Stabilization of trapped electron mode through effective diffusion in electron temperature gradient turbulence

Multi-scale gyrokinetic theory and simulations of a toroidal magnetized plasma have revealed the existence of cross-scale interactions of the trapped electron mode (TEM) and the electron temperature gradient (ETG) turbulence. Reduction of the TEM instability growth rate in the ETG turbulence is clearly identified, and is well represented in the form of effective diffusion. A theoretical model based on the stochastic forcing by the ETG turbulence well describes the turbulent diffusion coefficient observed in multi-scale turbulence simulations.


Introduction
Magnetized plasma involves multiple spatial and temporal scales characterizing plasma waves and instabilities, of which interactions beyond the gap of characteristic scales often play a key role in collisionless (or weakly collisional) plasma phenomena, such as the anomalous resistivity in the magnetic reconnection [1] or the turbulent viscosity in sheared plasma flows [2]. In the drift wave turbulence in magnetized plasma [3], multiple spatiotemporal scales are brought by the complexity of a charged particle motion in inhomogeneous electromagnetic fields as well as the large difference of ion and electron mass. Recent gyrokinetic simulations of drift * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. wave turbulence have revealed the importance of multi-scale turbulence interactions in the anomalous transport not only for understanding saturation mechanisms of turbulence [4,5] but also for elucidating the experimental results [6][7][8].
The trapped electron mode (TEM) of drift waves, the slow time scale of which is related to toroidal drift motion of trapped electrons, has attracted attentions in association with the isotopic ion mass dependence of transport through the sheared E × B flows, the non-adiabatic electron response [9], and stabilization effect due to finite collisionality [10]. Interplay between the TEM and the ion temperature gradient (ITG) modes has also been investigated [11]. However, the TEM turbulent transport has mostly been investigated in a singlescale (the ion gyroradius scale) limit, ignoring the cross-scale interactions with the electron gyroradius scale fluctuations, such as the electron temperature gradient (ETG) turbulence, while an interplay of TEM and ETG fluctuations via zonal flows was investigated in [12]. Recently, the gyrokinetic simulation of multiple ion species plasma has clarified that the TEM turbulent transport can be drastically reduced by the ETG turbulence for an optimum temperature ratio of electrons to ions, that is T e /T i ∼ 3 [13], where trapped electron orbits are influenced by the fine-scale turbulence.
In the present study, by means of gyrokinetic simulations covering the ion to electron gyroradius scales, we investigate the multi-scale interactions of the TEM and ETG turbulence, focussing on the turbulent diffusion caused by the smallscale ETG fluctuations. Our simulations, utilizing massively parallel supercomputers, demonstrate that the ETG turbulence rapidly growing in the electron transit time scale prevents the TEM instability growth. The reduced growth rate of TEM is modeled by means of a theoretical approach with the stochastic forcing [14,15]. This paper describes the existence of turbulent-diffusion type interactions in multi-scale plasma turbulence, and discusses the theoretical model in comparison with the simulation results. The remainder of the paper is organized as follows. We derive the model of turbulent diffusion in the next section, and describe the simulation results in section 3. Comparison with the numerical results is discussed in section 4. Obtained results are summarized in section 5.

Modeling of turbulent diffusion
We derive the effective diffusion model explaining stabilization of the TEM instability by the ETG turbulence from the gyrokinetic equation for the perturbed electron gyrocenter distribution function, f k ⊥ (z, ε, µ, t), ∥ + µB, and µ mean the perpendicular wavenumber, the field aligned coordinate, the kinetic energy, and the magnetic moment, respectively. The linearized collision term employed in the simulation is neglected in the theoretical analysis. The sum over k ′ ′ ⊥ is taken for k ⊥ = k ′ ⊥ + k ′ ′ ⊥ . The magnetic and diamagnetic drift frequencies are denoted by ω Dk ⊥ and ω * k ⊥ . The generalized potential fluctuation and the Maxwellian distribution are represented by ψ k ⊥ and F M , respectively. Here, the generalized potential ψ k ⊥ involves the electrostatic potential ϕ k ⊥ and the parallel component of the vector , including the zeroth order Bessel function, J 0 (k ⊥ v ⊥ /Ω), for the gyroaverage. The parallel derivative is given by i k ∥ =b · ∇, and e means the elementary charge. For simplicity of notation, k ⊥ and k ′ ′ ⊥ are denoted by a and b such as f a and N ab . When the ITG mode is stable, contribution of ion dynamics to the ETG modes is subsidiary, except for the ion polarization due to finite gyroradius effect which is introduced in the calculation of electrostatic potential fluctuations.
