Impact of fast ions on microturbulence and zonal flow dynamics in HL-2A internal transport barriers

The turbulent transport properties and dynamics of zonal flows (ZFs) in the presence of fast ions (FIs) are investigated for a typical internal transport barrier (ITB) plasma based on the gyrokinetic approach, focusing on the role of FI temperature and the effects of the toroidal rotation, including the E× B rotational shear, parallel velocity gradient (PVG) as well as the rotation velocity itself. Linear GENE simulations have shown that the core ITB plasma on HL-2A is dominated by ion temperature gradient (ITG) modes and trapped electron modes (TEMs), where the former is stabilized by FIs whereas destabilized by the PVG. Neither of the FIs or the PVG has observable effect on TEMs. The ion heat transport generally decreases at large FI temperature due to the nonlinear electromagnetic stabilization of turbulence with increased total plasma β until electromagnetic modes are excited. The transport fluxes peak around a certain FI temperature and the ZF shearing rate is significantly higher at such value compared with that in the absence of FIs, and the heat flux reduction is a result of the synergistic interaction between turbulence, ZFs and the external rotational shear. The E× B shear stabilizing and PVG destabilizing is not obvious at low normalized ITG R/L Ti, indicating they are less important in determining the stiffness level in the relatively low density and rotation scenarios regarding the HL-2A ITB discharges. The turbulence suppression is predominated by the nonlinear stabilization of ITG turbulence as well as enhanced ZFs simultaneously in the presence of FIs. These results have also provided the possible way to reduce the turbulence transport through increasing the FI temperature in the off-axis neutral beam heated plasmas such as in HL-2A.

