Effect of resonant magnetic perturbations including toroidal sidebands on magnetic footprints and fast ion losses in HL-2M

Externally applied resonant magnetic perturbations (RMPs), generated by magnetic coils located outside the plasma (referred to as RMP coils), provide an effective way to control the edge localized mode (ELM) in tokamak devices. Due to the discrete nature of the toroidal distribution of these window-frame coils, toroidal sidebands always exist together with the fundamental harmonics designed for ELM control. In this work, the MARS-F code (Liu et al 2000 Phys. Plasmas 7 3681) is applied to investigate the detailed features of the RMP spectra considering both the dominant harmonic (n = 2) and the associated sideband (n = 6), and the impact of the combined fields on magnetic footprints as well as on the fast ion losses for a reference double-null scenario in the HL-2M device. It is found that the sum of the n = 2 and n = 6 RMP fields splits the footprint and widens the footprint area, as compared to the single-n (n = 2) harmonic case. The resistive plasma response breaks the up–down symmetry of the footprint pattern on the outer divertor plates, which is otherwise symmetric assuming vacuum RMP fields. Considering fast ion losses, a threshold value exists for the initially launched radial position of test particles, as well as for the RMP coil current, before the loss occurs. When the threshold criterion is satisfied, the combined n = 2 and n = 6 RMP fields enhance the fast ion loss rate by ∼20% , as compared to that of the n = 2 component alone. These results illustrate the important role of the sideband of RMP fields on the magnetic footprints and fast ion losses in tokamak plasmas.


Introduction
Externally applied magnetic field perturbation was proposed to control the heat flux on plasma-facing components (e.g. the divertor plate) [1][2][3][4]. Resonant magnetic perturbation (RMP), produced by externally applied magnetic coils (referred to as RMP coils), is a mature method to control the type-I edge localized mode (ELM), with the purpose of reducing the transient heat flux on the divertor plate in tokamak devices. The efficiency of the RMP technique for suppressing and/or mitigating ELM has been demonstrated in many devices, such as DIII-D [5], JET [6], MAST [7], KSTAR [8], EAST [9], ASDEX Upgrade [10] and HL-2A [11]. The RMP coil efficiency depends on both the amplitude and spectrum of field perturbations, which are mainly determined by the plasma equilibrium parameters and by the RMP coil configuration [12][13][14][15][16][17][18][19][20][21][22]. The RMP technique will be applied to control ELMs in the International Thermonuclear Experimental Reactor (ITER) [23]. For a given equilibrium with fixed RMP coil geometry parameters, a significant aspect to consider when optimizing the RMP field spectrum is the toroidal phasing of the RMP coil current. The resistive plasma response usually plays an important role in optimizing the RMP field spectrum [24][25][26]. In addition, due to the discrete nature of the toroidal distribution of window-frame RMP coils, toroidal sidebands consistently coexist with the dominant harmonic designed for ELM control [27,28]. It has been demonstrated that the sidebands of perturbation play a part in controlling plasma behaviors such as magnetohydrodynamic (MHD) instabilities [29]. Hence, it is expected that RMP sidebands can also impact the magnetic footprints on the divertor target plate and/or fast ion redistribution/losses [27,[30][31][32][33][34][35][36][37][38][39] during ELM control.
Precise prediction of magnetic footprints is essential for controlling the heat load pattern on the divertor target surface [40]. The magnetic footprint pattern on the divertor plate surface is easily affected by 3D field perturbations [41][42][43][44][45][46][47][48], in which the plasma response plays an essential role. In addition to the heat flux induced by thermal particles, the losses of fast ions can also induce a heat load on the plasma-facing materials [33,49,50]. Furthermore, loss of fast ions degrades the plasma performance through decreasing the plasma heating power or by affecting MHD instabilities (e.g. the resistive wall mode [51]). Reduction of fast ion losses during the amplification of an RMP field to control ELMs is crucial for maintaining the high performance of tokamak plasma. Understanding fast ion loss mechanisms related to realistic RMP fields is important for simultaneously considering the issue of fast ion loss and the ELM control problem. However, systematic investigations of the RMP field features taking into account toroidal sidebands and their effect on the magnetic footprint and fast ion losses are relatively scarce. Such investigations are required for a deeper understanding, interpretation and optimization of experiments on current devices, which is the motivation of this study.
In this work, the MARS-F [52] code is applied to study the features of the RMP fields including both the dominant (i.e. n = 2) and sideband (i.e. n = 6) harmonics, and the effect of RMP fields on magnetic footprints and fast ion loss. The latter is carried out by utilizing the REORBIT module [53,54] developed within MARS-F. MARS-F, using the linearized resistive single-fluid MHD model in the full toroidal geometry, predicted plasma response fields that quantitatively agree with experimental measurements [55,56]. The optimized coil phasing for suppressing or mitigating ELMs obtained from MARS-F computations is consistent with the experimental values for many devices, such as MAST [24], ASDEX Upgrade [57], DIII-D [25] EAST [58] and HL-2A [28].
The remainder of the paper is organized as follows. Section 2 describes the formulation of the employed modeling approach. A brief introduction to the HL-2M tokamak is presented in section 3. Section 4 presents the reference equilibrium and the RMP coil configuration in the HL-2M device. Section 5 reports features of the RMP fields for the single n = 2 case and for the case of combined n = 2 and n = 6 spectra. The effect of RMPs on magnetic footprints is discussed in section 6. The influence of RMPs on fast ion losses is described in section 7. The numerical sensitivity test is reported in section 8. Section 9 draws the conclusion and discussion.

