Gyrokinetic simulation study of magnetic island effects on neoclassical physics and micro-instabilities in a realistic KSTAR plasma

Tokamak devices aim to confine high temperature plasmas using axisymmetric magnetic fields. However, there always exist non-axisymmetric magnetic field perturbations breaking the toroidal symmetry of the confining fields. The origins of the 3-dimensional (3D) perturbations are various: error fields caused by miss-aligned magnets, intentional perturbations by external coils, large scale MHD activities, etc. Depending on the experimental conditions, they cause a wide range of plasma responses with different topological configurations of the perturbed fields. Among them, 3D magnetic islands are the most commonly observed form, and the physics of magnetic islands and their impacts on plasma confinement and stability are one of the heavily studied subjects over the past decades.

The presence of a macroscopic magnetic island is usually detrimental to the confinement and performance of toroidal plasma. However, depending on the perturbation strength and plasma response, the magnetic island can also contribute to the enhancement of thermal confinement. Notable such examples are experimental cases showing the beneficial roles of magnetic island on the formation of internal transport barrier (ITB).^1–6 To optimize the performance of a tokamak plasma, it is important to understand the physics underlying these multifaceted roles of magnetic island. Among them, the physics of macroscopic flow associated with the magnetic island is crucial for the understanding.

There have been many progresses in measuring the detailed structures of flow and fluctuation near the magnetic island.^7,8 Recently, it was found that the magnetic island can drive macroscopic flow, which can impact on the confinement of ambient plasma.^14,15 Also, the interactive evolution of the magnetic island and turbulence were measured using BES on DIII-D.^12,13 These results are helpful for related numerical and theoretical studies. A recent KSTAR experiment is also expected to provide very useful information.⁹ In the KSTAR experiment, a resonant magnetic-field perturbation (RMP) was used to induce a stationary magnetic island in a neutral beam injection (NBI)-heated L-mode plasma. The magnetic island had $(m,n)=(2,1)$ mode structure (m and n denote the poloidal and toroidal mode number, respectively). By employing 2D electron cyclotron emission imaging (ECEI) diagnostics,^10,11 the RMP-induced magnetic island was measured with its ambient flow and fluctuation in a wide 2D poloidal window. Since the fluctuation and flow are closely linked to the transport changes around the island, those results can provide essential information to further the theoretical understanding of related physics, especially the confinement improvement by the magnetic island.

The detailed results can be found in Ref. 9. Here, we summarize some key aspects of the experiment to be addressed in this work. By applying RMP, a (2, 1) magnetic island was induced at the rational flux surface (r_s) with safety factor $q(rs)=2$⁠. It was observed that the electron thermal confinement was enhanced in an inner radial region neighboring the magnetic island. Due to this confinement enhancement, the core electron temperature showed little changes, even though there were large drops of the edge electron temperature and the overall degradation of ion thermal confinement. ECEI was used to measure the distributions of electron temperature fluctuation and its apparent poloidal flow around the magnetic island. It was found that the 2D patterns of the fluctuation and flow were inhomogeneous. The fluctuation level was the maximum near the magnetic X-point (but not exactly at the X-point) and it decreased as the observation point was moved toward the magnetic O-point. The poloidal flow showed an opposite trend, i.e., the apparent flow was small near the X-point, and the flow level increased as the observation channel moved toward the O-point. Also, very interestingly, the apparent flow direction was reversed across the magnetic island. In the inner and outer region neighboring the island, the flow was in the electron and ion diamagnetic direction, respectively. The flow inside the island could not be measured, because the fluctuation level was too low in the region.

In Ref. 9, it was interpreted that the shearing of the poloidal flow is responsible for the suppression of the fluctuation and enabling the enhancement of the core electron thermal confinement. However, the detailed physics of the flow drive and the nature of the fluctuations and their suppression mechanism were unanswered and left for future study.

