The role of collisionless trapped electron mode turbulence on removal of helium ash and transport of deuterium-tritium ions

Removal of helium ash and the anomalous transport of deuterium (D) and tritium (T) ions driven by collisionless trapped electron mode (CTEM) turbulence in tokamak plasmas with weak magnetic shear are studied. We derive the eigenvalue of CTEM with helium ash, and calculate the quasi-linear turbulent fluxes of helium ash, D and T ions simultaneously. Based on the analytical results, the parametric dependence of CTEM instability as well as the anomalous transport of helium ash and D-T ions is investigated, in order to explore the parameter region that is favorable for expelling more helium ash than D and T ions. It is found that helium ash with higher temperature and steeper density profile plays a role of destabilizing CTEM instability, and has higher transport level than that of T ions. We also find that increasing electron temperature and flattening electron density profile are favorable for exhausting helium ash. Isotopic effects (i.e. increasing the fraction of T ions) enhance the transport of both helium ash and D-T ions. Moreover, the trend of stronger transport level of helium ash than that of D-T ions is enhanced by raising electron temperature and flattening electron density profile as well as isotopic effects. Besides, the diffusivity is much larger than the convection. This indicates that the CTEM turbulence driven helium ash transport is favorable for removing helium ash under the parameter region used in the present paper. The possible relevance of our theoretical results to experimental observations is also discussed.