Formulation
The MARS-F code [52] is applied to compute the total magnetic field perturbations produced by the external RMP coils by solving the following linearized resistive single-fluid MHD equations in Eulerian form for the plasma region. For the vacuum regions and RMP coils, the curve-/divergence-free equations and electromagnetic equations are self-consistently numerically resolved, respectively [15].
Here, R is the plasma major radius.φ andẐ are the unit vectors along the toroidal angle and the vertical direction of the cylindrical coordinates (R,ϕ,Z), respectively. The variables B 1 , J 1 and P 1 represent the perturbations of the equilibrium magnetic field B, plasma current J and pressure P, respectively. ρ and Ω are the plasma mass density and toroidal rotation frequency, respectively. The plasma displacement and the perturbed velocity are denoted by ξ and v, respectively. κ is a coefficient describing the parallel sound wave damping strength [59,60]. n and m are the toroidal and poloidal mode numbers of the perturbation, respectively. v th,i is the thermal velocity of bulk ions. Parallel sound wave damping plays an important role in high-pressure plasmas (which is the case in our study). A large coefficient (κ = 1.5 as assumed in this work) allows proper capture of this damping physics. For plasmas at very low pressures, however, the parallel sound wave damping becomes less important. The plasma response in this case is insensitive to the choice of the damping coefficient κ.
In addition, the REORBIT module was recently implemented in the MARS-F code [53] to study the influence of field perturbations on magnetic footprints [22] on the divertor target surface and on the fast particle confinement/loss [21,37,61]. REORBIT is employed to trace the field lines and the guidingcenter drift orbit trajectories of fast ions in this work. For fast ion orbit tracing, the REORBIT module time-advances the guiding-center drift orbit equations for test particles directly in the MARS-F curvilinear coordinates. The magnetic perturbation fields are employed in the internal representation (the raw format) as computed by MARS-F without mapping to other coordinate systems, thus allowing high fidelity in treating the 3D perturbation by REORBIT. The module also allows inclusion of particle collision and electric field effects, but these are not included for fast ion tracing in this study, meaning that the particle energy and magnetic moment are both conserved during the tracing. The particle loss is therefore induced purely by the 3D RMP fields.

The HL-2M tokamak
HL-2M is a newly constructed copper-conductor tokamak device at the Southwestern Institute of Physics, with a plasma current exceeding 1 MA in 2022. The main goal of HL-2M is to achieve high-performance plasmas, thus providing support for the critical requirements of ITER operation and for the physics design and optimization of future reactor-scale devices. These include, e.g., the plasma physics related to fast ions, high beta high bootstrap current operations and flexible divertor configurations (snowflake, tripod).
The main parameters of HL-2M are presented in table 1. The designed maximum plasma current is 3 MA and the toroidal magnetic field is up to 3 T [62]. The present design value of the total auxiliary heating power is 27 MW (with future upgrades expected), including 15 MW neutral beam injection (NBI), 8 MW electron cyclotron resonance heating and 4 MW low hybrid current drive [63,64]. A brief summary of the first plasma experimental results and recent modeling progress for the HL-2M were reported in [64]. For the NBI system, the three beam lines are designed with 5 MW power per line, with the birth energy of the neutral beam being about 80 keV. The two lines are in the co-current direction, while the third one is in the counter-current direction [65].

Equilibrium and RMP coil configuration
Since HL-2M also aims to find power exhaust solutions for future reactors, the double-null divertor configuration is one of the primary operational scenarios. This motivates our choice of the double-null shape in this study. Furthermore, the kinetic profiles for the reference equilibrium are carefully designed utilizing the OMFIT code [66] in order to avoid internal kinks and to maximize the plasma current with the purpose of raising the plasma density limit. In addition, the toroidal magnetic field B 0 = 1.8 T is adopted with the purpose of providing support for future steady-state scenario discharges in HL-2M [63]. The radial profiles of the plasma pressure, density, toroidal rotation frequency and safety factor for the reference equilibrium are plotted in figure 1. The safety factor at the magnetic axis and plasma edge are q 0 = 1.27 and q a = 3.13, respectively, which exclude the unstable internal kink and edge-peeling instabilities. The plasma current is I p = 1.53 MA. The normalized plasma pressure is β N ≡ β(%)/[I p (MA)/a(m)B 0 (T)] = 2.42 with µ 0 = 4π × 10 −7 Hm −1 , minor radius a = 0.65 m and β = 2µ 0 < p > /B 2 0 being the thermal pressure normalized by the magnetic pressure. Note that β N = 2.42 is much smaller than the no-wall beta limit (β limit N ≃ 3.5) for the n = 2 external kink mode. <A> denotes the volume average of variable A. The on-axis toroidal rotation frequency is Ωτ A = 0.01, with τ A being the on-axis Alfvén time. The major radius of the geometric center is R 0 = 1.78 m and the on-axis density n e = 6.1 × 10 19 m −3 . In this work, the Spitzer model for plasma resistivity is assumed with the on-axis Lundquist number S = 1.67 × 10 8 , based on the assumed on-axis electron temperate T e0 = 5.85 keV.
