Multi-scale turbulence simulation suggesting improvement of electron heated plasma confinement

Abstract

Turbulent transport is a key physics process for confining magnetic fusion plasma. Recent theoretical and experimental studies of existing fusion experimental devices revealed the existence of cross-scale interactions between small (electron)-scale and large (ion)-scale turbulence. Since conventional turbulent transport modelling lacks cross-scale interactions, it should be clarified whether cross-scale interactions are needed to be considered in future experiments on burning plasma, whose high electron temperature is sustained with fusion-born alpha particle heating. Here, we present supercomputer simulations showing that electron-scale turbulence in high electron temperature plasma can affect the turbulent transport of not only electrons but also fuels and ash. Electron-scale turbulence disturbs the trajectories of resonant electrons responsible for ion-scale micro-instability and suppresses large-scale turbulent fluctuations. Simultaneously, ion-scale turbulent eddies also suppress electron-scale turbulence. These results indicate a mutually exclusive nature of turbulence with disparate scales. We demonstrate the possibility of reduced heat flux via cross-scale interactions.

Introduction

Plasma is an ionised gas coupled with electromagnetic fields. It is ubiquitously found in nature, laboratories, and industries, for example, in black-hole accretion discs, jets, the Sun’s core, and solar wind, the Earth’s magnetosphere, and magnetic fusion plasma. Magnetic fusion plasma characterised by a strong confinement magnetic field (~5 T) has steep density and temperature gradients (~10 keV/m) sustained by external heating of microwave and neutral beam injections in existing fusion experimental devices, and by fusion-born alpha particles in future burning plasma experiments. The magnetic fusion plasma is a non-equilibrium open system, self-organised with transport processes via micro-instabilities and associated turbulence driven by the steep density and temperature gradients¹. Turbulence in magnetic fusion plasma is regarded as a multi-scale problem involving wide temporal and spatial scales, from the radius of electron gyration (~0.1 mm) to that of ion gyration (~1 cm). The electron-scale and ion-scale turbulence were often analysed separately under the scale-separation assumption. Since large-scale eddies in ion-scale turbulence were often dominant, the conventional model was designed to reproduce ion-scale turbulent transport. However, recent gyrokinetic simulation studies revealed the existence of cross-scale interactions between electron-scale and ion-scale turbulence^{2,3,4,5,6,7,8}. Comparisons with experiments in Alcator C-Mod and DIII-D tokamaks in the United States suggest that the multi-scale interactions are necessary to explain the heat fluxes measured in the experiments³ and play an important role in the near-future ITER device⁴. Latest campaigns in TCV, ASDEX Upgrade and JET tokamak devices in Switzerland, Germany and the United Kingdom also report that the electron-scale turbulence is responsible for a stiff dependence of electron heat flux against electron temperature gradient (ETG) in ion-heated plasma⁸. Electron-scale effects are important not only in tokamak core plasma but also in spherical tokamaks^9,10,11 and tokamak edge plasma^12,13. We note that there exists another class of multi-scale problems in fusion plasma, namely, interactions between ion-scale turbulence and device scale (~1 m) or meso-scale fluctuations¹⁴, which is beyond the scope of this work.

Beyond the existing devices, electron heating is expected to dominate in ITER. In particular for the pre-fusion power operation 1 (PFPO-1) phase, the central ratio of electron to ion temperature T_e/T_i can be even larger than 3.0 by electron cyclotron heating¹⁵. Additionally, fusion-born alpha particles in burning plasma mainly heat electrons with keeping electron temperature T_e higher than that of ions T_i. Due to the relaxation by ion-electron energy exchange, a reactor-relevant temperature ratio is considered around 1 < T_e/T_i < 2. As the temperature ratio T_e/T_i increases, ion-scale instabilities tend to be destabilised^16,17, in contrast to electron-scale instabilities that tend to be stabilised¹⁸. Therefore, the extrapolation of multi-scale interactions toward future burning fusion plasma experiments is non-trivial. To improve the performance prediction of future fusion devices, it should be clarified whether the cross-scale interactions are needed to be considered for ITER-relevant electron heated plasma and future burning plasma at T_e/T_i > 1.

Here, we address this problem by means of numerical simulations of multi-scale turbulence in high electron temperature tokamak plasma with electrons (e), deuterium (D) and tritium (T) fuel and helium (He) ash, mimicking future experiments on burning plasma. The turbulent transport process in magnetised plasma is well described by non-linear gyrokinetic theory^19,20,21, which is widely used for the analyses of magnetic fusion plasma and is also applied to accretion-disc and solar-wind turbulence²² and auroral arc²³. The massively parallel computation resolving from electron to ion gyroradius scales is carried out by the gyrokinetic Vlasov simulation code GKV^24,25 on the Japanese national flagship supercomputer Fugaku. The time evolution of perturbed distribution functions and electromagnetic potential fluctuations in a tokamak magnetic configuration is solved based on the delta-f electromagnetic gyrokinetic equations in a five-dimensional phase space²⁶. See “Methods” for the numerical model and employed plasma parameters.