Introduction
Helium ash is an unavoidable impurity component in the deuterium (D)-tritium (T) plasmas. The accumulation of helium ash in the core will dilute the fuels and cool the plasmas resulting in the decrease of fusion power, and is unfavorable for the success of ignition. Therefore, removing helium ash from the core plasmas and controlling the concentration of helium ash in the core are necessary on the path towards successful achievement of economic fusion power production. For example, the International Thermonuclear Experimental Reactor (ITER) requires the concentration of helium ash should not exceed 10% [1]. Previous studies have pointed out efficiency of exhausting helium ash depends on not only the recycling and pumping in the plasmas edge but also the intrinsic transport of helium from the core to the edge [2][3][4]. In this regard, understanding the transport of helium ash [5][6][7][8] has been treated as one of the most crucial issues for fusion burning plasmas.
The transport of helium impurity has been extensively studied by external helium puffing or beam injection. The corresponding experimental results such as on PDX [9], TFTR [10,11], TEXTOR [12], DIII-D [13], JT-60U [14] and JET [4] clearly show the transport of helium is anomalous, which is caused by drift wave turbulence [15]. Up to now, only JET and TFTR have ever run the D-T experiments, and the transport of helium ash was also found to be anomalous in TFTR except in the very central region [16], where neoclassical transport might be dominant mechanism. Therefore, it is meaningful to study the anomalous transport of helium ash driven by drift wave turbulence.
Both the simulation [17] and our previous theoretical work [18] based on gyro-kinetic theory revealed that the ion temperature gradient (ITG) mode turbulence is helpful for expelling the helium ash. Recently, we also studied the transport of helium ash driven by parallel velocity shear (PVS) turbulence [19], which is developed from PVS driven Kelvin-Helmholtz instability [20,21]. However, collisionless trapped electron mode (CTEM) turbulence driven by the trapped electron precession drift resonance [22] could be also an important potential contributor to the anomalous transport of helium ash in fusion devices. It has been observed that CTEM turbulence is the dominant mechanism for the anomalous helium transport in MAST [23]. But, to our knowledge, the anomalous transport of helium ash driven by CTEM turbulence has not been theoretically studied. It is worth to note that dominant electron heating was found to be favorable for removing impurity, especially for low-Z impurities [24]. Here, Z is the charge number of impurity. So, it is also interesting to investigate the following questions: whether helium ash can be removed through CTEM turbulence? How is the parametric dependence of the transport of helium ash in D-T plasmas?
Moreover, most of existing works focus only on the anomalous transport of helium ash [18,19], and few works care the transport of both impurity and main ions simultaneously. Recent experimental results on Large Helical Device (LHD) studied the transport of carbon and hydrogen (H)-D ions, separately [25]. One may wonder whether the transport of helium ash is different from D-T ions or not. Indeed, during D-T operation on TFTR, it is found that the diffusivities of T ions and helium ash are similar in magnitude, but the convective velocities of them are different [26]. It would be great to find parameter regime to expel more helium ash than D-T ions. Thus, studying the anomalous transport of both helium ash and D-T ions driven by CTEM turbulence simultaneously is of great significance for future burning plasmas.
At the same time, understanding the isotopic effects (i.e. the effects of varying the effective mass number of hydrogen isotopes) on confinement and anomalous transport is a long-standing issue for the research of magnetic confinement fusion. Recently, this topic has attracted a lot of attention [27][28][29][30][31][32][33][34][35] due to the operation of D plasmas on LHD [36,37] and approaching to the JET-DTE2 [38]. The gyro-kinetic simulations by GKV verified the universal nature of isotopic effects on trapped electron mode (TEM) turbulence in both axisymmetric tokamaks and non-axisymmetric helical/stellarator systems [33,39]. The presence of impurity can modify the isotopic dependence of TEM instability [40]. But, the isotopic effects on helium ash transport driven by CTEM turbulence are still an indistinct issue.
