Turbulence stabilization in tokamak plasmas with high population of fast ions

This letter provides a new physical insight into the fast ion effects on turbulence in plasmas having a high fast ion fraction and peaked fast ion density profile. We elucidate turbulence stabilization mechanisms by fast ions that result in internal transport barrier formation in the fast ion regulated enhancement mode plasma. Both linear and nonlinear gyrokinetic simulations show that the dominant turbulence suppression mechanisms are the dilution effects. In particular, we find that turbulence can be sufficiently suppressed solely by an inverted main ion density gradient due to fast ions, for the first time. New physical findings reported here improve our understanding of fast ion effects on turbulence, essential for fusion energy production where . Moreover, they will open up a new methodology to control plasma turbulence applicable to a wide range of plasma confinement regimes.

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Introduction
In fusion plasmas, fast ions such as alpha particles generated from fusion reactions are considered to play a critical role in the mechanism of self-organization and self-sustainment.Therefore, understanding the fast ion relevant physics is essential for fusion power generation in ITER and beyond.Fast ions also have drawn attention to their effects on turbulence suppression, improving plasma performance [1][2][3][4][5][6][7][8][9][10][11][12].Turbulence driven by drift-wave instabilities is the main physical mechanism of particle and energy transport in fusion plasmas, which limits plasma performance.Therefore, understanding fast ion effects including their influence on turbulence becomes an issue of prime importance.
Apart from previous advanced plasma confinement regimes, a new self-organized operation regime was recently discovered in KSTAR Tokamak relying on the role of fast ions, called the fast ion regulated enhancement (FIRE) mode [13].The FIRE mode shows higher performance than the Hmode, the high confinement mode most commonly considered [14], and comparable performance to the hybrid mode [15,16].The FIRE mode achieves a high confinement operation through internal transport barrier (ITB) [17] formation, but it avoids the disadvantages of ITB operation reported in the past [18][19][20][21][22][23] in terms of impurity accumulation and sustainability.
One observation was that the ITB foot location is correlated with the fast ion fraction in the FIRE mode operation, implying that fast ions affect the ITB dynamics [13].The thermal energy fluxes of the FIRE mode predicted by gyrokinetic simulation using CGYRO [24] were significantly reduced as fast ions were included [13].These simulation results suggest that fast ions are responsible for ITB formation through the suppression of turbulence.However, the physical mechanism behind turbulence suppression and ITB formation has not been demonstrated.
Fast ion effects on turbulence suppression have been previously investigated by considering the wave-particle resonance interaction between fast ions and turbulence [5,6,9,12], the interaction between a fast ion driven mode and turbulence [8], fast ion-induced changes in the plasma such as increasing the zonal flow level [10,11] and increasing β * (≡ ( −8π /B 2 ) dp/dr, where B and p are the magnetic field and the plasma pressure) [2,3], and changes in the main ion density profile due to the addition of fast ions [1,4,7].In this letter, we report the physical mechanism behind ITB formation by fast ions in FIRE mode plasmas.
We investigate through linear and nonlinear gyrokinetic simulations the contribution of various possible mechanisms for the thermal energy flux reduction in the FIRE modeincreased β * effects, two dilution effects including the main ion fraction reduction and changes in the main ion density gradient, and changes in the zonal flow level.Based on the results, we identify that the dilution effects are the main physical mechanism behind ITB formation by fast ions.Since the FIRE mode has a relatively higher fast ion fraction and a more peaked fast ion density profile than other plasma operation modes analyzed in previous studies [1][2][3][4][5][6][7][8][9][10][11][12], this letter will provide a deeper understanding of the fast ion effects on turbulence.Furthermore, it is noteworthy that the two dilution effects were scrutinized separately in this work, in contrast to previous studies [1,4,7].We also demonstrate the importance of the inverted main ion density gradient in the turbulence suppression in the plasma having high fraction of fast ions, for the first time, compared to previous studies that focused on the importance of the inverted electron or impurity density gradient [25,26] or on the changes in the density gradients of all species together without separation of the effects of each species [27].

