The effect of plasma toroidal rotation on n = 1 resonant magnetic perturbation field penetration under low neutral beam injection torque in EAST

An experiment was conducted to study the mode penetration of n=1 resonant magnetic perturbation (RMP) in EAST under low neutral beam injection torque in the co-current direction (Co-NBI). The experimental results indicate that the threshold current IRMP,th for field penetration decreases with higher input torque TNBI . Furthermore, it is observed that the plasma mode frequency |fMHD| at counter-current direction is greatly reduced when the plasma toroidal rotation frequency fϕ increases. The theoretical scaling of mode frequency (IRMP,th∝|fMHD|0.70) predicted by the field penetration theory is in good agreement with the experimental observation ( IRMP,th∝|fMHD|0.53 ). The role of |fMHD| and fϕ on the mode onset threshold was separately investigated using the full toroidal geometry initial value code MARS-Q (Liu et al 2013 Phys. Plasmas 20 042503). The numerical scaling based on the experimental mode frequency is consistent with the experimental and theoretical ones. Numerical results suggest that evaluating the total mode frequency |fMHD| is crucial in field penetration analysis, in contrast to toroidal rotation frequency fϕ . With the increase of TNBI , the decreasing |fMHD| leads to a reduction in the field penetration threshold. This suggests that more attention should be paid to error field tolerance in low Co-NBI torque scenarios, where the electron diamagnetic frequency may be canceled out by NBI-driven toroidal plasma rotation.

