Toroidal modeling of plasma flow damping and density pump-out by RMP during ELM mitigation in HL-2A

Reduction of both the plasma density and toroidal flow speed, due to application of the predominantly n = 1 (n is the toroidal mode number) resonant magnetic perturbation (RMP) for controlling the edge localized mode in the HL-2A tokamak, is numerically investigated utilizing the quasi-linear initial-value code MARS-Q (Liu et al 2013 Phys. Plasmas 20 042503). Simulation results reveal that the neoclassical toroidal viscosity (NTV) due to three dimensional fields plays the key role in modifying the plasma momentum and particle transport in the HL-2A discharge. By comparing the modeling results with the measured density pump-out in the experiment, the electron NTV particle flux model, in combination with the free-boundary condition for the axisymmetric change of the density at the plasma edge, is found to yield the best agreement in terms of both the pump-out level and the overall time scale. Further sensitivity studies show that the simulated density pump-out level is reasonably robust against variations in the model assumptions, including the particle diffusion model and the non-ambipolar versus ambipolar NTV particle flux. The latter however affects the time scale for reaching the steady state solution. Finally, it is found that the plasma edge-peeling response, the NTV torque, as well as the plasma momentum and particle transport, all are sensitive to the toroidal phase difference between the upper and lower rows of the RMP coil currents in HL-2A, with the 30∘ coil phasing producing the minimal side effects on the plasma.


Introduction
Resonant magnetic perturbation (RMP) coils have been installed in many present-day magnetic confinement fusion devices, for the main purpose of actively controlling the edge localized mode (ELM) in H-mode plasmas [1][2][3][4][5][6][7]. This technique is also planned to be applied in ITER [8][9][10][11]. On the other hand, the toroidal asymmetry produced by three dimensional (3D) RMP fields can cause substantial side effects on the plasma transport, such as reductions of the plasma density and/or the toroidal flow speed, which in turn impacts the plasma energy confinement and stability [12,13]. Understanding physics mechanisms of the plasma density pump-out and rotation damping due to RMPs is therefore essential for the success of ELM control in future tokamak devices.
It has been recognized that the neoclassical toroidal viscosity (NTV) torque plays an important role in the plasma toroidal flow damping in tokamaks in the presence of 3D perturbations [12,[14][15][16][17]. The NTV torque arises from a radial current exerted by non-ambipolar radial particle flux due to the toroidal symmetry breaking [18,19]. Such a torque is generally small in tokamak geometry, but can be substantially enhanced by the Landau resonant effect between the 3D magnetic perturbations and the precessional drift of thermal particles in the lowcollisionality and slow-flow regimes.
The RMP-induced plasma density pump-out has not been fully understood. An analytic particle transport model, associated with strong magnetic field line stochastization near the plasma edge assuming vacuum field approximation, was proposed to account for this phenomenon [20]. Non-linear magnetohydrodynamic (MHD) simulations with the JOREK code revealed that the field line ergodicity and the X-point displacement can only partly explain the density pump-out observed in the ASDEX Upgrade experiment [21]. A cylindrical model showed that the formation of magnetic islands at the top and foot of the edge pressure pedestal may explain the density pump-out induced by the n = 2 (n is the toroidal mode number) RMP in low-collisionality DIII-D plasmas [22]. Recent toroidal modeling found that the neoclassical particle flux associated with NTV plays an important role in the density pump-out during ELM control by RMP in DIII-D [23].
In this work, we perform systematic toroidal modeling of both the plasma flow damping and density pump-out due to the externally applied n = 1 RMP field for the HL-2A tokamak experiment, where the type-I ELMs were successfully mitigated by RMP [24] as shown in figure 1(a). Figures 1(b) and (c) shows time evolution of the plasma density and toroidal rotation frequency, measured by the far infrared (FIR) interferometer polarimeter and the charge exchange recombination spectroscopy (CXRS) system, respectively. A global reduction for the plasma density due to the applied RMP field is observed ( figure 1(b)). On the other hand, the plasma toroidal rotation is mainly affected (reduced) in the edge region ( figure 1(c)). The main purpose of the present study is to identify the physics mechanisms associated with these transport processes due to 3D fields in the HL-2A ELM control experiment. The neoclassical effect is our focus of investigation.
The paper is structured as follows. The next section briefly describes the computational models. Section 3 describes the HL-2A plasma equilibrium and the RMP coil configuration. Section 4 reports the modeling results assuming the experimental plasma conditions. As an important part of this investigation, sensitivity studies with respect to various model assumptions are also carried out here in order to demonstrate the robustness of the simulation results. Conclusions are drawn in section 5.