In this paper, we perform numerical simulations to understand the physical mechanisms of flow drive and fluctuation suppression by the magnetic island. There have been numerical studies on related subjects: micro-instabilities in the presence of magnetic island,^16,18,19 effects of magnetic island on micro-turbulence and flow,^20–25 profile changes by magnetic island,¹⁷ etc. Though there were many interesting findings, we note that these studies were mainly focused on local physics based on a local flux tube or simplified geometries.

We want to emphasize that the magnetic island and poloidal flows observed in the KSTAR experiment are macroscopic ones, which require a global approach in toroidal geometry. For kinetic simulation, the simple adiabatic electron model cannot be employed, since the magnetic fields are no more in simple concentric forms and an accurate description of electron motion under perturbed magnetic field is critical. Since the dynamics involving macroscopic magnetic island and microscopic fluctuations are essentially multi-scale phenomena, the necessary computing cost easily exceeds manageable levels depending on the sophistications of numerical models. One soothing aspect of the KSTAR experiment is that the magnetic island was externally induced one. It is non-rotating and stationary at the fixed q = 2 rational surface. So the island can be simply modeled as a numerically prescribed one on the axisymmetric equilibrium, and the target physics can be studied on that background.

In this work, we choose a global approach in two steps. In the first step, using the KSTAR experimental condition, we perform a global neoclassical simulation using a gyrokinetic particle in cell (PIC) code XGC1.^28–30 With the elements for axisymmetric neoclassical physics, we allow the toroidal variations of the particle distribution functions and the self-consistent electrostatic field to accommodate 3D perturbations by magnetic island. From the simulation, we establish a global kinetic equilibrium with a static (2, 1) magnetic island. The most important physical quantity is the perturbed electrostatic potential which is self-consistent with the modified 3D equilibrium profiles.

In the next step, using the modified profiles from XGC1, we perform a set of global linear micro-stability analyses using a gyrokinetic δf PIC code gKPSP.^26,27 Through the δf simulations, we analyze the properties of the micro-instabilities ambient to the magnetic island and compare the growth rates of the instabilities with the E × B shearing rates obtained from the XGC1 simulation. From the comparison, we aim to understand the impacts of the flow on the instabilities and their implication on the transport around the magnetic island.

The remainder of this paper is organized as follows: In Sec. II, we explain the detailed simulation setups. The key features of the target KSTAR experiment are also summarized here. In Sec. III, the simulation results are presented with discussions on how they are compared with the experimental observations. Finally, conclusion and summary are given in Sec. IV.

The target experiment we analyze in this work is a NBI-heated L-mode discharge on a KSTAR tokamak. In the experiment, n = 1 resonant magnetic-field perturbation (RMP) was applied at t = 7.0 s to induce a (2, 1) magnetic island on the q = 2 surface, which was non-rotating and stationary. As the island was formed, the plasma rotation at the q = 2 surface dropped to zero level and the plasma was locked. Also, there was an overall degradation of the ion thermal confinement with a significant reduction of the edge electron temperature. However, the core electron temperature (T_e) showed little changes and an internal transport barrier (ITB) like profile of T_e was observed in the immediate inner radial region neighboring the magnetic island. It was conjectured that the poloidal flow linked to the magnetic island plays important roles in the formation of the quasi-ITB. The poloidal flow and fluctuation structure around the magnetic island were measured using 2D ECEI diagnostics. Further details of the experimental setup and ECEI measurement results can be found in Ref. 9.

We employ the global PIC code XGC1 to simulate the island induced changes of the plasma profiles and poloidal flow. The code is based on the total-f simulation algorithm,²⁹ which allows us to self-consistently treat kinetic physics and full profile dynamics without any scale separation assumption. Also, the algorithm can reduce discretize particle noises considerably as compared with the direct full-f method. This enables us to perform high quality simulations within reasonable computing resources. In the code, gyro-kinetic ions and drift-kinetic electrons are evolved with Coulomb collisions and self-consistent electrostatic potential, which is obtained by solving the gyrokinetic Poisson equation. To accommodate the non-axisymmetric perturbations by the (2, 1) magnetic island, the particle distribution functions and the electrostatic potential are allowed to vary toroidally with n = 1 Fourier harmonics in addition to the axisymmetric components.