Simulation setup
Various mechanisms of turbulence stabilization by fast ions were investigated via CGYRO for gyrokinetic analysis.The linear and nonlinear gyrokinetic simulations were performed at 5.35 s for one FIRE mode discharge (shot 22663) in KSTAR [13].
A local flux tube simulation was performed inside the ITB region (ρ = 0.4 or r/a = 0.47), where ρ is the normalized radius based on the square root of the normalized toroidal magnetic flux, and r/a is the radius normalized by minor radius, a. Electromagnetic effects were considered by including both perturbed electrostatic potential δϕ and vector potential δA ∥ .The simulation focused on the ion scale turbulence in k y ρ s ⩽ 1 and |k x ρ s | ⩽ 6.88, where k y and k x are poloidal and radial wave numbers, respectively, and ρ s is the ion gyro radius.The Miller equilibrium model [28] was used to describe the non-circular elongated flux surface and its geometric effects.Collisional effects were considered by applying the Sugama collision operator [29], and electron dynamics were considered using gyrokinetic equations.We included rotation and E × B flow shearing effects from the radial force balance based on the experimentally measured carbon toroidal rotation and neoclassical poloidal rotation from the NEO [30] calculation.Carbon was used as a single impurity species with a flat Z eff (=2, where Z eff ≡ ∑ j Z 2 j n j /n e , j is the ion species) profile, and fast ions were treated as an additional ion species with a Maxwellian distribution function.In comparison to a more realistic velocity distribution of fast ions, previous studies showed that the changes in the results when a more realistic slowing down distribution was used for fast ions were not significant [31][32][33].It is therefore expected that choosing the Maxwellian distribution as the distribution of fast ions generated by neutral beam injection will not change the main conclusion of this letter, although minor variations in the results may be possible.
The resolution parameters and calculation domain size used in the nonlinear gyrokinetic simulations were determined through convergence tests.The resolution parameters for radial, pitch angle, energy, poloidal, and toroidal grids [n radial , n ξ , n energy , n θ , n φ ] were chosen as [216, 16,8,28,16], and the domain size [L x , L y ] was set to [97.72 ρ s , 93.78 ρ s ].The simulation conditions required for accurately describing low-n fast ion modes could be more demanding than what is employed in this study.Nonetheless, it should be emphasized that the primary focus of this study is not centered on the investigation of fast ion mode and its impact turbulence.Rather, our is the significant reduction in turbulence, irrespective of the presence of the fast ion mode, as demonstrated in the subsequent sections of this letter.
Experimental profiles were used as input parameters for the gyrokinetic simulations.The main input parameters with (without) fast ions are as follows: safety factor, windingness of the magnetic field line, q = 1.264, magnetic shear s (≡ (r/q) (∂q/∂r)) = 0.611, n e = 1.862 × 10 19 m −3 , T e = 1.958 keV, T i /T e = 1.419,T f /T i = 9.8 (N/A), n i /n e = 0.395 (0.800), n f /n e = 0.405 (N/A), a/L ne = 1.067, a/L nc = 1.067, a/L ni = −1.494(1.067), a/L nf = 3.572 (N/A), a/L Te = 2.978, a/L Ti = 3.141, a/L Tc = 4.398, a/L Tf = 0.562 (N/A), and β * = 0.075 (0.024), a/L x denotes inverse gradient scale length (−a∇X/X), respectively.Figure 1 shows the time series data of the energy flux normalized to the gyroBohm (Q/Q GB ) of each species predicted by the nonlinear simulations for cases with and without fast ion species.Here, the gyroBohm energy flux used for normalization is defined as Q GB ≡ n e c s T e (ρ s /a) 2 , where c s is the ion sound speed (≡ √ T e /m i ), m i is the main ion mass, and time is normalized by a/c s .The thermal energy flux decreased significantly from 19.200 to 0.001 for Q i /Q GB , and from 9.614 to 0.067 for Q e /Q GB when the fast ion species was included.This result is qualitatively consistent with experiment showing the correlation between fast ion fraction and ITB foot location [13].

