-trivial pressure and islands and chaotic fields will be presented.
• 17th International Stellarator / Heliotron Workshop
Cantori, chaotic coordinates and temperature gradients in chaotic fields
S.R. Hudson, abstract, slides
[#s86: 12 Oct 2009]
Toroidal magnetic field line flow is a $1\frac{1}{2}$dimensional Hamiltonian system and, in the absence of symmetry, such systems are generally chaotic. The existence of local regions of irregular trajectories has profound implications for a variety of problems in plasma confinement: here we consider anisotropic heat transport. The study of heat transport in chaotic fields has a long history [1], but conventional approaches effectively treat chaotic fields as random, and thus overlook some important properties of chaotic fields, namely the hierachy of invariant dynamics comprised of regular, irrational trajectories. The invariant irrational sets, known as “cantori”, that persist after the destruction of the “KAM” surfaces can form effective partial barriers to anisotropic transport. We demonstrate using a model of heat transport with separate parallel and perpendicular thermal diffusion coefficients, $\kappa_\parallel$ and $\kappa_\perp$. For fusion plasmas the ratio $\kappa_\parallel / \kappa_\perp$ may exceed $10^{10}$, and the temperature adapts to the fractal structure of the magnetic field. This paper will show that temperature gradients coincide with the cantori. We develop chaotic-magnetic coordinates [2]: coordinates adapted to the invariant structures of the field line flow. By adapting the coordinate surfaces to the partial barriers formed by the cantori, the coordinate surfaces coincide with isotherms. The temperature written in chaotic coordinates is well approximated by $T = T(s)$, where $s$ is a radial coordinate, and an expression for the temperature gradient is derived: \begin{eqnarray}\frac{dT}{ds} = \frac{c}{\kappa_\parallel \varphi + \kappa_\perp G}, \end{eqnarray} where $\varphi \equiv \int \int d\theta d\phi \sqrt g B^2_n$ is the squared field-line flux across a coordinate surface, and $G \equiv \int \int d\theta d\phi \sqrt g g^{ss}$ is an averaged metric quantity.
[1] A.B. Rechester & M.N. Rosenbluth, Phys. Rev. Lett. 40, 38 (1978)
[2] S.R. Hudson & J. Breslau, Phys. Rev. Lett. 100, 095001 (2008)
• 22nd IAEA Fusion Energy Conference
Temperature Gradients are supported by Cantori in Chaotic Magnetic Fields
S.R. Hudson, abstract, slides
[#s85: 13 Oct 2008]
With the tantalizing prospect that localized regions of chaotic magnetic field can be used to suppress ideal instabilities in fusion devices, as suggested by the resonant magnetic perturbation (RMP) experiments on DIIID, it becomes necessary to understand the impact of chaotic fields on confinement, particularly so considering that RMP fields are being considered as an ELM mitigation strategy for ITER. Using a model of heat transport for illustration, this paper will show that chaotic fields can support significant temperature gradients, despite the fact that flux surfaces may be destroyed by applied error fields. The remnants of the irrational flux surfaces, the cantori, present extremely effective partial-barriers to field-line transport, and thus present effective barriers to any transport process that is dominantly parallel to the field. We extend the concept of magnetic coordinates to chaotic fields [1], and show that the temperature, generally a function of three-dimensional space, takes the simple form $T(s)$, where $s$ labels the chaotic-coordinate surfaces.
[1] S.R. Hudson & J. Breslau, Phys. Rev. Lett. 100, 095001 (2008)