To model the stationary (2, 1) magnetic island induced by RMP, we employ a simple magnetic perturbation model based on the following analytic form:

Here, θ and $φ$ represent the poloidal and toroidal angle, respectively. It induces a stationary (2, 1) magnetic island at the q = 2 rational surface. The width of the magnetic island is approximately given by $w=8qA0R0/B0ŝ$⁠, where R₀, B₀, and $ŝ$ correspond to the major radius, magnetic field at the center, and the magnetic shear at the q = 2 rational surface, respectively. We choose the amplitude A₀ to match the width of the magnetic island obtained from simulation with the experimentally measured one. By choosing $A0=3.5×10−4R0B0$⁠, we find that the widths are reasonably matched.

XGC1 uses unstructured triangular meshes to deal with arbitrary tokamak plasma equilibrium. The mesh sizes can vary in different regions according to simulation needs. In this work, we set fine meshes in an annular region containing the magnetic island as $Δr∼0.25ρi$⁠. Other regions are discretized with bigger meshes with $Δr∼2ρi$⁠. The total number of meshes covering a poloidal plain of the simulation domain is $N∼172 000$⁠. To represent n = 1 toroidal variations, 32 poloidal plains are placed along the toroidal direction uniformly. Marker particles are loaded to satisfy the number of particles per mesh as $Ni,e∼8000$ for both ions and electrons. The time step size for the simulation is set as $Δt≈4×10−8$ s.

Regarding the equilibrium geometry for the grid generation, we use the EFIT equilibrium reconstruction data at t = 6.858 s, which is immediate before the application of RMP in the experiment. Initial profiles for the simulation are also taken from the same time slice (t = 6.858 s) except the ion rotation profile. The rotation becomes almost zero as the plasma is locked by the RMP. In principle, the rotation breaking by the magnetic perturbation can be simulated by XGC1, though it requires very long simulation time. Since the toroidal rotation of the target plasma state is almost zero, its contribution to the poloidal rotation should be negligible. Considering this, instead of evolving the rotation, we simply set the toroidal rotation as zero from the start of the simulation.

After obtaining the 3D kinetic plasma equilibrium modified by the (2, 1) magnetic island, we perform a set of global linear gyrokinetic simulations using the equilibrium profiles to understand the effects of the magnetic island on micro-instabilities. In this aim, we employ a global δf gyrokinetic PIC code gKPSP. The code is based on the conventional δf algorithm,^32,33 and employs gyrokinetic ions and bounce-averaged drift-kinetic electrons. The bounce-averaged electrons allow very efficient simulation for trapped electron physics.^26,27,31 The code can be used for both linear and nonlinear simulation with experimental profiles and equilibrium geometry given in the GEQDSK format. In this work, we only perform global linear simulations to investigate the electrostatic ion temperature gradient (ITG)-TEM instabilities of the profiles modified by the magnetic island.

The same experimental GEQDSK file, which was used to generate the XGC1 grid, is employed for the linear analyses. The XGC1 simulation provides the initial equilibrium profiles for the analyses. The XGC1 simulation generates plasma states with poloidal and toroidal variations. The corresponding plasma profiles should be averaged over flux surfaces to be used in the δf simulations. Also, the equilibrium electric field obtained from the XGC1 simulation is not included in the δf simulations. The gKPSP simulation results presented in Sec. III were obtained with the number of marker particles for ions and electrons as $Ni∼107$ and $Ne∼106$⁠, respectively. The radial grid size is set to satisfy $Δr=0.2ρi$ and the time step size is chosen as $Δt=0.02R0/vti$⁠, where v_ti represents the ion thermal velocity. The simulations include Coulomb collisions.