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Introduction
Transport in magnetically confined devices is one of the key subjects in the area of fusion relevant plasma research.Both theoretical and experimental investigations have shown that the plasma confinement is largely controlled by drift wave fluctuations and associated anomalous transport across the magnetic field lines over the past several decades [1][2][3].The general description of transport or confinement requires consideration of the various instabilities including large scale (of the order of the minor radius of device) magnetohydrodynamic (MHD) modes [4] which determine the operation limits and short scale (of the order of ion/electron gyro radii) drift-wave instabilities [5] that dominate the transport, among whom the electrostatic (ES) ion temperature gradient (ITG) mode [6] or the electromagnetic kinetic ballooning mode (KBM) [7], trapped electron mode (TEM) [8] and electron temperature gradient mode (ETG) [9] are of great importance, which are suspected to be the primary candidates for explaining the ion heat transport, electron particle transport and electron heat transport in the plasma core, respectively.The suppression or mitigation of turbulence is one of the major challenges in plasma physics which is particularly important for approaching commercial viability for fusion energy in future devices such as ITER [10].
Over the last decade, lots of important progresses have been made in clarifying the impact of energetic particle on microturbulence and transport, which has been successfully applied in explaining the ion temperature profile stiffness in tokamaks where the ITGs are the dominate microinstabilities [11][12][13].Energetic particles, or fast ions (FIs), can play a significant role in stabilizing turbulent transport in tokamak plasmas and it has been demonstrated in many cases, the presence of FIs can effectively suppress turbulence [14].In fusion plasmas, FIs are present due to external heating methods such as neutral beam injection (NBI) or ion cyclotron resonance heating which is a component of the whole plasma that have a higher energy compared to the thermal energy of the bulk plasmas.These FIs can interact with the ambient turbulence through a number of mechanisms, depending on their energy and distribution [15].One of the key ways is through their effect on the background plasma density and temperature gradients known as the dilution effect [16], which is always present as long as another type of ions exist that differs from the main ion species due to the requirements of quasineutrality condition.The inclusion of FIs can modify the density and temperature profiles of the bulk plasma, which can in turn decrease the driving force of the ITG instability.This mechanism was identified by the reduced turbulent transport in the short lifetime ion internal transport barrier (ITB) on ASDEX-U and further confirmed by linear GS2 simulations.The second mechanism is the nonlinear electromagnetic (EM) stabilization [11] as well as the ES resonant interaction [17].While the former has been known as finite-β (β is the ratio of plasma pressure to magnetic pressure) effect [18] for a long time, the latter was firstly demonstrated by global GENE gyrokinetic simulations and proven to be the trigger mechanism of a novel type of transport barriers (FI-induced anomalous transport barrier), indicating that strong zonal flows (ZFs) were generated at the barrier boundary when FIs were included [19].The inclusion of FIs will increase the EM coupling hence the stabilization effect, which is enhanced with FI pressure.The above mechanisms were identified to be the important ingredients to explain the ion temperature (T i ) stiffness level observed in experiments and have provided encouraging suggestions for extrapolation to future reactors, where the fusion gained α-particles will occupy a large proportion of the total pressure.
At present it is widely accepted that in addition to the mean E × B shear flow, the nonlinearly self-generated poloidal flow in saturated turbulence states known as ZFs [20][21][22], can act as an effective role in reducing the turbulent transport.The former has been examined on many devices such as TFTR [23], JET [24], DIII-D [25] and JT-60U [26] and its shearing rate (ω E×B ) was found to be close to the linear growth rate of the ITG modes (γ ITG ) at the time of barrier formation when compared with several tokamaks [27].For instance, analysis of the DIII-D discharge has shown that ω E×B > γ ITG holds across the whole plasma [28].The relationship between the amplitudes of zero-frequency ZF and turbulence has been firstly investigated in ITB plasmas in the CHS device using dual heavy ion beam probe, showing that the turbulence amplitude is much lower and the magnitude of the ZF is larger in the plasma with ITB compared to that without it [29].In addition to the self-generated ZFs, the toroidal rotational shear [30][31][32] is another important factor that suppresses the turbulence transport and plays a key role in reducing the stiffness level [33].It is suggested that the latter provides the major shearing effect in the core plasma as the mean poloidal rotation is strongly damped in these regions, especially for the case of NBI heated plasmas.Although the shearing due to toroidal rotation is stabilizing, the impact of parallel velocity gradient (PVG) [34][35][36] destabilization is significant as well.The PVG drive increases with increasing geometrical factor q/ε, where ε = r/R is the local inverse aspect ratio.As the NBI can not only provide FIs with certain temperature distribution, but also drives various toroidal rotation velocity by adjusting the input power hence it is important to examine the role of FI temperature and the rotational shear as well as PVG.Previous simulations based on ASDEX-U [37] and JET experiments [38] have shown that an increased FI temperature generally has stabilizing effect on ITG turbulence due to increased αstabilization.However, the combined role of E × B shear and PVG seems to be different, i.e. the stiffness reduction due to increased E × B shear is partially compensated by the consistent increase of the PVG for JET plasmas [12] but the PVG has weak effect on ITG linear growth rate and turbulence levels do not drop significantly even including the E × B flow shear for ASDEX-U scenario.In addition, results from CGYRO simulations of DIII-D high confinement scenarios employing different NBI conditions shows that both the FIs and E × B are generally important for the qualitative prediction of stiffness levels [39].For the above reason, the interplay between PVG destabilization, E × B stabilization and FI effects is particularly critical and should be examined carefully for turbulent transport regarding tokamaks employing powerful NBI heating along with external momentum input which is the common situations for present and future devices.
Overall, the effects of FIs and rotations on the dynamics of ITG instability and ZFs are complex and multifaceted phenomenon.While some studies have shown that FIs can stabilize the turbulence and reduce energy transport, other factors can significantly alter the properties of the turbulence and increase transport together with the changes in ZF properties.Detailed research is needed to further understand the mechanisms by which they affect the ITG instability and to determine the conditions under which they may be beneficial or detrimental to plasma confinement in fusion devices, which is the major goal of the present paper.
The remainder of this paper is organized as follows.The typical parameters in HL-2A ITB plasmas are described in section 2. The GENE gyrokinetic simulation results of the ITG turbulence based on experimental data are presented in section 3, where the ZF dynamics, thermal transport and stiffness levels have been analyzed.Finally, concluding remarks are given in section 4.

