Role of turbulent separatrix tangle in the improvement of the integrated pedestal and heat exhaust issue for stationary-operation tokamak fusion reactors

The magnetic separatrix surface is designed to provide the final and critical confinement to the hot stationary-operation core plasma in modern tokamak reactors in the absence of an external magnetic perturbation (MP) or transient magneto-hydrodynamic perturbation, while diverting the exhaust heat to divertor plates. All the stationary operational boundary plasma studies and reactor designs have been performed under this assumption. However, there has been a long-standing suspicion that a stationary-operation tokamak plasma even without external MPs or edge localized modes (ELMs) activities may not have a stable closed separatrix surface, especially near the magnetic X-point. Here, the first gyrokinetic numerical observation is reported that the divertor separatrix surface, due to homoclinic tangles caused by intrinsic electromagnetic turbulence, is not a stable closed surface in a stationary operation phase even without MPs or ELMs. Unlike the MP- or ELM-driven homoclinic tangles that could cause deleterious effects to core confinement or divertor plates, it is found that the micro-turbulence driven homoclinic tangles could connect the divertor plasma to the pedestal plasma in a constructive way by broadening the divertor heat-exhaust footprint and weakening the pedestal slope to the ELM-safe direction. Micro-turbulent homoclinic tangles can open a new research direction in understanding and controlling these two most troublesome and non-locally connected edge-plasma issues in a tokamak fusion reactor.


Introduction
Empirical scaling and heuristic modelling using data from present tokamaks [1][2][3] predict extremely narrow SOL heat flux channel widths (λ q ≲ 1 mm) on ITER for high current, burning plasma operation.Although detached divertor plasmas using impurity seeding can still achieve tolerable target heat-fluxes under such conditions [4], the operational window is reduced and an accidental divertor reattachment poses a great threat to target integrity.An edge plasma condition which inherently yields a much wider λ q significantly relaxes this power handing problem both for ITER and future reactors operating at high plasma current, I p .Achievement of edge localized mode (ELM)-free pedestal is another critically important edge physics issue, which can be nonlocally related to the divertor footprint problem.
Gyrokinetic simulations with XGC [5,6] have predicted that the 15 MA ITER edge is in a different physics regime from those in present tokamaks due to much smaller ρ i * = ρ i /a.The weak ExB shearing rate in plasmas with such a low ρ i * allows the streamer-type electrostatic, electron-directional turbulence across the magnetic separatrix and hence spreads the heat-load footprint.XGC electrostatic simulation indeed found λ q ≃ 6 mm at 15 MA in ITER, an order of magnitude higher than predicted by the popular empirical scaling called Eich's multi-machine regression formula #14 [1].New electromagnetic simulations using XGC now find an even wider λ q (≃8 mm) for 15 MA ITER plasmas (figure 1, upper right orange symbol).The origin of this increased electromagnetic broadening is found to be mostly the microturbulence-driven homoclinic tangles making the magnetic separatrix surface to be unstable under stationary operation conditions without external magnetic perturbations (MPs) or edge magnetohydrodynamics (MHDs) instabilities.When the same electromangetic simulations are performed for present devices, this electromagnetic broadening of λ q is still within the regression error bar from Eich's λ q values.XGC is a 5D total-f gyrokinetic particle-in-cell code, combined with a continuum 5D grid method to conveniently describe sources/sinks and Coulomb/atomic collisional physics [7,8].XGC has recently been upgraded to include electromagnetic turbulence by lifting the delta-f method of mixed-variable/pullback electromagnetic implementation by Kleiber and Mischenko et al [9][10][11] to total-f [7].A deltaf electromagnetic version XGC also exists [12], directly importing the delta-f mixed-variable/pullback method.XGC is optimized for tokamak edge physics across the pedestal and magnetic separatrix to the divertor surface, even though the simulation domain is the whole volume all the way to the magnetic axis.XGC includes the neoclassical physics, the electromagnetic turbulence physics, the reduced-MHD physics, the neutral particle recycling and atomic interactions with plasma, the particle-torque-heat sources, the radiative heat loss models, the multispecies gyrokinetic impurity particles, and the conservative nonlinear Fokker-collision operator [13,14].To handle the complicated magnetic X-point and material wall geometry, XGC uses unstructured triangular mesh on cartesian cylindrical coordinate system.The mesh nodes follow the magnetic field lines, approximately, to reduce the number of mesh nodes in the toroidal direction in studying the strongly magnetized plasma that makes the relevant physics to be stretched along the magnetic field lines.
The magnetic separatrix surface is designed to provide the final, and critical, confinement to the hot core plasma, while diverting the exhaust heat to the divertor material plates.However, there has been a long-standing suspicion that the separatrix surface may not be a stable closed surface and will only exist in incomplete forms made stochastic by homoclinic tangle instabilities of Poincaré [15] characterized by the wild lobe-structured magnetic-field distortions even in the absence of edge MHD activities or external MPs.However, this has been difficult to prove experimentally or in first-principlesbased plasma simulations.In the presence of edge MHD activities or external MPs, the homoclinic tangle magnetic separatrix surfaces have been observed in numerical simulations and laboratory experiments [16][17][18].In the absence of these two events, the tokamak edge physics have been studied assuming that there is the closed separatrix surface.
Here, we report the first first-principles-based, selfconsistent numerical observation that the divertor separatrix surface, due to the intrinsic electromagnetic turbulence, is not a stable closed surface even in a stationary operation phase without the external MP coils or edge MHD instabilities.It is found that the fine-scale micro-turbulent homoclinic tangles resonate mostly with the electron dynamics only and, due to the ambipolarity requirement where the unaffected ions prohibit the electron loss, do not result in the significant loss of pedestal density as observed in the MP and MHD cases.Instead, it connects the divertor plasma to the pedestal plasma, may broaden the divertor heat-exhaust footprint and may weaken the pedestal slope towards the MHDsafe direction.Micro-turbulent homoclinic tangles can open a new research direction in understanding and controlling these two most troublesome and non-locally integrated edge-plasma issues in a tokamak fusion reactor: possibly lowering the peak heat-load density on divertor material and stabilizing the edge MHD modes.
We note here that the study presented here is more of baseline physics research in a deuteron plasma (to augment previous studies by Eich et al [1], Goldston [2] and others) than an actual prediction for the peak heat-load density on the outboard divertor plates that will be observed in a 15 MA ITER plasma since the simulations do not contain tritium, tungsten, and neon species in high-recycling condition.
This paper is organized as follows: A simple heuristic introduction to homoclinic tangle formation physics is presented in section 2. Electromagnetic gyrokinetic simulation of a full-current ITER plasma and its implication on edge physics are described in section 3 in which the intrinsic edge turbulence generates homoclinic tangle of divertor surface.Section 5 contains conclusion and some discussion on experimental connection.