As the XGC1 simulation starts with the magnetic field perturbation in Eq. (1), the electron temperature profile responds rapidly and strongly. The electron thermal transport is enhanced by the perturbed magnetic field. Figure 1 shows the temporal evolution of flux surface averaged T_e at $R≈2.11 m$ and $R≈2.15 m$ at the outer mid-plain, which roughly correspond to the inner and outer minor radial boundary of the (2, 1) magnetic island, respectively. After the very fast responses in the electron transit time scale, the T_e profile subjects to slower evolutions caused by changes of other profiles and Coulomb collisions. Other profiles such as electron density and ion temperature also respond to the magnetic island, though their responses are slower and weaker compared to the T_e response. The plasma establishes a quasi-steady state eventually, though the slower changes of the profiles continue throughout the simulation time.

As the profile evolutions indicate, the T_e-changes closely follow the magnetic island. We can identify the locations of X and O-points of the magnetic island by examining T_e profiles at different toroidal or poloidal directions. Figure 2 shows the T_e profiles across the X and O-point of the magnetic island. Other profiles across the points are also shown in Fig. 3. The T_e profile across the O-point has a flat region with a width $w≈3.4 cm$⁠. The difference between this profile and the T_e profile across the X-point is evident. Along with these changes, the inner radial region neighboring the island develops strong radial gradients of T_e, which resembles a quasi-ITB for T_e. The electron density profiles across the O and X-point also show a difference, though it is less pronounced than the difference of the electron temperature profiles (see Fig. 3). There is no noticeable difference between the ion temperature profiles.

It is interesting to compare these profile changes from the simulation with the experimental trends in Ref. 9. In the experiment, T_e changes measured by 2D ECEI indicate $w≈4 cm$ across the magnetic O-point. T_e profiles measured by ECE also show a flat region with $w≈2 cm$ in the high field side. In the experiment, the T_e level at the magnetic island location was $Te≈0.5 keV$⁠, which is much lower than the value from the XGC1 simulation $Te≈1.0 keV$⁠. In the experiment, in addition to the transport by the magnetic field perturbation and collisions, various other physical mechanisms contribute to the electron thermal transport. So it is natural to expect lower T_e values in the experiment. T_i and n_e profile were not available after the RMP application in the KSTAR experiment. However, the line averaged value of the electron density decreased about 10% after the island formation.

The magnetic island induces a non-axisymmetric perturbation on the electrostatic potential $Φ$⁠, and it contributes to the global force balance in addition to the axisymmetric neoclassical component. The total potential has a (2, 1) mode structure, which can be seen in the contour plots in Fig. 4. The potential has an odd profile across the center of the magnetic island as expected for a perturbation with tearing parity. The potential drives non-axisymmetric E × B flow. Since the perturbed potential has the (2, 1) mode structure, there should be O and X-points associated with the island of the potential. It is expected that the variation of the potential amplitude is the largest across the O-point and the smallest across the X-point. Since E × B flow is proportional to the E_r, it is natural to expect that maximum and minimum E × B flow should appear across the O and X-point of the potential island, respectively.

Figure 5 shows the E × B flow profiles along the radial directions crossing the O-point (the green curve) and the X-point (the red curve) of the potential. In this figure, E_r is calculated along the radial direction by fixing the poloidal and toroidal angle. The width of the magnetic island is denoted as the ellipse. The profiles clearly show the impacts of the magnetic island. As expected, the flow across the O-point is larger than the flow across the X-point. The flow direction changes across the magnetic island. In the inner region (⁠$R≤2.09 m$⁠), the flow has positive values, which corresponds to the electron diamagnetic direction. Near the boundaries of the island, the flow is in the ion diamagnetic direction (i.e., negative values). Figure 6 shows the E × B shearing rates across the O-point (the green curve) and the X-point (the red curve) of the potential. Evidently, the maximum and minimum shearing rates appear across the O and X-point of the potential, respectively.