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Introduction
Increasing attention has been paid over the past few decades to the effect of small 3D magnetic field perturbation on tokamak plasmas.These non-axisymmetric magnetic fields, known as error fields, exist in tokamak devices due to the engineering design imperfections and the coil misalignments [1].Error fields have been recognized to cause detrimental effects on tokamak plasmas.This often introduces magnetic perturbations that resonate with magnetic field lines at q (r s ) = m/n, where r s is the rational surface, and (m, n) denote poloidal and toroidal mode numbers.Once the resonant error field exceeds a certain threshold, the plasma flow at the rational surface rapidly goes to zero, leading to the onset of locked modes [2,3].This phenomenon, known as error field penetration, can be responsible for the degradation of confinement and even terminating discharges.In current tokamak experiments, the locked mode restricts the operating region, especially the ability to extend the operation region towards low-density ohmically heated plasmas.Consequently, the impact of error field penetration has been a matter of concern for ITER [4].The focus of the issue is making extrapolations to the error field tolerance in future burning plasma experiments.
The scalings of the error field penetration critical value, (b r /B T ) crit , with respect to plasma density n e , toroidal field B T , safety factor at the 95% poloidal magnetic flux q 95 and device size R 0 have been investigated on existing tokamaks [5][6][7][8][9][10].In recent experiments on EAST, the scaling of the experimental data show good agreement with the locked-mode theory in cylindrical geometry [11][12][13][14][15][16].
In future fusion reactors, sustained tokamak operation at high β N is required for economic feasibility.Here, β N = β (%) a (m) B 0 (T) /I p (MA), and β is the ratio of volumeaveraged plasma pressure to the toroidal magnetic field pressure.a, B 0 and I p are the minor radius, toroidal magnetic field and plasma current, respectively.The DIII-D high β N experiments have revealed an increased susceptibility to error field penetration when the plasma beta is close to the nominal beta stability limit [17].Recently, the detrimental effect of plasma β N on field penetration has also been observed in pure radiofrequency (RF) heating discharges in EAST [18].This suggests that the toroidal effect plays an important role in field penetration for high-beta plasma.Furthermore, in toroidally confined plasmas, the breakdown of toroidal symmetry can lead to the so-called neoclassical toroidal viscosity (NTV) torque, which also breaks the plasma flow [19,20].
One of the main concerns in error field penetration physics is plasma rotational effect.Theoretical results have shown that the mode penetration threshold b r is an increasing function of the plasma flow [11].The relationship between critical error field and plasma beta, and rotation had been observed in DIII-D [17].The experimental dependence between rotation and critical error field for penetration is in good agreement with a straight cylindrical model until higher beta values are reached.Here, it should be noted that the rotation is given by the plasma toroidal rotation frequency.
The plasma rotation is then employed in some investigations of error field penetration.A power scan including ion cyclotron resonance heating (ICRH) and neutral beam injection (NBI) heating was performed in JET [6].The experimental findings clearly indicate that the dominant influence on the penetration threshold is the NBI heating, which leads to an increase in plasma rotation, hence raising the threshold.Lazzaro et al derived a power-law scaling of the form b pen /B ∝ n an B aB ω η 0 based on JET's experimental data [21], where ω 0 denotes the initial toroidal angular frequency.The index of the scaling law shows good agreement with the empirical scaling of the field penetration threshold.In NBI heated plasma in NSTX [8].The reduction in the error field tolerance is explained by the error field amplification in higher plasma β N .The importance of the role of plasma toroidal rotation has also been discussed.
To clarify the effect of plasma rotation, a precise experimental design in TEXTOR was used to study the mode penetration threshold [22,23].Dynamic ergodic divertor coils can produce a static or rotating perturbed external field to excite an m/n = 2/1 locked mode.In addition, the combination of flexible NBI and ICRH systems enabled independent investigations into the effect of plasma rotation.Experimental results have shown that the penetration threshold current reaches its minimum value when the frequency of the external field is equal to the mode frequency.This demonstrates that the plasma toroidal rotation affects the mode penetration by altering the magnetohydrodynamic (MHD) mode frequency.From the theoretical analysis on experiment [24], it is derived that the MHD frequency is equivalent to the electron fluid frequency, which is determined by the sum of the electron diamagnetic drift and the background plasma toroidal rotation.In the NBI heating scheme, it gives rise to bulk toroidal plasma rotation.However, there is little or no bulk plasma rotation in ohmically heated tokamak plasmas, and the mode frequency stems from plasma diamagnetism [11].
In a subsequent study, Yu numerically investigated the error field penetration based on two-fluid equations [25].These results showed that the penetration threshold current reaches its minimum value when the external helical field frequency is equal to the mode frequency.Hu et al found that the minimum error field amplitude for field penetration occurs at ω E /ω e * = −1 [26], due to the canceling between the plasma rotation frequency ω E and the electron diamagnetic drift frequency ω e * .These results highlight the dominant role of mode frequency in the process of mode penetration.
Although many investigations can draw a rough conclusion on the effect of plasma rotation in error field penetration physics, a detailed comparison between theory and experiment is still essential to give a definite conclusion.The influence of variable multi-parameters on field penetration should be considered during experimental analysis.In addition, previous numerical modeling concerning error field penetration is established in cylindrical geometry, excluding the effect of toroidal coupling from numerical results.In recent years, it has been found that plasma toroidicity can significantly affect the penetration threshold [27,28].Moreover, the non-resonant components of the error field also affect the plasma rotation by giving rise to an NTV torque, which exerts a braking force.Hence, the extent to which toroidal geometry contributes to the penetration threshold cannot be evaluated in cylindrical modeling.
Compared to resonant magnetic perturbation (RMP) penetration experiments in ohmic and L-mode plasmas, the significance of H-mode discharge in low NBI torque heated plasmas has not been appreciated previously.In our previous investigations, MHD frequency is employed in error field penetration [14,16].Nonetheless, the impact of plasma toroidal rotation on experimental scaling on the basis of error field penetration theory has not been clarified.Therefore, an experiment on error field penetration threshold with low NBI torque was carried out in EAST to investigate the role of plasma rotation frequency.Furthermore, numerical modeling based on full toroidal geometry has also been carried out to reproduce the error field penetration in the experiment.
The rest of this paper is arranged as follows.Section 2 introduces the experimental setup and the analysis of the experimental results, section 3 gives the numerical simulation, followed by the summary in section 4.

