Penetration properties of applied resonant magnetic perturbation in HL-2A tokamak

Any arbitrary perturbation on a magnetic field separatrix can cause a structure named homoclinic tangle in tokamaks. Both an edge localized mode (ELM) and a resonant magnetic perturbation (RMP) can lead to a perturbation of the magnetic field on the separatrix. Under the appropriate circumstances, RMP could alleviate or even completely suppress a rapid collapse process of an ELM. The simulation results using the CLTx code, the extended version of the three-dimensional toroidal magnetohydrodynamic (MHD) code (CLT (Ci-Liu-Ti, which means MHDs in Chinese)) with a scrape-off layer, show the structure of the homoclinic tangle with a borderline stochastic region resulting from RMP in HL-2A tokamak. Strongly distorted magnetic field lines with the homoclinic tangle could connect to the tokamak divertors. The footprints of these magnetic field lines on the divertors are consistent with the energy deposit spots in the experiment. From Poincaré plots of escaped magnetic field lines, it is found that the depth of the plasma edge region penetrated by these field lines depends on the RMP coil current, the rotation frequency of the RMP field, and the plasma resistivity.


Introduction
Large type-I edge localized mode (ELM) could cause unacceptable damage to materials of the first wall of tokamaks [1]. Resonant magnetic perturbation (RMP) [2] has effectively controlled ELMs in an H-mode discharge. The suppression of ELMs by RMP has been achieved in many devices, * Authors to whom any correspondence should be addressed.
Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. such as DIII-D [3], KSTAR [4], EAST [5], and ASDEX-Upgrade [6]. In some other devices, ELM is mitigated instead of suppressed by RMP, such as JET [7], MAST [8][9][10], EAST [5,11], and HL-2A [12]. The feature of the ELM mitigation is that the ELM frequency of bursts increases while its amplitude reduces. The RMP system is also planned to install in future tokamaks, such as ITER [13] and HL-2M [14]. Therefore, understanding the mechanism of ELM suppression and mitigation is essential to design the RMP system for future devices. Some extensive modeling works and computational simulations have been carried out to study these problems, such as MARS-F [13,[15][16][17], M3D-C1 [18,19], JOREK [20][21][22], and NIMROD [23].
It is found that the separatrix of the magnetic field leads to a special structure named homoclinic tangle under any arbitrary perturbation on the separatrix [10,20,[24][25][26][27][28]. The magnetic field topology will undergo significant changes near the X point, and the invariant manifold will split into a stable manifold and an unstable manifold. The stable and unstable manifolds are corresponding to the forward and backward tracks. The tortuosity gradually approaches infinity when a magnetic field line is close to the X point. Thus, the invariant manifolds largely depend on magnetic field perturbations. The perturbations resulting from ELM and RMP on the separatrix could cause different formations of homoclinic tangles. Magnetic field lines near the separatrix become stochastic because of overlapped magnetic islands. These homoclinic tangles could prescribe how stochastic field lines are organized and decide the footprints on the divertors. The energy dispositions on the divertors in experiments could be determined by the field line footprints [10,20,25,29]. The shape of the plasma will strongly influence the result of the homoclinic tangle and the footprint on the divertor. The results aroused by RMP on many devices have been revealed, such as EAST [29], DIII-D [30,31], NSTX-U [18,32], JET [33], KSTAR [34], MAST [9,10,35], ASDEX [36], and ITER [13,37,38]. Many numerical methods are used in these researches. Some of them studied RMP field influence without plasma response [9, 10, 30-33, 35, 38], while others investigated the RMP field with plasma response through the eigenvalue method, such as MARS-F [13,29,37] and GPEC [34,39]. To construct the homoclinic tangle and footprint on the divertor, tools such as TRIP3D [40], ERGOS [35], EMC3-EIRENE [41], are commonly employed. The influences of the RMP field were also studied self-consistently with the inclusion of the X point through the initial value problem, such as M3D-C1 [18] and JOREK [21]. Our present investigation will self-consistently handle the scrape-off layer (SOL) and the X-point, which means that our simulation boundary is located in the vicinity of the first wall of tokamak instead of the last closed magnetic surface. This paper firstly investigated the structure of the homoclinic tangle with RMP in HL-2A. The prediction of the 2D footprint distribution matches the scale of the 1D measurement results of heat flux in HL-2A [42].