The plasma boundary of the reference equilibrium, the RMP coil location, the surfaces of the limiter and the vacuum vessel wall are all shown in figure 2(a). There are 2 × 8 in-vessel RMP coils in the HL-2M device. Each RMP coil  spans about 36 • along the toroidal angle ϕ, with two adjacent RMP coils being separated by a 9 • gap in ϕ. The total toroidal coverage by the RMP coils is thus about 80%. The current of a single RMP coil, consisting of four turns, is up to 10 kAt. The predominant n = 1, 2 and 4 field perturbations can be produced by the RMP coil system. The relative toroidal phase difference between the upper and lower RMP coil current ∆Φ ≡ Φ U − Φ L for a given toroidal harmonic n, is the essential freedom for controlling the edge plasma, resulting in tuning the spectrum of field perturbations [12,15,28]. Furthermore, toroidal sideband harmonics are intrinsically produced due to the discrete nature of the coil distribution for the RMP coil system as shown in figure 2 In the modeling, the exp(−inϕ) variation for the nth component of the coil current is assumed. The amplitude and toroidal phasing of the effective coil current for a specific toroidal harmonic n is obtained by Fourier decomposition of the assumed RMP coil current distribution (figure 2(b)) along the toroidal angle. Due to their likely weak effect on the plasma, sidebands with rather high harmonic numbers (i.e. n ⩾ 14) are not considered here. However, the main sideband (i.e. n = 6) is taken into account in this work, and a coil current of 10 kAt is assumed for all RMP coils. The RMP coil configuration, as shown in figure 2(b), yields ∼10.5 kAt and ∼5.7 kAt effective coil currents for the n = 2 and n = 6 components, respectively. This coil configuration produces the coil phasing ∆Φ being close to the optimal one, for the dominant n = 2 component, will be shown later. The computed resonant radial field amplitude per unit RMP coil current for (a) the dominant toroidal mode number (i.e. n = 2) and (b) the secondary harmonic (i.e. n = 6), while scanning the coil phasing ∆Φ n=2 and ∆Φ n=6 for the n = 2 and n = 6 components, respectively. The dotted and solid curves denote the vacuum RMP field and the total RMP field including the resistive plasma response, respectively. The red 'O' symbols show the corresponding coil phasing for the n = 2 (i.e. ∆Φ n=2 = −90 • ) and n = 6 (i.e. ∆Φ n=6 = 90 • ) components for the coil configuration in figure 2(b). The Lundquist number is S ≃ 1.67 × 10 8 on the magnetic axis, with the Spitzer resistivity model being adopted.

n = 2 and n = 6 RMP field perturbations
To characterize the features of the RMP field perturbations, three quantities b 1 res , b 1 m and b ⊥ are defined, which denote the resonant radial field perturbation at the corresponding rational surface, the poloidal Fourier harmonic m of radial field perturbation for the given toroidal component n and the field perturbation perpendicular to the flux surface, respectively, with b 1 m ≡ (J B 1 · ∇s) m and b ⊥ ≡ B1·∇s |∇s| . J = (∇s · ∇χ × ∇ϕ) −1 is the Jacobian of the chosen equal-arc flux coordinate system with s ≡ ψ 1/2 p , χ and ϕ being the radial coordinate, general poloidal angle and toroidal angle, respectively. ψ p is the normalized poloidal flux. b 1 res is defined as the amplitude of b 1 m at the corresponding rational surface. b 1 res at the last rational surface is often used as an indicator for optimizing the RMP coil configuration (i.e. poloidal location, coil phasing, etc) for ELM control [18].

Figures 3(a) and (b)
show the single-fluid resistive plasma response to the externally applied vacuum RMP fields, while scanning the toroidal phasing ∆Φ for the specific toroidal harmonics of n = 2 and n = 6, respectively. In both cases, the plasma response affects the dependence of the pitch-resonant radial field amplitude |b 1 res | at the last rational surface (i.e. q = 3) on ∆Φ and reduces the maximum value of |b 1 res | q=3 , as compared to the vacuum RMP fields. For the RMP coil configuration shown in figure 2, the computed field perturbations |b 1 res | q=3 for the predominant (n = 2) and secondary (n = 6) harmonics are 0.4 G kAt −1 and 0.3 G kAt −1 for the vacuum case, respectively. When the resistive plasma response is taken into account, the amplitude of |b 1 res | q=3 for the n = 6 harmonic (∼0.14 G kAt −1 ) is about 35% of that for the n = 2 component (∼0.4 G kAt −1 ). Here, the toroidal phasing difference between the upper and lower rows of the coil current ∆Φ n=2 = −90 • and ∆Φ n=6 = 90 • (labeled by the red circles in figure 3) are adopted for the n = 2 and n = 6 fields, respectively, based on the RMP coil current configuration shown in figure 2(b). In the following sections, the aforementioned toroidal phasing is fixed. In this work, Gauss (referred to as G) is taken as the unit of magnetic field perturbation, with 1 G representing 10 −4 T.