The experimental setup in EAST
The operation and research capabilities of EAST have been greatly improved after the upgrade of various auxiliary heating and diagnostic systems.The RMP system, which was installed in 2014 [29], has achieved full edge localized mode (ELM) suppression and divertor power flux control in dominant RF wave heating H-mode with low NBI torque and a tungsten upper divertor [30,31].The RMP system consists of a set of 2 × 8 helical coils mounted on the upper and lower sides of the low-field within the vacuum vessel.These coils can be connected in various ways to produce resonant perturbation fields with multiple phase and mode numbers (n = 1 − 4).In addition, the coils are supplied with a DC current resulting in a static field, with AC current yielding a rotating one.The frequency of the rotating field can reach 10 Hz, and one coil has four turns with a maximum current of 4 kA.In the past few years, the measurement of device intrinsic error field and the scaling of field penetration threshold have been investigated [14,16,32].Figure 1 shows typical error field penetration in EAST, with the plasma being heated by the lower hybrid wave (LHW) and NBI systems.NBI injection in the co-current direction (Co-NBI) is applied for both shots, but for shot 91709 an addition counter-current NBI (Ctr-NBI) is injected.The fixed RMP spectrum and ramp-up coil current I RMP in figure 1(a) are maintained, resulting in mode penetration, as shown by the magnetic signal b r in figure 1(b).The dotted lines indicate the occurrence of field penetration.The plasma electron density and plasma β N are shown in figures 1(c) and (d), respectively.For these two shots, before field penetration, the electron density and plasma β N are very close.
Figure 1(e) gives the temporal evolution of plasma toroidal rotation frequency at the q = 2 surface Ω ϕ ,q=2 , while the beam power of Ctr-NBI is shown in figure 1(f ).The plasma toroidal ratio is measured from CXRS.For shot 91414, we observed that the plasma rotation is non-zero when the magnetic field penetrates.We will discuss the reasons in a subsequent section.However, after the magnetic field penetration, the plasma rotation is strongly damped, which may be due to the dominant role of the NTV torque after field penetration [18,33,34].
The radial profiles of safety factor, electron density and temperature for two shots are shown in appendix A. For shot 91709, it can be clearly observed that the plasma toroidal Relationship between the error field penetration threshold current I RMP,th and plasma β N under various auxiliary heating conditions.Data were obtained during the EAST 2019 experiment using n = 1 RMP.Points marked with a plus sign will be used for the following theoretical analysis.rotation decreases with the application of Ctr-NBI.In addition, experiment results suggest that Ctr-NBI heating has a stabilizing influence on mode penetration.A similar experimental trend has also been found in TEXTOR [22].
Figure 2 illustrates the relationship between error field penetration threshold I RMP,th and plasma β N under various auxiliary heating scenarios in EAST.All experimental data were obtained during the EAST 2019 experiment, utilizing n = 1 RMP with ∆ϕ UL = 0 • .The gray symbols represent pure RF wave heating.In addition to RF heating, the blue symbols stand for the experiment containing Co-NBI.However, for the red symbols, both Co-NBI and Ctr-NBI were employed.In these discharges, the value is approximately between 4.5 and 5.5, and the plasma range increases with heating power from 0.35 to 0.95.The points marked with a plus signal will be used for the following theoretical analysis.
To obtain the empirical scaling required for extrapolation to ITER, heating effects were studied to examine the roles of plasma beta and rotation.In pure RF heating experiments, the error field tolerance improves with increasing heating power [18], mainly due to the stronger stabilizing effect of the kinetic profile than the destabilizing effect of plasma β N .However, in Co-NBI heating, the field penetration threshold I RMP,th decreases as plasma β N increases, and the error field tolerance with Ctr-NBI is higher than that with only Co-NBI.It is important to note here that the relationship between the plasma beta and penetration threshold current, shown in figure 2, is determined by various physical quantities during experiments, such as density, temperature, etc.In pure-RF experiments, we have investigated the effect of plasma beta on the field penetration threshold.In this paper, the range of plasma β N in the experimental data used for the analysis is not significant, as shown in the yellow shaded region in figure 2. Therefore, our main focus in the next step is to study the impact of plasma rotation on field penetration, rather than plasma β N .
Compared to pure RF heating experiments, the distinct discrepancy in NBI heating is due to the influence of the plasma mode frequency.In RF wave heating cases, an increase in mode frequency, mainly from f e * in the counter direction, can improve the error field tolerance.Therefore, from the perspective of preventing the error field damage, pure RF or Ctr-NBI heating is more desirable.
The present study aims at investigating the impact of NBI heating on the penetration process.Thus, the mode penetration induced by n = 1 RMP with low NBI torque is systematically studied.The experiments take advantage of the flexible RMP system and substantial magnetic diagnostics on EAST [32].Given the effect of plasma β N on the field penetration process, the abundant database in locked-mode experiments allows us to evaluate the influence of plasma toroidal rotation at constant plasma beta.Specific experimental data are presented in the following section.

Decreased error field tolerance in low torque input plasmas in EAST
Figure 3 shows the RMP current threshold for mode penetration (I RMP,th ) and plasma toroidal rotation frequency (f ϕ ) at the q = 2 rational surface, plotted against the injected beam torque (T NBI ) (calculated by NUBEAM code [35]).These data correspond to the solid symbols in figure 2. In the range of low-input T NBI ∈ [0.2, 1.0] Nm, it is observed that a lower penetration current threshold is associated with higher torque, indicating an increased sensitivity to error field in low-torque-input Hmode plasma.Moreover, we observe that the dominant factor influencing the reduced penetration threshold is the increasing plasma toroidal rotation in the co-current direction.In JET, however, the faster plasma rotation can increase the field penetration threshold [6].
Notably, if the field penetration threshold is positively correlated with plasma toroidal rotation, then increased plasma toroidal rotation should improve the penetration threshold, which is inconsistent with the experimental results.Therefore, the plasma flow determining the field penetration needs to be discussed next.

