Plasma rotation and diamagnetic drift effects on the resistive wall modes in the negative triangularity tokamaks

It was found previously that the negative triangularity (NT) configuration is more MHD-unstable for low n modes than the positive triangularity (PT) case, although the situation is reversed for intermediate n modes and the NT configuration becomes more stable for intermediate n modes ( n=3−10 ) (Zheng et al 2021 Nucl. Fusion 61 116014). Here, n is the toroidal mode number. In this work, we extend the studies to include the rotation effects, as well as the diamagnetic drift effects, to see how the resistive wall modes (RWMs) in the NT configuration are affected as compared with the PT configuration. This is particularly motivated by noting that the wall interface with the plasma is quite different between the NT and PT configurations. It affects the plasma rotation and diamagnetic drift effects on the low n RWM. We consider the DIII-D-NT-experiment equilibrium reconstructed by the EFIT code. Based on the equilibrium g-file, the extended equilibria are constructed with the VMEC code by varying the beta values while keeping the pressure and poloidal current flux profiles basically unchanged. The bootstrap current contribution to the equilibria is taken into account with the Sauter formula. The MHD stability is then computed using the AEGIS code with the rotation and diamagnetic drift effects taken into account. We found that, although the NT configuration is less stable for n = 1 MHD modes, the rotation and diamagnetic drift stabilization effects on RWMs are more effective in the NT configuration than in the PT one. Note that even in the PT case, the stabilization of RWMs by the rotation and kinetic effects is critical. Because the low-n RWMs in the regular NT case are more unstable, the rotation and diamagnetic drift stabilization effects found in this research are important for the NT tokamak concept.


Introduction
In magnetically confined fusion, strong magnetic fields are used to confine hot plasmas in the concepts like tokamaks.Confinement efficiency is directly related to plasma beta (β, the ratio of plasma pressure to magnetic pressure).In tokamaks, H-mode confinement is found to have longer energy confinement time and thus enables more efficient high beta confinement than L-mode [1].However, due to the building up of pedestal pressure gradient, the so-called edge localized modes (ELMs) can prevail in the H-mode confinement [1].The ELMs are the bursting magnetohydrodynamic (MHD) instabilities localized at the plasma edge.They cause particles and heat in the pedestal to be discharged into the scrape-off layer.Because the scrape-off layer is connected to the divertors, such a discharge can damage the divertor plates.This constitutes a huge challenge for tokamak engineering.
Since high beta confinement is preferred, extensive efforts have been made in the field of magnetically confined fusion to achieve H-mode confinement with the ELMs suppressed or mitigated.Alternatively, the negative triangularity tokamak (NTT) was proposed as a natural solution [2][3][4].It shifts the design priority from the high beta core to the stable edge.Compared with the positive triangularity tokamak (PTT), the divertors in the NTT case are moved from the high field side to the low field side.As pointed out in the earlier works [2][3][4], this change makes the NTT have great advantages in the divertor design, for example a larger separatrix wetted area, lower background magnetic field for internal poloidal field coils, and larger pumping conductance from the divertor plenum, etc.
Other than the advantages in divertor design, a natural concern for NTT falls in its stability.This is because NTT has a larger volume of bad curvature region than PTT.Indeed, the H mode confinement is seldem achieved in the NT configurations.Numerical studies of low n MHD modes also show that the NT configurations are more unstable than the PT ones [5][6][7], besides the advanced steady state scenario [6,8].Here, n is the toroidal mode number.Nevertheless, the NT experiments in TCV and DIII-D experiments have shown a promising result.It is found that the L mode discharges in the NT experiments can reach about the same level of normalized beta (with the definition detailed in section 2) as in the H mode confinement in the PT confinement.Since being L mode discharges, they are ELM-free.These make the NTT configuration extremely attractive.Furthermore, it is found that the turbulence level in NT experiments is remarkably lower than in the PT case [2,4,9].In DIII-D, this is confirmed by the comparison of a local measurement of density fluctuation level at the minor radius around 0.7 for a NT L-mode discharge and a PT H-mode discharge [4].In TCV, it is found that the electron heat transport level exhibits a continuous decrease with decreasing triangularity from positive to negative triangularities in the L mode discharges [2,9].Note that the turbulence spectrum observed in the NT experiments peaks in the low frequency spectrum where the MHD activities prevail.This is confirmed by the MHD stability analyses with the AEGIS and DCON codes for DIII-D-like equilibria.It was found that while the NT configuration is more MHD-unstable for low n modes, the situation is reversed for intermediate n modes (n = 3 − 10), for which the NT configuration is more stable [7].
In this work, we extend the NTT studies in [6] and [7] to include the rotation and diamagnetic drift effects in order to see how the resistive wall modes (RWMs) in the NT configuration are affected as compared with the PT configuration.In tokamaks, the unstable low n external kink modes can often be stabilized by the perfectly conducting wall.However, in reality, the wall cannot be perfectly conducting and usually has a finite resistivity.This makes the external kink modes stabilized by the perfectly conducting wall become unstable again.They are therefore referred to as the RWMs.RWMs usually are the slow growth modes and are affected by the nonideal MHD effects, such as the continuum damping in the presence of rotation, wave-particle resonance, the diamagnetic drift effects, etc [10][11][12][13][14].Note that the wall interface with the plasma is quite different between the NT and PT configurations.This modifies the plasma rotation and diamagnetic drift effects on the low n modes.Furthermore, noting that even in the PTT case the stabilization of RWMs by the rotation and kinetic effects is critical [10][11][12][13][14], this research is important because the low n RWMs in the regular NTT case are more unstable.
We found that, although the NT case is less stable for n = 1 MHD modes, the rotation and diamagnetic drift stabilization effects on RWMs are more effective in the NT configuration than in the PT one in the DIII-D-like L mode discharges.The results are therefore useful for considering NT as the future tokamak configuration.
The manuscript is organized as follows.In section 2 the equilibrium and numerical scheme are described; In section 3 the stability analyses are presented; The conclusions and discussion are given in the last section.