The poloidal spectrum including both the resonant and nonresonant parts of the n = 2 RMP is shown in figure 4. The contour plots in figures 4(a) and (b) represent the response field harmonics b 1 m (ψ p ) plotted as a function of the poloidal number m (x-axis) and the radial coordinate ψ p (y-axis). The symmetry breaking with respect to the m = 0 plane is enhanced by the resistive plasma response (figure 4(b)), as compared to the vacuum case (figure 4(a)). The plasma response results in an overall field amplification-a factor of three, when the maximum value of b 1 m in (ψ p − m) domain is compared with that for the vacuum case. For instance, the b 1 m amplitude of the nonresonant harmonic with the m = 10 is about 1.86 G kAt −1 , which is about three times larger than its vacuum counterpart 0.67 G kAt −1 . Figure 4(c) clearly shows that the amplitude of the resonant poloidal harmonics is also amplified by the plasma response. For the m = 6 harmonic, the maximum value of b 1 m along the radial coordinate is enhanced to be ∼1.5 G kAt −1 by the plasma response. For other resonant harmonics (m = 3, 4, 5), the amplification of the perturbation amplitude still occurs. The amplification of the amplitude of the poloidal harmonic is mainly induced by the finite plasma pressure [12]. On the other hand, the plasma also screens the pitch-resonant radial magnetic perturbation (i.e. |b 1 res |) at the corresponding rational surface due to the occurrence of the shielding current resulting from the plasma response. For example, at rational surfaces (i.e. q = 1.5, 2, 2.5), due to the strong screening effect from the toroidal plasma flow in the core and middle regions, |b 1 res | is greatly reduced by the resistive plasma response (figure 4(d)). However, the screening m (ψp) of the n = 2 RMP fields for (a) the vacuum radial field and for (b) the total radial field, including the resistive plasma response, plotted along the poloidal mode number m (x-axis) and the radial coordinate ψp (y-axis). The '+' symbols denote the resonant harmonics at the locations of the corresponding rational surfaces (i.e. q = m/n = 3/2, 4/2, 5/2, 6/2). (c) Comparison of the radial profiles of the resonant harmonics for the vacuum case (in red) and for the case including the plasma response (in blue). (d) The pitch-resonant radial perturbation at the corresponding rational surfaces denoted by the vertical lines in (c), for the vacuum (in red) and the total (in blue) fields. ∆Φ n=2 = −90 • is adopted here. effect of plasma response on |b 1 res | almost vanishes in the edge region (such as the computed |b 1 res | at the q = 3 rational surface shown in figure 4(d)) due to the combined influence of slow plasma flow, the relatively large plasma resistivity and the strong amplification of b 1 m=3 by the plasma response. The full poloidal spectrum of field perturbations for the n = 6 component is reported in figures 5(a) and (b). The overall amplitude of |b 1 m | in the 2D-spectrum domain is increased from 0.38 G kAt −1 for the vacuum case to 0.46 G kAt −1 for the case including resistive plasma response. The pattern of poloidal spectrum is not significantly changed by the plasma response, except for the pitch-resonant radial fields at the locations of the corresponding rational surfaces and the radial perturbations of non-resonant harmonics with m >∼ 18. For the resonant harmonics, the maximum values of |b 1 m | along the radial coordinate are slightly amplified by the plasma response ( figure 5(c)). This is due to the higher toroidal mode number and the associated stronger screening effects from more rational surfaces on the external vacuum field, as compared to the n = 2 case. The |b 1 m | value of the n = 6 perturbations in the core region is very small, as expected. In a pitch-resonant radial field, the plasma response plays an overall screening role at all rational surfaces. At the outermost two rational surfaces (q = m/n = 17/6,18/6), |b 1 res | is about 0.12 G kAt −1 and 0.14 G kAt −1 , respectively, which are about half of their vacuum counterparts. We note here that the total 151 poloidal harmonics with m = −75, . . . , 75 are adopted in the computations, in order to guarantee the numerical accuracy. We have checked that |b 1 res | vanishes at all corresponding rational surfaces, when the ideal plasma response is considered (not shown here). Figure 6 shows the total normal field perturbations b ⊥ inside the plasma in the (R, Z)-plane, including the plasma response, related to the cases shown in figures 4 and 5.  (f ) show that the plasma response field in the high-field-side (HFS) is weaker than that in the low-field-side (LFS), consistent with that discussed in [20]. For the n = 2 and n = 6 harmonics (figures 6(c) and (f )), the plasma response produces a finite field perturbation near the top and bottom of the torus. The plasma response fields near the upper null region are much stronger than that near the lower one. The maximum values of the overall amplitude of b ⊥ for the n = 2 and n = 6 components are ∼17.4 G kAt −1 and 7.0 G kAt −1 , respectively. Figures 6(g) and (h) present a mixture of the n = 2 and n = 6 components, which is defined as ∑ n=2,6 b ⊥ exp(−inϕ). Here, b ⊥ is in complex numbers and ϕ = 0 is assumed in figures 6(g) and (h). Figures 6(g) and (h) show that the overall patterns of the real and imaginary parts of the superposed perturbations are mainly determined by the n = 2 component. However, due to the local cancellation or enhancement at certain special locations, the overall pattern of magnitude of mixed |b 1 n | is slightly different from that for the n = 2 component. The maximum value of mixed |b 1 n | is slightly enhanced to 24.1 G kAt −1 (figure 6(i)).
In order to study the detailed features of the superposed b 1 n , the distribution of b 1 n on the last closed flux surface (LCFS) is reported in figure 7. As expected, the real (i.e. Re(b 1 n )) and imaginary (i.e. Im(b 1 n )) parts of b 1 n periodically vary along the toroidal angle for the case with a pure toroidal component (i.e. n = 2). The corresponding amplitude of b 1 n varies along the equal-arc poloidal angle χ and is a constant along ϕ at a fixed χ. On the LCFS, the field perturbation near the upper null (i.e. χ ∼ 100 • ) is much larger than that near the lower null (i.e. χ ∼ −100 • ), as shown in figure 7(c). The added n = 6 sideband significantly modifies the patterns of Re(b 1 n ) and Im(b 1 n ), as shown in figures 7(d) and (e). The sum of the n = 2 and n = 6 RMPs results in a substantial periodic variation of the b 1 n amplitude along ϕ, as expected. This implies that the effect of mixed total perturbations with two toroidal components on the magnetic topology differs from that with the single-n (n = 2) harmonic case.