In this paper, we will focus on the anomalous transport of helium ash and D-T ions induced by electron density gradient driven CTEM turbulence in tokamak plasmas, and try to elucidate the favorable parameter regime for expelling more helium ash than D-T ions. Although electron temperature gradient can be also the drive of CTEM turbulence, this is not the focus of this work. We firstly derive the eigenmode equation of CTEM instability in D-T plasmas with helium ash based on the gyro-kinetic and bounce-kinetic theories, and then study the parametric dependence of CTEM instability. For simplicity, the equilibrium distribution function for both helium ash and D-T ions are assumed to be Maxwellian. The temperature of helium ash used in this work is a free parameter, which could be slightly higher or lower than that of background plasmas. More importantly, we derive the quasi-linear fluxes of helium ash, D and T ions simultaneously, and study the parametric dependence of the transport of both helium ash and D-T ions based on the analytical results. The isotopic dependence of CTEM instability and particle transport is also discussed. However, the isotopic effects on collisions [33], bounce-kinetic equation [34], zonal flow [35] as well as the combination of isotopic effects and electromagnetic effects [31,32] are beyond the scope of this paper. The principal results can be summarized as: (a) Higher (Lower) temperature of helium ash destabilizes (stabilizes) the CTEM instability, and steeper density profile of helium ash can further enhance this trend. The critical temperature ratio between electrons and helium ash for distinguishing the role of destabilization and stabilization by helium ash is given. Besides, it is relatively easier to expel helium ash when its temperature is slightly higher than that of background plasmas (T e = T i < T z ). (b) Increasing electron temperature (T e > T i = T z ) and flattening electron density profile not only destabilize the CTEM instability with helium ash but also enhance the transport level of both helium ash and D-T ions. Moreover, the higher electron temperature and the flatter electron density profile we have, the easier to expel more helium ash than D-T ions it is.
(c) Isotopic effects enhance the transport level of both helium ash and D-T ions and the trend of stronger transport level of helium ash than that of D-T ions. (d) Under the parameters used in the present paper, the CTEM driven diffusion dominants over convection, which is favorable for expelling helium ash.
The remainder of this paper is organized as follows. In section 2, we will present the derivation of eigenmode equation and its solution, and explore how some key parameters affect CTEM instability in D-T plasmas with helium ash. In section 3, the quasi-linear fluxes and the corresponding transport coefficients (i.e. diffusivity and convective velocity) of helium ash, D and T ions are calculated simultaneously. Then, the parametric dependence (such as the temperature of helium ash, the density gradient of helium ash, the electron temperature, the electron density gradient and isotope fuel mixing ratio) of the anomalous transport of helium ash, D and T ions driven by CTEM turbulence will be studied in detail. Finally, summary and some related discussions are given in section 4.

CTEM instability in D-T plasmas with helium ash
In this section, we will derive the eigenmode equation for CTEM instability by employing gyro-kinetic (bounce-kinetic) theory for helium ash, D and T ions (electrons). After solving the eigenmode equation and getting the real frequency and linear growth rate in section 2.1, we mainly analyze the parametric dependence of CTEM instability with helium ash in section 2.2.