It is very interesting to compare the locations of this maximum and minimum E × B flow with the O and X-points of the magnetic island. Comparing those locations, we find that there exists a finite phase difference. The difference between the poloidal angle of the potential O-point (⁠$θΦO$⁠) and the magnetic O-point (⁠$θMO$⁠) is found to be $Δθ≡θΦO−θMO≈0.1π$⁠, and the magnetic X-point and the potential X-point are separated by a similar angle. Due to various physical mechanisms (e.g., finite compressibility, diamagnetic effects, collisionality, etc.), there can be systematic phase differences between different physical quantities. Though the global perturbations of the density and temperature are modulated by the magnetic island, their respective (2, 1) structures are not necessarily the same with the mode structure of the magnetic island.

From these observations on the XGC1 simulation, we can put an interesting conjecture: if the E × B flow shearing is sufficient enough to suppress ambient micro-instabilities, the region where the most efficient suppression should appear near the magnetic O-point but with a finite phase shift (and similarly the region with the least efficient suppression should be around the magnetic X-point but with a finite shift). In Sec. III B, we discuss this further with micro-stability analysis results.

The next step of the study is to investigate whether the E × B flow can affect the micro-stabilities of the profiles near the magnetic island. In this subsection, we present linear gyrokinetic simulation results and discus their implications on fluctuations around the island.

Figure 7 shows the gradients of the flux surface averaged profiles for the δf simulations. In this figure, the density and temperature values are normalized by $ne0=3.2×1019 m−3$ and $T0=1.14 keV$⁠. The x-axis represents $x=ψ/ψx$⁠, where ψ_X denotes the poloidal flux at the last closed flux surface. The center of the magnetic island is located at $x≈0.84$⁠. It is immediately noticeable that the T_e-gradient is significantly steepened in the inner region neighboring the magnetic island and drops inside the island. From the profiles, it is expected that trapped electron mode (TEM) can be excited in the inner radial region by the steepened T_e-gradients. In the outer region with (x > 0.86), the Ti-gradient becomes stronger with increasing electron-ion collisionality. So, as the minor radius increase, TEM can be stabilized while ion temperature gradient (ITG) instabilities can be destabilized.

Figure 8 shows the linear structures of micro-instabilities with different toroidal mode numbers on a poloidal plain. In the lowest mode number case (⁠$n=20,kθρi≈0.12$⁠), the mode center is located in the steep T_e-gradient region (⁠$x∼0.8$⁠) and the fluctuation propagates toward the electron diamagnetic direction, which confirms that the instability is $∇Te$-driven TEM. The radial extents of the TEM instabilities are limited by the magnetic island because of the flattened T_e inside the island. TEM is the dominant instability with increasing frequency and growth rate as the mode number increases. In the highest toroidal mode number case (⁠$n=140,kθρi≈0.8$⁠), the TEM instability becomes weaker due to the collisional stabilization of TEMs, which is more effective for higher mode numbers. Along with this trend, it is found that ITG instabilities are excited in the outer region neighboring the magnetic island. In this case (n = 140), TEM and ITG instability co-exist with different mode centers separated by the magnetic island.

The frequency and growth rate of micro-instabilities are plotted in Fig. 9. In the figure, the x-axis represents the poloidal wave number of the mode normalized by the ion gyro-radius. In the upper figure, the mode frequencies are plotted. The plus and minus values indicate modes propagating in the electron and ion diamagnetic direction, which corresponds to TEM and ITG, respectively. The mode growth rates are plotted in the lower figure. Here, the maximum and minimum E × B shearing rate in Fig. 6 are indicated as the horizontal lines with $|ωE×BO|∼3.6×105$ and $|ωE×BX|∼2.5×105$⁠, respectively.

We can see that the maximum shearing rate across the O-point of the island is comparable with the highest growth rate, which implies a significant portion of the instabilities can be suppressed near the O-point of the potential island. On the other hand, the lower shearing rate of the potential X-point is insufficient to suppress the instabilities in a wide range $0.2≤kθρi≤0.8$⁠. The intrinsic phase velocity of the instabilities can be estimated from the dispersion curves. The estimated phase velocity of TEM and ITG are $vTEMph∼+1.0 km/s$ and $vITGph∼−0.3 km/s$⁠, respectively.