Observation of mode frequency
The excitation of mode penetration depends on the electron fluid frequency (f e⊥ = f ϕ + f e* ) in two-fluid theory.Both the electron diamagnetic drift frequency f e* at counter-current direction and the background plasma rotation frequency f ϕ play significant roles in the process of mode penetration since they can either strengthen or weaken the total mode frequency depending on the direction of rotation.In this study, we designate the direction of rotation that opposes the plasma equilibrium current as the 'negative' direction.
Figure 4 depicts the Co-NBI heating beam power and mode frequency detected by the Mirnov probe.The y-axis symbol ('−f MHD ') denotes that the mode frequency is in the opposite direction.The color contour in figure 4(b) represents the amplitude of the magnetic field perturbation.The amplitude of the perturbation before the RMP field application is extremely smaller by two orders of magnitude than that due to the typically observed tearing mode in EAST.It might be forced oscillations of a marginal stable tearing mode.It often exists in stable plasmas in many machines, e.g.J-TEXT [9] and EAST [14].
Specifically, figure 4 Figure 5 shows the relationship between | f MHD | and | f e⊥ |, f ϕ , f e * .The electron fluid frequency f e⊥ is the sum of f e * and f ϕ , while the electron diamagnetic frequency, f e * , is determined using [36,37].It can be seen that the calculated value for f e⊥ closely approximates the experimentally observed f MHD .In addition, the faster plasma co-rotation, f ϕ , causes a downward shift in the mode frequency | f MHD |.For certain shots, the Mirnov probe fails to capture a clear mode frequency due to the D α emission.Hence, f MHD can be substituted with the value of f e⊥ for these scenarios.

Mode frequency scaling on n = 1 error field penetration threshold current
The above analysis also reveals that the mode frequency, f MHD , plays a crucial role in the field penetration process.The penetration threshold current, I RMP,th , increases with faster mode frequency (I RMP,th ∝ | f MHD | 0.53 ), as shown in figure 6(a).This means that increasing the plasma flow enhances the shielding effect against external perturbation fields.
The theoretical regime of field penetration is determined to be in the Waelbroeck regime [11], according to the method introduced in appendix B in [14]: ( The scaling between the penetration threshold, I RMP,th , and plasma flow, f 0 , is related to the interdependence of other relevant plasma parameters.Previous studies have emphasized the necessity of determining all plasma variables during single variable scanning [14][15][16].Appendix B shows the relationship between f 0 and key physical variables (such as n e , τ ν and T e ) by replacing the value of f 0 with experimental | f MHD | and f ϕ .In order to evaluate the theoretical scaling based on equation (1), data irrelevant to β N and q 95 are selected from the experimental database, as illustrated in figures B1(b) and (c).
Combining the experimental dependence in figures B1(d)-(f ), the theoretical scaling using the formula equation ( 1) is  Figure B2 shows the experimental scaling between f ϕ and plasma parameters (I RMP,th , n e , T e and τ E ).By incorporating the dependency relationship of f ϕ with (n e , T e and τ E ) into the theoretical formula in the Waelbroeck regime, the theoretical scaling for f ϕ can be obtained.
Therefore, the theoretical and experimental scaling with respect to 'f 0 = f ϕ ' can be compared.Notably, the trend predicted by theory (∝ f ϕ 0.60 ) is diametrically opposed to that observed experimentally (∝ f ϕ −0.43 ), as summarized in table 1.We conclude that the scaling of | f MHD | based on the error field penetration theory formula is more consistent with the experimental one in contrast to f ϕ .

Quasi-linear modeling of n = 1 RMP field penetration with full toroidal geometry
In our previous study [18], it was found that NTV effects are important for field penetration scaling with finite plasma beta.We are not sure how much of a contribution is made in this study since plasma beta is slightly different and the evolution of plasma rotation may be influenced by NTV.To better comprehend the discrepancy between plasma mode frequency | f MHD | and toroidal rotation f ϕ in experimental scaling, we employed full toroidal effect simulation (MARS-Q) as well as a theoretical formula from the layer physics to investigate the n = 1 RMP field penetration.The full toroidal coupling includes both the plasma response to the external field from resonant and non-resonant harmonics.