The eigenmode equation and its solutions for CTEM instability
Similar to the ITG and PVS instability [18,19], we deal with helium ash, D and T ions by gyro-kinetic theory. The equilibrium distribution function for both helium ash and D-T ions are assumed to be Maxwellian. We choose the sheared slab geo- Ls ⇀ e y , where x, y and z correspond to the coordinates of radial, poloidal and toroidal in the tokamak configuration, respectively, B is the magnetic field along z direction, and L s = qR/ŝ means the scale length of magnetic shear (R is the major radius, q is the safety factor,ŝ is the magnetic shear). After linearizing gyrocenter Vlasov equation and pull-back transformation, the perturbed density of species α (α = z, D, T represents the helium ash, D and T ions, respectively) can be given by Here, the first term in the bracket is called 'hydromagnetic' term, while the second term corresponds to the modifications due to both finite Larmor radius (FLR) effects and acoustic term. k 2 ⊥ is treated as k 2 ⊥ = k 2 y − ∂ 2 ∂x 2 with k y being the poloidal wavenumber. Therefore, the FLR effects include both b α = k 2 y ρ 2 α and ρ 2 α ∂ 2 ∂x 2 terms which correspond to y and x directions, and ρ α = v thα Ωα is the gyroradius with v thα = Tα mα and Ω α = ZαeB cmα being the thermal velocity and gyrofrequency, T α and m α being the temperature and mass of species α.
The acoustic term is proportional to where ω means the frequency of CTEM instability. We have expressed the electrostatic potential in the Fourier space as δϕ ⇀ r = k δϕ k (x) e −iωt+ikyy+ik ∥ z and taken the parallel wavenumber Ls and x being the distance away from the rational surface. The expressions of the other symbols in equation (1) are: ∇n0α is the density gradient scale length of species α with n 0α being the equilibrium density, Z α being the charge number (Z α = Z for helium ash and Z α = 1 for D and T ions), c is the light speed in vacuum, e is the elementary charge, and η α = Lnα LTα with L Tα = − Tα ∇Tα being the temperature gradient scale length of species α.
For electrons, bounce-kinetic theory is utilized to calculate the perturbed density of trapped electrons in the toroidal configuration, and passing electrons is assumed to be adiabatic. Then, the perturbed density of electrons is represented as [22,41] with the susceptibility of electrons being The first term on the right hand side of equation (3) is contributed by passing electrons, the second and third terms come from the contribution of non-resonant trapped electrons, while the fourth term is from the resonant electrons. Here, ε 0 = r R is the inverse aspect ratio with r being the minor radius, ω * e = ky Lne cTe eB is the electron diamagnetic drift frequency with L ne = − n0e ∇n0e representing the electron density gradient scale length and n 0e being the equilibrium electron density,ω de = Lne R ω * e means the bounce averaged magnetic drift frequency of trapped electrons, G av = 0.64ŝ+0.57 is obtained by averaging over the azimuthal angle of the turning point of a trapped electron [42], K e = 1 + η e , η e = Lne LTe and L Te = − Te ∇Te is the electron temperature gradient scale length with T e being the electron temperature.
Then, bringing equations (1) and (2) into the quasineutrality equation, we can get the differential eigenmode equation of CTEM instability and it is a Weber equation expressed as Here, TH is the temperature ratio between electrons and H ions, the fractions of D ions and T ions, respectively, and the larger f T , the larger A i,eff . f c = n 0z /n 0e is the concentration of helium ash, A α is the mass number of species α (A z = 4 for helium ash, corresponds to the normalized frequency of CTEM instability, is satisfied due to the quasi-neutrality condition of equilibrium densities. Moreover, we know that the solution of a Weber equation is Hermite function, i.e. the ℓth Here, the positive sign for σ is adopted due to the condition of waves with outgoing energy flux. due to the hypothesis of L nD = L nT and T D = T T (thus L eD = L eT and K D = K T ) used in this work. We can clearly see that F (ω) ≈ 1 when the concentration of helium ash f c is small. Thus, we will use the simplification F (ω) ≈ 1 in the following calculation in order to get analytical progresses. It should be noted that equation (4) is exactly the same as equation (7) of [19]. if we ignore the non-adiabatic electrons and PVS effects. However, we normalize the distance away from rational surface x by ρ H and the frequency by v thH /R in this work rather than by ion acoustic Larmor radius ρ s and ω * e as in [19]. This will be helpful for the analysis of parametric dependence of our results, since the effects of electron temperature, electron density gradient and effective ion mass number on the normalization are eliminated. Finally, the dispersion relation of CTEM instability in the presence of helium ash can be determined by solving equation (4), where ℓ = 0 has been chosen. If we further express the normalized eigenvalueω =ω r +iγ k (ω r = ωr v thH /R ,γ k = γ k v thH /R witĥ γ k ≪ω r ) and assume the ordering of S ∼ b H ∼γ k ωr ∼ω de ω * e ∼ ε 0 , the normalized real frequency and growth rate of CTEM instability in the plasmas with helium ash are calculated by neglecting the higher order terms, Equation (6) clearly shows that FLR effects can cause the downshift of real frequency, which is consistent with the case without helium ash. This downshift of real frequency destabilizes CTEM instability. In particular, the scale length of helium ash density gradient (i.e. R Ln z = L e z R Ln e ) as well as the temperature of helium ash (i.e. τ z with fixed T e ) can also affect the real frequency via influence onf z K, and thus affect CTEM instability. Since we have used the hypothesis of L n D = L n T and T D = T T ,f z K can be rewritten as Here, τ i (i = D, T) is temperature ratio between electrons and ions, and τ D = τ T = τ H = τ i is used in the present work.
It is worth to note that the Az Z 2 τz − Ai,eff τi term is directly proportional to ρ 2 z − ρ 2 i, eff , where ρ i, eff is the effective ion gyroradius defined by A i,eff . In addition, L n z , L n e and L T z are all positive in equation (8) in our work, and the stronger FLR effects for ρ z > (<) ρ i, eff destabilize (stabilize) CTEM instability due to the presence of helium ash. Thus, we can immediately identify a critical value of τ z determined by Az Z 2 τz − Ai,eff τi =0 for distinguishing the destabilization and stabilization role of the presence of helium ash, Moreover, steeper helium ash density profile, i.e. larger R/L nz can further stabilize (destabilize) the CTEM instability more for τ z > (<) τ z, cri .
From equations (6) and (7), we can also analyze the effects of the other parameters on CTEM instability, which will be shown soon. For example, how electron temperature and electron density gradient affect the CTEM instability with helium ash? This might be relevant to the effects of electron cyclotron resonance heating (ECRH) on the anomalous transport of 'non-trace' helium ash. Here, 'non-trace' means the concentration helium ash is not ignorable, and the presence of helium ash affects the micro-instabilities [43]. Meanwhile, isotopic effects can be analyzed by varying the fraction of T ions (i.e. f T ). More results can be found in section 2.2.

