Resonant mode effects on rotation braking induced by n = 1 resonant magnetic perturbations in the EAST tokamak

The spectrum effects on toroidal rotation braking, induced by n = 1 resonant magnetic perturbations (RMPs) in the discharges with q95=4.1 and q95=5.1 , are studied in the EAST tokamak. Here n is the toroidal mode number, RMP spectrum is varied by scanning δϕUL , the phase difference between the upper and lower rows of RMP coils. The toroidal rotation changes periodically with the periodic δϕUL scanning and such an effect is stronger in the discharge with lower q95=4.1 . The spectrum dependence of the neoclassical toroidal viscosity (NTV) torque, modeled by NTVTOK based on the magnetic perturbation obtained from MARS-F calculation, agrees well with that of the experimentally observed braking torques in both discharges. The modeled NTV torque is stronger in the discharge with lower q 95, which also agrees with the observations. The comparisons between the spectrum dependence of the NTV and magnetic perturbations show that the resonant mode of magnetic perturbations near the plasma edge mainly contribute the NTV torque. These agreements between modeling and experiments highlight the capability of NTV theory in explaining the experimental observation in the EAST tokamak.

The spectrum effects on toroidal rotation braking, induced by n = 1 resonant magnetic perturbations (RMPs) in the discharges with q 95 = 4.1 and q 95 = 5.1, are studied in the EAST tokamak. Here n is the toroidal mode number, RMP spectrum is varied by scanning δϕ UL , the phase difference between the upper and lower rows of RMP coils. The toroidal rotation changes periodically with the periodic δϕ UL scanning and such an effect is stronger in the discharge with lower q 95 = 4.1. The spectrum dependence of the neoclassical toroidal viscosity (NTV) torque, modeled by NTVTOK based on the magnetic perturbation obtained from MARS-F calculation, agrees well with that of the experimentally observed braking torques in both discharges. The modeled NTV torque is stronger in the discharge with lower q 95 , which also agrees with the observations. The comparisons between the spectrum dependence of the NTV and magnetic perturbations show that the resonant mode of magnetic perturbations near the plasma edge mainly contribute the NTV torque. These agreements between modeling and experiments highlight the capability of NTV theory in explaining the experimental observation in the EAST tokamak.
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Introduction
Tokamak magnetic field configurations are designed to be toroidally symmetric to confine fusion plasmas. However, the toroidal symmetry is not always preserved in realistic condition due to the existence of non-axisymmetric magnetic perturbations induced by magnetohydrodynamics (MHD) instabilities, intrinsic error field or resonant magnetic perturbations (RMPs). In recent years, RMP has received increasing attention as a result of its control effects on edge localized modes (ELMs) [1,2] and applications on ITER. However, toroidal rotation braking [3,4] occurs frequently when RMP is used. This is usually adverse for plasmas since toroidal rotation strongly influences not only the MHD instabilities such as the resistive wall modes [5] but also the confinement such as the L-H transition [6,7] and formation of internal transport barriers [8,9]. It is necessary to understand the mechanism of this braking effect induced by RMP and its role in toroidal rotation control.
Toroidal rotation braking in 3D magnetic field configurations induced by RMP has been widely observed in many present tokamaks such as NSTX [4], DIII-D [10], JET [11], TEXTOR [12], EXTRAP T2R [13], RFX-mod [14], ASDEX Upgrade [15], KSTAR [16] and EAST [17]. A widely accepted theory to explain the rotation braking is the neoclassical toroidal viscosity (NTV) [18] as a result of the radial electric field altered by non-ambipolar transport in 3D magnetic fields. The quadratic dependence of the NTV torque on the strength of 3D magnetic field are predicted by the NTV theories or simulations, and proved by the experiments [19][20][21][22]. It means that by adjusting the strength of the 3D fields, we can change the NTV and subsequently the plasma toroidal rotation. The NTV is consequently significantly influenced by the plasma response that reestablishes the 3D fields. The following examples from numerous reports also show this. Simulations in JT-60SA have reported that the NTV at resonant RMP is much stronger than that at non-resonant RMP [23]. In DIII-D, it has been shown that the modeled and measured NTV torque has a similar dependence on the δϕ UL with linear plasma response [24]. Experiments in KSTAR with the n = 1 [16] and n = 2 [25] RMP proved the importance of the kink response on rotation braking, which can be altered by the q 95 and δϕ UL . However, all the above experimental results are the comparisons of different discharges with fixed δϕ UL , which is discontinuous and has some uncertainties between different discharges.
In this paper, RMP spectrum effects, conducted by continuous δϕ UL scanning with n = 1 RMP in the discharges with q 95 = 4.1 and q 95 = 5.1, on the rotation braking are studied in the EAST tokamak. Section 2 includes the experiment observations of the rotation braking effects by scanning the δϕ UL of RMP. In section 3, the experimental torque is calculated through momentum transport analysis and compared with NTV torque modeled by NTVTOK [26][27][28]. The plasma response modeled by MARS-F [29] and its comparison with NTV torque are discussed in section 4. Finally, conclusion and discussions are given in section 5.