Homoclinic tangle of divertor separatrix surface
A brief introduction to the homoclinic tangle formation is presented here in this section before the ITER simulation result is presented in the next section.Homoclinic tangles form due to the magnetic flux conservation near the single magnetic X-point, which is a hyperbolic fixed point, from MPs (for double null separatrix, this phenomenon is called 'heteroclinic' tangle).As the poloidal magnetic field-line distance approaches zero toward the X-point per toroidal field-line turn, the radial lobe length must approach infinity to preserve the magnetic flux.The wild lobes form at each side of the X-point (belonging to two different manifolds) and get tangled without touching each other due to the property ∇ • B = 0.
Figure 2 shows several magnetic flux surfaces inside the magnetic separatrix surface on a 'poloidal plane' crosssection at a constant toroidal angle, which confine the charged particles in an ITER plasma equilibrium.Red line is the approximate boundary surface for the material wall and blue line is the stable axisymmetric magnetic separatrix surface containing an X-point that is supposed to provide the last confinement to the plasma.The bottom material surface area where the two X-point legs of the separatrix meet (called 'divertor' surface) is made of tungsten plates in ITER.'Outer leg' denotes the separatrix leg that meets the divertor plates at a greater major radius than the 'inner leg.' Details of the homoclinic tangle formation mechanism is as follows.The poloidal magnetic field strength vanishes toward the magnetic X-point, which is a hyperbolic singular point: A magnetic field line started on a separatrix surface other than the X-point itself can never reach the X-point (or, equivalently, will take infinite number of toroidal turns to reach it). Figure 3(a) presents a coarse-grained cartoon story of the hyperbolic nature of the magnetic X-point.If a non-axisymmetrically perturbed magnetic field-line that started from the X-point in the clockwise direction crosses the separatrix on a poloidal plane at the point k at the kth toroidal turn, then the separatrix-puncture step size on the same poloidal plane in the subsequent toroidal turns become smaller and smaller.Due to the hyperbolic singularity nature of the tokamak X-point, the number of magnetic field points intersecting the poloidal plane between a finite poloidal distance l along the separatrix surface to the X-point position l X is infinite.This fact, together with the magnetic flux conservation, becomes the basis for the thin 'homoclinic lobe' formation of the perturbed separatrix surface.
Continuing with the cartoon description, figure 3(b) shows the behaviour of radial component of a perturbed magnetic field line on a poloidal plane, which started its journey clockwise from the X-point and arrived at the position k on the original stable separatrix (called a homoclinic point).After a complete toroidal rotation, the magnetic field line intersects with the stable separatrix at the next homoclinic point k + 1 on the same poloidal plane.Due to the hyperbolic singularity nature of the X-point, the distance between the subsequent homoclinic points becomes shorter and shorter, approaching zero.Due to the toroidal magnetic flux conservation, the lobes formed by the magnetic field line becomes longer and longer, diverging to infinity as the homoclinic points approach the Xpoint.
The same phenomenon occurs for the magnetic field that travels in the poloidally counterclockwise direction, as shown in figure 3(c).As a result, two groups of thin and wildly varying unstable lobes tangle together stochastically near the X-point, but without crossing each other (in the absence of magnetic reconnection and dissipation), which is allowed by another dimension in the toroidal direction.