Equilibrium and numerical scheme
In this section, the equilibrium construction and numerical scheme in this study are described.In the equilibrium, we mainly review the NTT equilibria used in the earlier studies without the rotation and nonideal MHD effects in [7].In the stability, we review the formalism for studying the rotation and diamagnetic drift effects on the RWMs in the PT case in [13] and [14].Investigation of the rotation and nonideal MHD effects on the RWMs in NTTs is necessary to advance the NTT concept noting that NTT tends to be more unstable than PTT to the RWMs in the ideal MHD case without rotation in the L mode discharges.The current work is actually among the first efforts in this direction.
To study the rotation and diamagnetic effects on the MHD modes in the positive and NT configurations, the same type of DIII-D-like equilibria as in [6] and [7] are used, in which the MHD mode stability for positive and NT cases has been investigated.Since the plasma rotation is assumed to be toroidal and subsonic, i.e. the toroidal rotation frequency is much lower than the ion acoustic frequency, the rotation effects on the equilibrium are neglected.This is because the kinetic energy of plasma rotation is smaller than the thermal energy in the subsonic rotation case.As described in [6] and [7], the equilibrium is reconstructed from the g-file from the DIII-D NT experiments using the EFIT code [15].To compare the stability properties on the positive and negative configurations, the equilibrium based on the g-file from DIII-D experiments is extrapolated to generate various equilibria in the positive and NT configurations.The equilibria with various beta in each triangularity configuration are obtained by varying the beta value while keeping the pressure and poloidal current flux profiles basically unchanged.The pressure and current profiles are generated from the density and temperature profiles self-consistently.The bootstrap current effects are taken into consideration.The Sauter formula in [16] is used for computing the bootstrap current.The MHD equilibria are constructed numerically by solving the Grad-Shafranov equation.The VMEC code is used as the Grad-Shafranov equation solver [17].
In this work, we study the subsonic rotation and diamagnetic effects on the RWMs in NTTs.As in the PTT case [13,14], we use the perpendicular MHD equations with the subsonic rotation and diamagnetic drift effects taken into account.The basic equation for describing plasma region is as follows: ( Here, ω is mode frequency, Ω represents toroidal rotation frequency, ω * denotes the ion diamagnetic drift frequency, ξ is the fluid displacement, with subscripts ⊥ and ∥ denoting respectively the perpendicular and parallel components of the magnetic field line displacement, vectors are denoted by boldface, B denotes equilibrium magnetic field, δB = ∇ × ξ × B represents the perturbed magnetic field, J is the equilibrium current density, µ 0 δJ = ∇ × δB denotes perturbed current density, µ 0 is the magnetic constant, P is equilibrium pressure, the perturbed pressure is given by δP = −ξ • ∇P, and the perturbed quantities are tagged with δ except for ξ.The mass density ρ m in equation ( 1) is total mass density, which is the sum of perpendicular and parallel mass (i.e. the so-called apparent mass) densities, The plasma compressibility effect −ΓP∇ • ξ is taken into account to give rise to the so-called apparent mass effect in the total plasma mass ρ m [18].Here, Γ is the ratio of specific heats.Since the rotation is assumed to be subsonic, from the MHD theory with plasma rotation in [19] one can see that the rotational effects of centrifugal and Coriolis forces on the stability can be neglected [20,21].We then include the rotational effects only through the Doppler frequency shift in the stability analysis, i.e. one has ω → (ω + nΩ).As the diamagnetic drift effects (ω * ) can also be important, adding them together, we then have ω(ω − ω * ) → (ω + nΩ)(ω + nΩ − ω * ) in equation (1).Note that the current work only treats the n = 1 RWMs.For high n modes, the rotation and rotation shear effects may need to be included [20,22,23].
