Stabilization of ion-temperature-gradient mode by trapped fast ions

Siqi WANG (王思琪),Huishan CAI (蔡辉山),Baofeng GAO (高宝峰) and Ding LI (李定)

1 CAS Key Laboratory of Geospace Environment,School of Nuclear Science and Technology,University of Science and Technology of China,Hefei 230026,People's Republic of China

2 Institute of Physics,Chinese Academy of Sciences,Beijing 100190,People's Republic of China

Abstract Understanding and modeling fast-ion stabilization of ion-temperature-gradient (ITG) driven microturbulence have profound implications for designing and optimizing future fusion reactors.In this work,an analytic model is presented,which describes the effect of fast ions on ITG mode.This model is derived from a bounce-average gyro-kinetic equation for trapped fast ions and ballooning transformation for ITG mode.In addition to dilution,strong wave-fast-ion resonant interaction is involved in this model.Based on numerical calculations,the effects of the main physical parameters are studied.The increasing density of fast ions will strengthen the effects of fast ions.The effect of wave-particle resonance strongly depends on the temperature of fast ions.Furthermore,both increasing density gradient and the ratio of the temperature and density gradients can strengthen the stabilization of fast ions in ITG mode.Finally,the influence of resonance broadening of wave-particle interaction is discussed.

Keywords: fast ions,ITG instability,gyro-kinetic

1.Introduction

Improving plasma confinement is beneficial for designing future nuclear fusion devices and optimizing the performance of present devices.An important limiting factor of plasma confinement in fusion devices is microturbulence [1].As a significant driver of plasma turbulence,ion-temperature-gradient (ITG)instability[2-5]is principally responsible for the degradation of ion energy confinement.Therefore,it is extremely valuable to study the mechanisms that limit its development.

Fast ions[6]are mainly generated by fusion reactions,as well as auxiliary heating systems,such as neutral beam injection [7] and ion cyclotron resonance heating [8].Understanding the behavior of fast ions is important since they are an essential component of fusion plasma and play a major role in sustaining fusion-relevant bulk temperatures.In addition,fast ions carry large power,which implies that even small fast-ion losses can damage the first wall of a fusion device.Therefore,the confinement of fast ions is worth studying.

The study of fast-ion interaction with plasma turbulence has recently attracted particular interest.On the one hand,background plasma turbulence can induce the transport of fast ions and lead to the redistribution or losses of fast ions[9-12].On the other hand,fast ions can affect plasma turbulence in turn [13-15].In some experimental and numerical studies[16-20],the suppression of plasma turbulence has also been observed,which is linked to the presence of fast ions.Generally,the effects of fast ions can be classified as electrostatic effects and electromagnetic effects.In [14],the effect of the dilution of the main ions on stabilizing ITG turbulence is investigated.However,the fast-ion effect can be observed even under low density,which suggests other interaction mechanisms in addition to dilution [21].In [22],a strong dynamic effect of fast ions on suppressing plasma turbulence has been observed in an electrostatic setup.Significantly,the wave-particle resonance mechanism is taken into account[23,24].With regard to electromagnetic stabilization of ITG turbulence,there are many experimental and numerical studies[25-30].As shown in[22],the linear growth rate exhibits the same behavior in the electromagnetic and electrostatic framework,while the growth rate is lower in the electromagnetic framework than that in the electrostatic framework for the same parameters.Mostly,the effects of the fast-ion stabilization of ITG turbulence are studied by both linear and nonlinear numerical simulation methods.

In this paper,a theoretical interpretation for the observed impact of fast ions on ITG mode is offered.Both analytical and numerical calculations are presented in this work.The physical mechanisms of fast-ion stabilization on ITG mode are studied in detail.The interaction of fast ions with ITG mode is investigated in the framework of gyro-kinetic theory[31].The dispersion relation consisting of ITG mode and fast ions is normally derived by utilizing the quasi-neutrality condition and ballooning transformation.It is discovered that fast ions can interact with ITG instability via wave-particle resonance.Based on numerical calculations,the effects of density,temperature,density gradient and the ratio of the temperature and density gradients of fast ions on both the real frequency and growth rate of ITG mode are investigated.The effect of dilution is also presented in numerical calculations.It is found that,in addition to dilution,the resonant fast-ion stabilizing effect plays a significant role.

The rest of the paper is organized as follows.In section 2,adopting proper approximations,the dispersion relation including ITG mode and trapped fast ions is established.In section 3,based on numerical calculations,the dispersion relation is solved and different physical parameters controlling the stabilizing effect are investigated.Furthermore,the influence of resonance broadening of wave-particle interaction is discussed.Our conclusions are given in section 4.