Let us consider a formal solution of equation (1) obtained by integration along an unperturbed particle trajectory, , (2) and take an iterative approach employed in formulation of the quasi-linear diffusion [14,15]. Dependence on the initial condition f a (0) can be neglected in the following discussion. Postulating the ETG turbulence as stochastic forcing, the ensemble average, denoted by ⟨· · · ⟩, extracts the TEM dynamics, such that, ψ a = ⟨ψ a ⟩ +ψ a and N ab = ⟨N ab ⟩ +Ñ ab , where ⟨· · · ⟩ and· represent the TEM and ETG fluctuations, respectively. The ensemble average of equation (1) reads Substitution of equation (2) into the last term on the right hand side (r.h.s.) of equation (3) leads to an iterative solution, which results in the two parts, where we omitted other two terms symbolically expressed as ⟨Ñ abÑbcfc ⟩ and ⟨Ñ ab ⟨N bc ⟩f c ⟩. The former is neglected because of the independency ansatz ⟨Ñ abÑbcfc ⟩ = ⟨Ñ abÑbc ⟩⟨f c ⟩ = 0. The latter involves the ensemble average of nonlinear coupling of ETG fluctuations modulated by the TEM potential perturbations through ⟨N bc ⟩. Supposing the fluctuation amplitude of ETG turbulence is provided by the multi-scale interactions which include the feedback from the TEM to the ETG turbulence, the contribution of ⟨Ñ ab ⟨N bc ⟩f c ⟩ is not added to the effective diffusion model. The first term in equation (4) describes the stochastic forcing by the ETG potential fluctuations, providing the quasilinear diffusion of the background distribution F M in M b and its projection on the wavenumber a. Since this term does not include the TEM fluctuations, it is not responsible for the exponential growth of the TEM instability. Also, it can be neglected if the TEM fluctuation amplitude (with a) is much larger than that of the ETG turbulence. Therefore, we ignore the indirect interaction of TEM and ETG turbulence in the modeling of effective diffusion. The second one includes interactions of the ETG turbulence (denoted by the quadratic form ofÑ ) and the TEM fluctuations (⟨ f c ⟩), and can be consistent to the exponential growth of the TEM instability in the ETG turbulence. It is noteworthy that, in contrast to the first term, the second term is not necessarily negligible even if the TEM has a much larger amplitude than that of the ETG turbulence.
Let us further evaluate the second term on the r.h.s. of equation (4) by assuming slower space-time dependence of TEM than that of ETG, such that ⟨ f c (t − τ )⟩ ≈ ⟨ f c (t)⟩, |k ⊥a | ≪ |k ⊥b | and τ ac ≪ t, where τ ac means the auto-correlation time of the ETG turbulence. Since interactions among TEMs are dominated by the nonlinear term (3), the indirect nonlinear coupling via ETG fluctuations is supposed to be minor so that only the term with c = a is retained. Ignoring the first term in equation (4), thus, where the ensemble average is replaced by the space-time average for the ETG turbulence. In the last step, we introduced the auto-correlation time τ ac of the ETG turbulence, Then, in the electrostatic limit, neglecting A ∥k ⊥ , one finds The operator D k ⊥ represents an anisotropic dissipation which in turn provides D k ⊥ = −k 2 y δ eff for the TEM with k ⊥ = (0, k y ).
Here, it is noted that the TEM dynamics is mainly controlled by electrons, where contribution of the perturbed ion gyrocenter distribution function is considered to be subsidiary. Also, the effective diffusion by the ETG turbulence does not directly influence the ion dynamics because of the finite gyroradius effect. It means that, if the effective diffusion term with a constant δ eff is added to the electron gyrokinetic equation, the linear eigenfrequency ω of the TEM is simply shifted by −i ω → −i ω − k 2 y δ eff . Thus, the effective growth rate is given by γ eff = γ lin − k 2 y δ eff . (Indeed, we have confirmed the stabilization of TEM growth by means of single-scale linear gyrokinetic simulations including the effective diffusion term, −k 2 y δ eff f (0,ky) .)