Computational model
In this work, the 3D-field induced momentum and particle transport modeling is achieved by solving the linear plasma response equations together with the axisymmetric perturbations in a self-consistent manner. The resulting model is referred to as the quasi-linear model since only the non-linear interaction between the n = 0 and a given n ̸ = 0 perturbations are taken into account [25]. The linear plasma response is described by a single fluid resistive MHD model, which includes the equilibrium plasma toroidal flow in full toroidal geometry (1) where the variables ρ, v, b, j and p denote the perturbed plasma density, velocity, magnetic field, plasma current and pressure, respectively. η is the plasma resistivity. V 0 = RΩ∇ϕ is the toroidal equilibrium flow speed assumed to be subsonic in the model, where R is the plasma major radius, Ω the toroidal rotation frequency, and ϕ the geometrical toroidal angle. Γ = 5/3 is the ratio of specific heats. ∆ρ n=0 represents the surfaceaveraged change to the equilibrium plasma density ρ eq . The RMP field is produced by the ELM control coil current j RMP that satisfies at the coil location in the vacuum region outside the plasma.
Note that the coil current is assumed to be a surface current in our model, with the exp(inϕ) dependence along the toroidal angle ϕ. For the plasma response computation, the above MHD equations and the coil equation, together with the vacuum equation outside the plasma and a resistive wall equation (with the thin-wall approximation), are numerically solved by the MARS-F code [26] in a toroidal flux coordinate system (s, χ, ϕ), where s ≡ ψ 1/2 p labels the radial coordinate with ψ p being the normalized equilibrium poloidal flux. χ is the (generic) poloidal angle.
The above model equations are augmented by transport equations for the surface-averaged (i.e. the n = 0 component) plasma density and toroidal rotation, in order to investigate the self-consistent interplay between the RMP field and the axisymmetric quantities. More specifically, the following n = 0 toroidal momentum balance equation is also solved where L ≡ ρ⟨R 2 ⟩Ω is the surface averaged toroidal momentum of the plasma, and D(L) the momentum diffusion operator with G ≡ F⟨1/R 2 ⟩ denoting a geometric factor. Here, F is the equilibrium poloidal current flux function. χ M represents the toroidal momentum diffusion coefficient, and V pinch the velocity pinch term which we shall ignore in our further modeling. As for the 3D-field induced momentum sink, we include three toroidal torques, i.e. the NTV torque (T NTV ), the resonant electromagnetic torque (T JXB ) and a torque associated with the Reynolds stress tensor (T REY ). Detailed expressions for these torques can be found in our previous work [27]. T source here denotes the momentum source. In the present model, we assume that the toroidal momentum balance, D(L(t = 0)) + T source = 0, has already been established at the time (t = 0) when the RMP field is applied. We also assume that the RMP field does not modify the momentum source. This allows us to solve for the change of the toroidal momentum △L ≡ L(t) − L(0) relative to the initial value L(0), without considering the detailed form of the momentum source term Effectively, the momentum source term enters into our model via the initial flow velocity. Similarly, we derive an equation for the radial balance of plasma thermal particles. Assuming that the particle balance has already been achieved before the application of RMP, the change of the plasma density, ∆ρ n=0 , due to 3D perturbations is then obtained by solving where the first term from the right hand side of equation (10) is associated with mass conservation within the MHD model. The second term is associated with the neoclassical effect (due to 3D perturbations) which is beyond the standard single-fluid MHD model. The last term describes the particle diffusion.
The above quasi-linear model equations (1)-(6) and (9), (10) are solved by the MARS-Q code [23,25] as an initial value problem. At the magnetic axis, free boundary conditions are assumed for the change of the n = 0 plasma momentum (9) and mass density (10). The choice of proper type of boundary conditions at the plasma edge is less certain. Both free (Neumann) and fixed (Dirichlet) boundary conditions for the plasma edge are implemented in MARS-Q, and considered in this study as part of the sensitivity investigation versus model assumptions. A semi-implicit, adaptive time advance scheme is devised for solving the above quasi-linear equations. The MARS-Q formulation has been well validated against experiments in MAST-(U) [15,28], ASDEX Upgrade [17], and DIII-D [23].
In view of the important role played by the neoclassical particle flux (and toroidal torque) in this study, we provide below a more detailed description of this physics model. The radial NTV particle flux Γ NTV and the associated toroidal torque T NTV satisfy the so-called flux-force relation ⟨Γ j NTV · ∇ψ⟩ = (−1/e)T j NTV , where j = i, e specifies the particle species (i.e. thermal ions or electrons) and where n j , e j and ω tj represent the number density, electric charge and thermal frequency of the given particle species, respectively. ε is the inverse aspect ratio,ψ p the (unnormalized) equilibrium poloidal flux, and V the plasma volume enclosed by the flux surface. As a key factor that is relevant to our study, λ n is proportional to the resonant operator 1/(ω E×B − ω d ) in the low-collisionality plasma, where ω E×B represents the E × B drift frequency and ω d the precessional drift frequency of particles due to ∇B and the magnetic curvature. Ω j nc,n denotes the neoclassical offset rotation [29]. An approximate analytic expression for the NTV torque has been obtained in [30,31], where various plasma collisionality regimes are smoothly connected. The above semi-analytical NTV particle and torque models have been implemented in MARS-Q.
We also emphasize that the (particle and momentum) transport considered in the present work mainly relies on the NTV physics, which is different from the transport induced by the field line stochasticity [20,32,33]. The NTV-induced transport is largely associated with the non-resonant spectrum of the applied 3D field perturbation. The field line stochasticity, mainly due to the response of the resonant spectrum (i.e. formation and overlapping of magnetic islands), often plays a limited role in the RMP field induced transport. This is because the plasma response in a rotating plasma typically significantly reduces the size of the magnetic islands (the so-called plasma screening effect) and hence, in most RMP experiments, does not result in field line stochasticity. We also mention that, although the additional transport due to (limited) field line stochasticity can be captured by MARS-Q via modification of the diffusion and pinch terms (e.g. equation (8)) in the particle and momentum balance equations, we choose to ignore this effect in this study by assuming that these diffusion and pinch terms do not change before and after application of the RMP field. Instead, we focus on studying the roles of NTV.