Experimental characteristics of the HL-2A ITB plasmas
The profiles and physical parameters used in this paper are described here.The ITB experiments were performed in the NBI heated deuterium plasmas in the HL-2A tokamak (major radius R = 1.65 m and plasma minor radius a ≈ 0.36 m) [40] with plasma current I p ≈ 150 kA, toroidal magnetic field B T ≈ 1.27 T and central line-averaged density ne ≈ 1.2 × 10 19 m −3 .It is noted that the ne was relatively low and almost kept constant even the ITB was formed, which is suggested to be important in governing the stiffness levels.Shown in figure 1 are the profiles of electron density n e , electron temperature T e , ion temperature T i and toroidal angular rotation frequency Ω t of shot #25733 deuterium plasma, which is a typical ITB discharge [41].The barrier starts to form at ∼30 ms and is well developed at ∼50 ms after NBI is turned on.The T i together with Ω t are measured with a 32-channel charge exchange recombination spectroscopy diagnostic system with spatial and temporal resolutions about ∼1.5 cm and 12.5 ms [42].A 32-channel fast electron cyclotron emission system provides the T e with temporal and spatial resolutions up to 0.8 µs and 1 cm [43].The density profiles n e are reconstructed from the formic acid (HCOOH) laser interferometer [44] measurements through Abel inversion method.All of these profiles are mapped on to the flux surface coordinates.It is clearly seen that the T i and its gradient have large values in the ITB region, where the locations of the largest gradient and ITB foot are around ρ ≈ 0.36 and ρ ≈ 0.42, respectively.It was observed that the T e has also increased slightly as the NBI heats electrons and ions at the same time, whereas the n e is shown to be slightly decreased during the ITB formation.The safety factor (q) profile shown in figure 2(a) is calculated by the kinetic equilibrium and reconstruction fitting code (kinetic EFIT) in the framework of OMFIT integrated modeling [45] averaged over 50 ms during the well-developed ITB with a resolution of 10 ms, i.e. 5 time slices are calculated and averaged as donated by the errorbars.It is discovered that the plasma shape generally shows a weak magnetic shear (ŝ) configuration in the core region due to the off-axis NBI heating.The E × B shearing rate due to toroidal rotation calculated as γ E = (r/q)dΩ t /dr in units of c s /R 0 is shown in figure 2(b), where the maximum value inside the ITB locates at the same position of the largest ITG because the gradient of the rotation also reaches maximum at this point.Here c s = √ T e /m i is the ion sound velocity and R 0 is the major radius of the device.In the present paper the simulations were performed at ρ ≈ 0.36; however, it should be pointed out that the electromagnetic microinstabilities such as KBMs or Alfvénic ITG (AITG) modes [46] are suggested to be easily excited in the weak magnetic shear regions [47] of ρ ≈ 0.1-0.3 which may also crucial in determine the ion heat transport and the stiffness levels.Detailed analysis of the effect of FIs on electromagnetic turbulence will be discussed in the following work in the near future.
The profiles of FIs are shown in figure 3 which is calculated by the NUBEAM module incorporated in the ONETWO transport solver in the OMFIT framework, during which the experimental ion heat transport is also estimated using power balance technique [48].The neutral beam energy is set as 45 keV.The position of maximum FI density (n f ) is near ŝ ∼ 0 as can be concluded from figures 3(a) and 2(a).The FI temperature (T f ) is almost constant across a wide region corresponding to the ITB region, indicating a strong relation between energy deposition of FIs and the ITB, as shown in figure 3(b).

Simulation setup
In this section, the linear and nonlinear simulations of the effects of FIs on the dynamics of ZFs and transport are investigated using gyrokinetic code GENE [49,50], focusing mainly on that of FI temperature and PVG.The parameters used in the simulations are chosen at ρ = 0.36, corresponding to the maximum FI concentration and rotational shear as can be inferred from figures 2 and 3. GENE is a δf formulated gyrokinetic code that solves the gyrokinetic Vlasov equation coupled self-consistently to Maxwell's equations [51].Field line coordinates were applied where x is the radial coordinate, z is the coordinate along the field line and y is the binormal coordinate [52].Collisions are modeled using a linearized Landau-Boltzmann operator.All the simulations performed here were local, electromagnetic and with collisions.The radial position was chosen as ρ ≈ 0.36 corresponding to the location of both the maximum values of ITG and rotational shear.Besides, the FI gradients are set to R/L Tf = 0 and R/L nf = 12 unless otherwise stated.An analytical ŝ-α equilibrium [53] is used and it is believed that other choices such as Miller geometry [54] would not affect the nature of transport and ZFs as the configuration of the HL-2A is well characterized by a circular flux surface shape and the Shafranov shift due to rotation is generally small.The plasma beta β = Σβ j and the stability threshold parameter in the fluid limit in terms of consistently, where j donates the jth species and L nj and L Tj are the density and temperature gradient scale lengths of the corresponding species.β ′ is the radial derivative of β and is related to the local pressure gradient.Such treatment is important especially for the cases where the magnetic shear is low while the pressure gradient is relatively large which will lead to the onset of KBM/AITG although β is not very high.In addition, typical grid parameters were as follows: perpendicular box sizes [L x , L y ] = [161, 105] in units of ρ s = c s /Ω ci with Ω ci = eB/m i being the ion Larmor radius, perpendicular grid discretization [n x , n ky ] = [128, 24], 24 point discretization in the parallel direction, 32 points in the parallel velocity direction, and 8 magnetic moments.Convergence tests were carried out for typical linear simulations and a reduced electron-to-deuterium mass ratio of m e /m i = 1/800 is used in all simulations in order to reduce the computational effort.The GyroBohm normalized transport fluxes are calculated as the velocity space moments of the fluctuating part of the distribution function ( f 1 ) averaging over the whole simulation domain, namely, the particle and heat fluxes are written as i /(eBR) 2 , respectively.The wavenumbers k y and k x is in units of 1/ρ s while the eigenvalues γ and ω are in units of c s /R 0 .The pure rotational plasma is assumed and the effects of impurities are neglected unless otherwise stated.