Effect of n = 2 and n = 6 RMP fields on magnetic footprints
Next, the Poincaré maps of magnetic field lines for the case of the n = 2 RMPs alone and for the case with the sum of the n = 2 and n = 6 RMPs are shown in figure 8. Here, the RMP perturbations are computed assuming the resistive plasma response (with the Spitzer resistivity model). For the case of including the n = 2 RMPs, the dominant magnetic islands occur at the q = 2.5 and q = 3 surfaces, which correspond to resonant harmonics m/n = 5/2 and m/n = 6/2, respectively. Furthermore, the secondary island chain with helicity of m/n = 11/2 occurs at ψ p ≃ 0.96. When the n = 6 component is added, the magnetic surfaces near the edge (i.e. ψ p ≃ 0.98) are further broken and the stochastic region forms. The n = 6 fields produce additional magnetic island chains at the rational surfaces q = 2.66 and q = 2.83, which correspond to resonant harmonics m/n = 16/6 and 17/6, respectively. In addition, at the q = 5.5 (i.e. ψ p ≃ 0.96) surface, corresponding to m/n = 11/2, the n = 6 component induces a distortion of the island structures generated by the n = 2 RMP fields. Note that small island structures exist between the ψ p = 0.97 and the ψ p = 0.98 surfaces when the n = 6 RMPs are included. The n = 6 RMPs play a remarkable role in modifying the magnetic Figure 6. Real (left three panels), imaginary (middle three panels) and amplitude (right three panels) of the normal perturbations for the total RMP fields including the resistive plasma response. The top, middle and bottom three panels correspond to the n = 2, n = 6 and superposition of the n = 2 and n = 6 components, respectively. We adopt the coil phasing ∆Φ n=2 = −90 • and ∆Φ n=6 = 90 • for the n = 2 and n = 6 harmonics, respectively. Here, the toroidal angle ϕ = 0 is chosen for plotting. field topology near the edge region. Due to the difference of the Poincaré maps of field lines between the aforementioned two cases, the n = 6 component is expected to impact magnetic footprints on the divertor target plate as studied in the following. Note that a 1 kAt RMP current is assumed here in order to clearly display the individual magnetic islands in the plasma edge region. If a 10 kAt coil current were to be adopted, magnetic surfaces in the edge region would be substantially broken without individual islands being visible.    figure 10(d), the plasma response significantly reduces the minimum value of ψ p , compared with that for the vacuum RMPs, which implies an increase in the particle loss region inside the plasma.
When the resistive plasma response is included, the additional n = 6 sideband also has a substantial influence on the trajectories of the field lines, as shown in figure 10(b). In addition, for the convenience of describing magnetic footprints in the following, the distance along the limiter (denoted by L) away from the reference point (labeled by 0 in figure 10(g)) is defined.
For the cases studied here, the computations show that the RMP fields mainly impact the magnetic footprints on the upper outer and lower outer divertor plates, referred to as 'FT1' and 'FT2', respectively. Hence, the effects of RMP field perturbations on 'FT1' and 'FT2' footprints are presented Here, the RMP spectra in figure 6 are adopted, assuming 10 kAt RMP coil current. We choose the coil phasing ∆Φ n=2 = −90 • and ∆Φ n=6 = 90 • for the n = 2 and n = 6 harmonics, respectively.  distance' defined in figure 10(g) and the toroidal angle, respectively. Figure 11 reports the influence of vacuum RMP fields on the magnetic footprints on the outer divertor plates. For the case with n = 2 RMPs, the overall minimum value of ψ p,min reached by field lines is about ψ p,min = 0.95 ( figure 11(a)). However, when the n = 6 sideband is added, ψ p,min is reduced to 0.93, as shown in figures 11(b) for 'FT1'. In addition, the n = 6 sideband extends to the width of the footprint in terms of the 'limiter' distance L by 75% from ∼4 cm (figure 11(a)) to ∼7 cm ( figure 11(b)). The sum of the n = 2 and n = 6 field perturbations results in an additional secondary periodic variation of the magnetic footprint along the toroidal angle ( figure 11(b)). The 'FT2' footprint pattern is similar to that for 'FT1' for the chosen RMP coil configuration and the reference equilibrium with an up-down symmetric boundary shape. The overall minimum value of ψ p,min for 'FT2' is almost the same as that for 'FT1', implying the symmetry of heat load for the upper outer (figures 11(a) and (b)) and lower outer divertor (figures 11(c) and (d)) plates when the vacuum RMP fields are considered. However, due to the very slight up-down asymmetry of the limiter surface, the width of the footprint for 'FT2' is slightly different from that for 'FT1'. For 'FT2', the width of the footprint is ∼4.8 cm ( figure 11(c)) for the case with n = 2 vacuum field perturbation, which is extended to ∼7.7 cm as the n = 6 sideband is added ( figure 11(d)).
The magnetic footprints for the case of including resistive plasma response are plotted in figure 12. For 'FT1', the width of the footprint is ∼8 cm as the n = 2 perturbation ( figure 12(a)) is assumed, which is about double that for the vacuum case. In addition, the minimum value of the overall ψ p,min is ψ p,min ∼ 0.92, which is much smaller than the corresponding value for the vacuum case (ψ p,min ∼ 0.95). When an additional n = 6 component is taken into account, the 'lobe' structure is split ( figure 12(b)), which implies the extension of footprints in the toroidal direction at the 'distance parameter' L ∼ 418 cm. For footprints near the lower-outer divertor plate (figures 12(c) and (d)), the additional n = 6 sideband induces an extension of the magnetic footprint width and modifies the periodic variation in the toroidal angle similar to that for the vacuum field case ( figure 11(d)). Interestingly, the plasma response yields the symmetry breaking of the footprint pattern on the two outer divertor plates. This symmetry breaking is mainly due to the asymmetric features of the total RMP field perturbations in the top and the bottom regions as shown in figures 6(c), (f ) and (i). For the lower outer divertor plate, the overall pattern of magnetic footprint is similar to that for the vacuum case, due to the weak plasma response fields in the bottom region of the torus. Furthermore, the plasma response generally reduces the minimum value of overall ψ p,min researched by field lines.