Numerical models
In this study, the plasma response to external n = 1 RMP is computed based on a single fluid resistive MHD approximation.The model is described by the following equations: where ρ, B, J and P denote the equilibrium plasma density, magnetic field, current and pressure, respectively.The model contains magnetic field b, plasma current j, fluid velocity v, plasma displacement ξ and plasma pressure p.The plasma is assumed to have a toroidal equilibrium flow V 0 = RΩ φ , where R is the plasma major radius, Ω is the toroidal plasma flow frequency and φ is the unit vector of the toroidal angle.Ω in V 0 = RΩ φ is scalar and is a function of the magnetic flux variable in the model.Γ = 5/3 represents the adiabatic heating coefficient.η = 1/S denotes the dimensionless plasma resistivity, where S = τ R /τ A is the Lundquist number.Here, τ R = µ 0 a 2 /η is the resistive decay time, and τ A = R 0 √ µ 0 ρ 0 /B 0 is the Alfvén time.The Spitzer model is used to determine the plasma resistivity.The parallel sound wave damping is included in the model via the last term of equation ( 3).κ ∥ is the numerical damping coefficient that determines the damping strength, and κ ∥ = 1.5 is assumed in our paper.k ∥ = (n − m/q) /R represents the parallel wave number.Here, m and q denote the poloidal harmonic number and safety factor, respectively.In addition, v th,i = 2T i /M i represents the thermal ion velocity, where T i is the thermal ion temperature and M i is the thermal ion mass.Note that equations ( 2)-( 6) are solved in the plasma region.Equation (7) describes the RMP coil current, which is solved together with equation ( 8) in a vacuum outside the plasma region.The above equations ( 2)-( 8) are numerically solved using MARS-F, with a toroidal flux coordinate [38].
The quasi-linear plasma response model solves the linear response equations mentioned above, together with the following toroidal momentum balance equation to study the interaction between the RMP field and the plasma flow: where L = ρR 2 Ω is the flux surface-averaged toroidal moment.The first term of the right-hand side in equation ( 9) represents the momentum diffusion operator: where G ≡ F⟨1/R 2 ⟩ denotes the geometric factor.Note that F is the equilibrium poloidal current flux function, and χ M represents the toroidal momentum diffusion coefficient.The neoclassical toroidal viscous torque T NTV , electromagnetic torque T JXB and Reynolds stress torque T REY act as momentum sink terms in equation (9).T source represents the momentum source term.The model for T NTV is based on a semi-analytic formula that smoothly connects various collisionality regimes [39,40].This formula is derived from the solution of the bounce-averaged drift kinetic equation.The T JXB and T REY come from magnetic field perturbations produced by RMP coils and the inertial term ρ (v•) v, respectively.A detailed explanation for these torques can be found in previous works [34,41,42].
In the MARS-Q model, it is assumed that the momentum balance is established before applying the 3D perturbation at t = 0, i.e.D (L (t = 0)) + T source = 0. Therefore, the change in the momentum (∆L = L (t) − L (0)) is solved in the following form: We solve the perturbed momentum evolution equation ( 11) for δΩ with additional forces from 3D effects, and show that the rotation at a certain time can be written as Ω = Ω 0 + δΩ.The different amplitude of Ω with different initial rotation did influence the plasma response described in equations ( 2)- (6).
A semi-implicit, adaptive scheme is applied in the MARS-Q model for time advance.Free boundary and Dirichlet conditions are assumed at the magnetic axis and plasma boundary, respectively.The mode penetration excited by the RMP field has been discussed in recent works [34,41,43].