Effect of FI temperature and E × B shear on turbulence and ZFs
The linear stability properties of the microinstabilities in the presence of FIs are illustrated in figure 4, where the FI population is treated as a separate plasma species with hot isotropic Maxwellian.The ratio of FI temperature inferred from NUBEAM to experimental electron temperature is T f /T e ≈ 16.It is quite clear that the plasma is dominated by ITGs and TEMs whose wavenumbers are in the range of 0.05 ≲ k y ρ s ≲ 0.6 and k y ρ s ≳ 0.7, respectively.Here positive ω is defined as the ion diamagnetic direction in GENE.It is shown that the intermediate scale ITG modes are stabilized once the FIs are introduced.However, at low T f /T e = 4 corresponding to T f /T i ∼ 2.4, there is a slight increase in the ITG growth rate at several wavenumbers since low temperature FIs behave similar to that of the thermal ions so that indeed both the FIs and bulk ions can contribute to the ITG drive.The ITG modes are stabilized with the increasing of the FI temperature, whereas only very weak effect is found on the TEMs.The results are also consistent with previous simulations of the TEM dominated hybrid scenarios on JT-60U [55], as shown in figure 4(a).The stabilization effect of ITG is enhanced as T f becomes larger, which can be partially explained by the increasing of FI beta hence the total plasma β as well as the α MHD [56].An electromagnetic mode having wavenumber k y ρ s = 0.05 and rotating in the ion diamagnetic direction is found when T f /T e = 32 which is identified to be the FI driven BAE/KBM by its significantly higher frequencies than that of the ITGs, as depicted in figure 2(b).It is obvious that a further increasing of T f will destabilize KBMs as well because of the total higher β, hence the values in the simulations are limited to T f /T e ⩽ 32 in order to avoid these strongly unstable EM modes, who are suggested to cause rather rapid ion heat transport level in simulations.
In addition to the effect of FIs, the rotational shear is another important mechanism that suppresses the turbulent transport.and the value is set to γ E = 0.15.It is seen that both of the E × B and the inclusion of FI can reduce the main ion heat transport Q i and the latter is much effective than the former, as shown in figure 5(a).Besides, in our simulations the energetic particle driven modes (EPMs) [57] are not observed thus the FI transport is determined by the stability and transport of the bulk plasma, i.e. the long wavelength ITGs and mediumsized TEMs.The FI transport Q FI is also decreased by the E × B shearing effect due to the stabilization of the background microturbulence, as can be found in figure 5(b).The heat transport induced by FIs is also important in determine the total confinement, who is almost half of that of the energy flux caused by the main ion species.Moreover, it is generally accepted that the EPMs can be excited as long as the FI pressure is large enough, which will provide a channel of FI loss and result in the degradation of confinement.In the present simulations none of such modes were driven unstable thus we have ignored the role of EPMs, which will be taken into account in the future.
Although the linear stability results and the heat fluxes from the nonlinear simulations differ in most cases, the shapes of the flux spectrums in wavenumber (k y ρ s ) space are generally similar in all cases.There are two comparisons for which the spectra of the ion heat fluxes showing the role of FI and E × B shear on transport in the ITG dominated plasma.The result is illustrated in figure 6, where the spectrums have been extracted in the quasi-states of the nonlinear simulations for a time duration typically larger than 50 R 0 /c s .It is shown that very similar predictions for the level of ion heat fluxes are found in the simulations regardless of the inclusion of FI or E × B.Moreover, it is clear that the FI dramatically affects the heat transport, which is regarded as the nonlinear stabilization of the ITG turbulence, who would lead to a drop of 60% in the total ion heat flux in our cases.The effect of the rotation is weak compared to that of the FI, as can be found in figure 6(a).A continuous reduction of the total flux is found with the increasing of the FI temperature, which can be seen in figure 6(b); however, it is believed that the suppression of turbulence by FI temperature is only valid once the EM modes were not excited or weakly unstable, which can also be concluded from the linear stability analysis shown in figure 4. The results excluding E × B shear and PVG are also shown for a comparison by the dashed curved at T f /T e = 32, when the KBM with k y ρ s = 0.05 is linearly unstable.It is obvious that even the KBM is only present at a single wavenumber, the heat fluxes increase significantly.However, once the E × B is taken into account, the transport induced by BAE/KBM is almost totally suppressed partly because the flow shearing rate, dΩ t /dr = q/r * γ E ≈ 0.44 c s /R 0 is close to the KBM growth rate γ = 0.377 c s /R 0 so that it has significant impact on large scale electromagnetic turbulence eddies.The total electron heat flux Q e = Q e,es + Q e,em shown by figure 6(c) also demonstrates that the background turbulence is generally suppressed by the FIs, where Q e,es and Q e,em are the heat fluxes induced by the ES potential fluctuation and magnetic flutter, respectively.The results have indicated that the ion heat transport dominated by ITG turbulence is mainly suppressed by the FIs in HL-2A ITB plasmas with finite rotational shear which is efficient in suppressing the electromagnetic turbulence localized at limited wavenumbers.