Results

Micro-instabilities at ion and electron scales

In this simulation, two types of micro-instabilities can linearly grow with time because of steep electron temperature and density gradients. The first one is destabilised by toroidal precession resonance of electrons trapped in a weak magnetic field side, which is called the trapped electron mode (TEM)²⁷. The other one is destabilised by the compression of the poloidal magnetic drift in torus magnetic curvature, which is named the toroidal ETG mode²⁸. TEM typically possesses long wavelengths in the ion gyroradius scale and low frequencies, whereas the ETG typically has short wavelengths in the electron gyroradius scale and high frequencies. From the linear stability analysis (See Fig. 1), the poloidal wavenumber of the most unstable TEM is k_y = 0.25 ρ_ti⁻¹ with the real frequency ω_r = 1.50 v_ti/R₀ and linear growth rate γ = 0.145 v_ti/R₀, where ρ_ti, v_ti and R₀ are respectively the ion (the hydrogen mass is used as a reference) thermal gyroradius, ion thermal speed, and tokamak major radius. The most unstable ETG mode is found at electron-scale k_y = 4.5 ρ_ti⁻¹, ω_r = 24.3 v_ti/R₀ and γ = 1.16 v_ti/R₀. Although the wavenumbers and frequencies of TEM and of ETG are different, the wave phase velocities in the toroidal direction are similar, that is, (ω_r/k_tor)_TEM = 42.7 v_tiρ_ti/R₀ and (ω_r/k_tor)_ETG = 45.1 v_tiρ_ti/R₀, with the toroidal wavenumber k_tor = ε_rk_y/q, where the ε_r = r/R₀ is the inverse aspect ratio of the torus and q is the safety factor. Using an estimate of the toroidal precession drift velocity for deeply trapped electrons v_pre ~ m_ev²q/(2erB₀), where m_e, e, r, and B₀ are the electron mass, electric charge, tokamak minor radius, and equilibrium magnetic field strength, the resonant condition of TEM (v_pre = ω_r/k_tor) is satisfied when the particle velocity v = 1.9 v_te. TEM has been considered relevant for experimental density scaling or electron temperature profile based on non-linear analytic theory^29,30. Recent gyrokinetic simulation studies have also attracted researchers’ attention because of the impact on the tokamak edge transport³¹ and the long-standing mystery of the hydrogen isotope effect³². Nevertheless, the cross-scale interactions between TEM and ETG have not been fully investigated yet.

Fig. 1: Linear dispersion relation.

Linear growth rates γ (blue) and real frequencies ω_r (orange) in the employed plasma parameters are plotted as functions of the poloidal wavenumber k_y (for k_x = 0). Low-wavenumber modes at k_y < ρ_ti⁻¹ are TEMs, whereas high-wavenumber modes at k_y > ρ_ti⁻¹ are the ETG modes.

Tracer particle trajectory analysis

The multi-scale turbulence simulations show that large-scale TEM and small-scale ETG fluctuations can co-exist in non-linearly developed turbulent flows. A typical trapped electron trajectory bounces on the weak magnetic field side of the torus outer region and makes a drift motion in the toroidal direction. The trapped electrons experience the fluctuating electric fields of TEM and ETG turbulence during the toroidal drift motion, as plotted in Fig. 2a. The poloidal electric fields of TEMs affect particles close to the resonant velocity (v = 2 v_te, v_pre = 47.3 v_tiρ_ti/R₀) over a toroidal path length yq/(ε_rρ_ti) < 400, which causes radial motion by the E × B drift (Fig. 2b). Further, an off-resonant particle (v = 2.5 v_te, v_pre = 73.9 v_tiρ_ti/R₀) travels faster than waves, feels positive and negative electric fields alternately, and has no net radial displacement. Since the ETG modes also propagate with the phase velocity close to that of TEMs, small-scale ETG electric fields are also averaged out for the off-resonant particles (seen as short spikes of the blue line in Fig. 2a). However, the effects of the ETG modes on resonant particles are significant because the ETG modes also propagate along with the toroidal drift of the resonant particles. Further, the trajectory of a resonant particle is disturbed by small-scale turbulence (Fig. 2b). A secular displacement is observed (the red line in Fig. 2b after yq/(ε_rρ_ti) > 700) because the tracer particle trajectory is modified by small-scale turbulence and moved in an oppositely rotating large-scale eddy. Since the statistical correlation between turbulent flows and perturbations of plasma distribution functions determines the resultant turbulent transport, the impacts of small electron-scale turbulence on turbulent transport are quantitatively investigated in the following analyses.