Effect of RMP fields on fast ion losses
In addition to their influence on the magnetic footprints, RMP fields also have significant effects on fast ion confinement and losses. Fast ion losses further impact the heat load on the plasma facing materials. In the following sections, the synergistic effect of the n = 2 and n = 6 RMP fields on the fast ion losses is investigated.
Externally applied RMPs commonly induce flux surface breaking near rational surfaces and the formation of magnetic islands in the presence of resistive plasma response. Island overlapping often causes magnetic field line stochasticity inside the plasma. It is therefore important to study how the flux surface breaking impacts the fast ion losses. To contrast this, we also consider the ideal plasma response in figure 13. In an ideal plasma, RMPs mainly induce distortion of magnetic field lines without changing the flux surface topology. Consequently, we expect differences in the fast ion losses due to RMPs, by assuming ideal versus resistive plasma response models. These differences indeed appear in figure 13. Figure 13 reports the final positions of 8100 test particles and fast ion losses along the poloidal angle after ∼5 ms timescale simulation, in the presence of RMP fields including either ideal or resistive plasma response. Numerically, we find that most of the losses occur within the first 5 ms, which is why we choose this timescale in our simulations to study the key features of fast ion losses. We note that the same timescale (5 ms) was also chosen to study the effect of RMPs on fast ion transport through the full orbit simulation [39]. Here, the test fast ions are launched at a given radial position (labeled by the normalized poloidal flux poloidal flux ψ p0 ) and from the outboard mid-plane, with a uniform distribution on a 90 × 90 phase-space mesh at the particle pitch (0.3< λ <1) and energy (10 keV < E <80 keV). The initial fast ions thus cover the majority of the particle population produced by the co-current NBI, which is assumed for the reference equilibrium design in this work. Figure 13 shows that the fast ion losses induced by the RMP fields including resistive plasma response are much larger than that caused by the case of including ideal plasma response. For instance, the resistive plasma response enhances the fast ion losses from ∼0.086% to ∼1.3% at the poloidal angle χ ∼ 0 • . The final distribution of test particles in the poloidal cross section for both of the above cases is radially extended. The lost fast ions mainly concentrate at four poloidal locations: the upper outer divertor plate, the outboard mid-plane and the lower two target plates. This indicates that the resistive plasma response induced breaking of flux surfaces is essential for fast ion losses here. This also implies that the n = 6 sideband of RMP fields with resistive plasma response may impact fast ion losses, since the additional n = 6 component enhances the breaking of magnetic surfaces near the edge and widens the stochastic region, as shown in figure 8. However, for the case of an ideal plasma response, the pitch-resonant radial fields from the vacuum RMP fields vanish at the rational surfaces, which does not change the magnetic topology and only induces the distortion of flux surfaces. The fast ion redistribution in (R-Z)-domain is similar for these two considered response models, which suggests that the field perturbations excluding the pitch-resonant component mainly induce fast ion redistribution or transport inside the plasma for the studied case. The particle loss mechanism related to the fractional resonance between the RMP fields and fast ions is not analyzed here; it has been reported in [33] and the references therein. In figure 13, only the n = 2 RMPs is considered to distinguish the difference between the ideal and the resistive plasma responses, in terms of affecting fast ion losses.
For a detailed analysis of the effect of RMP fields on fast ion losses, the trajectories of three representative test particles are shown in figure 14. This shows that the sum of the n = 2 and n = 6 perturbations enhances the radial drift of test fast ions, as compared to the case of single-n (n = 2) harmonic. It is evident that the field perturbation can change the type of particle orbit (e.g. from an initially trapped particle to a passing one), as shown in figure 14(a). Test particles #1 and #2 are still well confined in the presence of the n = 2 total field perturbation, which are lost when the n = 6 sideband is taken into account, as shown in figures 14(a) and (b), respectively. Figure 14(c) shows that the added n = 6 sideband affects the deposition of lost fast ions. For the cases of the n = 2 RMP field and of the combined n = 2 and n = 6 components, the lost fast ions hit the outboard middle-plane and the upper-outer divertor plate, respectively. Figures 14(d) and (e) show that the extreme radial positions (i.e. s 2 ≡ ψ p ) reached by the test particle in the HFS and LFS periodically vary due to the n = 2 RMP fields, at the simulation timescale. These two confined fast ions are lost on the timescale of ∼1 ms, when the additional n = 6 component is included. Before being lost, the added n = 6 component induces a gradual increase in the maximum of ψ p reached by the particle and the final loss as the particle moves across the LCFS. Figure 14(f ) shows that the n = 6 sideband induces the loss of the fast ion in a shorter timescale than that with the n = 2 component alone. The fast ion can move from the plasma region to the vacuum region and/or enter back into the plasma region, due to the magnetic drift. Figure 14 shows that the n = 6 sideband induces additional radial drift of fast ions, as compared to the n = 2 RMP alone case. This is mainly due to the enhanced field line stochasticity in the plasma edge region, as shown in figure 8.
The additional radial drift due to the n = 6 sideband as discussed above does not significantly impact the final configuration positions of test fast ions inside the plasma as shown in figure 15(a). However, figure 15(b) shows that the fast ion losses greatly rise due to the additional n = 6 RMPs, as compared to the case of considering the n = 2 RMPs alone. For instance, the fast ion loss is enhanced from 0.95% for the latter to 8.8% for the former, near the upper-outer divertor plate (i.e. χ ∼ 100 • ).