Plasma equilibrium and RMP coil configuration
The experimental data used in this modeling are extracted from the HL-2A ELM control discharge 36 965. The on-axis vacuum toroidal magnetic field is B 0 = 1.4 T with the plasma major radius of R 0 = 1.65 m. The plasma equilibrium is reconstructed at 1150 ms. The fixed boundary equilibrium code CHEASE [34] is then utilized to produce the input data for MARS-Q. The radial profiles of the key equilibrium parameters, including the safety factor, plasma pressure, plasma density and the toroidal rotation frequency, are shown in figure 2. Note that the plasma pressure shown here is normalized by B 2 0 /µ 0 . The normalized beta (β N ) is about 2 for this equilibrium. The plasma density is normalized to unity at the magnetic axis. The toroidal rotation profile was measured by the CXRS system. The plasma resistivity is assumed to follow the Spitzer model in this work. Figure 3 shows the plasma boundary shape for this discharge, together with the poloidal location of the RMP coils in HL-2A. A 2 × 2 coil system is applied in the experiment, with an upper and a lower row (with respect to the outboard mid-plane) each consisting of two window-frame coils located on the opposite sides along the toroidal angle. The maximum coil current is 5 kAt. The coil current is set to be of opposite signs between the upper and lower rows (i.e. with odd parity) in discharge 36 965. In experiment, the dominant RMP field component (in the plasma region) is the n = 1 component. This will also be the case assumed in both the linear and quasi-linear simulations reported below.