Noting that the growth rate of RWMs , γ, is much smaller than the Alfvén frequency, the small γ effects on the inertia term are unimportant [24].γ in equation ( 1) mainly plays a role in healing the Alfvén singularity.This resembles to that used to evaluate the Landau damping with a small imaginary parameter [25].The solution of equation ( 1) is then matched to the vacuum and wall solutions to form the eigenvalue problem.The thin-wall assumption is adopted and conformal wall is considered in our investigation.This is basically the formulation for studying the RWMs described in [24,26,27].In this formulation, the resistive-wall-mode growthrate can be estimated as follows [24,26,27] where γ RWM is the RWM growth rate, τ w = µ 0 σdb denotes the resistive wall time, σ is the wall conductivity, and d denotes the wall thickness.Here, δW ∞ and δW b correspond respectively to the plasma energies without a wall and with a perfectly conducting wall at the wall position b.In the toroidal geometry, they are matrices [27].In the cylinder geometry, δW ∞ and δW b are just scalars.In analyzing the RWMs, the concept of the critical wall position (b c ) needs to be introduced as shown in [24].The RWM instabilities occur when the system is stable with a perfectly conducting wall (δW b > 0) but unstable without a wall (δW ∞ < 0).The so-called critical wall position (b c ) for the perfectly conducting wall case is defined as follows: When the wall position is smaller than the critical wall position, the system is stable for external kink modes, otherwise unstable.
When the system contains the resonance effects, δW ∞ and δW b becomes complex.As shown in [26], using W r ∞,b for real parts and δW i for imaginary part equation (2) in the cylinder limit becomes From equation (3) one can see that δW i 2 > 0 plays a stabilizing role.This effect is especially effective in the case with the wall being near the critical wall position (i.e.δW b = 0).This will also be seen in the numerical results to be described in the next section for NT configuration.We use the AEGIS code for stability analyses [27].AEGIS code was benchmarked with GATO [28] when it was developed [27].In our current studies of NT tokamaks, DCON [29] is often used for further benchmark studies.Agreement between AEGIS and DCON has been found constantly.AEGIS was also benchmarked with other MHD codes, for example, a good agreement with MISHKA [30] is found for Alfvén eigenmode calculation.
To proceed to study the subsonic rotation and diamagnetic effects on the n = 1 RWMs, let us summarize the equilibrium and stability results from earlier studies in the ideal MHD formalism without rotation as follows.The equilibrium cross sections are given in figure 1 of [7].The typical plasma beta (pressure) and safety profiles are given in figure 2 of [7].The stability conditions as shown in figure 3 of [7] are summarized in table 1, in which three new NTT cases (NT2 -NT4) with the same critical wall positions respectively as the PTT ones are added.They will be used to discuss the effectiveness of rotation and diamagnetic drift effects in the next section.The critical wall position b c versus β geo N are given in this table for both DIII-D-like equilibria with the negative and positive triangularities (δ).Here, the normalized beta is defined as where ⟨β⟩ is the volume-averaged ratio of the plasma pressure to the toroidal vacuum magnetic pressure at the geometric center of plasma column, I N = I/aB is the normalized toroidal current, I is the toroidal plasma current, B is the vacuum toroidal magnetic field at the geometric center, and a is the minor radius.This definition is used in the DIII-D experiments [31].