2.Dispersion relation

The dispersion relation for ITG mode including fast ions and the bulk plasma is given by quasi-neutrality equation:

Here,the bulk plasma is perceived to be composed of deuterium and electrons.Zfis the charge number of fast ions.For the electron response,an adiabatic approximation is adopted for k‖vTe≫ω and the electron perturbation density δneis given by,

However,an adiabatic approximation is not appropriate for both main ions and fast ions.The perturbed ion distribution function can be written as,

Here,j refers to the species of particles and δgjis the nonadiabatic part of the perturbed distribution function,which is determined by the gyro-kinetic equation [31]:

Here,vdj=,ωcj=and K=v2/2,where b is a unit vector parallel to the magnetic field,κ is the magnetic curvature,Mjis the mass of species j and(r,θ,ξ) denote the minor radius,poloidal angle and toroidal angle,respectively.The equilibrium distribution function is assumed to be Maxwellian.Obviously,in the above equation,the first term in the square bracket denotes the space-gradientdriven term and the second term denotes the energy-gradientdriven term.

First,the dynamics of background main ions is presented.According to the gyro-kinetic equation,the perturbed ion distribution function δgiis obtained as,

where,

In previous works [3,4],the above equation has been studied under some limits.By considering the limits≪1,≪1 and keeping the leading contributions inand,the perturbed density of the main ions is expressed as,

withbi=.Here,ρTiis the Larmor radius of the background main ions.Note that,under the limits≪1,≪1),the growth rate of ITG mode in this work is larger than that in simulation studies [22].

For the fast-ion dynamics,both of δgfand δφ are expanded in toroidal and poloidal Fourier harmonics:

Then,the Fourier transform of equation (4) is,

The ordering relationship between the terms in equation (9) is as follows:

Here,Δr is the radial distance from a reference mode rational surface and Δrmis the distance between the two adjacent rational surfaces,implies,where ωtfis the transit frequency.It is also true thatfor ITG mode.kθρTf＜1 requires.Note that the upper and lower limits of Tf/Tiare given as.Subsequently,krρTf＜1 for kr～skθ.denotes the adiabatic part of the perturbed distribution function of fast ions.Then<1 and<1,where[1+ηf(E Tf-3 2)].

Based on the above ordering,equation (9) can be expanded as,

Equation (11) can be solved as,

Substituting equation (13) into equation (12) and taking the bounce average [32,33],we get:

According to equation (14),the expression for the nonadiabatic part of the perturbed trapped fast-ion distribution function can be derived as,

Significantly,the precession motion of fast ions can resonate with ITG mode when.Here,ωris the real frequency of ITG mode.This implies that fast ions can stabilize ITG mode via wave-particle resonance.

The equilibrium distribution function of fast ions is assumed to be a Maxwellian distribution function [11,34],wherec1=is the normalized coefficient.

Substituting the Maxwellian distribution function into equation (15),the perturbed density of trapped fast ions is written as,

Combining equations(2),(7)and(16)and employing the ballooning transformation [35]δφ=,the quasi-neutrality condition in equation (1) can be expressed as,

where,

with y=skθx,σ=,ι=,,τ1=Te/Tiand τ2=ZfTe/Tf.Ω=is the normalized frequency andis the normalized energy.=with Ω=Ωr+iγ.Here,Ωris the normalized real frequency and γ is the normalized growth rate.The bracket 〈…〉θrepresents a θ-average value.The notation for the angle part of integration is expressed as,where Q is an arbitrary function of α.

Note that,λ1expresses the response of the background electrons.λ2comes from the background main ions.λ3,stem from fast ions.λ3expresses the adiabatic part of fast ions.All ofandarise from the non-adiabatic part of fast ions.Since the Bessel functionis expanded as≈,〈λ4〉θstems from the principal part of the Bessel function.〈λ5〉θ,〈λ6〉θcome from the poloidal component and the radial component of the Bessel function,respectively.Namely,〈λ5〉θand〈λ6〉θdenote the finite Larmor radius effect.