Multi-scale turbulence simulation
The theoretical analysis shown above suggests that the influence of the ETG turbulence on the TEMs is modeled in the form of effective diffusion. For comparison with the theoretical model, we have carried out multi-scale simulations of the TEM and ETG turbulence by means of the flux tube gyrokinetic code, GKV [16], for the axisymmetric toroidal configuration with cocentric circular flux surfaces. The used parameters are the same as those in [10], where R 0 /L n = 3, R 0 /L TH = 1, T e /T H = 3, ν * eH = 0.05, but with steeper ETGs (that is, R 0 /L Te = 8.7, 9.342, or 10, while R 0 /L Te = 8 in [10]) presuming stronger electron heating for future applications to high electron temperature discharges towards ITER. Here, the gradient scale lengths of density L n , hydrogen ion temperature L TH , and electron temperature L Te are normalized by the major radius R 0 . The ratio of electron to hydrogen ion temperature is T e /T H , and the electron-ion collision frequency is normalized as ν * eH = qR 0 / √ 2(r 0 /R 0 ) 3/2 v te τ eH with the characteristic electron-ion collision time τ eH , the inverse aspect ratio r 0 /R 0 , the electron thermal speed v te , and the safety factor q. We used the Cyclone base case-like parameters of q = 1.42, r 0 /R 0 = 0.18, the magnetic shearŝ = 0.8, the ion mass m H = 1837m e , and the electron beta β e = 5 × 10 −4 . From the singlescale linear gyrokinetic simulations, one finds the maximum growth rates of the TEMs and ETG modes, that is, 0.79 and 2.9 v tH /R 0 at k y ρ tH = 0.4 and 5, respectively. Here, k y and ρ tH denote the wavenumber in the field-line-label coordinates and the thermal gyroradius of hydrogen ions with the thermal speed v tH = √ T H /m H . The simulation box size in the direction perpendicular to the confinement field is L x = 62.5ρ tH and L y = 20π ρ tH on the equatorial plane, and the poloidal angle is −π ⩽ z ⩽ π. The maximum parallel velocity for the particle species s is 4.5v ts , and the maximum magnetic moment is 10.125 m s v 2 ts /2B 0 . The number of grid points employed in the multi-scale simulation are (N x , N y , N z , N v , N µ ) = (1024, 1024, 40, 64, 16) in the radial, field line label, field aligned, parallel velocity, and magnetic moment coordinates, respectively. Figure 1 shows snapshots of the y component of the perpendicular electrostatic field E = −∇ ⊥ ϕ on the equatorial plane at (a) t = 10 and (b) t = 15R 0 /v tH , respectively, where E is normalized as eER 0 /T H in the gyro-Bohm units. Starting from the initial condition with small amplitudes of perturbations with random phases, the ETG instability rapidly grows and reaches the saturated turbulent state with streamer-type structures. Saturation of the ETG instability growth is followed by the TEM instability with the longer wave lengths and the smaller growth rates. The saturated ETG turbulence co-exists with the TEM fluctuations as seen in figure 1(b).
Time-histories of the linearly most unstable mode of TEM instability and the zonal flow are plotted in figure 2, where purple and cyan lines indicate the electrostatic field energy W Ek ⊥ integrated over k x for k y ρ tH = 0.4 (TEM), and 0 (zonal flows), respectively. Time-evolution of the ETG fluctuations summed for k y ρ tH > 2 is plotted by a green curve. The electrostatic field energy is defined as where [[· · · ]], ϕ k ⊥ , s, and e s mean the average along the field line, the electrostatic potential, the particle species, and the electric charge, respectively. Γ 0sk = I 0 (k 2 ⊥ ρ 2 ts )e −k 2 ⊥ ρ 2 ts with the zeroth order modified Bessel function, I 0 . The black solid lines represent the linear growth rates for k y ρ tH = 0.4 (TEM)  Time history of the electrostatic field energy, W Ek ⊥ , for a trapped electron mode (TEM) (purple), zonal flows (cyan), and ETG fluctuations summed for kyρ tH > 2 (green). The solid black lines represent the linear growth rates of the ETG and the TEM instabilities, respectively. and 5.0 (ETG) obtained by the single-scale linear gyrokinetic simulation. The initial exponential growth of the ETG mode is well fitted by the linear growth rate. After the saturation of ETG instability growth, the field energy of the ETG modes gradually decreases, and reaches a quasi-steady state. In contrast to the ETG instability, it is clearly found that the exponential growth of the TEM with k y ρ tH = 0.4 is slower than that of the single-scale linear growth shown by the black line. Then, the zonal flow generation leads to saturation of the TEM instability growth. After t = 20 R 0 /v tH , one finds the quasi-steady turbulence with the TEM, ETG and zonal flow fluctuations.