Experimental observations
The RMP system has been successfully applied to control the ELMs in EAST tokamak [2,30]. It consists of two arrays of coils with up and down symmetrically located at the last closed flux surface on the low field side. The coil system can generate different toroidal mode numbers up to n = 4 in the static perturbation case and n = 3 in the rotating case [31]. The n = 1 RMP with 1.5 kA with four turns coil current is employed in this experiment. The RMP spectrum is varied by scanning the δϕ UL of the RMP coils. Temporal evolution of the main plasma parameters is shown in figure 1 for discharge #69635, in which q 95 is 4.1. This plasma is heated by 3 MW NBI (neutral beam injection) and 2 MW LHW (lower hybrid wave), and the heating power are kept constant during the application of the RMP from 5 s to 7 s. It is obvious that the D α , density and stored energy change periodically with the δϕ UL scanning. Note that there are two periods of δϕ UL scanning from 5 s to 7 s as shown by the blue dotted line in figure 1(e). Toroidal rotation velocity and ion temperature at different radius (ρ t = √ Ψ t /πB 0 , where Ψ t is the toroidal flux and B 0 is the central magnetic field) measured by charge exchange recombination spectroscopy [32] are shown figures 1(d) and (c). The rotation braking induced by RMP also changes periodically with the δϕ UL scanning, where the toroidal rotation returns to the level before the application of RMP at about 5.6 s, also shown in figure 3(a).
Similar discharge #71202 with q 95 = 5.1 is also discussed. The n = 1 RMP with 1.5 kA coil current was also employed in this experiment. It is also heated by NBI and LHW with the same power as #69635 and the heating power are kept constant during the application of RMP from 4 s to 7 s. Temporal evolution of the main plasma parameters is shown in figure 2. With the RMP spectrum scanning, the D α , density and stored energy change periodically. The rotation braking also changes periodically with the δϕ UL scanning, however the maximum rotation braking effect is weaker than that of #69635. This can be demonstrated by the comparisons of profiles of the toroidal rotation as shown in figure 3. The blue solid lines in this figure are the fitted radial profiles of the maximum rotation during RMP application corresponding to the minimum rotation braking effects induced by RMP, while the red solid lines indicate minimum rotation profiles corresponding to the maximum rotation braking effects, and the black solid lines are the rotation profiles before the RMP application. The maximum rotation braking effect of #69635, about 8 krad s −1 at ρ t = 0.3, is significantly stronger than that of #71202 as shown in figure 3, about 5 krad s −1 at ρ t = 0.3. It should be noted that the maximum toroidal rotation of #71202 during the RMP application is slightly higher than that before the RMP application.