Electromagnetic simulation of ITER edge plasma and physics implications
Starting from the deuteron plasma equilibrium for the 15 MA ITER case reported in [5] and [6], an electromagnetic simulation is performed.To be more specific, the starting point of the present electromagnetic simulation is the end point of the previous electrostatic simulation that has a electrostatic-turbulence modified/relaxed plasma edge profile instead of the original MHD-limited 'ITER standard scenario' profile.About two trillion total ion, electron and neutral particles are used on ∼1 mm grid on the full-size Summit leadership class computer at ORNL for about 20 h, with the neutral recycling rate R = 0.99. 100 MW of central heating is applied, mimicking the alpha particle heating.No external torque source is used and conservative nonlinear Fokker-Planck collision operator, and charge exchange and ionization cross-sections are utilized.
We caution here that the artificial central heating power of 100 MW is simply to avoid possibility of fast reduction in the plasma temperature during the present total-f gyrokinetic simulation of baseline physics.It does not have much other meaning.Effect of self-consistent addition of deuteron, triton, alpha particles, and helium ash particles to the present problem could be important but beyond the scope of present work.Also, the central heating at 100 MW does not mean that the power flow from core to pedestal and separatrix is 100 MW.The core-edge power equilibration has not been achieved by the time the edge turbulence and divertor heat-flux width have saturated and measured.The turbulence growth in the core is much slower than in the edge due to their large difference in the driving force.
Figure 4 shows a snapshot Poincare plot of the timefluctuating total magnetic field at saturation around the magnetic X-point, which include both positive and negative directional manifolds.Turbulence correlation time is peaked around 200 kHz near the original stable separatrix.The Poincaré puncture plot starts from radially and poloidally uniformly distributed points within 0.94 < Ψ N < 1.0, where Ψ N is a poloidal-flux radial coordinate normalized to be 0 at the magnetic axis and 1 on the stable separatrix surface (green dashed line).The magnetic field lines-equilibrium and fluctuating fields together-are actually followed using parallel electron motion (v ∥ = ±1) without magnetic or ExB drift motions.Initial launch positions of the field-line following electrons are equally spaced at 32 poloidal angles within the radial domain, but from the same toroidal position.12 800 electrons are launched from each poloidal angle.Electrons are launched to trace the magnetic field lines instead of using the grid values since the thin and long lobe structures near the X-point have subgrid width.In a particle-in-cell code, the sub-grid physics resolution can be provided by individual particle dynamics.
In figure 4, the long and thin lobes are seen at both high and low field vicinity of the X-point across the stable separatrix surface, signifying the action of two unstable manifolds that will lead to mixing of plasmas between pedestal and the divertor regions.A close examination of the turbulence data shows that the turbulence modes are strongly peaked at the toroidal mode number n = 6.This mode is found to have a microtearing mode structure in the very early stage, as can be seen from figure 5, before the nonlinear structure mixing or the development of the significant homoclinic tangle structure.
In figure 6, we show the dissipated nature of the timedynamical homoclinic tangle turbulence.The same radial and poloidal electron initialization has been used as in figure 4. In the left-hand-side figure, figure 6(a), electrons with only v ∥ = +1 are initiated without ExB or magnetic drift, corresponding to the counterclockwise rotating homoclinic-tangle manifold (see figure 3(c)).Poloidally counter traveling electrons experience dissipation by the time-dynamical turbulence actions, some of them transit into the other manifold near the X-point, and they end up in both inner and outer divertor chambers.On the other hand, in the right-hand-side figure where a timesnapshot magnetic field is used, we see that the v ∥ = +1 electrons follow only the counterclockwise directional manifold and lost to the inner divertor chamber only, keeping its independence from the other manifold rotating in the clockwise direction.Thus, in a turbulent homoclinic tangle, the strict independence of the magnetic field manifolds is broken by the turbulent dissipation.We have observed that the independence is further broken in the electron dynamics by Coulomb collisions and E×B and magnetic drift actions.This observation implies that the turbulent homoclinic tangle can have some equipartition effect on the electron heat-load between outer and inner divertors, which will become more distinct in a future tokamak reactor with a higher-β (=plasma energy/magnetic energy) edge plasma.