To facilitate analysis,〈λ4〉θis divided into〈λ41〉θ,〈λ42〉θ,〈λ43〉θand〈λ44〉θ,i.e.〈λ4〉θ= 〈λ41〉θ+ 〈λ42〉θ+〈λ43〉θ+ 〈λ44〉θ,

Corresponding to equation(4),〈λ41〉θimplies the effect of the energy-gradient-driven term of fast ions.〈λ42〉θrepresents the effect of the density-gradient-driven term.Both of〈λ43〉θand〈λ44〉θstem from the temperature-gradient-driven terms and are denoted simply as the-3ηf/2 term andterm.They are opposite in sign,which signifies different effects between them.The ordering relationship between〈λ42〉θ,〈λ43〉θand〈λ44〉θis about 1: 3.The same division is applied equally to〈5λ〉θand〈λ6〉θ.

Following reference [4] and proceeding to perform the strong coupling approximation cosη+sηsinη=1+,equation (17) can be written as,

We find that equation(20)is just the familiar Weber-Hermite equation,as shown in [4,36].The eigenfunction solution is the Hermite function.Considering only the lowest eigenstate and seeking the solution of the form=exp(-ζη2),the dispersion equation is obtained:

with,

Here,bs=τ1biθ.Distinctly,let nf0=0 and equation (21)returns to the eigenvalue equation of ITG mode in [4].

It is important to note that a significant fraction of trapped fast ions can resonate with ITG mode when the precession frequency of fast ions is close to the frequency of ITG mode,i.e.when.Here,suggests the normalized energy of fast ions that resonate with ITG mode at a frequency Ωr.Distinctly,the resonant condition depends on τ2,i.e.Tf/Te.According to equation (18),the contributions to the nonadiabatic part of fast ions stem from the energy,density and temperature gradients of fast ions.If ηf≫1,the temperaturegradient-driven terms (〈λ43〉θand〈λ44〉θ) will be dominant since the ordering relationship between〈λ42〉θ,〈λ43〉θand〈λ44〉θis about.Subsequently,there is a threshold condition between〈λ43〉θand〈λ44〉θ.At relatively low temperature,the energy of resonant fast ions,i.e.is relatively high.Therefore,.When the temperature of fast ions increases,the resonant energydecreases and then.The threshold condition implies that wave-fast-ion resonance may play different roles(stabilizing or destabilizing) in ITG mode at different temperatures.

3.Numerical results

In order to obtain further understanding of fast-ion stabilization on ITG instability,the main physical parameters including the density nf0/ne0,temperature Tf/Te,density gradientand the ratio of the temperature and density gradients ηfof fast ions are investigated in more detail.Significantly,the effects of the background-driven terms(energy and space gradients) are studied separately.

In numerical calculation,equation (21) is solved without any approximation.The plasma parameters are mainly taken from a JET L-mode discharge 73 224[22,28]and are listed in table 1.To facilitate the analysis,there is a single fast particle species,fast helium-3,presented in the calculation and the bulk plasma is composed of deuterium and electrons.

Figure 1.(a) Normalized growth rate and (b) normalized real frequency of ITG mode with different densities of fast ions.Red and black dashed lines show the case without fast ions and the dilution,respectively.Solid blue and orange lines show the case with fast helium at Tf=5Te and Tf=30Te.

Table 1.Parameters for the JET discharge 73 224 with fast helium.

3.1.Effects of fast-ion density,nf0/ne0

In this subsection,the effects of nf0/ne0on both growth rate γ and real frequency Ωrof ITG mode are presented in figures 1(a) and (b),respectively.The temperatures of fast helium are at Tf=5Teand Tf=30Te.

From figure 1(a),it is found that fast ions play a stabilizing role on ITG mode and the growth rate reduces with the increasing nf0/ne0.As shown by the solid blue line in figure 1(a),relative to dilution,fast ions destabilize at Tf=5Te.The dominant effect of fast ions is dilution at Tf=5Te.However,the growth rate is lower than dilution at Tf=30Te.In addition,as shown in figure 1(b),the dilution leads to a reduction in the real frequency of ITG mode.Oppositely,the real frequencies of the two cases with fast helium rise as nf0/ne0increases.A negative value of the real frequency Ωrsignifies a mode propagating in the ion direction.