Experimental torque calculation
The experimental braking torques induced by the n = 1 RMP with δϕ UL scanning in EAST are determined by momentum transport analysis. The toroidal angular momentum transport equation used in this paper can be written as [11] Here, L ϕ = ⟨ R 2 ⟩ Ψ n i m i ω ϕ is the toroidal angular momentum density, where R is the major radius, ⟨. . .⟩ denotes the fluxsurface average, and n i , m i , ω ϕ are the ion density, mass and toroidal rotation respectively. G = RB ϕ ⟨ 1/R 2 ⟩ is a geometric factor. ρ t = √ Ψ t /πB 0 is the flux-surface label (Note that ρ t is normalized to ρ b in this paper, where ρ b is the value of the ρ t at last closed surface.). T is the toroidal torque density, and the radial momentum flux Γ m can be written as where χ m is the radial momentum diffusion coefficient, V inwm is the inward pinch velocity. In this paper, the convection term in equation (2) is neglected due to previous study has shown that the braking torque profile is not sensitive to whether include the convection term V inwm or not [11]. Then, we have The effective diffusion coefficient χ m [33] can be obtained by solving equation (3) before the application of RMP, at which time there is only NBI torque T = T NBI that can be modeled by the NUBEAM code [34]. χ m is assumed to be unchanged during the application of RMP if the n i is assumed to be constant, which has been verified in EAST [17]. Then the total torque T can be obtained by substituting the constants χ m , n i and the time varying ω ϕ into equation (3). The braking torque T brak can be obtained from T − T NBI . Note that T NBI is assumed to be constant during the application of RMP. By using above momentum transport analysis, the volume integrated braking torque under different δϕ UL has been calculated and shown in figure 4. The experimental torque T exp , which reflect the strength of rotation braking effects in the experiments, is the T brak mentioned above. The fitting function is [A (cos2π/360 + δϕ UL0 ) + a] 2 + b, which is the function of the resonant components of the magnetic perturbation. By the way, the fitting line is just to show the differences of the experimental torques between these two discharges more clearly. And we just use the original date instead of the fitting data for the comparison in the following sections. In discharge #69635, as shown in figure 4 with red color, the experimental torque shows a sinusoidal dependence on the δϕ UL and is minimized at about 185 • . Similar to #69635, the experimental torques of #71202 also shows a sinusoidal dependence on the δϕ UL , which is plotted in blue in figure 4. However, the δϕ UL corresponding the minimum experimental torque is about 225 • . And the maximum experimental torque of #69635 is larger than that of #71202, which is consistent with that the stronger rotation braking effects are observed in #69635. These results prove that the rotation braking has different dependence on δϕ UL with different q 95 . Note that the experimental torque in figure 4 is multiplied by a minus sign, which directly reflects the strength of the braking effect. The minimum experimental torque of #71202 is below 0 because the maximum toroidal rotation during the application of RMP is slightly higher than that before the application of RMP as shown in figure 3(b).

NTV modeling
RMPs can induce the toroidal symmetry breaking, which will cause an additional nonambipolar radial particle transport flux. The electron fluxes are quite different from the ion fluxes, resulting in a torque called NTV torque (T NTV ). It is a possible reason to explain the rotation braking presented in section 2. T NTV is modeled by NTVTOK [26,27] code by solving the drift kinetic equation. Typical radial profiles and equilibrium configuration used in this modeling are shown in figure 5. The equilibrium profiles are different for these two discharges. The 3D fields using these modelings are calculated by MARS-F [29], details will be shown in section 4.
The modeled volume integrated NTV torque of #69635 are plotted in figure 6(a) with red diamonds. It shows similar dependence on δϕ UL as the experimental torque plotted with black plus, where the experimental torque is the integrated barking torque from figure 4. The δϕ UL of the minimum NTV toque is about 185 • which is close to δϕ UL of the minimum experimental torque. Similar results also shown in figure 6(b), proving that the NTV torque is the dominant role in rotation braking with different RMP spectrum. However,  the amplitude of NTV torque is not identical to the experimental results. For #69635, the maximum and minimum NTV torque is about half of that of experimental torque as shown in figure 6(a). For #71202, the maximum NTV toque is also about half of the maximum experimental torque, while the minimum NTV torque −T NTV > 0 and the minimum experimental torque −T exp < 0. The reason is that the rotation at the minimum NTV torque δϕ UL is slightly higher than that before the RMP application as shown in figure 3(b), which may be due to the diagnose error or the slightly changes in confinement. In the NTV modeling, the NTV torque is mainly contributed by ions, i.e. −T NTV > 0, the sigh of which is almost not influenced by the RMP spectrum in our modeling.
Though there are some quantitative difference between the NTV torque and experimental torque, they are comparable in most RMP δϕ UL . The phasing dependence of the NTV torque and experimental torque is almost the same for both discharge. The phasing shift of the experimental torque between these two discharges can be also repeated in the NTV torque, and the maximum NTV torque is stronger at lower q 95 discharge consisting with the observations shown in section 2.
The above NTV modeling based on the MARS-F calculation is well consistent with the experimental observations. Actually, there are big differences between the experimental torque and NTV torque based on the vacuum fields. As shown in the figure 6(a), the phasing dependence of the NTV modeling based on the vacuum fields is not completely consistent with that of the experimental torque. And the magnitude of the NTV based on vacuum fields is much stronger than that of the experimental torque. Note that the NTV based on vacuum fields shown in figure 6(a) divide by 60. These results also prove that the plasma response is crucial to the 3D fields induced by RMP, and therefore we do not discuss the NTV modeling based on the vacuum fields in the following section.
The profiles of the NTV torque based on MARS-F are shown in figure 7. It is obvious that volume integrated NTV torque is mainly contributed by the NTV torque at plasma edge due to much larger volume and much stronger NTV torque at the plasma edge. And the difference between the maximum NTV torque and minimum NTV torque in the edge region (ρ > 0.8) is more significant compared to the core region (ρ < 0.8). It should mention that the comparisons between local experimental torque and NTV torque is still a challenge due to the limitation of the time and spatial resolution of the diagnosis.