Figure 7 gives the dependence of transport of the two ion species (bulk ions and FIs) on FI temperature.The results without and with E × B are shown by red and blue curves, respectively.It is identified that the transport of the main ions as well as the total transport indicated by the dashed curves generally decrease with FI temperature at T f /T e ⩾ 12, while they are almost unaffected when T f /T e is relatively small, which can be discovered in figure 7(a).It is noted that the degree of transport reduction is not consistent with the linear results, which shows a relatively strong stabilization of ITG modes even at low T f /T e (see figure 4(a)) whereas weak stabilization when the latter is large enough.Such discrepancy between the linear and nonlinear simulations is suggested to be resulted from the nonlinear electromagnetic stabilization of the ITG modes, which is further enhanced by increasing the T f /T e thus the total β until EM modes take over.It is also noted that the transport is largely suppressed by the inclusion of FIs and the E × B stabilization is weakened as long as the T f /T e becomes large.Furthermore, the FI transport, who is dominated by the background microturbulence, shows a more complicated dependence on T f /T e in addition to E × B shear.Except for the very high FI temperature case, T f /T e = 32, the FI heat fluxes shows the similar dependence on T f /T e , i.e. they generally peak around a certain T f /T e and the E × B shear has weak effect.The underlying reason is that the shearing effect by E × B generally acts on the ion scale turbulence whose time scale is comparable to the ion sound waves such as ITGs.Besides, the scale separation in the drift frequencies between FI and bulk ions is small when the T f /T e is low, leading to the fact that the motion of the two ion species couples with each other thus a larger transport is expected under such situations.At high T f /T e , the dynamics of FI and main ions decouples hence both the ion and FI transport are decreased due to the stabilization of ITG turbulence.It is also found that in the absence of rotational shear, the FI transport is increased once the EM modes are excited at T f /T e = 32, indicating that the FI β is crucial in determining the FI transport, as shown in figure 7(b).The FI transport is largely suppressed when the E × B shear is taken into account.However, it should be pointed out that only KBM with very small wavenumber is destabilized in this case and the transport induced by the EM modes is very low thus it will not affect the whole transport properties dominated by ITG turbulence, which can also be concluded from figure 6(b).The effect of FIs on electromagnetic turbulence will be left as another work.
It has been recognized for a long time that the ZF plays an important role in suppressing or regulating the turbulence and its saturation, especially for the case when the external mean field such as E × B shear and radial electric field is not large.In the present simulations, the zero-frequency ZF, defined as the m = n = 0 (with m and n being the poloidal and toroidal numbers, respectively) component of the ES potential is identified to be the main saturation mechanism for the case of ITG dominated HL-2A ITB plasmas hence the shearing rate by ZF is important in determining the transport level.The ZF shearing rate ω ZF E×B (in unit of c s /R 0 ) and its ratio between the maximum linear growth rate γ max is plotted as a function of the T f /T e .Both the ω ZF E×B itself and ω ZF E×B /γ max have similar dependence on T f /T e regardless of the inclusion of rotational shear, such as can be found in figure 8(a).It is seen that the ω ZF E×B shows a relatively complex dependence on T f /T e : it is minimum at T f /T e = 8, implying that the turbulence would reach maximum under such condition which is consistent with the heat transport shown in figure 7(a).The ω ZF E×B shows a continuous decrease in the range of 12 ≲ T f /T e ≲ 24, however, the value of ω ZF E×B /γ max keeps almost constant or slight decreases for the cases of without E × B shear and in the presence of rotational shear, respectively, as shown in figure 8(b).The ZF rate is significantly higher at medium T f /T e compared with that in the absence of FIs (shown by the dashed lines) and it is noted that the ω ZF E×B shows opposite trend at T f /T e = 32.The high level of ZF in the presence of E × B shear is resulted from the suppression of ultra-long wavelength EM modes by finite rotational shear who are suggested to be less effective in driving the ZFs.Moreover, the ZF in terms of ω ZF E×B is also suppressed by the external shear which is ascribed to by the suppression of background turbulence due to the mean flow shear, i.e. the Reynolds stress due to the eddy tilting is weakened because of the broken of the eddy structures thus the driving force of the ZF is decreased, leading to a weaker ZF as a consequence [58][59][60].The nonlinear heat flux reduction in the cases of 12 ≲ T f /T e ≲ 24 is a result of the synergistic interaction between turbulence, ZFs and the external rotational shear.A clear explosive grow in the ZF shearing rate is found at T f /T e = 32 whereas it further decreases when E × B shear is present and absent, respectively, which can be seen by comparing the last point in figure 8(a).In this case, the total heat flux will be strongly reduced which is consistent with figure 7.For higher T f /T e the situation may become complicated as the turbulence is dominated by KBMs, ITGs and TEMs.However, we can reasonably infer that although the ZF will increase, the destabilization of KBMs will also lead to a much stronger ion heat flux and the shearing by the former cannot effectively suppress it, hence the total transport level would be enhanced accordingly.