kyρ H spectrum.
Firstly, the wave number spectrum of eigenvalue is shown in figure 1. The normalized real frequencŷ ω r does not change significantly with the initial increase of k y ρ H due to the competition between the increased coefficient C 1 and the downshift by FLR effects. While, with further increase of k y ρ H , the downshift of the normalized real frequency dominates, and the CTEM instability is destabilized synchronously. This is because lower real frequency makes more electrons resonate with the waves and thus increases the growth rate. The monotonic increase of CTEM instability with wave number is also found in [44]. Besides, the different lines in figure 1(b) also indicate that helium ash with T z = T i = T e stabilizes the CTEM instability. As mentioned in the introduction, the temperature of helium ash could be slightly lower or higher than that of background ions. This may result in different effects of helium ash on CTEM instability.

Effects of Tz and R/Lnz.
In figure 2, we plot the variation ofω r andγ k as a function of τ z . It needs to be  stressed that varying τ z here actually means variation of T z with fixed T e = T i . With the increase of τ z , it clearly shows that ω r increases, while synchronouslyγ k decreases. Moreover, as compared to the case without helium ash (f c = 0), the presence of higher (lower) temperature of helium ash, i.e. smaller (greater) τ z destabilizes (stabilizes) CTEM instability. Equation (9) gives the critical value of τ z for distinguishing the role of destabilization and stabilization τ z, cri =0.4 for 1:1 D-T plasmas with helium ash, which is consistent with the results in figure 2.
As noted in [45][46][47], the density profile of impurity can also significantly influence the TEM instability. Different from other types of impurity, the source of helium ash is from the core plasmas, thus its density profile can be only inwardly peaked (R/L nz >0). In figure 3, we illustrate the effects of density profile of helium ash R/L nz on CTEM instability for different τ z .γ k in figure 3(b) is increased (decreased) with steepening of helium ash density profile when τ z < (>) 0.4. The critical value of τ z for distinguishing the destabilization/stabilization role of R/L nz is also determined by τ z, cri = τ i Az Z 2 Ai,eff as elaborated in equation (9). Here, we notice that the decrease ofγ k by steepening the density profile of helium ash with τ z =1 is qualitatively consistent with the result in [45]. with low-Z carbon, but not consistent with that in [48]. with heavy impurity.

Effects of Te and R/Lne.
ECRH effects on impurity transport are widely observed in recent years [14,[49][50][51][52][53]. The electron density profile with additional ECRH in the core can be either flattened [14] or steepened [53,54]. Motivated by these experimental observations, we study the effects of T e as well as R/L ne on CTEM instability with helium ash in the following.
In figure 4, the concentration of helium ash in 1:1 D-T plasmas is set to be 10%, and we fix T i =T z and R/L nz = 10. Thus, varying τ z actually means the variation of T e . For fixed R/L ne , we can see that increasing T e (i.e. increasing τ z ) upshifts the normalized real frequency and destabilizes the CTEM instability with helium ash. This seems different from the analysis in previous subsection where electron temperature is fixed. This is because the upshift of real frequency by increasing T e is not as strong as that of precession drift frequency due to FLR effects on real frequency. Then, increasing T e results in more rather than less resonant electrons, which destabilizes CTEM instability. Moreover, for fixed τ z , it exhibits that flattening the electron density profile (decreasing R/L ne ) downshiftsω r , but the growth rateγ k is clearly increased. The destabilization by flattening electron density profile in this range is qualitatively consistent with the results in [22]. Combining the effects of increasing electron temperature and flattening the electron density profile may destabilize the CTEM instability synergistically. figure 5, we display how isotopic effects influence the CTEM instability by varying the fraction of T ions f T . It shows thatω r is decreased but the growth rateγ k  is increased when we increase f T (i.e. increase A i, eff ). This can be also explained by the stronger downshift of real frequency due to the stronger FLR effects for larger A i, eff resulting in larger growth rate. Furthermore, the presence of helium ash weakens the destabilization role of isotopic effects on CTEM instability due to its smaller gyroradius as compared to ρ i, eff , and this trend is enhanced with increasing the concentration of helium ash. However, the destabilization of CTEM instability by isotopic effects (increasing A i, eff ) here is not consistent with figure 1 of [40], where the growth rate of CTEM instability satisfies γ H > γ D > γ T with normal helium impurity (not helium ash) and strong magnetic shear (ŝ = 1.5). It is also worth noting that the isotopic effects on PVS instability with helium ash were found to be weak [19], and on ITG instability in plasmas with tungsten [55] or carbon [18] is stabilizing. These indicate the isotopic effects on different kinds of micro-instability are different.