Linear plasma response
In what follows, we compute first the resistive plasma linear response to the n = 1 RMP field for this HL-2A discharge. Figure 4 compares the poloidal spectra of the radial field perturbation between the vacuum approximation and that including the plasma response. The radial field here is defined as where B eq is the equilibrium magnetic field. Note that the Fourier harmonics of b 1 are calculated in a straight-field-line flux coordinate system in these plots.
The computed results show two distinct effects introduced by the plasma response. One is the overall field amplification and the other the resonant field screening effect. This HL-2A plasma significantly amplifies the m > 0 non-resonant harmonics of the applied RMP field-the largest amplification factor is about 5. This plasma response induced field amplification is important for obtaining large NTV torque and NTV particle flux, which play large roles in the plasma momentum and particle transport due to 3D fields as we will show later on.
On the other hand, the resistive plasma response also introduces significant screening of the resonant spectrum, as is already evident from figure 4(b) and even more clear from figure 4(c). For this HL-2A plasma, the amplitude of the resonant harmonics is reduced by about one order of magnitude at the radial location of the corresponding rational surfaces. We mention that similar screening effects have also been reported in modeling of other devices [17,35]. Figure 4(d) shows magnitude of the normal displacement of the plasma, |ξ n | = |ξ · ∇s|/|∇s|, resulted from the plasma response to the RMP field. We find that the normal displacement strongly peaks near the X-point, indicating that the RMP field induces a strong edge peeling response. This type of plasma response, associated with the n = 1 RMP in odd parity configuration in HL-2A, has been found to strongly facilitate ELM control in other devices [36][37][38][39] as well as in HL-2A [35].

Quasi-linear simulations of toroidal flow damping
Having made a good understanding of the plasma response spectrum for the RMP field, we now perform the MARS-Q initial-value quasi-linear simulations for this HL-2A discharge, with the first set of results summarized in figure 5.  Here, we focus on the plasma toroidal flow damping. We found that the toroidal rotation is mainly damped in the plasma region outside the radial coordinate of s = 0.5 (figures 5(c) and (d)). The core rotation is however merely affected by the RMP field. An important observation here is that both the overall damping level and the detailed radial shape of the damped flow velocity agree reasonably well with that measured in experiment, the latter being shown as the dashed line in figure 5(c).
We emphasize that a well saturated steady state solution is obtained by MARS-Q simulation in this case. Figure 5(a) shows that the largest resonant field component (m/n = 5/1) saturates to a level of several Gauss. Because the plasma flow remains finite at saturation, no full penetration of the field occurs and the resonant spectrum of the applied RMP field remains well shielded by the plasma. Figure 5(b) compares time traces of the net torques (i.e. torques integrated over the plasma volume) among the NTV contribution, the resonant electromagnetic contribution (JXB) as well as that contributed by the Reynolds stress tensor. It is evident that the NTV torque plays a dominant role in the flow damping simulated for this HL-2A discharge. This motivates a closer look into the NTV torque as shown in figure 6.
In our model, the NTV torque comes from both the thermal ion and thermal electron contributions. For each particle species, the torque is divided into the resonant and non-resonant portions. Figure 6 shows radial profiles of all these individual contributions. The thermal ion-induced NTV torque density strongly peaks near the radial location of s = 0.92 ( figure 6(a)). This peaking is in turn associated with the resonant contribution, which is known to significantly enhance the NTV torque. As mentioned earlier, the perturbation-particle Landau resonance occurs when the E × B rotation frequency of the plasma (for a static perturbation) matches the precessional drift frequency of trapped thermal ions. On the other hand, most of the electron contribution comes from the non-resonant NTV torque (figure 6(b)). As a consequence, the overall torque amplitude due to thermal electrons is much smaller (by one order of magnitude) than that of thermal ions. The toroidal flow damping in this HL-2A discharge is thus mainly facilitated by the resonant NTV torque from thermal ions. The lack of resonant NTV torque from electrons is due to high electron collisionality, which weakens the resonant effect. Note that the (non-resonant) electron NTV torque density peaks only near the very edge of the plasma. Note also that the ion and electron contributions have opposite signs as theoretically expected.