The value of ξ X /ξ M was found to correlate closely with the strength of ELM mitigation and a density pumping-out [15,16,43,44], where ξ X is the plasma displacement near the X-point and ξ M is the plasma displacement at the midplane in the low-field side. But, this value will lose its efficacy in some situations (one example is shown below in figure 9(d)). However, based on the homoclinic tangle of magnetic field lines and the Poincaré plots of escaped magnetic field lines in HL-2A, it is found that the thickness of the escaped magnetic field lines in the √ ψ n coordinate could be an additional criterion. The escaped field lines are defined accurately in this paper as the field lines travel start from the area in the initial separatrix to the divertor.
This work presents simulation results of RMP penetrations in HL-2A using CLTx [45], an extended version of the threedimensional, toroidal, nonlinear, and compressible magnetohydrodynamic (MHD) code CLT (Ci-Liu-Ti, which means MHDs in Chinese). The additional feature is that in the CLTx, SOL is taken into account self-consistently. This paper will be organized as follows. Section 2 will introduce the equations used in CLTx and how we set RMP and equilibrium in CLTx. Section 3 presents primary results about discharge 29 676 in HL-2A, including a benchmark with MARSF. Section 4 discusses the influences of the RMP coils' current, the rotation frequencies, and the plasma resistivity on the homoclinic tangle, the Poincaré plot of the escaped magnetic field lines, and the footprints of the escaped magnetic field lines on divertors. Finally, a summary is given in section 5.

Simulation model for CLTx
In CLTx, the same set of the single fluid, resistive, compressible MHD equations, including dissipations, is used as in CLT [46]. The simulation model and the normalization of CLTx are given in Appendix.
The normalized parameters used in this work are as follows: D = 1 × 10 −6 and κ = 1 × 10 −5 . For the purpose of numerical stability, we choose these larger than experiment values. A series of tests with different diffusivity D and conductivity κ has been carried out. It is found that D and κ hardly influence the behaviors of the RMP penetration. The Prandtl number is chosen to be ν/η = 1 near the boundary. All of these three values are uniform both in the plasma and SOL region. The spatial distribution of the time-independent resistivity is determined by the initial plasma temperature T with η = η 0 * T −3/2 , where η 0 is the resistivity at the magnetic axis. D, κ, and ν are uniform both in the plasma region and the SOL region. The plasma density and pressure in the SOL region is chosen to be the same as the value at the separatrix.
The coordinate in CLTx is (R, Z, ϕ ) in the simulation, and the coordinate of (ψ n , θ, ϕ ) or √ ψ n , θ, ϕ are used sometimes in the diagnosis. The Spline interpolation method is used for the conversion between the coordinates. The normalized magnetic flux ψ n comes from the initial equilibrium and remains unchanged with time. The poloidal angle θ is generalized by tracing field lines from the initial equilibrium. The toroidal angle ϕ is the same in the two coordinate systems.

Equilibrium and RMP setup in CLTx
HL-2A has a 2 × 2 coil system for the RMP system, located on the upper ('U') and lower ('L') outer region, as shown in figure 1. The angle width (ϕ ) of the coils in the toroidal direction is 11.4 • . The simulation boundary with the fixed boundary conditions and the magnetic separatrix (or the last closed surface) are indicated in figure 1(a). The difference between the phases of the currents in the upper and lower coils is defined as ∆Φ = Φ U,+ − Φ L,+ that could be 0 • or 180 • for HL-2A where, '±' means the positive/negative coil current, and 'U/L' means the upper/lower coil. More details on the RMP system in HL-2A can be found in [47].  Initial profiles of (a) the safety factor q (blue line) from CLT-EQ and EFIT (green circle), and the toroidal rotation velocity from CLT-EQ. Initial profiles of (b) the density and the resistivity. (c) Initial profile of pressure in CLT-EQ and EFIT. (d) The electronic temperature profile measured in experiment. ψ n is the normalized magnetic flux ψ .