In this subsection, we discuss the implications of the simulation results and how they are compared with the ECEI observations on the KSTAR experiment.⁹ 

Based on the simulation results presented in Secs. III A and III B, we can anticipate the following trends of fluctuation and transport in the experiment. Once the (2, 1) magnetic island is formed, the electron temperature profile is significantly modified by direct electron heat transports following the perturbed magnetic fields, and the T_e-gradient in the inner region neighboring the island will be significantly steepened. The increased T_e-gradient would drive TEM instabilities, which may enhance electron heat transports in the inner region and reduce the electron temperatures. However, the E × B shearing driven by the magnetic island can suppress the TEM instabilities and therefore maintain the steepened T_e-gradient, which can be viewed as a quasi-ITB for electron thermal transport. Indeed, in the KSTAR experiment, such a transport barrier like region was observed in the inner region, and though the electron temperatures in the outer region were reduced by the island formation, the central electron temperature was sustained due to the barrier.

As seen in Fig. 6, the shearing rate is inhomogeneous and the fluctuation suppression by the shearing should follow a similar inhomogeneous pattern. There were also such evidences in the KSTAR experiment. The ECEI measurement showed that the fluctuation level is appreciable near the magnetic X-point but it almost vanishes as the ECEI observation window moves toward the magnetic O-point. The measured poloidal flow showed the opposite trend, i.e., the flow was strong near the O-point and weak near the X-point.

One very interesting observation in the experiment was the shift of the region with maximum measured fluctuation from the magnetic X-point. The phase difference between the magnetic island and the potential island can give an answer to this observation, i.e., the potential X-point, where the flow shearing is minimum, should be located in a poloidally shifted angle as in the simulation (⁠$Δθ≈0.1π$⁠).

Another interesting point to discuss is the pattern of the apparent poloidal rotation of fluctuation measured by ECEI. The measured poloidal velocity $vθ exp$ is the summation of ambient E × B velocity and the intrinsic phase velocity of fluctuation. The estimated range of the summed velocity from the simulation is $−3.8 km/s≤vθsim≤+3.4 km/s$ in the inner region (⁠$R<2.10 m$⁠) and $−4.7 km/s≤vθsim≤+1.5 km/s$ in the outer region (⁠$R>2.15 m$⁠) from the island. There are uncertainties and limitations in calculating the summed velocity in both the experiment and simulation. So it is difficult to compare them directly. However, we would like to note that the simulation result implies the enhanced locality of fluctuations, i.e., TEM and ITG in the inner and outer region are separated by the magnetic island. The different fluctuation types may explain the experimentally observed trend of the apparent poloidal flow reversal across the magnetic island.

In this paper, we performed gyrokinetic simulations to study the effects of a stationary magnetic island on macroscopic poloidal flow and micro-fluctuation. From the simulation results, we found that the E × B flow enhanced by the magnetic island can suppress ambient micro-instabilities and contribute to the enhancement of the electron thermal confinement in core plasma. The details of the flow were analyzed with their implications on the spatial structures of the fluctuations around the magnetic island. Through this, we could explain the key observations from the KSTAR experiment,⁹ which were mentioned in the introduction. Here, we rephrase them with the underlying physical mechanisms found from the numerical simulations in this paper.

• Q1. What is the physical mechanism of the T_e confinement enhancement near the magnetic island?

• A1. The electron thermal transport along the perturbed magnetic fields leads to flattening of the T_e profile inside the magnetic island and steepening of T_e in the inner region neighboring the island. Due to the island, the equilibrium E × B flow is enhanced and the resulting E × B shearing is strong enough to suppress a significant portion of micro-instabilities in the steepened-$∇Te$ region. So the quasi-ITB for T_e can be sustained in the inner region neighboring the island.

• Q2. Why are the measured fluctuation and poloidal flow pattern poloidally inhomogeneous?