From figure 4, again, we already visualized that the microturbulence-driven separatrix lobes protrude deep into the divertor plasma, hence giving the possibility for widening the target heat-load footprint.In figure 7(a), we present the parallel heat-load footprint on the outer divertor plates, with the x-axis mapped to the outboard midplane distance from the stable magnetic separatrix.Orange line is the new electromagnetic XGC-data and the blue line is its fit to the Eich's shape formula, yielding λ q XGC-EM ≈ 8 mm as shown with top right mark in figure 1. Considering that the electrostatic value reported in [5,6] is λ q XGC-ES ≈ 6 mm, as shown in figure 7(b), we have about 33% enhancement of the divertor heat-load width by the electromagnetic effect.This is a significant enhancement over the data-regression prediction given in [1].In fact, Eich's multi-machine scaling formula #14 predicted λ q Eich(14) 0.5 mm for the 15 MA ITER plasma equilibrium.
The peak parallel heat-flux near the outer divertor leg (midplane distance at ∼0 mm) in figure 7(a) corresponds to 22 MW m −2 peak power density on the divertor surface, with the total power deposition of 68 MW over the outer divertor plates.This level of peak power density is not compatible with the tungsten material limit, hence the ITER baseline scenarios must assume usage of seeded impurities such as neon to dissipate the exhaust power in SOL and divertor chamber [4,19,20].Figure 7 also shows an extra-sharp peak in the divertor footprint near the separatrix leg, which is indicative of the extra electron heat-load density due to the more active homoclinic tangle activity there.Thus, in a future highertokamak edge plasma, the peak exhaust power density may be higher than that in today's lower-β edge.In passing, we mention here that the total power deposition on the inner divertor plates is measured to be 53 MW, which is higher than the conventional 50% estimate in comparison with the outer divertor power deposition in a conventional aspect-ratio, pure plasma tokamak.
To study the question of how far the homoclinic-tangle lobes intrude into pedestal (and thus impact the pedestal transport and slope), we first use the snapshot magnetic field (B = B 0 + δB) of figure 4 and launch, in a post processing step, a statistically meaningful distribution of 1 keV electrons (a typical electron energy just inside the magnetic separatrix in the 15 MA ITER pedestal) with pitch angle v || /v = 1 without the ExB or magnetic drifts, thus mapping the magnetic field lines.A total of 4.096 × 10 5 electrons are launched evenly across the entire poloidal angle and in the radial range 0.9 <Ψ N <1.0.These electrons are then time-advanced for 1 ms in physics time in the static snapshot magnetic field of figure 4, and the results are depicted in figure 8. Vertical axis is the fraction of particles that show up outside the magnetic separatrix surface (orange colour, mostly covered by green colour that represents the electrons ended up on divertor plates) at any time within 1 ms.
Figure 8 suggests that the homoclinic tangle lobes intrude at least into ψ N ≈ 0.96 in 1 ms.We thus find that the divertor electrons and the pedestal electrons deep into ψ N ≈ 0.96 can be connected by the snapshot homoclinic tangle lobes in less than 1 ms.The green region in figure 8 is the fraction of the electrons which end up on the inner divertor plates in 1 ms, meaning that most of the pedestal electrons which start from 0.96 <ψ N <1.0 and move with v || /v = 1 in the counterclockwise direction along the snapshot homoclinic tangle lobes are lost to the inner divertor plates in 1 ms.We find a similar result by using the electrons flowing in the opposite direction v || /v = −1, thus, lost to the outer divertor plates.We note here that when we use trapped electrons with ExB and magnetic drifts, some fraction of the electrons are also lost to the other divertor plates even in the snapshot homoclinic tangles.
To find a more realistic level of electron heat diffusivity, we repeat the electron dynamics analysis of figure 8, but in the space-time fluctuating electromagnetic fields that include the turbulent homoclinic tangles and the actual E×B and magnetic  drift motions of electrons (but without Coulomb collisions in the postprocessing analysis).As can be seen from figure 9, the electron transport is now not limited to the homoclinic tangle region ψ N ≳ 0.96, but also extends deep into the pedestal top due to synergy between the turbulent homoclinic-tangle transport at ψ N ≳ 0.96 and the usual electromagnetic turbulent transport.The average effective electron diffusion coefficient in the pedestal is estimated to be χ e ≈ 2 m 2 s −1 when the electrons at all pitch-angles are considered.χ e is greater towards the pedestal foot due to the nonlocal homoclinic tangle transport.We note here that the homoclinic tangle turbulence is from resonance with electron dynamics and does not affect the ion particle transport much.Since the electron particle transport is held back to the ion level due to the ambipolarity requirement, the density pumping seen in a homoclinic-tangle Figure 9.The same case as in figure 3, but with the electrons moving in the time-dependent electromagnetic turbulence field from XGC and with E×B and magnetic drifts.We can see that the usual electromagnetic turbulence working together with the fluctuating homoclinic tangles, enhances the electron transport in the entire pedestal region.
driven by external MP is therefore not observed in the selfconsistent XGC gyrokinetic simulation.