The main physical mechanisms can be simply explained.When positively-charged fast ions are added to the background plasmas,the electromagnetic fields will have less response to the main thermal ions[14,18].Consequently,the growth rate arising from the bulk ions is reduced.Compared to the pure dilution case,both the growth rates and real frequencies in the two cases with fast helium are different.This fact demonstrates that there is another kinetic effect of fast ions in addition to dilution.Furthermore,the two cases at different fast-ion temperatures imply that the kinetic effect depends on the temperature of fast ions.Distinctly,this is the resonance mechanism that conforms to the last analysis in section 2.At low temperature (Tf=5Te),two temperaturegradient-driven terms are assumedand waveparticle resonance plays a destabilizing role.However,at relatively high temperature (Tf=30Te),and resonance leads to stabilization of ITG mode.Moreover,the resonant effect is weak at low temperature since only a small fraction of fast ions can resonate with ITG mode.As the temperature of fast ions rises,the fraction of resonant fast ions increases.Thus,the resonant effect of fast ions in the case at Tf=30Teis stronger than that at Tf=5Te.

3.2.Effects of fast-ion temperature,Tf/Te.

In this subsection,the effects of Tf/Teon both growth rate γ and real frequency Ωrof ITG mode are given in figures 2(a)and (b),respectively.Subsequently,the results are explained in detail using figures 3-5.The density of fast ion is nf0/ne0=0.07.

In figure 2(a),it is found that,for Tf/Te＜7,wave-fastion resonance leads to destabilization of ITG mode and the dominant effect is dilution.When the temperature of fast ions exceeds a critical value(Tf/Te～7),the growth rate decreases with the increasing Tf/Te.Meanwhile,resonance plays a stabilizing role on ITG mode.At higher temperature,i.e.Tf/Te＞30,as the temperature rises,the growth rate changes slowly.In figure 2(b),it can be observed that,as Tf/Terises,the real frequency Ωrincreases and then moves towards the frequency in the dilution case.

Figure 2.ITG (a) normalized growth rate and (b) normalized real frequency as a function of Tf/Te.

Figure 3.ITG normalized growth rate as a function of Tf/Te:(a)adiabatic response,(b)response of energy-gradient-driven term,(c)sum of adiabatic response and response of energy-gradient-driven term and (d) response of space-gradient-driven term.

First,to understand the results,in figures 3(a)-(d) both the adiabatic response and non-adiabatic response driven by energy and space gradients are depicted separately.In figure 3(a),the adiabatic response is shown by solving equation (21) with λ3of fast ions only,namely,let〈λ4〉θ,〈λ5〉θand〈λ6〉θbe zero.As can be seen,the adiabatic part of fast ions stabilizes the ITG,but this stabilizing effect quickly decays to zero as the temperature increases.The effect of the energy-gradient-driven term is shown in figure 3(b) by retaining〈λ41〉θ,〈λ51〉θand〈λ61〉θfor fast ions only.In contrast to the adiabatic response,the energy-gradient-driven term destabilizes the ITG mode.Similarly the destabilizing effect also rapidly weakens to zero with the increasing Tf/Te.Significantly,as shown in figure 3(c),the effects of the adiabatic part and energy-gradient-driven term of fast ions almost cancel each other so that the effect of fast ions mainly results from the space-gradient-driven term,which is shown in figure 3(d).

Second,the effects of the fast-ion space gradient including density and temperature gradients are investigated in more detail.In figure 4(a),the magnitude of the imaginary part of〈λ42〉θ,which represents the effect of density gradients,is shown.Here,to facilitate analysis,a certain angle is assumed which satisfies κ2=0.6.The imaginary part of-〈λ42〉θis basically positive,which implies that the densitygradient term plays a destabilizing role on ITG mode.Similarly,-Im(〈λ44〉θ)in figure 4(c) is also mainly positive,which suggests that one part of temperature-gradient-driven termsis destabilizing on ITG mode.However,as shown in figure 4(b),another part (-3ηf/2 term) plays a stable role on ITG since-Im(〈λ43〉θ)is mostly negative.Distinctly,both| Im(〈λ43〉θ)|and| Im(〈λ44〉θ)|are much larger than| Im(〈λ42〉θ)|,which conforms to the ordering relationship between〈λ42〉θ,〈λ43〉θand〈λ44〉θ,i.e.1: 3ηf2:Therefore,the temperature gradient of fast ions expressed by〈λ43〉θand〈λ44〉θis mainly responsible for the kinetic effect of fast ions on ITG mode.

Figure 4.Imaginarypart offastions driven by (a) density-gradient term (〈λ42〉θ),(b) temperature-gradient term (〈λ43〉θ) and (c)temperature-gradientterm(〈λ44〉θ).

Figure 5.Imaginary part of fast ions:(a)principal part of Bessel function(〈λ4〉θ),(b)finite Larmor radius effect in θ direction(〈5λ〉θ)and(c)finite Larmor radius effect in r direction (〈λ6〉θ).