Quasi-linear simulations of density pump-out
Next, we analyze the role of the NTV flux in the plasma particle transport in the MARS-Q quasi-linear simulation. As mentioned earlier, various model assumptions can be made on the NTV particle flux (electron, ion, or ambipolar contribution), the particle radial diffusion as well as the boundary condition for the plasma density. A systematic sensitivity investigation will be pursued with results reported in the next subsection. In what follows (figure 7), we show a base case assuming the thermal electron NTV flux and the free boundary condition. This is also one of the cases where the simulated density variation agrees well with that measured in the HL-2A experiment. Figure 7(a) here plots the time trace of the net density pump-out fraction defined as where s ≡ √ ψ p and n eq is the initial (equilibrium) density before application of RMP. The simulation finds a global reduction of plasma density, with the pump-out fraction reaching about 25% towards steady state. This overall density pump-out level is in good agreement with the experimental measurement. The simulated time scale for reaching saturation exceeds 200 ms which is longer than, but not too far from, that observed in experiment (∼100-150 ms). Figure 7(b) shows the simulated time evolution of the radial profile of the plasma density during RMP. Note that all the profiles here are normalized by the initial on-axis value. The saturated density profile after RMP as measured in experiment (dashed pink line), subject to the same normalization, is also plotted here for comparison. Note also that  only 41 simulated profiles are shown during the profile evolution, at equally spaced time slices. We find a good quantitative agreement between the modeling and the experiment, not only for the overall pump-out level but also for the saturated density profile. In this simulation, the partial radial diffusion coefficient χ D is assumed to be about 0.1 m 2 s −1 at the magnetic axis, with a radial profile that scales as T −3/2 e (to be referred to as the Spitzer-like model hereafter). As will be shown next, a similar pump-out level is found by assuming a uniform particle diffusion coefficient.

Sensitivity studies
In what follows, we investigate how the simulated plasma density pump-out depends on various model assumptions, including the particle diffusion coefficient mentioned above, the boundary condition for the density change ∆ρ n=0 , the role of (non-)ambipolar NTV particle flux, as well as the current phasing (∆ϕ U/L ) between the upper and lower rows of the ELM control coils.

Effect of particle diffusion model.
The particle diffusion coefficient (χ D ) is an input parameter of the MARS-Q quasi-linear model, that can either be a constant or adopt a radial profile. Figure 8(a) compares two simulation results, assuming either the uniform model (red curve) or the aforementioned Spitzer-like model (blue curve). In both cases, the same on-axis value of χ D = 0.1 m 2 s −1 is assumed at the magnetic axis. Assumed are also the same HL-2A plasma and the RMP coil configuration. We find that the density pumpout level is not very sensitive to the particle diffusion profile. However, the Spitzer-like model produces faster saturation of the pump-out process, which agrees better with experiments in terms of the time scale. On the other hand, the overall profiles of the simulated plasma density, after ∼450 ms, are similar between the two models and both agree reasonably well with the experiment, as shown in figure 8(b) (for the uniform diffusion model) and figure 7(b) (for the Spitzer-like mode). These results indicate that the particle diffusion profile does not play a major role in the simulated density pump-out for the HL-2A discharge.

Effect of boundary condition.
Next, we examine the influence of the boundary condition for ∆ρ n=0 at the plasma edge on the simulated density pump-out. The results are reported in figure 9 where the fixed boundary condition ∆ρ n=0 = 0 is assumed, with all other model parameters being exactly the same as that for figure 7. Compared to the free-boundary case with ∂∆ρ n=0 /∂s = 0 (blue solid curve in figure 9(a)), the fixed boundary condition (red dashed curve in figure 9(a)) results in lower level of density pump-out and much slower time scale (∼1 s) for the particle transport. Interestingly, the lack of pump-out, as compared to the experimental measurement ( figure 9(b)), occurs in the outer region towards the plasma edge. On the other hand, the pump-out level near the magnetic axis matches well with the experiment. Nevertheless, experimental evidence here (in particular the radial profile evolution near the plasma edge) does not generally support the fixed boundary model for the density pump-out.