Stability analyses
In this section, we describe the stability analyses of the toroidal rotation and diamagnetic drift effects on the n = 1 RWM stability using the AEGIS code [27], especially the Alfvén continuum damping effects are included [13].
The cases without the rotation and diamagnetic drift effects have been reported in [6,7].The results are summarized in table 1 for n = 1 modes with Cases NT2, NT3, and NT4 being newly added.It gives the dependence of critical wall position (b c ) on the normalized beta.The results show that at a given critical wall position the normalized beta is lower in the NTT cases than that in the PTT cases, or the conducting wall needs to be closer to the plasma torus for n = 1 mode stability in the NT case.This indicates that the n = 1 RWMs in the NT configuration are more unstable than in the PT configuration.We then introduce the plasma rotation and diamagnetic drift effects to see how the stability conditions are affected.
Figure 1 shows the typical rotation effects on the n = 1 RWM growthrate with Ω denoting the rotation frequency normalized by the Alfvén frequency.The NTT and PTT cases are shown respectively in Parts a) and b).To compare the rotation effects on NTT and PTT, we have studied respectively Cases NT2 and PT1 in table 1.Both cases have the same critical wall position b c = 1.51 so their RWM growthrates in the absence of rotation are almost the same.From figure 1 one can see that the reduction of the RWM growthrate by the rotation is more substantial for the NT configuration than for the PT configuration.This indicates that the plasma rotation can have a stronger stabilizing effect on the n = 1 RWMs in the NTT case than in the PTT case.We also examine the other two sets of cases: NT 3 versus PT2 and NT4 versus PT3 in table 1.It is also found that the plasma rotation can have a stronger stabilizing effect on the n = 1 RWMs in the NTT case than in the PTT case.In figure 2, we particularly plot out the comparison between Cases NT3 and PT2 with the critical wall position b c = 1.28 and rotation Ω = 0.05 to confirm our conclusion.beta, i.e. the upper curve delineating the magenta region.The external kink modes are unstable in this corner (δW b < 0).Under the curve of critical wall position, the external kink modes are stable when there is a perfectly conducting wall (δW b > 0).The RWMs are concerned with the instabilities where the system is unstable without a wall (δW ∞ < 0) but stable with a perfectly conducting wall (δW b > 0), i.e. the region under the curve of critical wall position.This can be seen from equation (2) for the RWM growthrate.Therefore, to study the rotation or diamagnetic drift effects on RWMs one only needs to consider the region below the critical wall position curve.In Parts a) and b) of figure 3 the stable regions due to the plasma rotation are marked in magenta, which indicates that the plasma rotation plays a stabilizing effect in the parameter domains where RWMs are unstable in both NTT and PTT cases.The magenta region is bound on the upper side by the critical wall position curve and on the lower side by the condition δW r ∞ δW r b + δW i 2 = 0 according to equation (3).At the critical wall position one has W r b = 0. Therefore, a small δW i due to the plasma rotation can stabilize the modes.As the wall position is taken to be smaller, W r b increases and the instability driving term δW r ∞ δW r b can become larger than the stabilizing term δW i 2 .Comparing Part a) and Part b) in figure 3, one can see that at a given normalized beta the magenta region in the NTT case is larger than that in the PTT case, which can also be seen in figures 1 and 2. Also, as shown in figures 1 and 2, it is found that the RWM growthrate is smaller in the lower left corner for the NTT case as compared with the PTT case.These results indicate that the plasma rotation can play a stronger stabilization role for n = 1 RWMs in the NTT case than in the PTT case, although the NTT case is more unstable in the absence of plasma rotation for having a lower beta limit.We also investigate the diamagnetic drift effects both in the negative and positive triangularity cases.Because the RWM growthrate is inversely proportional to the plasma energy with a perfectly conducting wall, δW b , as shown in equation ( 2), which tends to zero at the critical wall position, one can expect that the diamagnetic drift effects can also play a significant role in the case with the wall position close to the critical wall one (δW b = 0).This is because the diamagnetic drift effects can make δW b ̸ = 0 at the critical wall position.The RWM growthrate versus the wall position with and without the diamagnetic drift effects is plotted in figure 4 respectively for the NT and PT cases and with and without plasma rotation for the case with the critical wall position b c = 1.51.The diamagnetic drift frequency is computed self-consistently according to the pressure profile and mode number.From figure 4 one can see that the diamagnetic drift effects reduce the RWM growthrate, especially near the critical wall position.However, the diamagnetic drift alone cannot fully stabilize the RWMs.This can be seen from equation ( 3) by noting that the inertia effects, where the diamagnetic effects reside, are minimized when the mode frequency vanishes.When the plasma rotation with the direction being the same as the diamagnetic one is included, however, the diamagnetic drift effects are enhanced.Comparing figures 4(a) and (b), one can see that the diamagnetic drift effects in the NT are more effective than in the positive triangularity case.We also examine the other two sets of cases-NT 3 versus PT2 and NT4 versus PT3 in table 1.It is also found that the plasma rotation and diamagnetic drift effects can have a stronger stabilizing effect on the n = 1 RWMs in the NTT case than in the PTT case.In figure 5, we particularly plot out the comparison between Case NT 3 and Case PT2 with the critical wall position b c = 1.28 and rotation frequency Ω = 0.05 to confirm our conclusion.
The typical eigenmodes are plotted in figure 6, in which the real (ξ r ) and imaginal (ξ i ) parts of radial field line displacement are plotted against the poloidal flux (χ) normalized by its edge value.Compared to the ideal MHD case without the rotation and diamagnetic drift effects in [6] and [7], the imaginary of the eigenfunction appears in the current case, which reflects the Alfvén continuum damping [13].
In figure 7, we plot the stability diagram in the coordinate space of normalized beta versus wall position.The NTT case is plotted in Part a) and PTT in Part b).In these plots, the rotation frequency is assumed to be Ω = 0.05 and the diamagnetic drift effects have been taken into account.Comparing Part a) and Part b) in figure 7, one can see that at a given normalized beta the magenta region in the NTT case is larger than that in the PTT case, which can also be seen in figures 4 and 5. Also, as shown in figures 4 and 5, it is found that the RWM growthrate is smaller in the lower left corner for the NTT case than that in the PTT case.These results indicate that the plasma rotation and diamagnetic drift effects can play a stronger stabilization role for the n = 1 RWMs in the NTT case than in the PTT case, although the NTT case is more unstable in the absence of plasma rotation and the diamagnetic drift effects for having a lower beta limit.