The initial equilibrium is reconstructed by CLT-EQ [48], using the EFIT [49] data in the HL-2A ELM mitigation discharge 29 676 at 800 ms [47], before the RMP turns on. This discharge in HL-2A achieves ELM mitigation by RMP. After turning RMP on at 830 ms, the frequency of ELM increased, but the amplitude decreased. In this discharge, the major radius R 0 = 1.65 m, the minor radius a = 0.4 m, the plasma current I p = 155 kA, the toroidal magnetic field B 0 = 1.37 T, the safety factor q 95 = 3.7, the NBI power P NBI = 1MW, and I RMP = 4.6 kA. The time trace can be found in figure 10 of [47]. The equilibrium generated by CLT-EQ will include the SOL and the X point self-consistently, which include a 2D distributing of pressure P and magnetic field B. The density and velocity distribution can be artificially assumed as needed and they are set base on experiment data in this work. The initial profiles of the safety factor q (both in EFIT and CLT-EQ) and the toroidal rotation ω t are given in figure 2(a), and the density and resistivity profiles are given in figure 2(b). The pressure profiles (both in EFIT and CLT-EQ) are plotted in figure 2(c). The electronic temperature profile at 800 ms measured in the experiment is shown in figure 2(d), which was used to set the resistivity profile shown in figure 2(b). The distribution of the radial component of the RMP field B r rmp added in the simulation is shown in figure 1(a). The radial component of magnetic field is defined as B r = B · ∇ψ / |∇ψ |. The maximum RMP coils' current is 4.6 kA, and ∆Φ is 180 • . The vacuum magnetic field engendered by the RMP coils is calculated by the Biot-Savart law. The magnetic vector potential A is used as an intermediate variable to keep divergence-free of B in the calculation. We assume that the plasma will strongly prevent the RMP field from penetrating across the separatrix at the initial time, so we make the magnetic vector potential A decrease to zero rapidly when reaching the separatrix. This method was also used in previous work in EAST [45]. The amplitude of the vacuum RMP field ramps up from zero to maximum in 400 τ A .

The properties of the magnetic field in vacuum and with plasma response
In this subsection, ∆Φ is chosen to be 180 • , and the gird mesh is set to 256 × 32 × 256 in (R, ϕ , Z). Figure 3(a) shows the resulting magnetic field in vacuum (without the plasma response). It was calculated directly by the initial equilibrium magnetic field plus the vacuum RMP magnetic field. A field line tracing module of CLTx has been developed to trace magnetic field lines. The magnetic island with m/n = 3/1 and 4/1 are evident, but the 5/1 magnetic island is hard to recognize because it is too close to the separatrix. There are several chaotic regions separated by magnetic islands. In the vacuum field, it is evident that there are two stochastic regions in the surrounding area of the m/n = 3/1 and 4/1 magnetic islands that are isolated by chains of small n = 3 magnetic islands as shown in figure 3(a). The magnetic islands and the chaotic region are significantly reduced with the plasma response, as shown in figure 3(b). Then the magnetic islands gradually grow up and finally become comparable with in vacuum, as shown in figure 3(c).
The spectra of the radial magnetic perturbations B r n=1 and B r n=3 in vacuum are given in figure 4. The resonant lines m = nq √ ψ n are plotted with light blue lines. The red lines denote the location and the width of magnetic islands. The island width can be calculated from [50], (1) where S = ρ p q ′ /q is the global magnetic shear at the resonant surface. The spectra of the n = 3 component are the same as for the n = 1 component. The n = 1 magnetic islands are well stimulated because the n = 1 resonant line coincides with the strongest power spectra. Nevertheless, the n = 3 resonant line m = nq is far from the strong power spectra, which results in the smaller n = 3 magnetic islands. Naturally, magnetic islands resulted from the n = 3 perturbation is very small and can be ignorable. Thus, we will only focus on the result of the n = 1 component. It should be noted that all of other components are automatically included in the simulation system.