Role of PVG on ion heat transport and ZF dynamics
In addition to the shearing effect by toroidal rotation, it has been demonstrated that the PVG can destabilize the ITG modes, particularly in the case of NBI heating which provides an external momentum source.Recent gyrokinetic simulations have demonstrated that the turbulence will be suppressed by the PVG through the enhanced ZF activity, while the E × B shear directly reduces the turbulence amplitude or the eddy size.Meanwhile, the effect of the latter is more effective than the former, indicating that the possible nonlinear synergetic effect between parallel and the perpendicular E × B flow shear [36].The effect of FI and PVG on the linear stability of the microinstabilities is illustrated in figure 9, where the simulations in the presence of PVG have also take the parallel rotation into account with the value of Mach number M || ≈ 0.61 c s /R 0 .It is obvious that the inclusion of FI or PVG will not change the nature of the dominate microinstabilities which are consist of ITG modes with wavenumbers 0.05 ≲ k y ρ s ≲ 0.6 and TEMs with k y ρ s ≳ 0.7, respectively, as can be concluded by the positive and negative real frequencies shown in figure 9(b).The ITG modes are destabilized by PVG, whereas the FI has stabilization effect.Moreover, both of them have rather weak effect on TEMs, as shown in figure 9(a).From the linear calculations, it is quite clear that both the effect of FI and PVG should be considered when making a quantitative calculation of the ion heat transport in a ITG dominated plasma, such as that in the HL-2A ITB regions.
In order to obtain the feature of ion transport stiffness, nonlinear simulations at various normalized ITG R/L Ti are carried out.Figure 10 shows the time traces of transport fluxes for different R/L Ti and the effect of PVG is also examined.Here the FI temperature is set to T f /T e = 16, corresponding to the ratio of FI temperature calculated from NUBEAM to the experimentally measured T e .It is obvious that the flux generally increases with R/L Ti as the ITG modes are destabilized no matter the PVG is included or not.The threshold value of R/L Ti for the nonlinear transport is around R/L Ti ≈ 6, whereas the linear threshold of the ITG modes is predicted at R/L Ti,c ≈ 3.4 [61].The difference between the linear threshold and the nonlinear upshift is known as the Dimits shift [62].The inclusion of FIs would further increase the nonlinear threshold due to the suppression of ITG turbulence.In addition, the heat flux shows a more obvious oscillating feature in the presence of PVG, especially for large R/L Ti which can be found by comparing figures 10(a) and (c) at R/L Ti = 18.The underlying physics is that the intermittent transport is suppressed by PVG while the ZF amplitudes are higher at larger R/L Ti , hence the regulation of transport by ZF becomes more distinct.Besides, the FI transport becomes larger once the PVG is present due to the destabilization of the ITG turbulence, as shown in figure 10(d).
The sensitivity of ion transport stiffness level to R/L Ti , namely, the predicted gyroBohm normalized ion heat fluxes from the R/L Ti scans are shown in figure 11, where the cases are examined when including and neglecting the contribution from E × B shear stabilization and PVG destabilization.The reduced level of stiffness is observed when E × B shear is included whereas it is enhanced by the PVG.The ion heat transport is close to the experimental observation as long as both of the two effects are included, as seen by the comparison with the data from experimental power balance in figure 11(a).The competition between E × B shear stabilizing and PVG destabilizing is invisible at low R/L Ti and the difference in the bulk ion transport also becomes less obvious for larger gradient.However, the transport induced by FIs becomes the dominate factor that influence the total heat flux which can occupy up to a fraction of ∼30% of the total heat flux at large R/L Ti , as shown in figure 11(b).From the discussions above, it is strongly suggested that the PVG contributes significant  in the FI transport thus the total flux, which plays an important role in determining the stiffness levels especially for the cases of deep gradient regions.At moderate R/L Ti , the PVG destabilization of the ITG turbulence increases the stiffness, whereas the decrease of stiffness level due to the ITG stabilization by E × B flow shear dominate over the PVG at higher R/L Ti .Note that for pure toroidal rotation, the relative degree of PVG destabilization versus E × B stabilization is sensitive to the geometric parameter q/ε, as discussed previously in [12].In conclusion, the results do not predict a significant reduction in stiffness due to flow shear or PVG in the experimental parameter space.The major reason of the reduction of flux is due to the nonlinear stabilization of ITG turbulence and the enhanced ZF in the presence of FIs.The required E × B shearing rate for suppressing turbulence is beyond the experimental value and PVG destabilization is less important under experimental conditions.For the typical ITB plasmas on HL-2A, it is implied that the main mechanisms of the turbulence suppression are the nonlinear, electromagnetic stabilization and increased ZFs due to the presence FIs while subdominated by the mean flow shear.The reduced stiffness level is only observed at large R/L Ti once both the E × B shear and PVG are considered.Nevertheless, we recall the attention that the presence simulations have neglected the role of fast particle driven modes which are frequently observed in HL-2A ITB discharges and have close relation to the triggering and sustaining of ITB [63].More importantly, the important role that the fast particle driven modes such as toroidal Alfvén modes [64] can enhance the ZFs through complex nonlinear interactions among them hence improved confinement has been identified [65][66][67].Dedicated simulations exploring the role of EPMs will be carried out in the next work.
Shown in figure 12 is the comparison of the effect of PVG on the nonlinear evolution of transport for different values of T f /T e , during which the toroidal rotational shear has always been considered.It can be easily discovered that both the ion and FI transport shows a more smoothed time trajectories in the presence of PVG combined with the rotation itself, while they are characterized by more intermittent events without the two effects, as can be found by the comparisons between figures 12(a) and (d) or figures 12(c) and (f ).The significant reduction of FI transport by rotational shear is observed once the E × B shear is turned on, as depicted in figures 12(b) and (e).The underlying mechanism is suggested to be the suppression of BAE/KBM turbulence localized at a single wavenumber, while the primary reason of transport reduction of thermal ions and electrons by FI temperature is the enhanced nonlinear electromagnetic stabilization of ITG turbulence due to the larger total plasma β which can also be inferred from figure 6.Moreover, it is also suggested that when the strong electromagnetic turbulence such as KBM is fully suppressed, the ZF amplitude would become larger as a large magnetic fluctuation may erode the ZFs [68].The turbulence transport suppression is only obvious when T f /T e is relatively large for both situations, for example, T f /T e = 24.It is noted that a longer simulation time is need for reaching the steady state in the absence of PVG because of the intermittency.The total heat flux in the quasi-state state as a function of FI temperature is shown in figure 11 averaged over time period typically larger than 60 R 0 /c s .It is seen that the total heat flux does not show a significant difference for T f /T e ⩽ 16 even with the inclusion of PVG when considering the statistical errors.The obvious turbulence reduction occurs only when T f /T e ⩾ 24.Besides, although the PVG has destabilization effect on linear ITG modes, it seems that it has weak effect on the transport from nonlinear simulations, as shown in figure 13(a).Moreover, as the ZF amplitude and turbulence intensity is strongly correlated, the ZF shearing rates are almost constant T f /T e ⩽ 24 and the PVG would suppress the ZFs, as can be found in figure 13(b).The transport reduction at T f /T e = 32 is suggested to be caused by the increase in the amplitude of ZFs, which has also been discovered previously in figure 8. Double the FI energy, i.e.T f /T e = 32 versus T f /T e,exp = 16 for the present scenarios for the relatively low density and rotation ITB plasmas on HL-2A, the transport is expected to drop by half.The simulations have suggested that increasing the FI temperature is favorable for improving the confinement, unless the electromagnetic modes become important and take over.