Modeling the dependence of the plasma response on δϕ UL
On one hand, the amplitude of NTV torque is determined by the plasma parameters such as the collisionality, precession or bounce frequencies [19,21,27,35]. However, their relationship is very complicated due to the complicated resonances between different motions [17,36] and the various transport regimes with different collisionality scaling [18,[37][38][39]. On the other hand, the amplitude of NTV torque is directly proportional to the square of the displacement of the magnetic perturbations [19][20][21][22]. In this section, the dependence of the plasma response modeled by MARS-F [29] on δϕ UL and their comparisons with NTV torques are presented.

Effect of the δϕ UL on the spectrum of the magnetic perturbations
From the modeling of NTV torques shown in section 3, the δϕ UL corresponding to the minimum NTV torque of #69635 and #71202 are about 185 • and 225 • respectively. And the difference of the δϕ UL between the maximum torque and minimum torque is about 180 • for these two discharges. Therefore, the comparisons of the radial profile of the n = 1 normalized perturbation field b ρ /B ξ (%) between the δϕ UL of minimum torque and δϕ UL of maximum torque are shown The position of the integral q surfaces, the maximum integral q value and the q profiles can also be seen in figure 8. The radial range of figure 8 is set from 0.5 to 1 owing to the volume integrated NTV torque is mainly contributed by the local NTV torque near the plasma edge. The comparisons of the vacuum fields is not shown due to the NTV modeling based on the vacuum magnetic fields is not consist with the experimental observations. As shown in the figures 8(a) and (d), the island width near the plasma edge (m/n > 3) at the maximum NTV torque δϕ UL is longer compared with the minimum NTV torque δϕ UL for both discharges. The non-resonant components at the maximum δϕ UL are also stronger than that of the minimum NTV torque δϕ UL for both discharge as shown in figures 8(b), (c) and (e), (f ). That means that both the resonant and the nonresonant components at the maximum NTV torque δϕ UL are stronger than that of the minimum NTV torque δϕ UL , which is a feature of kink-like response. The difference of the NTV torque or the experimental torque between different δϕ UL may be explained by the different strength of plasma response with different δϕ UL .