• A2. Since the electrostatic potential driving the E × B flow has a (2, 1) mode structure, the associated flow shearing should follow the same spatial structure. The flow shearing is the strongest and the weakest near the O and X-point of the potential, and the fluctuation level becomes the largest and the lowest near the O and X-point, respectively. However, due to the phase difference between the potential island and the magnetic island, the regions with the largest and the lowest fluctuation should appear in spatially shifted regions from the magnetic O and X-points, respectively.

• Q3. What is the physical mechanism behind the observed poloidal flow reversal across the magnetic island?

• A3. TEM and ITG instabilities are excited in the inner and outer region neighboring the magnetic island. The directions of their intrinsic poloidal velocities are opposite, which are important components in determining the poloidal flows. Though the direct comparison between the simulation and the experiment is difficult, the estimated values are in similar ranges.

Generally, this mechanism of confinement enhancement by the magnetic island can also contribute to the formation of ITB by external heating. The threshold power for ITB formation would be set by the amount of external heating to increase the core plasma temperature and resulting equilibrium flow shearing to a level which is strong enough to suppress a significant portion of ambient micro-turbulences, and thereby to start the positive feedback loop for better confinement and stronger equilibrium flow. As found in this work, the island driven flow can be strong enough to suppress the ambient micro-instabilities, more specifically TEM. However, other micro-instabilities such as ITG can be also suppressed by the flow shear. So, our findings imply that the magnetic island and its associated flow can lower the threshold powers to obtain an electron or ion thermal ITB by strengthening the required E × B flow shearing in the inner radial region neighboring the magnetic island. We note that such a trend of the threshold power reduction in the presence of the magnetic island was found in previous experiments.⁴ 

As explained in the introduction, a resonant RMP was used to induce the (2,1) magnetic island in the KSTAR experiment. Since the island can contribute to the core confinement enhancement and also to the formation of ITB, it is interesting to compare this with the effects of resonant RMP on the edge transport barrier (ETB) and H-mode transition. Recently, there were dedicated experiments in KSTAR to investigate the effects of RMP on the H-mode power threshold.³⁴ Very interestingly, it was found that resonant RMP can increase the threshold power significantly. This trend seems to be contradictory to our finding of the beneficial role of resonant RMP on core confinement enhancement. However, it should be noted that the plasma response to resonant RMP can be completely different depending on the resonant location. In the plasma core, a resonant RMP can induce an isolated magnetic island, which can change the plasma flow and improve the confinement as we discussed. In the plasma edge, on the other hand, a resonant RMP can induce multiple magnetic islands because rational surfaces are more densely distributed. So, a resonant RMP is more likely to induce stochastic magnetic field layers in the edge, which can alter the plasma flow and other profiles in completely different ways as compared to those in the core. We would like to emphasize that the roles of resonant RMP on ITB and ETB formation can be completely different.

Before closing this paper, we would like to discuss potential future studies. In this work, we studied the effects of a stationary magnetic island on flow and fluctuation. Simulations with other forms of magnetic perturbations are also needed. For examples, similar studies with stochastic magnetic fields will give important clues for understanding RMP related physics such as H-mode power threshold dependence on resonant RMP.³⁴ To study the dynamic evolutions of fluctuation and flow around the magnetic island, nonlinear simulations are necessary including both micro-turbulence and global neoclassical physics. Such simulations will be helpful to understand the spreading of fluctuations near and across magnetic island, which was observed in recent DIII-D experiments.^12,13

On the experimental side, it will be very interesting to measure and compare the mode structures of different physical quantities directly. For example, separate measurement of electron temperature and density can provide a more direct evidence for the existence of the phase difference between the magnetic and potential island, since the potential is expected to be more closely aligned with the electron density. Ultimately, these studies will contribute to the development of a comprehensive model to understand multi-scale interactions between a macroscopic magnetic island and microscopic turbulence, and their impacts on tokamak plasma performance.

This work was supported by the U.S. Department of Energy under Contract No. DE-AC02-09CH11466. This work was also supported by the R & D Program No. NFRI-EN1841-4. The XGC1 simulations presented in this paper used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.