Concluding remarks
The simulations presented in this paper have investigated the role of FIs on turbulence transport and ZF dynamics in the typical HL-2A ITB plasmas, where the rotational shear and PVG have also been examined as well.It is shown that the plasma is dominated by ITGs and TEMs in the core ITB region at where the E × B shear is maximum at the same time.The main findings of this work are summarized as follows.
(1) Linear simulations have revealed that the ITGs are stabilized in the presence of FIs whose effect is enhanced with the FI temperature.The ITGs are linearly destabilized by the PVG, however, both the FIs and PVG have negligible effect on TEMs, implying that the ion heat transport reduction in the ITB region is caused by the suppression of long wavelength ITGs.(2) Massive nonlinear simulations have demonstrated that the E × B further stabilize the turbulence but its effect is weaker compared with that of FIs.The transport generally decreases with FI temperature larger than T f /T e ⩾ 12, while they are almost unaffected when T f /T e is relatively small, which is ascribed to the increasing of the total plasma β.The E × B stabilization is weekend when T f /T e becomes large.
(3) The heat fluxes generally peak around a certain T f /T e and the ZF shearing rate is significantly higher at the moment compared with that in the absence of FIs.As a result, the nonlinear heat flux reduction is a result of the synergistic interaction between turbulence, ZFs and the external rotational shear.(4) The PVG has weak effect on the heat transport in nonlinear simulations; however, it reduces the intermittent transport events, leading to a more obvious oscillating feature in the presence of PVG especially at large R/L Ti .The competition between E × B shear stabilizing and PVG destabilizing is not obvious at low R/L Ti , indicating that flow shear or PVG do not predict a significant reduction in stiffness level in the present experimental parameter space.(5) Although previous experiments and linear simulations of the HL-2A ITB plasmas have suggested that the turbulent transport suppression is mainly caused by the E × B shear, these simulations have not taken the effect of FIs into account, which is identified to be the main mechanism observed in the present nonlinear simulations.The major reason of the reduction of transport is due to the nonlinear stabilization of ITG turbulence as well as enhanced ZFs simultaneously in the presence of FIs.
The above results have also suggested that increasing the FI temperature might be an important way in suppression the transport and improving the confinement in a low rotation and low density plasma once the electromagnetic turbulence is not excited such as in HL-2A.
Appendix.Description of some physical quantities used in the main text.
respectively.The drift velocity v D is approximated by the generalized E × B velocity while the GyroBohm units are defined as Γ GB = T e 3/2 n e m 1/2 i /(eBR) 2 and Q GB = T e 5/2 n e m 1/2