Growth rate reduction of the TEM instability is summarized in figure 3(a), where the purple line shows the linear growth rates (γ lin,ky ) of the single-scale TEM instability as a function of k y . The green squares are plotted for the exponential growth rates of TEMs observed in the multi-scale turbulence simulation for t = 8-13 R 0 /v tH . Stronger reduction of the TEM growth rate is found for higher k y components, which is well fitted by an effective diffusion. The cyan curve in figure 3(a) defined by γ lin − δ eff k 2 y for δ eff = 1.9 ρ 2 tH v tH /R 0 agrees fairly well with the TEM growth rates obtained from the TEM/ETG simulation, suggesting the effective diffusion of TEM fluctuations by the ETG turbulence, while the real frequencies are found to be hardly influenced. We also evaluated the growth rate reduction for the multi-ion species plasma with mixture of deuterium (D), tritium (T), and helium (He) components, where the same physical and numerical parameters are employed but for different ion species with the density fractions of n D = n T = 0.45 and n He = 0.05 [13]. The single-scale linear TEM growth rates shown by the purple line in figure 3(b) are about five or six times smaller than those in figure 3(a) because of the heavier ion mass [10]. The reduced growth rates of TEMs in the presence of ETG turbulence can also be well fitted by the effective diffusion of δ eff = 2.75 ρ 2 tH v tH /R 0 . The electron heat transport found in the multi-scale TEM/ETG turbulence simulation of the H-plasma with R 0 /L Te = 9.342 (which corresponds to the case shown in the second column of table 1) is about χ e = 64.3ρ 2 tH v tH /R 0 (timeaveraged from t = 20 to 50R 0 /v tH ), which has shown no clear difference from χ e = 65.6ρ 2 tH v tH /R 0 obtained by the singlescale TEM simulation with the maximum wavenumber of k y ρ tH = 1. On the other hand, we have observed transport reduction in the multi-scale turbulence simulations for the multiple ion species case [13] from χ e = 56.1ρ 2 tH v tH /R 0 (the single-scale run) to χ e = 9.4ρ 2 tH v tH /R 0 (the multi-scale run, that is, given in the fourth column of table 1), where the greater impact of the effective diffusion by the ETG turbulence is identified (see also figure 3(b)). As is well known, however, saturation level of the drift wave turbulence is not merely determined by the instability growth rates but also depends on interactions with zonal flows. It is still an open issue how strongly the zonal flows are generated in (or influenced by) the multi-scale turbulence, and remains for further investigations.

Comparison with numerical results
Ansatz of the scale separation and the form of equation (5) indicate that the direct interaction of ETG turbulence and ion dynamics should be negligible, because of the finite Larmor radius effect provided by the zeroth order Bessel function J 0 (k ⊥ v ⊥ /Ω H ) for k ⊥ ρ tH ≫ 1. Linearization of equation (3) for TEMs (that is, ⟨N k ⊥ ,k ′ ′ ⊥ ⟩ = 0) coupled with the gyrokinetic Poisson and Ampere's equations provides suppression of the TEM instability growth, where the growth rate is reduced as γ lin,ky − k 2 y δ eff . Thus, the theoretical model derived above is consistent with the simulation results shown in figure 3.