Numerical scaling between the field penetration threshold and T NBI
The input parameters for the MARS-Q model are adopted from experimental data.The Lundquist number S is calculated from the electron temperature via the Spitzer formula.The electron density n e alters the Alfvén time, while the momentum diffusion coefficient χ M is estimated from the experimental viscosity diffusion coefficient.In numerical simulations, the phase difference between the upper and lower coils is ∆ϕ UL = 0 • , and the RMP current is set to ramp up with time.To avoid uncertainties arising from other effects and obtain a precise comparison with theoretical results, we utilize the same equilibrium files in simulations.
Figure 7 presents the radial profile of n = 1 poloidal harmonic spectrum of the normalized perturbation magnetic field b ρ /B ζ (%) in a straight-field-line flux coordinate system (PEST) [29], which is used for the evaluation of the NTV (%) in a straight-field-line flux coordinate system (PEST).Symbols represent the position of q = m/n rational surfaces.Blue dashed line represents the q-profile.
torque.The symbols in the contour represent the location of q = m/n rational surfaces.The blue dashed line represents the q-profile.The radial coordinate is defined by ρ t = 2ψ t /B 0 .Here, ψ t = Ψ t /2π , where Ψ t is the toroidal flux in real units.
To enhance our understanding of the experimental findings for varying the NBI torque, we investigate the scaling relationship between T NBI and the field penetration threshold.Numerical simulations are performed using two distinct sets of parameters: | f MHD | and f ϕ .The numerical case of T NBI corresponds to different | f MHD | and f ϕ .To be specific, conditions f 0 = | f MHD | and f 0 = f ϕ are obtained by setting the initial values of Ω (q = 2) in the model as the value of | f MHD | and f ϕ , respectively.
MARS-Q is a single-fluid model.f MHD includes the diamagnetic effect in the two-fluid model.However, both the analytical theory in [44] and numerical simulation in [25] indicate that the penetration threshold obtained from the two-fluid model is similar to that from the single-fluid model, when the mode frequency due to background plasma rotation is replaced by that due to the diamagnetic Figure 8 shows the temporal evolution of the simulated radial m/n = 2/1RMPs b 2/1 and the plasma rotation frequency at the q = 2 rational surface f  appendix B. These values can be used as numerical parameters for each case.In fact, T NBI is used as the name of a numerical case, and different T NBI corresponds to different numerical input parameters (f 0 , n e , τ E and T e ) in the model.We set the value of plasma flow Ω (q = 2) in the MARS-Q model to be 1.0 kHz as an input for the condition of f 0 = 1.00 kHz.The pink line in figure 8 represents the RMP current that increases with time.It should be noted that the goal of our simulation is to determine whether the variation trend of the permeation threshold at different f 0 is consistent with the experimental observations, rather than a quantitative comparison with experiments.Considering the time efficiency and numerical accuracy of the simulation, we have set a faster ramping rate of RMP in the simulation compared to the experimental setup.The perturbed magnetic field strength suddenly increases while the RMP current reaches a threshold value.At the same moment, the plasma rotation Ω 2/1 goes to zero.This is a dynamic process known as mode penetration, which arises from the forced magnetic reconnection of an error field [9,16,26,45].The vertical lines denote the moment of penetration, with the threshold value I RMP,th being directly proportional to it.
The scaling between T NBI and penetration threshold current I RMP,th in experiment, theory and simulation is shown in figure 9. Based on the relationship between T NBI and (n e , T e , τ ν , f 0 ) in the experiment, it is possible to derive a theoretical scaling between T NBI and I RMP,th .The numerical scaling for the f 0 = | f MHD | case, which is numerically simulated following the full experimental data, aligns with the experimental and theoretical results.However, when considering the f 0 = f ϕ scenario, the numerical scaling conforms to the theoretical one but exhibits a completely opposite trend compared to the experimental observations.In summary, the above numerical results suggest that the reduction in I RMP,th during low torque input NBI heating experiments is primarily attributable to the decreased total mode frequency.Therefore, the evaluation of the total mode frequency, | f MHD |, is crucial in the analysis of error field penetration.