Third,the resonant fast-ion stabilizing mechanism is studied.As shown in figures 4(b) and (c),at relatively low temperature,the magnitude of both| Im(〈λ44〉θ)|and| Im(〈λ43〉θ)|is small.As Tf/Terises,the magnitude of| Im(〈λ44〉θ)|and| Im(〈λ43〉θ)|increases and becomes maximum around Tf～12Te.Then,the magnitude of| Im(〈λ44〉θ)|and| Im(〈λ43〉θ)|decreases with increasing Tf/Te.This result conforms to the analysis of resonance.When the temperature of fast ions is low,only a small fraction of fast ions can resonate with ITG mode.Therefore,the resonant effect of fast ions is weak at relatively low temperature.As Tf/Terises,the fraction of resonant fast ions increases,which implies that the fast-ion resonant effect strengthens.When the temperature exceeds a certain value (Tf～12Te),the fraction of resonant fast ions decreases and the resonant effect weakens with increasing Tf/Te.In addition,at relatively low temperature,| Im(〈λ44〉θ)| ＞| Im(〈λ43〉θ)|,which suggests the destabilizing effect of theterm is stronger than the stabilizing effect of the-3ηf/2 term.When the temperature exceeds a critical value,conversely,the stabilizing effect of the -3ηf/2 term is stronger.This result conforms to the last analysis of resonance in section 2.As the fast-ion temperature increases,decreases.When the threshold condition<3 2is satisfied,stabilization of ITG mode by fast ions realized.

Finally,the finite Larmor radius effect is studied in figures 5(a)-(c).In figure 5(a),as the principal part from the expansion of,for low temperatures,-Im(〈λ4〉θ)is positive,which suggests the destabilizing effect.When the temperature exceeds a certain value,-Im(〈λ4〉θ)becomes negative and fast ions stabilize ITG via wave-fast-ion resonance.Significantly,as shown in figures 5(b) and (c),the finite Larmor radius effect stabilizes ITG mode since both-Im(〈λ5〉θ)and-Im(〈λ6〉θ)are negative.

3.3.Effects of fast-ion density gradient Lne /Lnfand ηf

In this subsection,the effects of ηfandon both growth rate γ and real frequency Ωrof ITG mode are presented in figures 6(a)-(d) separately.The density of fast ions is nf0/ne0=0.07 and the temperature of fast ions is Tf=30Te.In figures 6(a) and (b),the density gradient of fast ions remains unchanged,i.e.=1.6.In figures 6(c) and (d),ηfis fixed,i.e.ηf=14.4.

Figure 6.ITG (a) normalized growth rate and (b) real frequency as a function of ηf.ITG (c) normalized growth rate and (d) real frequency versusL neLnf.

First,from figures 6(a) and (c),it is observed that both increasing ηfandwill reduce the growth rate of ITG mode,which implies that the increasing ηfandwill strengthen the stabilizing effect of fast ions.Second,from figure 6(b),it is found that,as ηfincreases,the real frequency of ITG mode increases and then moves towards the frequency in the dilution case.Finally,as shown in figure 6(d),the real frequency of ITG mode rises with increasing.

The results in figures 6(a) and (c) can be simply explained.At first,in section 3.2,it is shown that fast-ion resonance leads to the stabilization of ITG mode at Tf=30Te.Bothand ηfare unrelated to the resonance condition.Therefore,wave-particle resonance always plays a stable role at different ηfandIn equation(18),it can be seen that the temperature-gradient-driven terms of fast ions are proportional to ηf,i.e.〈λ43〉θand〈λ44〉θare proportional to ηf.Thus,the increasing ηfwill increase fast-ion resonant stabilization.From equation (18),it is also found that the spacegradient-driven terms including density- and temperaturegradient-driven terms are also proportional to,namely,〈λ42〉θ,〈λ43〉θand〈λ44〉θare proportional to.Therefore,the stabilizing effect of fast ions strengthens asincreases.

In summary,both the increasing ηfandcan strengthen the stabilization of fast ions on ITG mode.Therefore,there are two ways to improve the stabilizing effect of fast ions on ITG mode.First,keepand increase the ratio of the temperature and density gradients of fast ions,i.e.increase ηf.Second,keep ηfand increase the density gradient of fast ions,i.e.increase.

3.4.Influence of resonance broadening of wave-particle interaction

The influence of resonance broadening of wave-particle interaction is discussed in this subsection.As described above,the resonance condition is written as,i.e..However,the resonance is broadened due to the growth rate of ITG mode.