Anomalous transport of helium ash, D and T ions
In this section, we will firstly derive the quasi-linear fluxes of helium ash as well as D and T ions, and divide the fluxes into diffusive and convective parts. Then, the parametric dependence of anomalous transport of helium ash, D and T ions driven by CTEM turbulence will be analyzed in section 3.2, where we may find the clue about how to expel more helium ash than D and T ions.
Here, δϕ * k is the conjugation component of δϕ k . Then, putting the perturbed density of helium ash from equation (1) into equation (10), we get the normalized fluxes of helium ash where the diffusivity and convective velocity arê and In equations (12) and (13), we have modelled the averaged square of normalized width of modes around the rational surface as For D and T ions, the transport coefficients have the same formation and physical meaning as those for helium ash, thus it would be very easy to get the detailed expressions of every term presented in equations (14) and (15) by replacing A z by A i and taking Z=1 in the corresponding terms of equations (12) and (13). So, we do not repeat them here.
The interesting point is that the lowest order of diffusivitŷ D α0 =γ k ω 2 r are exactly the same for helium ash and D-T ions. However, the modifications to transport coefficients due to the FLR effects and acoustic terms, which depend on mass number, charge number and temperature of species α, will be different for helium ash and D-T ions. Thus, the total transport coefficients and corresponding turbulent fluxes will be also different for helium ash and D-T ions. In the next part, we will explore the parametric dependence of this difference, and try to identify the parameter regime for higher transport level of helium ash than that of D-T ions.
Before next part, we would like to expound the validation of ambipolarity of the radial particle transport. Taking equation (2), i.e. the perturbed density of electrons into the turbulent electron flux Γ e = Re δn e δv ⃗ E× ⃗ B , which is also divided into diffusive and convective parts, the normalized electron transport coefficients (normalization rule is similar with previous parts, i.e.D e = De Rv thH|φk| 2 andV e = Ve v thH|φk| 2 ) can be calculated, with b H = k 2 y ρ 2 H . Then, the validation of radial ambipolar relationship of turbulent fluxes Γ D + Γ T + ZΓ z = Γ e requires the linear growth rate beinĝ Here, the term 3 √ 2ε 0 τ H k y ρ H G av K e C1 ωr in the denominator can be approximated as 3 √ 2ε 0 τ H k y ρ H G av K e , if the higher order terms ∼O ε 2 0 are neglected. Thus, the linear growth rate from the constraint of ambipolar turbulent flux is exactly the same as that from solving eigenvalue equation, i.e. equation (7). In other words, the ambipolarity of turbulent particle transport is self-consistently satisfied.

Parametric dependence of anomalous transport of helium ash, D and T ions
Based on the analytical results in section 3.1, we are going to study the parametric dependence of the anomalous transport of species α. As described in section 2, α = z, D, T represents helium ash, D and T ions, respectively. The values of various parameters used in this subsection are the same as those in section 2.2 if there is no additional statement.

Effects of
Tz and R/Lnz. Firstly, we study the dependence of the anomalous transport on the temperature of helium ash T z (varying τ z here actually means T z is varying with fixed T e =T i ). Figure 6(a) shows that the increase of τ z brings a significant decrease inD α , which is mainly due to the upshift of real frequency and the stabilization of instability resulting in the reduction ofD α0 =γ k ω 2 r . For convective velocities in figure 6(b), bothV z andV D are outward and decrease with increasing τ z , whileV T increases from slightly inward to outward with the increase of τ z . However, it is worth to notice thatD α ≫V α , which means diffusion mechanism dominates over the convection. This is especially important for removing helium ash. Moreover, the difference betweenD z and D D is relatively small than that betweenD z andD T . This is because the difference is resulted from FLR effects and acoustic terms. The good point isD z >D T and the difference is slightly enhanced by lowering τ z .  In section 2, it is found that when τ z < (>) τ z, cri = τ i Az Z 2 Ai,eff = 0.4, increasing R/L n z destabilizes (stabilizes) CTEM instability. This indicates the effects of R/L n z on the associated anamolous transport of species α also depend on τ z . Thus, we show how R/L n z affects the anomalous transport of helium, D and T ions for two cases in figure 7.
For the first (second) case with τ z = 0.3 (τ z = 1) in figures 7(a) (figure 7(c)),D α is increased (decreased) when we steepen the density profile of helium ash. This is again because the downshift (upshift) ofω r and the corresponding destabilization (stabilization) ofγ k (as shown in figure 3) by larger R/L n z increase (decrease) the lowest order of diffusivity  From figures 6 and 7, we conclude that the diffusivity dominants over convection. This is favourable for exhausting helium ash especially when its temperature is slightly higher than that of background plasmas. Moreover, higher transport level of helium ash than that of T ions can be achieved for a wide parameter regime for the density profile of helium ash. But, the diffusivity of helium ash can be either greater or smaller than that of D ions, which depends on the temperature ratio of electron (ion) to helium ash.