Summary and conclusion
We investigate the field penetration induced by n = 1 RMP low input NBI heating plasmas in EAST.The results reveal that the threshold current for mode penetration I RMP,th decreases with increasing torque T NBI .In addition, experimental results further indicate that the faster co-current plasma toroidal rotation frequency f ϕ increases with T NBI , leading to a significant decrease in the total mode frequency | f MHD | at the countercurrent direction.
Based on the locked-mode empirical formula, we further find that the theoretical scaling between penetration threshold and mode frequency exhibits good consistency with the experimental one when all experimental physical parameters are considered.However, theoretical and experimental results are not consistent for the f ϕ scaling.
Numerical simulations from the initial value code MARS-Q are used to distinguish the effect of | f MHD | and f ϕ on penetration threshold scaling.Two groups of settings with different types of plasma flow are numerically studied.
The results indicate that the numerical scaling based on | f MHD | shows good agreement with both theoretical and experimental results but less agreement with the scaling using toroidal rotation f ϕ .These findings suggest that the total mode frequency, | f MHD |, is a significant physical quantity for the evaluation of field penetration in contrast to f ϕ .In low-input torque experiments, the greatly reduced | f MHD | plays a crucial role in decreasing I RMP,th as T NBI increases.
We find that the penetration threshold dependence of the MHD frequency based on the layer physics has a similar one due to the simulation of the NTV effect.The prediction is supported by the comparison to the experimental results.In error field penetration studies related to layer physics, as long as the correct type of plasma flow is considered, the scaling power index obtained through simulation based on toroidal geometry remains close to both theoretical and experimental results when the toroidal effects are kept constant.In EAST, a previous error field experiment showed that the field penetration threshold in ohmic plasma is not related to the total mode frequency since it is kept almost constant [14].The total mode frequency increases with the increasing heating power in the EAST pure-RF heating experiment [18].Hence, the faster total mode frequency enhances the screening effect on the external field, and improves the error field tolerance.However, the additional beta effect needs to be considered to explain the observed scaling.The NBI heating experiment can directly distinguish the effect of different rotations on field penetration scaling, and indicate that the layer physics is sufficient to explain the experimental scaling between electron fluid frequency and penetration threshold when the plasma beta remains constant.RMP field penetration is more relevant to f e⊥ from the two-fluid theory, rather than f ϕ from the single-fluid theory.The f e⊥ is close to the observed f MHD , which means that the two-fluid theory gives a better description of the mode frequency.Hence, we can also directly use the observed MHD mode frequency.In low input torque cases (such as ITER), a slight increase in the NBI torque may reduce f e⊥ and the mode frequency.Hence, it may not increase the error field tolerance that is observed in high-NBI torque cases.
Finally, the reduced tolerance to error fields observed in low input torque heating scenarios poses a significant challenge for ITER.Alternative heating methods, such as pure-RF and Ctr-NBI, may be more desirable to mitigate the adverse effects of field penetration.Moreover, it is essential to recognize the importance of error fields in low input Co-NBI experiments.The plasma equilibrium is constructed on the basis of EFIT from magnetic diagnostic.The radial position of q = 1 is determined by the reverse surface of the sawtooth oscillations.The density and temperature profile are measured by POINT [46] and Thomson scattering diagnostic [47], respectively.The q 95 values of these two shots are similar, and the position of the q = 2 surface is also close.However, there are differences in the safety factor at the core.For shot 91709, higher stored energy corresponds to higher temperature distribution.Before field penetration, the electron density at the q = 2 surface and plasma β N are very close for both shots.In addition, the influence of n e and T e has been considered in field penetration scaling.Figure B1 illustrates the correlation between mode frequency | f MHD | and various plasma parameters, including plasma toroidal rotation f ϕ , plasma normalized beta β N , q 95 , electron density n e , energy confinement time τ E and electron temperature T e .It is worth noting that the energy confinement time τ E is computed as W/ (P tot − dW/dt), where W is the plasma stored energy and P tot denotes the total absorbed power.In this study, the energy confinement time is assumed to be proportional to the viscosity diffusion time, i.e. τ ν ∝ τ E .Figure B2 shows the experimental scaling between f ϕ and plasma parameter (I RMP,th , n e , T e and τ E ).Based on these relationships, theoretical and experimental scaling between f ϕ and I RMP,th can be obtained.Figure B3 shows the experimental scaling between | f MHD | and T NBI .

Figure 1 .
Figure 1.Error field penetration experiments with low NBI torque plasmas.Time trace shows (a) RMP coil current I RMP , (b) n = 1 magnetic perturbation signal br, (c) electron density, (d) plasma β N , (e) plasma toroidal rotation frequency at the q = 2 surface, which is measured from charge exchange recombination spectroscopy (CXRS), (f ) beam power from Ctr-NBI.Dotted lines indicate the moment of locked mode.

Figure 2 .
Figure 2.Relationship between the error field penetration threshold current I RMP,th and plasma β N under various auxiliary heating conditions.Data were obtained during the EAST 2019 experiment using n = 1 RMP.Points marked with a plus sign will be used for the following theoretical analysis.

Figure 3 .
Figure 3. Penetration threshold current I RMP,th (left y-axis, red line) and the plasma toroidal rotation frequency f ϕ at rational surface (right y-axis, blue line) versus beam torque T NBI .Shots correspond to solid symbols shown in figure 2. Solid lines in the graph represent the experimental scaling relationship.

Figure 4 .
Figure 4. (a) Heating Co-NBI beam power, RMP current and (b) mode frequency f MHD detected by Mirnov probe for shot 91414.Solid line with circles on the mode frequency shown in the contour plot represent the variation of toroidal plasma rotation frequency at the q = 2 surface (δf ϕ ,q=2 = | f MHD (t = 3.5 s)| − f ϕ ,q=2 ) obtained from CXRS.Vertical solid line represents the moment of field penetration.