Turbulence stabilization mechanisms by fast ions
Among the possible mechanisms for turbulence suppression by fast ions, resonant interaction with ambient microinstabilities [5,6,12] typically occurs only when a/L Tf ≫ a/L nf .This is usually the case when fast ions are generated by ion cyclotron resonance heating, but not in the FIRE mode in which fast ions are generated by neutral beam injection.In the FIRE mode, a/L Tf ≪ a/L nf is satisfied.Resonant interaction, therefore, does not play a dominant role in turbulence suppression in FIRE mode plasmas.

β * effects
When the fast ion species is included, its contribution increases the pressure gradient.It is known that a higher pressure gradient expressed as β * contributes to the suppression of turbulence [34][35][36].Results consistent with this were also checked through the linear stability analysis of the previous FIRE mode study [13].We performed nonlinear simulations by varying β * without fast ion species.Here, the case A in figure 2 corresponds to figure 1(a).The case A and B in figure 2, which exclude fast ions, show the results of the dependency of β * level excluding other effects by fast ions.The main ion and electron energy fluxes were reduced as β * increased, consistent with the previous linear stability analysis results [13].However, the reduced energy fluxes (Q i /Q GB ∼ 4.567 and Q e /Q GB ∼ 2.192) were still higher than the levels obtained in the case with fast ions (Q i /Q GB ∼ 0.001 and Q e /Q GB ∼ 0.067), indicating that the effect of the increased β * is not sufficient to explain the reduced thermal energy flux in figure 1.

Dilution effects I, reduced main ion fraction
The main ion fraction and its density gradient will change as the fast ion species is included to maintain quasi-neutrality in the plasma.These two dilution effects were analyzed separately in this study.The effect of the main ion fraction change was investigated by decreasing n i /n e when including the fast ion species while the inverse density gradient scale lengths of all species were fixed to the level used in the case without fast ions (a/L n = 1.067).The case C in figure 2 shows the gyroBohm normalized energy flux for the case where n i /n e = 0.395 (and n f /n e = 0.405), in which the main ion fraction is diluted compared to the case without fast ions (n i /n e = 0.8) shown in case A in figure 2. Thermal energy fluxes decreased to Q i /Q GB ∼ 1.075 and Q e /Q GB ∼ 0.880, which are closer to the level of figure 1(b) compared to the case with increased β * without fast ions, indicating that the reduced main ion fraction due to the addition of fast ions is more effective than the increased β * on turbulence suppression.However, the thermal energy fluxes shown in the case C in figure 2 are still higher than the flux with fast ion species shown in figure 1(b).The thermal energy flux can be reduced to Q i /Q GB ∼ 0.836 and Q e /Q GB ∼ 0.769 when both reduced main ion fraction and increased β * effects are considered (not shown here).This result indicates that these effects are not linearly additive in the simulation results.