Effect of ∆Φ on resonant harmonics at the q = 4 rational surface
The relative phase ∆Φ of the currents in the lower and upper coils significantly influences the RMP field and its penetration. Although ∆Φ was fixed to 180 • or 0 • in the HL-2A experiments, it is worthwhile to investigate the contribution of the radial component of the resonant magnetic field B r n=1 near the separatrix by scanning the relative phase ∆Φ , as did in [16]. The same equilibrium reconstructed from discharge 29 676 in HL-2A is adopted. Figure 5 shows the amplitude of the m/n = 4/1 resonant harmonics of the radial magnetic perturbation versus ∆Φ at the q = 4 rational surface. The dashed blue line with squares is the results without plasma response, and the red line with circles indicates the results with plasma response. The dashed blue (solid red) arrow indicates the location of the minimum B r without (with) plasma response calculated by MARS-F. The maximum B r 4/1 appears at ∆Φ = 150 • without plasma response and at ∆Φ = 180 • with plasma response while its minimum value happens at   The inversion symmetry of magnetic field lines on the separatrix will be broken with any nonaxisymmetrical perturbation [27,51]. A magnetic field line will be intensively twisted in the vicinity of the X-point and not close, and the forward and backward trace of the magnetic field line from a point will get completely different results. The magnetic field perturbation fully determines the separatrix's deformation. The forward or backward traces of magnetic field lines are performed to study the properties of the homoclinic tangle. Figure 6 shows the Poincaré plots of the n = 1 RMP without plasma response. The total 5000 points are initially sampled randomly in the region of 0.9 < ψ n < 1, and the maximum turns of the trace is 200. When a magnetic field line escape to the divertor, we stop the trace and mark it as a 'escaped magnetic field line'. The minimum ψ n of the points on this orbit is ψ min and we let δ Esc = 1 − √ ψ min , which may be regarded as the deepest radial depth where magnetic field lines can be escaped.  clings on the separatrix of the magnetic field or on the ψ = 1 surface. The first forward/backward lobe with a toroidal number n = 1 moves close to the X-point in a clockwise/anticlockwise direction and is largely stretched with the change of the toroidal angle ϕ . Stratified structures appear in the first lobes.

Escaped magnetic field lines with RMP
The footprints of the strike points have been proven to be a good indicator of the heat flux distribution on the divertor in experiments [33,52]. Figures 6(c) and (d) show the spatial distribution of the footprints of the backward/forward traced field lines in the inner/outer divertors. The color codes of the points indicate the value of δ Esc . The footprints of the backward/forward traced field lines also show n = 0 and n = 1 structures, but the n = 1 structures are antiphase. The stratified structures are also evident. Particles from the inner layer of the plasma region carry more energy and cause enormous heat flux on the divertor. The stratified structures of magnetic footprints are consistent with the observation of the heat flux splitting on the divertor in the experiment [33]. The simulation boundary is approximate to the actual divertors, but it will only cause the topological deformation in the R coordinate. Therefore, it will not influence the understanding and conclusions of this paper. Moreover, because the patterns of the inner and outer footprints are symmetrical, we only focus on the outer footprints in the following section.