Conclusion and discussion
In this work, first numerical observation from a first-principles based total-f edge gyrokinetic code of the existence of intrinsic homoclinic tangle of divertor separatrix surface caused by electromagnetic turbulence is reported, in the absence of the external MPs or edge MHD activities.Unlike the externally driven or edge-MHD driven homoclinic tangles that could cause deleterious effects to core confinement or divertor plates, it is found that the micro-turbulence driven homoclinic tangles could connect the divertor plasma to the pedestal plasma in a constructive way by broadening the divertor heat-exhaust footprint and increasing the pedestal electron heat-transport, thus widening the electron-temperature pedestal slope to the MHD-safe direction.
The present observation could offer answer to another question.In many tokamak experiments anomalously large electron heat transport has often been observed at the pedestal foot just inside the stable separate surface where the mild electrontemperature gradient could not drive the required level of driftwave turbulence [21].The homoclinic tangle effect may be able to explain this anomaly.Quantitative analysis of specific examples is needed for a more definitive resolution of the question and is left for a future study.
ITER baseline scenarios assume usage of seeded impurities such as neon to dissipate the exhaust power in SOL and divertor chamber.Thus, the actual peak power density on the divertor plates in ITER will be much lower than what is described in this paper.We refer the readers to some [4,19,20] that contain more comprehensive list of publications on this topic.
Experimental validation of the separatrix homoclinic tangle could include visible camera imaging around the divertor Xpoint at ≳1 kHz time resolution and spectral analysis of turbulence in front of the divertor plates in the absence of external MPs and ELM activities, or between ELMs.
A possible experimental suggestion can be made to actively control and improve the integrated pedestal/heat-exhaust problem, which could benefit DEMO/FPP performance.The proposal is to enhance the micro-turbulence driven homoclinic tangle amplitude via the electron turbulence resonance with a low frequency, tuneable, evanescent-wave antenna (f = 100 kHz-200 kHz, n ≲ 10).If the main chamber wall is made of low-resistivity tungsten in DEMO/FPP, the tuneable antenna structure can be built into the wall design.Since only a low level non-axisymmetric MP is needed (δB/B ∼ 10 −3 ) on the midplane separatrix surface and the needed antenna current can be distributed toroidally over many antenna modules, the needed antenna current or wall heating may not be significant.We mention here that an enhanced homoclinic tangle transport could lower the impurity radiation requirement in solving the pedestal relaxation and the heat-exhaust broadening issue.A more careful quantitative study is needed.
Extension of the present study to advanced divertor geometries where secondary X-points are employed, hence secondary homoclinic tangles could be formed for further spread of the heat-load, could also be of interest.
We add a note here that there has been a continuous search for experimental evidence in the low-edge of present tokamaks for broadening of the divertor heat-load footprint by electron-directional kinetic turbulence, discovered in [5] and [6].Recently, such a phenomenon has been observed on DIII-D when the wide-pedestal QH mode plasmas transit to the broad-band turbulence QH mode plasmas with significantly enhanced electron directional turbulence [22].The divertor heat-load width broadens by almost factor of two.XGC simulation agrees and shows that the heat flux carried by electrons emerges to broaden the heat flux profile.