Scale separation between the ETG turbulence and TEM fluctuations presumed in derivation of the turbulent diffusion model is confirmed by a time-averaged spectrum of |E k ⊥ | 2 shown in figure 4 obtained by the multi-scale gyrokinetic simulation for the H-plasma. Here, one finds a nearly isotropic spectrum of the ETG fluctuations centered at around k y = 4.24ρ −1 tH (≡ k y0 ) with the standard deviation of σ = 1.36ρ −1 tH (where we employed the data in a spectral window of |k x ρ tH | < 4 and 1 < k y ρ tH < 9 for the statistics, and the time-average is taken from t = 8 to 13R 0 /v tH ), while the TEMs grow around k x ≈ 0 and |k y ρ tH | < 0.6. For quantitative evaluation of δ eff , we assume that the autocorrelation time τ ac is roughly approximated by inverse of the maximum linear growth rate of the ETG instability, τ ac ≈ γ −1 ETG . If we employ the flux surface average of the power spectrum of radial electric field component, the effective diffusion coefficient δ eff for k x = 0 is estimated as whereṽ Eyk ′ ⊥ denotes the y component ofṽ Ek ′ ⊥ . From the linear ETG simulation for the H-plasma case with R 0 /L Te = 9.342, γ ETG ≈ 2.9 v tH /R 0 . The sum over wavenumber space in equation (9) is taken for |k ′ ⊥ − k y0ŷ | < 2σ (where more than 84% of the ETG fluctuation energy in the spectral window is involved) to avoid unphysical contributions of small amplitude fluctuations but with high wavenumbers. Then, we find δ eff ≈ 2.7 v tH ρ 2 tH /R 0 , which is 1.4 times larger than that estimated from the reduction of the TEM growth rate, that is, δ eff ≈ 1.9 v tH ρ 2 tH /R 0 . We have also carried out the multi-scale TEM/ETG simulations for cases with R 0 /L Te = 8.7 and 10, and summarized the estimated diffusion coefficients δ eff in table 1. Although there is the three times difference in γ ETG between the cases of R 0 /L Te = 8.7 and 10, increase of δ eff observed in the simulation is only about 10%, of which tendency is well reproduced by the theoretical model. This is because the increase (decrease) in γ ETG is partly canceled by enhancement (reduction) of the ETG turbulence intensity in qualitatively consistent with the mixing length estimate suggesting |v Ek ⊥ | 2 ∝ γ ETG /k 2 ⊥ . In the multi-ion species case, where the TEM growth tH /R 0 from the simulation data of TEM growth rates. Although the rough estimate of auto-correlation time τ ac ≈ γ −1 ETG is used here, we have found the reasonable agreement of δ eff within the factor of 1.5 for all of the multi-scale simulations shown in table 1.

Summary and discussion
In the present study, multi-scale gyrokinetic simulations, covering the ion to electron gyroradius scales, have revealed reduction of the TEM instability growth rate due to the effective diffusion caused by the ETG turbulence, which is well approximated by γ lin,ky − k 2 y δ eff . Theoretical modeling based on the iterative solution of the gyrokinetic equation with the stochastic forcing successfully captures the property of effective diffusion found by the multi-scale simulation. A quantitative comparison between the theoretical and the multi-scale turbulence simulation results shows a reasonable agreement, while more elaborate evaluation of the auto-correlation time may improve the model estimate of δ eff .
In the present theoretical analysis, the ETG turbulence is regarded as stochastic forcing with a given power spectrum, which is validated in the growth phase of TEM instability. Suppression of ETG turbulence by ion gyroradius scale fluctuations is necessary to be taken into account in the steady state of multi-scale turbulence. Extension of the effective diffusion model explicitly including the feedback from the TEM to ETG turbulence will be pursued in future works, including time-dependence of the micro-scale turbulence intensity. Nevertheless, the present analysis is meaningful in the transport studies, since the quasilinear transport model is mainly based on the linear growth rates of ion-scale instabilities. Applying the effective diffusion model to the single-scale linear TEM growth rates may partly extend the quasilinear transport model.
Since the ion dynamics does not play a major role in TEM instability, we considered the effective diffusion only in the electron gyrokinetic equation, and found reduction of the TEM growth rate represented by −k 2 y δ eff . This is because the ion dynamics is hardly influenced by the ETG turbulence that is averaged out along ion gyro-orbits. Thus, the effective diffusion due to ETG turbulence may affect the ITG mode mainly through the electron gyrokinetic equation, resulting in a different model of cross-scale turbulence interactions.
The present result of TEM stabilization in the ETG turbulence is consistent to the previous simulations of multiscale interactions, such as ITG/ETG turbulence and the interplay with the micro-tearing mode (MTM). The micro-scale ETG turbulence may disturb the electron dynamics even in the ion gyroradius scale, as found in the present study, and leads to destruction of the electron current sheet of the MTM turbulence [5]. Our previous analysis of ITG/ETG turbulence [4,17] has shown that the enhancement of ITG modes in cases with the ETG fluctuations is attributed to damping of the sub ion gyroradius scale zonal flows with electron density perturbations by the ETG turbulence. Application of the effective diffusion model to other types of multi-scale drift wave turbulence remains for future studies.