With the plasma response, the simulation result shows that the resonant field amplification happens [53]. The magnetic field perturbation inside the plasma region will keep growing with time and the magnetic islands could be comparable with in vacuum. Thus, the growing perturbation causes the broadening of the first lobes and the extension of the stochastic field region. Figures 7(a) and (b) show the right lobe structure at two different stages on the section of ϕ = 270 • . The extent of the right lobe and δ Esc increase. The results will be more precise in the figure of the footprints of the escaped magnetic field lines, as shown in figures 7(c) and (d). The shape of footprints also extends with time. Both the density of footprints on the outer divertor and δ Esc increase with time, which may suggest that the heat flux on the divertor will increase. The results of the left lobe and the footprints on the inner divertor are similar. The divertor in fact is along theẐ direction. But for convenience in simulation, we set the divertor along theR direction at Z = −0.6 m. According to figure 1(e) in [42], the width of heat flux when ELM is mitigated by RMP is about 10 mm inẐ. Based on the angle of the ψ = 1 line near the divertor (the blue line shown in figures 7(a) and (b)) at Z = −0.6 m  (∼144 • ) in the experiment, the corresponding width of heat flux on the divert in our simulation should be around 7 mm that is consistent with the distribution range of the footprint width in figure 7(d).
The RMP system in HL-2A also can produce the n = 2 magnetic perturbation by setting the coils' current as { ++ −− } (see in figure 2 of [47]), which is coming from the third harmonic of the n = 0 RMPs. The RMP with a higher toroidal number generally causes a broader stochastic field region than that with the n = 1 RMP if their resonant regions are the same. But actually, the n = 2 resonant line deviates from the stronger poloidal power spectrum of the perturbated magnetic field aroused by RMP in HL-2A, similar with the n = 3 component of RMP with the coil combination of { +− −+ }. Therefore, the resonant perturbation from the n = 2 RMP is much smaller than that from the n = 1 RMP. The extent of the secondary lobes and the footprints are also smaller. Figure 8 shows the Poincaré plot at the Φ = 135 • section and the footprints on the outer divertor. Two second lobes appear, but their extent is not large enough to reach the boundary. Both the first and second lobes cyclically change in the toroidal direction with a toroidal number n = 2. The footprints of the zeroth and first forward lobes on the outer divertor are only presented. The footprints of the first lobes have the same period in the toroidal direction. After the same long period of the evolution, the shape of the footprints with the n = 2 RMP is narrower than that with n = 1 RMP, and δ Esc is also smaller. The wider spread of footprints benefits to spread the thermal deposition on the divertors, and the bigger δ Esc means the RMP can lead to the losses of the plasma in the larger area. Therefore, the n = 1 RMP is more effective in HL-2A.

The influence of the current in RMP coils
In this subsection, we carry out investigation in the influence of the RMP coil current on the homoclinic tangle and the footprints of escaped magnetic field lines. Since the RMP penetration may be largely screened due to plasma response with the plasma rotation [17,[53][54][55][56], we investigate individual influences of different RMP currents on magnetic field perturbations inside the plasma. Figure 9 Since the current screening effect strongly depends on the ramping speed of the RMP coil current, the larger screening effect resulted from the shorter ramp time could prevent the RMP field from penetrating into the inner region. After some time, the RMP field restrained in the outer region penetrates into the inner region and then causes the second growth of the 4/1 magnetic island. It explains why the 4/1 island shows much larger overshoot than the 5/1 island. When the RMP field reaches to its maximum in the 4/1 rational surface, the 4/1 island becomes saturated. The 5/1 island exhibits continuous growth, which could be driven by the larger 4/1 island due to the toroidal coupling. As we can see, the overshoot of the island width reduces with the increase of the ramp time and completely disappears with t up = 4000 as shown in the figure 10. It should also be noted that the island size W 5/1 is very difficult to be measured accurately since the its original flux surface, ψ 5/1 = 0.9993, is almost located in the separatrix. Thus, we will only pay attention on the tendency of the m/n = 5/1 mode, especially in nonlinear stage. The Chirikov criterion, which is famous for magnetic island overlapping or stochasticity [50,57] is defined as, σ ≡ Wm 1 ,n 1 +Wm 2 ,n 2 2|ρ2−ρ1| ⩾ 1 (2) where σ is the Chirikov parameter determined by two adjacent magnetic islands. ρ 1 , ρ 2 and W m1,n1 , W m2,n2 are the locations and widths of the islands, respectively. Figures 9(b) and (c) shows the time evolutions of the maximum σ and the radial distribution of σ at t = 4081τ A (∼1.36 ms) with different RMP currents. The similar tendency of the time evolutions of and the spatial distributions associated with the change of the RMP current clearly indicates that there is no critical value of the RMP current at which the magnetic field perturbations inside the plasma region exhibit an evident difference. The value of the ratio of ξ X /ξ M is also used to measure the effect of RMP on ELM in some research [13,15,16]. We found that these values nearly have same time tracing with different RMP current, as shown in figure 9(d).