Effects of
Te and R/Lne. Similar to figure 4, increase of τ z here actually means T e becomes larger due to fixed T i =T z . Figure 8 clearly demonstrates that the increase of T e significantly enhances the diffusivityD α and it satisfiesD z >D D >D T andV z >V D >V T >0 as shown in figures 8(a) and (b), respectively. Enhanced anomalous diffusivity of helium ash here due to the increase of T e is qualitatively consistent with the prediction given by GKW in [8]. Besides, figure 8(a) also shows that the larger τ z it is, the more significant difference betweenD z andD i as well as betweenV z andV i we have.
Meanwhile, it reveals that the diffusivities of helium ash, D and T ions in figure 9(a) as well as the outwardV z andV D in figure 9(b) are all enhanced by decreasing R/L ne (i.e. flattening the electron density profile) as demonstrated in equations (14) and (15). But, for T ions,V T does not change too much with R/L ne . Since convective part is still much weaker than diffusive part, the key point here is that the diffusivity of helium is enhanced more than that of D-T ions by flattening the electron density profile.
Incorporating the results in figures 8 and 9, we may conclude that flattening the electron density profile and increasing T e are favorable for expelling helium ash through CTEM turbulence. The exhaust of argon in the ECRH heated plasmas (with increase of T e and flattening of electron density) has been observed in high β p mode plasma of JT-60U [14],

Isotopic effects.
Last but not the least, we study the isotopic effects by varying the fraction of T ions f T . As shown in figure 10(a), more T ions in D-T plasmas clearly enhanceŝ D α , which is mainly due to the enhancedD α0 by stronger FLR effects on instability as explained before. Moreover, the trend forD z >D D >D T is further strengthened by increasing the fraction of T ions. Again, the convection is much smaller than the diffusion. It indicates that the isotopic effects may also help remove more helium ash than D-T ions.

Summary
In the present work, we have investigated the anomalous transport of helium ash, D and T ions driven by CTEM turbulence under weak magnetic regime. The eigenvalues of CTEM instability are derived and the parametric dependence of CTEM instability with helium ash is studied. Moreover, the quasi-linear turbulent fluxes of helium ash, D and T ions are calculated and divided into diffusion and convection parts. Then, the detailed parametric dependence of the anomalous transport of helium ash, D and T ions is studied in order to find parameter regime to exhaust more helium ash than D-T ions. The main results can be found in tables 1 and 2. It needs to be stressed that, under the parameter used in this paper, we findD α ≫V α , and thus just list how to enhance the difference betweenD z andD i in table 2. The main conclusions are worth to be stressed again as (a) The diffusivity dominates over convection, which is favorable for exhausting helium ash especially when its temperature is slightly higher than that of background plasmas; (b) The trend of expelling more helium ash than D-T ions is evidently strengthened by increasing T e and flattening electron density profile as well as increasing f T .
It would be interesting to test our theoretical results by gyrokinetic simulation, and explore the nonlinear process, which may bring new physics in the burning plasmas. Extension of this work to predict helium ash profile by including helium ash source is worth of exploring in the future. Because the energetic alpha particles mainly heat electrons, the electron temperature gradient driven CTEM turbulence could be another important candidate for helium ash transport. It may also need considering the other type of microturbulence driven anomalous transport or even neoclassical transport in the very central region, where helium ash is produced. This is because CTEM turbulence may not be dominant mechanism for helium transport due to small fraction of trapped electrons in the very central region. There are also many other interesting related works left for future. For example, it is found that resonant magnetic perturbation (RMP) decreases the confinement time of helium impurity [56], and the removal of helium ash is also identified by ICRH-driven ripple transport [57]. However, the physical mechanism about why those non-axisymmetric perturbed magnetic field helps expel helium ash is not very clear, and it is worth to explore. Electromagnetic (EM) effects on impurity transport have attracted attention [44,58]. Therefore, studying the helium ash transport driven by EM turbulence such as kinetic ballooning mode (KBM) turbulence in the future is of great significance. Besides, the TEM turbulence is also found to evidently influence the transport of other impurities [59] such as tungsten (W) [60,61], which is another important impurity in ITER. Our ongoing work is to investigate how ECRH affects the transport of W in the D-T plasmas.