Non-ambipolar versus ambipolar NTV particle flux.
We have so far assumed the electron NTV particle flux in the MARS-Q density pump-out simulations. At steady state, the plasma particle transport is generally considered an ambipolar process. During the transient phase, however, a time-varying radial electric field exists which drives a radial current. This radial current J r , which is also ultimately responsible for producing the NTV torque, is related to the non-ambipolar particle flux where the J i r , J e r and Γ i , Γ e are the radial currents and particle fluxes associated with bulk ions and electrons, respectively. The net radial current vanishes as the particle flux becomes ambipolar, i.e. Γ i = Γ e . During the transient phase (i.e. before the plasma rotation reaching the state steady in the MARS-Q Figure 10. Radial profiles of the NTV particle fluxes for the HL-2A discharge 36 965 at 1150 ms and based on the linear plasma response. Compared are the thermal ion flux (solid green), the thermal electron flux (solid red), and the ambipolar portion (dashed blue) among the two particle species. Assumed is the n = 1 RMP configuration in odd parity at 5 kAt coil current. Vertical dashed lines indicate the radial location of the n = 1 rational surfaces. simulation), we expect the non-ambipolar particle flux Γ i ̸ = Γ e . We note that even in the transient phase, an 'ambipolar' portion of the particle flux can still be defined, as the minimum flux among Γ i and Γ e . This is illustrated by the following figure. Figure 10 shows the MARS-F computed radial profiles of the NTV particle fluxes from the thermal ion (solid green curve) and thermal electron (solid red curve) contributions. The 'ambipolar' portion of both contributions is indicated by the dashed blue line. Note that as an example, the computed Figure 11. The simulated density pump-out for the HL-2A discharge 36 965, assuming the ion NTV particle flux due to the n = 1 RMP field and free boundary condition at the plasma boundary. Plotted are (a) the time trace of the net pump-out fraction ∆n/neq ( dash-dot red curve) compared that assuming electron NTV particle flux (solid blue), and (b) the (normalized) radial profile evolution of the plasma density. The arrow in (b) indicates the time flow of the simulation. Shown in (b) are only 41 equally spaced time slices during the profile evolution, with the initial (thick blue) and the final (thick red) profiles being highlighted. The dashed pink curve shows the FIR-measured profile at 1260 ms (corresponding to the maximal density pump-out) in this discharge. Vertical dashed lines indicate the radial location of rational surfaces associated with the n = 1 mode. Assumed are RMP coils in odd parity at 5 kAt coil current. linear plasma response is used here to evaluate the particle fluxes. The latter are normalized by a factor Γ 0 = n 0 V A , where n 0 and V A represent the (equilibrium) particle number density and the toroidal Alfvén velocity at the magnetic axis. We observe a similarity in the radial profile shape between the NTV particle flux and toroidal torque density shown in figure 6-a large narrow peak occurs for the ion contribution near the plasma edge, mainly due to the resonant enhancement effect discussed earlier. This resonant effect is largely suppressed for the thermal electron contribution, due to much higher particle collision frequency for the latter. As a consequence, the ambipolar portion of the NTV particle flux is relatively small everywhere along the plasma radius. As will be shown later, this affects the simulated density pump-out level and the time scale of the quasi-linear response.
Before reporting the simulation results with the ambipolar NTV particle flux, it is interesting to consider one more example where we assume the thermal ion particle contribution alone. The results are reported in figure 11, where the free-boundary condition is again assumed for the plasma density at the last closed flux surface. It is interesting that the simulated pump-out level (at steady state) with the thermal ion flux is similar (∼25%) to that with electron flux ( figure 11(a)). But it takes longer time (∼1 s) for the ion flux model to reach steady state. Figure 11(b) also shows that the final density profile, on the other hand, agrees well with the experimental measurement.
The reason for the longer time scale (for reaching steady state) associated with the ion NTV particle flux, as compared to that with the electron flux, can be traced to a cancellation effect illustrated in figure 12(a). Plotted here are the particle fluxes at the simulation time of 50 ms, due to the thermal ion NTV contribution and the particle diffusion. It is evident that the diffusion flux, which depends on the radial profile of the change of the particle density ∆ρ, is negative (i.e. inward) in this case as opposed to the outward particle flow due to NTV. This cancellation effect slows down the density pumpout for the modeled case. Such cancellation does not occur  for the case with the electron NTV particle flux ( figure 12(b)). Furthermore, the generally more global radial distribution of the electron NTV flux also helps enhance pump-out.
Finally, we consider the ambipolar particle flux model discussed earlier, with the MARS-Q simulation results summarized in figure 13. The same (free) boundary condition is assumed. Compared to the ion or electron NTV particle flux model, the ambipolar contribution results in a slightly lower level of density pump-out, but still significant at ∼20%. The simulation time required to reach the steady state is however much longer (∼1.5 s) as compared to experiments. This means that the ambipolar model, which neglects the particle contribution during the transient phase, under-predicts the particle radial transport in the modeled HL-2A discharge.