Figure 3 .
Figure 3. Profiles of fast ions calculated by NEBEAM: (a) density n f (red) and the ratio of n f /ne (black) and (b) temperature T f .The data was averaged over 5 time slices and the shaded areas donate the relative errors.

Figure 4 .
Figure 4. Wavenumber spectrums of the normalized (a) growth rate γ and (b) real frequency ω for different values of T f /Te.The case without FIs is shown by the dashed curves.

Figure 5 Figure 5 .
Figure 5.Comparison of the effects of FI and E × B shear on heat transport.Time evolutions the heat flux of (a) bulk plasma and (b) fast ions.The dashed lines donate the start time of E × B shear and the FI temperature is set to T f /Te = 16.

6 .
Comparison of (a) kyρs spectrum of ion heat flux Q i for the cases of with/without E × B and FI, (b) total heat transport Q i + Q FI and (c) electron heat flux for various FI temperature.The results excluding E × B shear and PVG at T f /Te = 32 when KBM is linearly unstable are also plotted for a comparison by the dashed curves in (b) and (c).

Figure 7 .
Figure 7. Dependence of (a) ion (marked curves) as well as total heat flux (dashed curves) and (b) fast ion heat transport on FI temperature.The results for the cases without and with E × B shear are donated by red and blue curves, respectively.

Figure 8 .
Figure 8. Dependence of (a) ZF shearing rate ω ZF E×B and (b) its ratio to maximum linear growth rate ω ZF E×B /γ max on fast ion temperature T f /Te.The results for the cases without and with E × B shear are donated by red and blue curves, respectively.The values without FI or E × B are shown by the dashed lines.

Figure 9 .
Figure 9. (a) Growth rate and (b) real frequency as a function of poloidal wavenumber.The cases without and with FI are shown by red and blue curves while the results without and with PVG are donated by hollow and solid marks, respectively.The value of PVG is set to γ pfs = 0.15 and FI temperature T f /Te = 16.

Figure 10 .
Figure 10.Time evolutions of ion and fast ion transport at different values of R/L Ti .Figures (a) and (b) are the results without PVG while (c) and (d) are those in the presence of PVG, respectively.

Figure 11 .
Figure 11.Comparisons of the sensitivity of the transport of (a) bulk ions and (b) fast ions to R/L Ti including and neglecting E × B and PVG.The experimental power balance of ion transport is calculated by ONETWO in the framework of OMFIT integrated modeling.

Figure 12 .
Figure 12.Time traces of transport for different values of T f /Te.Left panel: heat flux of (a) thermal ions, (b) fast ions and (c) electrons in the absence of parallel rotation and PVG.Right panel is similar to the left except for that the two effects are taken into account.

Figure 13 .
Figure 13.(a) Total heat flux and (b) zonal flow shearing rate as a function of T f /Te.The results neglecting and including rotation and PVG are donated by red and blue curves, respectively.