We traced magnetic field lines and plotted the Poincaré plots of escaped magnetic field lines in the θ − √ ψ n coordinate, as shown in figure 11. Five thousand initial points in the region of 0.94 < √ ψ n < 1 are used in the field line trace. All initial points are the same in different cases, and the maximum turns are set to be 200. It is clearly indicated that δ Esc increases with the increase of the RMP coil current. It should be noted that δ = 0.0 at the plasma boundary or the magnetic field separatrix. Especially, escaped field lines exhibit a distinct expansion towards to the inside plasma when I RMP increases from 1 kA to 2 kA as shown in figures 11(a) and (b). Figure 12 shows the time evolutions of the deepest radial depth (δ Esc ) of escaped magnetic field lines with different RMP coil currents (the first 10% points in δ were excluded to avoid the noise error). When the depth reaches δ Esc = 0.01, the depth exhibits a rapid increase to δ Esc = 0.025 at t = 2400τ A (0.8 ms) for I RMP = 2 kA and at t = 740τ A (0.25 ms) for I RMP = 4.6 kA as shown in figure 11. The depth δ Esc = 0.01 is coincidently located at the q = 4 rational surface, which suggests that the RMP field will pile-up in the front of the q = 4 rational surface at the early stage due to the current screen effect. When the RMP field penetrates across the q = 4 rational surface, the depth of the RMP penetration shows a sudden increase due to formation of larger magnetic islands. The escaped magnetic field lines enhance the radial plasma transport along the magnetic field lines. With a low coil current, the radial transport caused by RMP only exists in the narrow edge region. Therefore, in order to the deep enough magnetic field lines to be escaped for the ELM crash, a minimum RMP coil current is required for the ELM mitigation. Furthermore, we should also notice that all the escaped  magnetic field lines are around the m/n = 4/1 magnetic island. Thus, the large radial plasma region is also strongly influenced by the equilibrium, which will also decide whether the ELM mitigation happens.

The influence of the RMP rotation
As the spatial distributions of the footprints of escaped field lines on the divertors are shown in figures 6 and 7, these field lines are located on the divertor statically with static RMP. It would be harmful to the divertors if the heat flux accumulates on the same location. Therefore, it is important for investigation into diffusive and dynamic striking points of the losing particles on the divertor by rotating the RMP field. However, the RMP rotation will influence the permeation of the RMP field which is equivalent to adding an additional plasma rotation. In this subsection, the RMP field is assumed to rotate smoothly by forcibly changing the phase of the applied magnetic field. Figure 13 shows the time evolutions of the widths of the m/n = 4/1 and 5/1 magnetic islands. For the RMP rotation frequency below 50 Hz that is usually adopted in nowadays experiments, it is found that the m/n = 4/1 and 5/1 magnetic islands remain almost unchanged, especially for the m/n = 5/1 magnetic island. We want to figure out what happens when the rotation frequency increases even more. For the rotation frequency above 50 Hz, the development of the magnetic islands shows different tendencies. The m/n = 4/1 magnetic island increases, but the m/n = 5/1 magnetic island slightly decreases with the increase of the rotation frequency. Since the plasma itself is fixed without rotation at ψ n = 1, the RMP rotation can be regarded as adding an additional rotation on the plasma that can be positive or negative. With a counter rotation of the RMP phase that we used in our simulations, the plasma equivalently experiences a contrarotation in the narrow range near the separatrix, as shown in figure 14. Then, with the increase of the RMP rotation, the equivalent plasma rotation ω plasma − ω * RMP at ψ 5/1 increases, but it will first decrease and then increase at ψ 4/1 , which leads to the screen effect at ψ 5/1 is always enhanced. On the contrary, at ψ 4/1 it will first decrease and then increase, which is evident as shown in figure 13. ω * RMP is assumed to be equal with ω RMP = 50 Hz everywhere in figure 14. The equivalent plasma rotation at ψ 4/1 will be zero under this assumption, which means that the RMP rotation can weaken the current screen effect and cause the quick penetration of the RMP field to reach the q = 4 rational surface. From figure 14, it is indicated that the weakening effect on the  current screen presents until ω RMP = 1 kHz. The Hall effect is not considered in the presented study, but it should be nonignorable because the diamagnetic drift frequency ω e,⊥ associated with Hall effect could largely affect the RMP penetration.