Figure 5 .
Figure 5. Mode frequency detected from Mirnov probe | f MHD | versus electron fluid frequency f e⊥ , plasma toroidal rotation frequency f ϕ and electron diamagnetic frequency fe * .
(b) shows a sudden drop in the m/n = 2/1 mode frequency at t = 3.5 s with the injection of Co-NBI, followed by a gradual decrease in the mode frequency with the implementation of ramp-up RMP current until the occurrence of mode penetration.The solid line with circles in the frequency spectrum indicates the variation of toroidal plasma rotation frequency at the q = 2 surface (δf ϕ ,q=2 = | f MHD (t = 3.5 s)| − f ϕ ,q=2 ), as measured from CXRS.Notably, in NBI heating experiments, it is observed that the increasing f ϕ ,q=2 correlates with the decrease in total mode frequency | f MHD |.

Figure 6 .)
Figure 6.(a) Experimental and theoretical scaling between the penetration threshold current I RMP,th and mode frequency | f MHD |.(b) Value of I RMP,th / ( n 7/16 e τ −7/16 ν T 9/32 e ) as a function of | f MHD |, in order to compare experimental results and theoretical predictions.
70 shown in figure6(a), and closely matches the experimental one.

Figure 6 ()
b) illustrates the functional relationship between I RMP,th /(n and | f MHD |, further indicating a qualitative consistency between experimental results and theoretical expectations.

Figure 7 .
Figure 7. MARS computed the radial profile of n = 1 poloidal harmonic spectrum of the normalized perturbation magnetic field ( b ρ /B ζ )(%) in a straight-field-line flux coordinate system (PEST).Symbols represent the position of q = m/n rational surfaces.Blue dashed line represents the q-profile.
2/1 in the f 0 = | f MHD | setup, considering three different NBI torques.Based on experimental results in figure 3 and appendix B, we can obtain the relationships between all the physical quantities (T NBI , | f MHD |, f ϕ , n e , τ E and T e ).In the numerical simulation, we set | f MHD | to be 1.00, 1.25, 1.5, 1.75 and 2.00 as five cases.Based on the experimental fitting functional relationship between | f MHD | and T NBI , n e , τ E and T e , we can obtain the corresponding values of T NBI , n e , τ E and T e .Among them is the experimental function relationship between | f MHD | and T NBI is T NBI = 0.97 × | f MHD | −1.11 , as shown in figure B3 of

Figure 8 .
Figure 8. MARS-Q simulated time traces for radial resonant magnetic field at the q = 2 surface b 2/1 (left y-axis, solid line) and the plasma flow frequency f 2/1 estimated from the mode frequency 'f 0 = | f MHD |' (right y-axis, dashed-dotted line), considering three different NBI torques, i.e. f 0 = 1.00 kHz (red), f 0 = 1.50 kHz (blue) and f 0 = 2.00 kHz (black).Parameter dependencies are based on all experimental observations, and the vertical dashed lines indicate the moment of m/n = 2/1 field penetration.Pink line represents the RMP current that increases over time.

Figure 9 .
Figure 9.Comparison of the scaling of penetration threshold with T NBI is shown for experimental results (red line), theory (black line) and simulation (blue line) at two distinct frequencies: f 0 = | f MHD | (solid line) and f 0 = f ϕ (dashed line).Other plasma parameters, including ne, Te and τν , are varied using all experimental data.Numerical setup for f 0 = | f MHD | and f 0 = f ϕ uses the experimental mode frequency f MHD and toroidal rotation f ϕ , respectively, to replace the plasma flow Ω in the MARS-Q model.

Figure A1 .
Figure A1.Radial profiles of (a) safety factor, (b) line-averaged electron density and (c) electron temperature for EAST discharges 91414 (blue line) and 91709 (red line).Dotted lines indicate the location of the q = 2 surface.
Appendix B. Relationship between the mode frequency | f MHD | and plasma parameters (f ϕ , β N , q 95 , n e , τ E , T e and T NBI )

Figure B1 .
Figure B1.Experimental scaling of (a) the plasma toroidal rotation f ϕ , (b) plasma normalized beta β N , (c) q 95 , (d) electron density ne, (e) the energy confinement time τ E and (f ) the electron temperature Te at the q = 2 surface with the mode frequency | f MHD |.

Figure B2 .
Figure B2.Experimental scaling of (a) penetration threshold current, (b) electron density ne, (c) energy confinement time τ E and (d) electron temperature Te with the plasma toroidal rotation f ϕ .

Figure B3 .
Figure B3.Experimental scaling of T NBI with the mode frequency | f MHD |.

Table 1 .
f 0 scaling of field penetration threshold current I RMP,th .