Conclusions and discussion
The rotation and nonideal MHD stabilization effects on RWMs in the PT tokamaks have been extensively studied in this field, for example in [10][11][12][13].Nevertheless, this question is still open for the NT tokamaks.This is important since the n = 1 RWMs are more unstable in NTTs than in PTTs.To justify the feasibility of NTT as a confinement concept to address the divertor heat load issue, the rotation and nonideal MHD stabilization effects on RWMs in NTTs need to be clarified.We obtained important results, showing that the rotation and diamagnetic stabilization effects on the RWMs are more effective in the NT tokamaks than in the PT tokamaks.The results are new and give a strong support to the NT tokamak concept.Nevertheless, we point out that the current comparison of the rotation and diamagnetic drift effects is based on the DIII-Dlike L mode discharges.It is assumed that both NTT and PPT have the same density and temperature profiles with the current computed self-consistently.Since RWMs are investigated in this work, only the cases with the wall positions less than the critical wall position are treated.The unstable external kink modes with a perfectly conducting wall can give a beta limit through the critical wall position.In the current DIII-D-like configurations, at a given beta the critical wall positions are different for NTT or PTT, or at a given critical wall position the beta limits are different for NTT or PTT.This makes us unable to compare the rotation and diamagnetic drift effects on NTT and PTT with the same beta for RWMs.Note however that RWMs are characterized by their growthrates, i.e. −δW ∞ /δW b .This leads us to make the comparison between NTT and PTT with nearly the same RWM growthrates by imposing the same critical wall positions, although the PTT case has a higher normalized beta.The rotation and diamagnetic drift effects make the RWM growthrate in the NTT case smaller than that in the PTT case.This leads us to conclude that the rotation and diamagnetic drift effects on NTT are more effective.Since the plasma confinement involves multiple parameters, other scenarios remain to be explored in the future.
We first study the toroidal plasma rotation effects on the n = 1 RWMs in the NT configuration as compared with the PT one.This is motivated by the earlier MHD mode studies without rotation [5][6][7], which show that the n = 1 modes in the NT case is more unstable than in the PT case.While the wall interface with plasma is quite different between the negative and PT cases, it is naturally to ask if the rotation effects on the n = 1 RWMs can be different.The NT configuration has a larger bad curvature volume.But, it also has a larger plasma-wall interface on the bad curvature side.These motivate our investigation.We indeed found that, although the NT case is less stable for n = 1 MHD modes, the rotation stabilization effects on the RWMs are more effective in the NT configuration than in the PT one.
We next studied the diamagnetic drift effects.It is also found that the rotation stabilization effects with the diamagnetic effects included on the RWMs are more effective in the NT configuration than in the PT one.
Based on similar reasoning, one may expect that the kinetic stabilization effects on the n = 1 modes can be more effective in the NT configuration than in the PT configuration.Certainly, this needs to be confirmed by future computations.Note that even in the PTT case the stabilization of RWMs by the rotation and kinetic effects is required [10,11,13].Because the low-n RWMs in the regular NTT case are more unstable, the results in this research are important for the NTT concept.