As δ Esc shown in figure 15, for the RMP rotation frequency with 500 Hz, the time of the abrupt increase of δ Esc appears in a late time. Furthermore, when the frequency comes to 1 kHz, δ Esc remains at a very low level around 0.01. The stochastic magnetic field lines limit in the range from √ ψ n = 0.98 to the separatrix when ω RMP =1 kHz. In order to mitigate ELM or ensure enough radial plasma transport along escaped magnetic field lines, a favorable response of RMP on the resonant surfaces close to the separatrix is usually required.
The characteristic time of the ELM crash is shorter than 1 ms, and the characteristic time between the ELM crash is about 3 ms in HL-2A [58]. In order to achieve the striking points of escaped particles diffusively over the divertor of a single ELM crash, the RMP rotation frequency should be over 1 kHz. But, even in the ideal condition shown in the simulation results, the RMP rotation frequency will strongly limit δ Esc of escaped magnetic field lines. Thus, for a single ELM crash, diffusive striking points of escaped particles on the whole divertor by RMP are unachievable. However, diffusive striking points on the divertor are feasible for several ELM crashes. The statistical distribution of escaped magnetic field lines in the Rcoordinate is presented in figure 16. The main conditions are similar for all cases when δ Esc is larger than 0.02. Moreover, a point should be emphasized is that the distribution will not change with the rotation frequency of the RMP field or the RMP coil current. Although the RMP rotation can spread the particle striking points in the ϕ coordinate, they are still largely concentrated in the R coordinate. According to the Poincaré plot of escaped magnetic field lines shown in figure 7, the lobes become narrower when they are far away from the X-point, suggesting that the striking points are distributed in a broader region when the divertor is closer to the X-point.
As for the actual condition of the HL-2A tokamak, because there are only two pairs of fixed RMP coils, the phase of the RMP field only can switch between 0 • and 180 • , which leads to that the striking points are usually located at a fixed and narrow region.

The influence of the resistivity
With the improvement of plasma temperature confinement, the plasma resistivity will decrease. Whether RMPs still bring the ELM mitigation will be a fundamental problem in future installations. Therefore, we investigate the effects of the resistivity on the striking positions of escaped particles with the RMP coil current I = 4.6 kA and without the rotation of the RMP field. Figure 17 shows the time evolutions of δ Esc with different plasma resistivities. With the decrease of the resistivity, the evolutions are slowed down. The normalized resistivity on the separatrix η b is less than 10 −7 (the corresponding resistivity at the center is 10 −9 ), magnetic field lines only in the vicinity of the separatrix can be escaped from the inner plasma region. For similar parameters of HL-2A in discharge 29 676, when the electron temperature T e at the magnetic axis comes to 5 keV [59], the normalized classical resistivity at the center η 0 will be 2 × 10 −9 and the normalized resistivity near the separatrix will be about 2 × 10 −7 . On the other hand, the rotation is expected to be lower in future installations, which will make the penetration of RMP easier. The competing results of the two effects need further study, but a bigger RMP coil current must be helpful. Recently, the RMP coil current of HL-2A can attain 10 kA [60].