Figure 2 .
Figure 2. A poloidal cross-sectional view of magnetic flux surfaces in ITER at a constant toroidal angle.The red line shows the material wall boundary and the blue line is the last confinement surface formed by a supposedly stable magnetic separatrix surface that forms one X-point.The material wall areas that meet two legs of the separatrix surface just below the X-point are divertor plates.

Figure 3 .
Figure 3.A coarse-grained cartoon story of homoclinic tangle formation: (a) if a non-axisymmetrically perturbed magnetic field-line that started from the X-point in the clockwise direction crosses the separatrix on a poloidal plane at the point k at the kth toroidal turn, then the separatrix-puncture step size in the subsequent toroidal turns become smaller and smaller.Due to the hyperbolic singularity nature of the tokamak X-point, the number of magnetic field points intersecting the poloidal plane between a finite poloidal distance l along the separatrix surface to the X-point position l X is infinite.(b) This fact, together with the magnetic flux conservation, becomes the basis for the radially stretched 'homoclinic lobe' formation of the perturbed separatrix surface.(c) When we add a magnetic field-line rotating in the counterclockwise direction to figure (b), we have tangled homoclinic lobes.

Figure 4 .
Figure 4. XGC prediction of the turbulent homoclinic tangles in the 15 MA ITER edge in the stationary operation phase without ELM activity or magnetic perturbation (MP) coils.Shown here is a snapshot Poincare puncture plot of turbulent magnetic field lines that started poloidally uniformly from the pedestal region (0.94 <Ψ <1.0).

Figure 6 .
Figure 6.(a) Field-line following electrons launched in the counterclockwise direction in the time-dynamic turbulent field jump between two manifold lobes by the time-dynamics turbulence action.(b) On the other hand, the field-line following electrons launched in the counterclockwise direction in a snapshot turbulence field keep the independence from the other manifold and all end up at the high-field side lobes.Magnetic field lines are traced using 1 keV electrons that started in the purely counterclockwise direction without experiencing E × B or magnetic drift motions.

Figure 7 .
Figure 7. Parallel heat-flux footprints on outer divertor plates, mapped to outboard midplane, from (a) the electromagnetic XGC simulation and (b) from the previous electrostatic XGC simulation [6] of 15 MA ITER.The electromagnetic result shows about 33% enhancement of the heat-flux width over the electrostatic result.Reproduced from [6].© 2017 IAEA, Vienna.All rights reserved.

Figure 8 .
Figure 8. Fraction of the v || /v = 1 electrons, initiated evenly over all poloidal angle and over 0.9 <Ψ N <1.0, that travel to outside ITER's stable separatrix surface manifold (orange) and eventually to inner divertor plates (green) in a snapshot homoclinic tangle structure in 1 ms of physics time.Electrons in this test are collisionless and without ExB or magnetic drifts.Electrons with v || /v = −1 are lost to the outer divertor plates (not shown here).