Effect of coil phasing.
We have so far considered the odd parity (∆ϕ U/L = 180 • ) RMP coil current configuration, which has been experimentally found to be most effective for ELM control in HL-2A. This is largely because the odd parity configuration (at the same coil current amplitude) yields a strong edge-peeling response as shown in figure 4(d). The magnitude of the edge-peeling response is sensitive to the coil phasing ∆ϕ U/L between the upper and lower rows of the ELM control coils. This is in turn quantified by the plasma boundary displacement (|ξ n | = |ξ · ∇s|/|∇s|) near the X-point ( figure 14(a)). It is evident that the RMP configurations with ∆ϕ U/L ∼ 180 • yield the strongest edge-peeling response. On the other hand, the coil phasing of ∆ϕ U/L ∼ 30 • would produce the weakest response. The NTV torque, computed using the linear response field, varies in a similar manner with the coil phasing ( figure 14(b)). This is not surprising, since the NTV torque is proportional to the square of the magnetic field perturbation in the Lagrangian frame. The latter in turn roughly scales with the plasma displacement. Note also the negative values of the net NTV torque, indicating the rotation braking effect in general.
From figure 14, we expect the weakest effect of the RMP field on the plasma transport with the coil phasing of ∆ϕ U/L ∼ 30 • . Figure 15 reports the MARS-Q modeling

Summary and conclusion
We have numerically investigated effects of the n = 1 RMP field on the plasma density and toroidal flow evolution during the type-I ELM control on HL-2A, utilizing the quasi-linear initial-value code MARS-Q. We emphasize that the particle and momentum transport effects that we consider in this study are directly due to the macroscopic MHD response of the plasma (i.e. not due to the RMP induced plasma turbulence). As a key finding, the simulation results, when compared with the experimental data, reveal that the NTV contribution plays a crucial role in the observed plasma density pump-out and the edge rotation damping in the HL-2A ELM control discharge considered. With inclusion of the NTV physics, the level of the modeled flow damping agrees well with that measured in experiments. Near the plasma edge, the NTV torque is substantially enhanced due to Landau resonance between the precessional drift motion of thermal ions and the applied 3D magnetic perturbation. This resonance enhancement regime has previously been identified in other devices, but is shown in the present study to occur in the HL-2A plasmas as well.
The outward radial particle flux associated with NTV also leads to the plasma density pump-out in HL-2A. The MARS-Q initial-value simulations show that the thermal electron NTV particle flux, combined with the free boundary condition at the plasma edge, provides results that are in good agreement with experiments in terms of (i) the net pump-out level, (ii) the detailed radial profile for the steady state density, and (iii) the time scale for achieving the steady state. Furthermore, extensive parameter scans have been carried out in the simulation, in order to understand robustness of the numerical results against variations in the model assumptions. We find that (i) The simulated density pump-out level is not very sensitive to the particle diffusion model (e.g. the uniform versus Spitzer-like model) within the typical confinement time scale. (ii) The assumption of free-boundary condition (at the plasma edge) for the n = 0 density perturbation appears to yield better agreement with experiments. The fixed-boundary condition results in slightly less pump-out level but much longer time scale for reaching steady state.
(iii) Both ion and electron NTV particle flux models result in good agreement with the measured pump-out level. Including only the ambipolar portion of the particle flux, which neglects the transient effects, yields somewhat weaker pump-out (but still significant). The major difference between these assumptions is in the simulated particle transport time scale, with the best result (as compared to experiments) achieved assuming the electron particle flux. (iv) The density pump-out (as well as flow damping) is sensitive to the coil phasing of the ELM control coils. The ∆ϕ U/L ∼ 30 • phasing produces the weakest side effects on the plasma transport in HL-2A, but also likely has the minimal effect on ELMs.
Overall, this study points to the important role of the neoclassical effect in the momentum and particle transport in the presence of external macroscopic 3D fields. This effect can be even more pronounced in ITER plasmas, because of both factors-lower particle collision frequencies and slower toroidal flow of the plasma-enhance the resonant NTV contribution.