Comparisons between magnetic perturbations and NTV torques
Though the above comparisons show that the strength of plasma response induced by RMP with different δϕ UL may determine the magnitude of NTV, we still do not know the dominant role and just compare the maximum NTV torque δϕ UL and minimum torque δϕ UL . In order to get a much clearer relationship between the NTV torques and magnetic perturbations, the comparisons between the different Fourier components of magnetic perturbations with plasma response and the volume integrated NTV torques are shown in figure 9. As shown in figures 8(a) and (d), the maximum integer q value for #69635 and #71202 are 5 and 6 respectively. The m/n = −3/1, m/n = −4/1 components and m/n = −5/1 located at q = 3, q = 4 and q = 5 respectively are resonant Fourier components, and the m/n = −7/1 are the strongest non-resonant component at q = 5, where near the last closed flux surface. The reason that we select this non-resonant component, near the edge, is the integrated NTV torque is dominated by much larger local NTV in the edge region. It is apparent that the dependence of the NTV torques on δϕ UL is closer to that of the m/n = −5/1 resonant Fourier component magnetic perturbations, where near the plasma edge, for these two discharges. Not that the phasing dependence of resonant m/n = −6/1 component is similar to that of the resonant m/n = −5/1 for 71202, it is not shown in figure 9(b) for simplicity. Actually, the key is that the δϕ UL of the maximum and minimum NTV can match that of the m/n = −5/1 since ratio scales between the NTV and the resonant components are difficult to obtain and change with δϕ UL .
Though the non-resonant components often drive NTV, the dependence of the non-resonant component on δϕ UL is not consistent with that of the NTV as shown in figure 9(a). This means the dependence of NTV on δϕ UL is mainly determined by the dependence of resonant components of the magnetic perturbations near the edge region on δϕ UL . The reasons are as follows. As we known, the NTV is directly proportional to the square of the displacement of the magnetic perturbations ξ ρ mn [19][20][21][22]. In our modeling, the ξ ρ mn contributed from the resonant m/n = −5/1 is much larger than that from the nonresonant component m/n = −7/1 at the q = 5, as shown in figure 10, due to ξ ρ mn ∝ √ δB/B for the resonant components in the vicinity of a magnetic island [40] and ξ ρ mn ∝ δB/B [18] for the non-resonant components. Actually, the square of the nonresonant m/n = −7/1 is better for this comparison, it does not influence the main results, i.e. the square of the non-resonant components do not change the position of the peak and valley values. The difference of the ξ ρ mn between the maximum NTV phasing and minimum NTV phasing in the resonant component is much stronger than that in the non-resonant component. Furthermore, both the analytic results and the modeling results from NTVTOK [27] show that the NTV torque from each harmonic of the perturbations reaches the maximum value at the corresponding rational surface. Therefore, the integrated NTV torque, dominated by the large local NTV in the edge region, is mainly determined by the resonant components of magnetic perturbations in the edge region. And the different dependence of the NTV torques on δϕ UL between these two discharges can be attributed to the different dependence of resonant components of magnetic perturbations in the edge region on δϕ UL .

Conclusion and discussion
In summary, toroidal rotation braking induced by the n = 1 RMP, while scanning the coil phasing δϕ UL at different q 95 values, has been observed and compared with NTV torque in EAST. The experimental torque has been calculated by the momentum transport analysis. The obtained T brak in two discharges with different q 95 changes periodically with periodic δϕ UL scanning, while the δϕ UL value for achieving strongest T exp is different between these two discharges. The maximum experimental torque is smaller at lower q 95 , which is consistent with the experimental observation that the rotation braking effect is stronger at lower q 95 . The modeled NTV torque, by NTVTOK [26,27], also change periodically with periodic δϕ UL scanning. The δϕ UL value for the computed maximum NTV torque is also different in these two discharges. The maximum NTV torque is stronger in the lower q 95 discharges, agreeing well with the experimental observation. The experimental and NTV torques show similar dependence on the RMP spectrum for these two discharges. These prove that the NTV torque play an important role in magnetic braking introduced by RMP. The plasma response field is modeled by MARS-F [29]. Results show that the kink responses at δϕ UL for maximum NTV torque is much stronger than that for minimum torque for both discharges. Comparisons between the NTV torque and perturbation fields show that the resonant components near the plasma edge play a dominant role in the magnitude of NTV torque. The above results not only provide a validation of NTV theory using experiments in the EAST tokamak, but also prove it is a good way to use linear plasma modeling to predict the relative strength of the NTV torque with different δϕ UL .
However, disagreement between the NTV torque and experimental torque still exists for both discharges. One of the possible reasons is the uncertainty of the edge pressure gradient. Because the edge resonant components of the perturbation field, which determine the volume integrated NTV torque that mainly contributed from the NTV torque at edge, are sensitive to edge pressure gradient. This is a big challenge for both the modelling of the plasma response and the measurement of the edge plasma parameters. Another reason is that the J × B torque is not considered in this paper. The direct effect of the RMP on the plasma rotation damping and density variation is also important which will be considered in future works.