The growth rate of ITG mode without fast ions is denoted as γITGhere.Assuming γITG=0,and the wave-particle resonance contribution without resonance broadening is then evaluated.The response of the resonant fast ions is calculated by replacing〈λ4〉θwith res〈λ4〉θ.Following Landau’s prescription and res〈λ4〉θis given by,

Here,the same angle κ2=0.6 is also assumed.

Figure 7.Imaginary part of fast ions from wave-fast-ion resonance as a function of Tf/Te.

In figure 7,the magnitude of res〈λ4〉θis depicted as a function of Tf/Te.It is found that wave-particle resonance plays a stabilizing role on ITG mode since-Im (res〈λ4〉θ)is negative.As Tf/Teincreases,the magnitude of-Im (res〈λ4〉θ)decreases rapidly and the fast-ion resonance stabilization effect weakens.At high temperature,i.e.,Tf～40Te,the resonance effect is probably negligible,which implies that fast ions may be modeled by pure dilution [17].

Comparing figure 5(a) with figure 7,it is found that the resonance broadening (from γITG) actually reduces the resonance effect since the magnitude of-Im(〈λ4〉θ)is smaller than-Im (res〈λ4〉θ).Moreover,in figure 5(a),at Tf/Te＞30,-Im(〈λ4〉θ)regains slowly with increasing Tf/Teand the magnitude will not reduce to zero at high temperature(Tf～40Te).The strong resonance broadening effect is the main reason for the difference between our results and the simulation ones[22].The expression of〈λ4〉θin equation(18)can be used to explain this result.Fast ions can resonate with ITG mode when.However,the denominator of〈λ4〉θis not zero since there is an imaginary part,i.e.This imaginary part makes a difference to the resonance effect.In addition,the growth rate (γITG) makes a difference to the adiabatic response of fast ions.When γITG=0,the adiabatic part of fast ions no longer contributes an imaginary part and cannot offset the effect of the energy-gradient-driven term.

In summary,the growth rate of ITG mode brings resonance broadening which reduces the effect of the resonance and makes a difference.As a result,the stabilizing effect of fast ions on ITG mode is underestimated in our work,since γITGin our background model is much larger than the actual growth rate of ITG mode.

4.Conclusion

The stabilizing effect of trapped fast ions on ITG mode has been studied,based on both analytical and numerical calculations.The relevant physics mechanisms have been explained.

It is found that fast ions can strongly affect ITG mode through a wave-particle resonance mechanism when the precession frequency of trapped fast ions is close enough to the frequency of ITG mode.The fast-ion stabilizing effect depends on density,temperature,and the density and temperature gradients of fast ions.

Fast-ion resonance destabilizes ITG mode at very low temperature,but is stabilizing as soon as the fast-ion temperature exceeds a certain value.By investigating the effect of the fast-ion temperature in more detail,it is found that the effects of the adiabatic part and energy-gradient-driven term of fast ions almost cancel each other.Thus,the effect of fast ions mainly results from the space-gradient-driven term.The space-gradient-driven term is derived from density and temperature gradients.The density-gradient term of fast ions plays a destabilizing role on ITG mode.Moreover,one part of the temperature-gradient-driven term (ˆEηfterm) is destabilizing on ITG mode,but another part (-3ηf/2 term) plays a stable role on ITG.When the threshold conditionis satisfied,stabilization of ITG mode by fast ions is realized.Increasing the density of fast ions can enhance their effects.In addition,both increasing ηfandcan strengthen the fast-ion resonant stabilization effect.

These findings contribute to the understanding of stabilization of ITG mode by trapped fast ions and suggest a means for improving ion energy confinement in fusion devices.However,in our analytic model,the growth rate of ITG mode without fast ions,i.e.γITGis large since the resonant effect of background main ions is ignored.This large γITGweakens the wave-fast-ion resonant effect in our work.The improvement of this issue will be shown in our future work.In addition,the electromagnetic effect is different to the resonance effect.It will be complex if the mode coupling effect is considered.In this work,the electromagnetic effect is not considered,which is left for future research.

Acknowledgments

This work is supported by National Natural Science Foundation of China (Nos.11822505,11835016 and 11675257),the Youth Innovation Promotion Association CAS,the Users with Excellence Program of Hefei Science Center CAS (No.2019HSC-UE013),the Fundamental Research Funds for the Central Universities (No.WK3420000008) and the Collaborative Innovation Program of Hefei Science Center CAS(No.2019HSC-CIP014).