For a more detailed analysis of the dependence of fast ion losses on the pitch and kinetic energy of particles, figure 16 reports the effect of RMP fields on the fast ion loss region in phase space. The fast ions initially launched in the 'loss region' are finally lost due to the RMP fields. This shows that the n = 2 RMP fields mainly induce the losses of fast ions near the topright triangular area in phase space, where the particles are passing ones. Fast ions with a relatively higher kinetic energy E > 40 keV and a larger pitch λ > 0.6 are likely lost. This suggests that the motions of passing fast ions are more likely to be affected by the RMP field than the trapped ones. The combined n = 2 and n = 6 field perturbations significantly extend the loss region. The kinetic energy boundary of 'loss region' is extended to ∼10 keV and the pitch boundary is slightly modified when the intrinsic n = 6 sideband fields are included.
Since the fast ion orbit drift increases with the particle energy, higher energy particles are more prone to loss if the particles drift outwards. Moreover, the 3D trajectory of the particle drift orbit can also resonate with the magnetic perturbation. Such a resonance induces a radial transport or even  The corresponding fast ion losses (in percentages with respect to the total number of test particles) along the poloidal angle on the limiter surface. We adopt the coil phasing ∆Φ n=2 = −90 • and ∆Φ n=6 = 90 • for the n = 2 and n = 6 harmonics, respectively. The 10 kAt RMP coil current is assumed. particle loss [67]. Since the fast ion drift orbit depends on both the particle energy and pitch, the resonance condition can only be satisfied in certain regions of the particle phase space, for a given magnetic perturbation structure and magnitude. This is a possible reason for the occurrence of the threshold reported in figure 16(a). Furthermore, the additional n = 6 RMP enhances the stochastic region near the plasma edge, resulting in a larger loss of fast ions with relatively lower energy. This can explain the results in figure 16(b). The effect of the increased stochastic region on the fast ion transport/loss is similar to that for the thermal particles [68]. The loss map method, introduced in [69], has been shown to be useful for detailed analysis of fast ion losses related to the distribution in the pitch-energy space and will also considered in our future studies.
In the above simulations, a 90 × 90 particle mesh in the pitch-energy space is adopted. However, simulations with a Figure 16. The lost fast ion initial locations (in yellow) in the pitch-energy phase space for the case of (a) including only the n = 2 RMP fields and of (b) including both the n = 2 and n = 6 components, corresponding to the two cases in figure 15. The initial radial position ψ p0 = 0.95 is assumed. Here, the ∼5 ms timescale simulation is carried out. Figure 17. Loss rate of fast ions as a function of the initially launched radial position (a) and of the assumed RMP coil current, for the case with only an n = 2 component (in blue) and with both the n = 2 and n = 6 components (in red). Here, the resistive plasma response is included in the RMP fields. A 10 kAt RMP coil current is assumed in (a), and the initial position of test particles ψ p0 = 0.95 is adopted in (b). The coil phasing ∆Φ n=2 = −90 • (∆Φ n=6 = 90 • ) for the n = 2 (n = 6) RMPs is fixed during the parameter scan here. 20 × 30 particle mesh in phase space also capture the key features of fast ion losses due to RMP fields in this work, including: (i) the fast ion loss rate (section 8); (ii) the threshold in particle energy and pitch for fast ion losses; and (iii) the loss pattern along the limiting surface. A reduction in the total number of test particles is computationally much more efficient, especially for the following parameter scans.
Dependences of the fast ion loss rate on the initial radial position ψ p0 and on the RMP coil current are studied assuming 600 test particles (i.e. on the 20 × 30 phase space grid), with results reported in figure 17. Here, the loss rate is referred to as the ratio of the number of lost fast ions to the total number of test particles. Figure 17 clearly shows that a threshold (denoted by ψ 0,t ) of ψ p0 exists for fast ion losses induced by the RMP perturbations. For the case of the n = 2 RMP fields, ψ 0,t is about 0.9, which is larger than that (ψ 0,t ≃ 0.875) assuming the combined n = 2 and n = 6 perturbations. This shows that the intrinsic n = 6 sideband extends the radial region in which the fast ions can be lost by the RMP fields. For the fast ions initially located in the region ψ p0 < ψ 0,t , the RMP fields mainly induce redistribution, instead of losses, for the studied case here. Studies of fast ion redistribution in phase space and in configuration space will be carried out in the future. In this work, we mainly focus on the features of fast ion losses. The fast ion loss rate is significantly increased by the combined field perturbations, as compared to the case of the single n = 2 RMPs. In the case of particles deposited at ψ p0 = 0.98, adding the n = 6 sideband increases fast ion losses from ∼18% to ∼48%.
Another key parameter affecting fast ion loss is the magnitude of the RMP coil current (I RMP ) at a given toroidal phasing between the upper and lower coil current. The threshold of the RMP coil currents I RMP,t for fast ion loss is about 5.3 kAt for the combined n = 2 and n = 6 RMP fields, which is smaller than that (∼7 kAt) with the n = 2 harmonic alone. The occurrence of the threshold in the RMP coil current for fast ion loss is likely due to the fact that the radial extension of the stochastic region near the plasma edge reaches the initial position of test particles. Assuming 10 kAt RMP coil current, the fast loss rate reaches ∼30% for the superposed perturbations, which is much larger than that (∼10%) for the n = 2 field case. The sideband-induced enhancement of the fast ion loss rate is consistent with that observed in the DIII-D tokamak [27]. The threshold signature of I RMP,t agrees with the experimental observations in the KSTAR device [35,70]. For various initial positions of launched test particles or different magnitudes of RMP coil current, the n = 6 sideband significantly enhances the fast ion loss rate, as ψ p0 or I RMP exceeds a critical value.