Dilution effects II, inverted main ion density gradient
In the FIRE mode discharge, the main ion density gradient gets inverted [13] (a/L ni = −1.494)when including fast ions as its fraction gets higher and the profile strongly peaks.To consider the effect of the main ion density gradient change without reduced main ion fraction, a linear stability analysis was performed with n f /n e = 0.01 (n i /n e = 0.79) and compared with the case excluding fast ions (n i /n e = 0.8).As shown in figures 3(a) and (b), the results of the case with n f /n e = 0.01 and the case without fast ions (yellow diamonds, red empty circles, blue empty triangles) are well overlapped, indicating that the effect of 1% dilution of the main ion fraction is negligible to the linear growth rate.Here, the horizontal blue dashed line in figure 3(b) denotes the mean E × B shearing rate, which is the shearing rate of equilibrium E × B flow, ω E×B (≡ (−r/q) dω 0 /dr with angular frequency ω 0 ).Figures 3(a) and (b) also show that the linear growth rate is reduced significantly as the main ion density gradient is inverted.
However, in the k ρ s ⩽ 0.27 domain of case when n i /n e = 0.79 and T f /T i = 9.8, the real frequency moves to the ion diamagnetic direction with a discontinuous jump, and linear growth increases, unlike the k y ρ s ⩾ 0.27 regions.Since the peak in the linear growth rate in the k y ρ s ⩽ 0.27 domain does not exist in either case without fast ions or with a/L nf = 1.067, this low k y mode is most likely to be a fast ion relevant mode.We also observed that this mode was destabilized with higher a/L nf and stabilized with lower T f /T i .It is found that this mode is stabilized with lower T f /T i (=5) compared to experimental value (∼9.8) in the experimental condition used in figure 1(b).It is noteworthy that the changes in the thermal energy flux levels in this case are within 0.01 Q GB compared to the experimental case.This indicates that the effects of the fast ion relevant mode on turbulence is not significant in this study.Moreover, in the case with n f /n e = 0.01, which is not a realistic condition but utilized to investigate the sole effect of the main ion density gradient inversion on the turbulence, we observed the highly destabilized fast ion relevant mode, which will not be observed in the real experiment.To exclude the effect of this artificial fast ion relevant mode in investigating the effect of the main ion density gradient inversion on the turbulence, linear and nonlinear simulations were performed with n f /n e = 0.01 and T f /T i = 5.The linear growth rate in this case, shown as the red lines in figures 3(a) and (b), show that the fast ion relevant mode is suppressed significantly compared to the case with T f /T i = 9.8 (exp), shown as the blue lines.We can also see that the linear growth rate in k y ρ s ⩾ 0.27 is increased as this mode is suppressed, implying the effect of the fast ion relevant mode on the turbulence suppression.Figures 3(c) and (d) show gyroBohm normalized energy fluxes of the ions and electrons of a/L ni scan with varying a/L nf while fixing n i /n e (= 0.79), T f /T i (= 5), and β * (= 0.024).As the main ion density gradient becomes inverted, the main ion and electron energy fluxes are reduced significantly to Q i /Q GB ∼ 0.030 and Q e /Q GB ∼ 0.055, similar to the level shown in figure 1(b).This indicates that the effect of fast ion relevant mode on the saturated level of the turbulent energy flux is not significant, consistent with the aforementioned observation in the case with experimental conditions.Considering the reduction in the energy flux between a/L ni = 1.067 and − 1.494, the results in figure 3 show that turbulence can be suppressed significantly, sufficient for ITB formation, by the inverted density gradient of the main ion only, without the reduced main ion fraction.These results are consistent with previous studies showing favorable roles of inverted density profiles for ion temperature gradient stability [37] and turbulence level [38].But in this work, the sole effect of the inverted main ion density gradient is shown for the first time.Although an exact quantitative comparison of the importance between two dilution effects is difficult, the current results indicate that the dilution effects, including both factors, are essential to explain the reduced thermal energy flux by fast ions.
The real frequency and linear growth rate considering each fast ion effect so far are shown in figure 4. The linear growth rate reduction due to the dilution effects with fixed β * is much larger than the increased β * effects.The linear growth rate including only the dilution effects also becomes similar to that including fast ions except for the fast ion relevant mode for k y ρ s ⩽ 0.27.It follows that dilution effects are mainly responsible for the turbulence suppression in FIRE mode discharge.These linear results are consistent with the nonlinear simulation results shown in figures 2 and 3.