Summary and discussion
The SOL is included in the recently upgraded CLT code, named CLTx. We simulated the evolution of the RMP field in HL-2A based on discharge 29 676. When the SOL is included in the code, the homoclinic tangle and the footprint on the divertor can be traced. With field line trace under the RMP field, we are able to obtain the escaped magnetic field lines, the homoclinic tangle structure, and the footprints with RMP in HL-2A.
The simulation results show that the footprints show stratified structure and cluster in a narrow range in the R coordinates. With the help of the Poincaré plots of escaped magnetic field lines, we can identify the deepest radial depth where magnetic field lines can be escaped, δ Esc = 1 − √ ψ min that is generally associated with the radial transport caused by RMP or other perturbation near the separatrix, and could be a critical factor in ELM mitigation with RMP. The RMP current plays a crucial role in the time evolution and other properties of the escaped magnetic lines. When the RMP coil current reaches a critical value, δ Esc exhibits an abrupt increase in a very short time. In the meantime, it is found that the numbers of escaped magnetic field lines remain almost the same when the sudden increase of δ Esc occurs, which could be suggested that when the number of escaped field lines exceeds a certain amount, a barrier is broken through. This barrier is generally located at a rational surface where the m/n = 4/1 magnetic islands are usually present.
The rotation of the RMP field can diffuse the locations of the striking points of lost particles, but it will screen the response of the RMP on the resonant surface close to the separatrix. The favorable frequency of the RMP rotation is between 500 Hz and 1 kHz with the parameters chosen in our simulation. The plasma resistivity decreases with the improvement of plasma confinement. The decreasing resistivity slows down the RMP penetration, similar to the decrease of the RMP coil current.
According to the simulation experiment results obtained from HL-2A, it was observed that a simple 2 × 2 RMP coil could also be used to mitigate the ELM. In other devices, such as EAST, the ELM mitigation was also achieved by imposing the n = 1 RMP with more coils, as seen in [5]. However, comparing the escaped magnetic field, HL-2A performed better. Nonetheless, as shown in figures 11(b) and (c), it is easy for the bigger island on the resonant surface resulting from the n = 1 RMP to form hollows of the escaped magnetic field, which could reduce the radial transport. And the location of n = 1 resonant surface can strongly limit the δ Ecs . Although RMPs may penetrate deeply into the plasma, the sparse distribution of n = 1 resonant surfaces presents a challenge for magnetic islands to overlap. RMPs with higher toroidal numbers have been found to generate smaller yet more layered magnetic islands on a greater number of resonant surfaces. This effect can be advantageous in creating a broader stochastic field region. In DIII-D [31], it is found that the depth δ Esc of an escaped magnetic field with the n = 3 RMP is bigger than 0.1, which represent for a strong radial transport of the plasma and a powerful capability of the ELM suppression.
where ρ, p, v, B, E, and J are the plasma density, the thermal pressure, the plasma velocity, the magnetic field, the electric field, and the current density, respectively. The subscript '0' indicates the initial equilibrium part. Γ (=5/3) is the specific heat ratio of the plasma. All variables in CLTx are normalized as x/a → x, ρ/ρ 00 → ρ, v/v A → v, t/τ A → t, p/ B 2 00 /µ 0 → p, B/B 00 → B, E/(v A B 00 ) → E, η/ µ 0 a 2 /τ A → η, J/(B 00 /µ 0 a) → J, where, 'a' equals to 1 m and B 00 equals 1 T, v A = B 00 / √ µ 0 ρ 00 is the Alfvén speed, and τ A = a/v A is the Alfvén time. ρ 00 is the initial plasma density and magnetic field at the magnetic axis. There is a point to note here that the normalizing condition of 'a' and 'B 00 ' are different in CLTx and CLT. The Hall term in Ohm's law is not included in this work.