Numerical sensitivity test
In the above computations, the initial poloidal angle location for test particles is fixed at χ = 0 (i.e. at the outboard middle-plane). However, the fast ion loss rate is also related to the initially poloidal location, in addition to the initial radial location. In order to test the dependence of the main conclusions on the initial poloidal location, figure 18(a) shows an example of launching test field lines at different poloidal angles (χ = −100 • , 0 • , 50 • ), while the initial radial position (i.e. ψ p0 = 0.95) is fixed. At χ = 50 • , the field line directly connects to the limiter, before passing one whole period in the toroidal direction in the plasma, which implies that the open field line occurs at this location. For the case of χ = 0 • , the field lines are in the marginal region to form closed field lines. For the case with n = 2 RMP fields, the fast ion loss rate for the case of launching fast ions at χ = 50 • is about 34%, which is much larger than that with χ = 0 • ( figure 18(b)). This is mainly due to the fact that the magnetic topology at χ = 50 • differs from that at χ = 0 • . For the case with χ = −100 • (i.e. region below the outboard middle-plane), the field lines are at the boundary of forming open field lines. While for the fast ions, the equivalent initial radial position ψ p,0 at the outboard middle-plane is larger than 0.95 due to the particle's outward magnetic drift in the LFS. As a result, the fast ion loss rate for the case with χ = −100 • is much larger than that for the reference case with χ = 0 • . However, we emphasize that the intrinsic n = 6 sideband enhances the fast ion loss rate by ∼20%, which does not depend on the choice of initial poloidal locations for launching test particles ( figure 18(b)). Furthermore, this implies that the relative enhancement of the fast ion loss rate by the combined n = 2 and n = 6 components is insensitive to the initial toroidal location of the test particles, due to the toroidal periodic feature of Poincaré maps of field lines. However, for a given RMP, the absolute loss rate of fast ions will depend on the initial 3D positions of test particles.
Further numerical sensitivity tests are carried out to study the dependence of the fast ion loss rate on the assumed number of test particles ( figure 19). The loss rate of fast ions with 8100 particles becomes 9.1%, which is slightly lower than the value of 9.5% obtained with 600 particles. This shows that the loss rate very slightly depends on the number of test particles. Hence, the chosen number (N = 600) of test particles in figure 17 is sufficient to capture the key features of the fast ion losses induced by the RMP fields. For quantitative predictions of fast ion loss rate for the HL-2M scenario, a realistic fast ion distribution (e.g. from TRANSP modeling) should be considered and large-scale modeling by launching a large number of test particles is required, which is left for a future study.

Conclusion and discussion
In this work, the MARS-F code is applied to study important aspects of the plasma response to the dominant toroidal component (n = 2) and the secondary sideband (n = 6) RMP fields for the double-null scenario for the new HL-2M tokamak device. The amplitude of the n = 6 pitch-resonant radial field perturbation reaches 35% of that for the n = 2 component at the q = 3 rational surface for the reference case. The total RMP fields, including the resistive plasma response, in the top region of the torus are much larger than that in the bottom region, for both the n = 2 and n = 6 components. We find that the n = 6 sideband modifies the 3D structure of the field perturbations in the configuration space, as compared to the case of including the n = 2 harmonic alone. In particular, the additional n = 6 harmonic induces substantial periodic variation of the amplitude of the total perturbations along the toroidal angle, which is constant for the n = 2 RMP field.
The n = 6 sideband induces secondary magnetic islands and extends the stochastic region near the plasma edge, as shown in the Poincaré maps of field lines. As a result, adding the n = 6 harmonic significantly modifies the pattern of magnetic footprints, as compared to the case of considering the n = 2 fields alone. When the plasma response is included, adding the n = 6 harmonic results in the splitting of 'lobe structure' of the upper outer magnetic footprints and extends the width of the lower outer footprints. Interestingly, the plasma response yields the symmetry breaking of the patterns of the footprints on the two outer divertor plates.
Furthermore, we show that the fast ion losses are sensitive to the magnetic topology near the plasma edge, by comparing cases of the ideal and resistive plasma response. The n = 6 sideband of the RMP fields enhances fast ion losses. For the studied case, a threshold value exists for inducing fast ion losses, in terms of the initial radial position of test particles and of the RMP coil current. The existence of the n = 6 sideband reduces the aforementioned threshold values and greatly enhances the fast ion loss rate as the threshold criterion is satisfied, as compared to the case of including the dominant component n = 2 alone. The n = 6 sideband also extends the 'loss region' in phase space where the fast ions will be lost in the presence of field perturbations.
Although a relatively small number of test particles is assumed in this work, the role of the sideband of RMP fields on fast ion losses and threshold features of the RMP coil current for fast ion losses are consistent with the experimental observations on the DIII-D and KSTAR devices, respectively [27,35]. However, quantitative prediction of fast ion loss properties (the loss rate, the phase space dependence as well as the deposition location of lost particles) requires knowledge of the realistic equilibrium distribution in both the particle phase and 3D configuration spaces. We leave this to a future study using more dedicated particle tracing codes such as ASCOT [71]. Moreover, this study does not account for the influences of electric field, recombination and charge exchange on fast ion confinement and losses. These aspects are left for future investigations, especially concerning their quantitative comparison with experimental data. In addition, we emphasize that the qualitative findings from this study, being that the toroidal sidebands of the RMP field play important roles in divertor magnetic footprints and fast ion losses in HL-2M, should also generally apply to other tokamak devices. In particular, these new findings can have significant implications for RMP applications in future devices, where linear plasma response modeling efforts have often neglected sideband effects.

Disclaimer
This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.