Zonal flow
Turbulence has also been shown to be reduced by zonal flows generated by fast ion driven mode and turbulence itself [8,10,11,[39][40][41].One quantitative measure of zonal flow strength is the residual flow level [42] in the absence of damping from collisions and turbulence.Cho and Hahm [43] indicated that fast ions can enhance the residual level of zonal flows at moderate wavelength k x ρ s ∼ 0.1.
Figure 5(a) shows an enhancement of the residual zonal flow level (R zf ), which was calculated using equation (62) in [43], due to fast ions as a function of k x ρ s .The shaded region denotes the simulation-relevant region (|k x ρ s | ⩽ 6.88).At k x ρ s ∼ 0.2, the residual zonal flow increases by 20% when fast ions are included.Figures 5(b) and (c) show that the zonal flow velocity (V zf ≡ ∑ k x δϕ n=0 ) and the zonal shearing rate (ω Zf ≡ ∑ k 2 x δϕ n=0 ) increase when fast ion species is included, consistent with the changes in the residual zonal flow.While one might have expected a reduction of zonal flow level due to aforementioned stabilization of turbulence, the observed enhancement of zonal flow and its shearing rate in the presence of fast ions shown in figures 5(b) and (c) can be understood by noting the changes in shielding properties leading to a higher residual zonal flow level [43] and the enhanced zonal flow generation due to the presence of fast ions [44].The zonal flow shearing rate increase caused by the inclusion of fast ions is ∼30% of the mean E × B shearing rate in the presence of fast ions.Observing the changes in thermal energy fluxes by varying the mean E × B shearing rate in the case without fast ions reveals that when ω E×B increases by 30%, the thermal energy fluxes decrease by ∼20%, which is lower than the reduction by the other effects such as increased β * and dilution.This result indicates that the effect of increased zonal flow by including fast ions is not the main mechanism responsible for turbulence suppression, according to a conventional rule-of-thumb based on the shearing rates.In addition, the effect of the mean E × B shearing rate is evaluated by varying its amplitude in the case with fast ion species.The thermal energy fluxes in the absence of the mean E × B shearing effect stay below 0.2 times Q GB , much smaller than the fluxes without fast ions, as can be seen in figure 5(e).This result also supports that the increased flow shearing rate due to the addition of fast ions is not the dominant mechanism behind the thermal energy flux reduction in FIRE mode discharge compared to other effects.

Conclusion
In this letter, we investigated possible mechanisms of turbulence stabilization by fast ions in the magnetized plasmas having a high fast ion fraction and a highly peaked fast ion density profile through gyrokinetic simulation using CGYRO.Based on simulation results, we conclude that the dilution effects, including a reduced main ion fraction and its density gradient inversion, due to fast ions are the main physical mechanisms driving the thermal energy flux reduction and subsequently ITB formation in the FIRE mode.We also showed that turbulence can be suppressed sufficiently for ITB formation by the changes in the main ion density gradient only.This result is particularly important since it can be utilized to develop new experimental methodologies to make a transition to and sustain the high-performance mode by manipulating the main ion density gradient, e.g.not only by fast ions but also by impurity injection.Moreover, adjusting the main ion density gradient based on quasi-neutrality is applicable to any types of plasmas, not just for fusion plasmas.This new physical finding can provide a new method to control plasma turbulence, beneficial for the whole plasmas field, including turbulence suppression for plasma propulsion and stable plasma source development for plasma processing as well as fusion energy.

Figure 1 .
Figure 1.Time series data of the gyroBohm normalized energy flux of each species for the cases (a) without and (b) with fast ions.Time is normalized by a/cs.

Figure 2 .
Figure 2. GyroBohm normalized energy fluxes of ions and electrons for the case A excluding fast ion species, case B excluding fast ion species with increased β * , and case C including fast ion species with a reduced main ion fraction but fixed β * .The inverse density gradient scale length of all species are also fixed at 1.067 (a/Ln = 1.067).

Figure 3 .
Figure 3. (a) Real frequency and (b) linear growth rate of the most unstable mode by inverted main ion density gradient effects.Yellow diamonds denote the case without fast ions, red circles the case with n f /ne = 1% but T f /T i = 5, and blue triangles the case with T f /T i = exp (=9.8).The empty and filled symbols indicate a/L ni = 1.067 and − 1.494, respectively.The horizontal blue dashed line in (b) shows the mean E × B shearing rate, ω E×B .(c) GyroBohm normalized energy flux of ions and (d) that of electrons as a function of a/L ni .

Figure 4 .
Figure 4. (a) Real frequency and (b) linear growth rate of the most unstable mode by fast ion effects.Yellow denotes the case without fast ions, red shows also the case without fast ions but with increased β * to the level in the case with fast ions.Green is the case with fast ions but including only dilution effects without increased β * , and black denotes the case with fast ions, i.e. including both the effects of the increased β * and dilution.

Figure 5 .
Figure 5. (a) Ratio of the calculated residual zonal flow level, (b) zonal flow velocity (V zf ), and (c) zonal flow shearing rate (ω zf ) from nonlinear simulations for the cases with and without fast ions.(d) GyroBohm normalized energy flux with fractional change of ω E×B for the cases fast ions and (e) that without fast ions.