Figure 1 .
Figure 1.The growthrate of n = 1 resistive wall modes versus the wall position normalized by the minor radius on the mid-plane on the low field side.The critical wall position (bc = 1.51) is plotted with the vertical dash-dotted line.The normalized toroidal rotation frequency Ω is used as a parameter.Part (a) is the NTT case, Part (b) is the PTT case.

Figure 2 .
Figure 2. The growthrate of n = 1 resistive wall modes versus the normalized wall position.The critical wall position (bc = 1.28) is plotted with the vertical dash-dotted line.The normalized toroidal rotation frequency Ω is used as a parameter.The NTT case is plotted in blue and the PTT case in red.

Figure 3 .
Figure 3. Stability diagram in the coordinate space of normalized beta versus normalized wall position with the rotation frequency Ω = 0.05.The NTT case is plotted in Part (a) and PTT in Part (b).The stable regions are marked in magenta.

Figure 4 .
Figure 4.The growthrate of n = 1 resistive wall modes versus the normalized wall position.The critical wall position (bc = 1.51) is plotted with vertical dash-dotted lines.The normalized toroidal rotation frequency Ω is used as a parameter.The diamagnetic drift effects are taken into account in this figure.Part (a) is the NTT case, Part (b) is the PTT case.

Figure 5 .
Figure 5.The growthrate of n = 1 resistive wall modes versus the normalized wall position.The critical wall position (bc = 1.28) is plotted with the vertical dash-dotted line.The normalized toroidal rotation frequency Ω is used as a parameter.The diamagnetic drift effects are taken into account in this figure.The NTT case is plotted in blue and the PTT case in red.

Figure 6 .
Figure 6.The typical eigenfunction in the presence of plasma toroidal rotation (Ω = 0.05) and diamagnetic drift effects for the NTT case with the wall position b = 1.38 and the critical wall position bc = 1.51.

Figure 7 .
Figure 7. Stability diagram in the coordinate space of normalized beta versus normalized wall position with the rotation frequency Ω = 0.05 and the diamagnetic drift effects taken into account.The NTT case is plotted in Part (a) and PTT in Part (b).The stable region marked in magenta.

Table 1 .
Critical wall position versus β geo N .