Fig. 2: Effects of electron-scale turbulence on resonant particles.

a Turbulent poloidal electric field experienced by trapped electrons, and b radial displacement of trapped electrons during the toroidal drift motion. Red and blue solid lines plot the trajectories for resonant (v = 2 v_te) and off-resonant (v = 2.5 v_te) particles, respectively, which are calculated as tracer particles under the perturbed electric field obtained by multi-scale plasma turbulence simulations of burning Fusion plasma in the time range of 162.4 < tv_ti/R₀ < 179.4. For comparison, black and green dashed lines show the particle trajectories traced under the low-pass-filtered (k_x < 4ρ_ti⁻¹, k_y < 1ρ_ti⁻¹) electric field for resonant (v = 2 v_te) and off-resonant (v = 2.5 v_te) particles, respectively.

Turbulent fluctuation profiles

Perturbed electron pressure and the streamlines of turbulent E × B flows are plotted in Fig. 3a. Radially elongated fluctuations of TEMs are observed in the torus outer region, so-called the bad curvature region. Streamlines of the E × B flows are superimposed on the colour map, namely, turbulent E × B flows cause electron thermal transport. According to the flow directions, high-temperature fluctuations go outward and low-temperature fluctuations inward. Velocity–space dependence of turbulent flux in Fig. 3c shows that trapped particles (surrounded by green lines) satisfying the precession drift resonance condition (on black dashed line) are responsible for the turbulent thermal transport. In the magnified picture of perturbed electron pressure and streamlines in Fig. 3b, small-scale ETG turbulent eddies are observed to co-exist with large-scale TEM fluctuations. This means that the small-scale ETG turbulence disturbs the streamline of E × B flows, modifies the correlation between turbulent flows and perturbed electron pressure, and possibly affects electron heat transport.

Fig. 3: Mechanisms of electron heat transport in multi-scale turbulence.

a Poloidal cross-section of perturbed electron pressure p_e (colour contour) and streamlines of E × B drift flows (solid black lines). b Magnified picture showing E × B drift velocity by arrows. c Velocity–space-dependent turbulent energy flux of electrons Q_e^v at the poloidal angle θ = 0, averaged over time 100 < tv_ti/R₀ < 200. The green solid and black dashed lines show trapped-passing boundaries and the precession drift resonance condition, respectively, for the TEM phase velocity (ω_r/k_tor)_TEM = 42.7 v_tiρ_ti/R₀.

Turbulent transport spectra

To examine the effect of ETG turbulence on TEM-driven turbulent transport, multi-scale simulation results are compared with an ion-scale simulation resolving only TEM scales and an electron-scale simulation resolving only ETG scales (for details, see “Methods”). The resultant poloidal wave number (k_y) spectra of electron energy flux are shown in Fig. 4. The multi-scale spectrum shows two peaks attributed to low-k_y TEM and high-k_y ETG turbulence. Low-k_y TEM components in the multi-scale simulation are reduced compared with those in the single ion-scale simulation, suggesting the suppression of TEM by ETG turbulence. The cross-scale interactions modify ion-scale turbulent fluctuation amplitude, which affects turbulent transport levels of not only electrons but also fuel D, T, and He ash. The turbulent heat fluxes in multi-scale TEM/ETG turbulence [Q_e(TEM/ETG) = 5.66 Q_gB, Q_D + Q_T(TEM/ETG) = 0.24 Q_gB, Q_He(TEM/ETG) = 0.02 Q_gB] are reduced compared with those in single-scale TEM turbulence [Q_e(TEM) = 33.63 Q_gB, Q_D + Q_T(TEM) = 1.11 Q_gB, Q_He(TEM) = 0.09 Q_gB] because of the suppression of TEM turbulence by ETG turbulence, where Q_gB = n_eT_ec_aρ_a²/R₀² is the gyro-Bohm unit with the speed of acoustic wave c_a = √(T_e/m_i) and the acoustic gyroradius ρ_a. The agreement of high-k_y ETG peaks of turbulent flux spectra in multi-scale and single electron-scale simulations seems coincident. Although ETG turbulence initially saturates at a higher transport level, ETG-driven zonal flows suppress turbulent transport after a long time tv_ti/R₀ ≥ 50 in the single electron-scale simulation (not shown), which is consistent with a recent study on single-scale ETG turbulence¹⁰. Therefore, the suppression of ETG turbulence by ETG-driven zonal flows dominates the single electron-scale simulation result, whereas no significant zonal flow is observed in the multi-scale simulation, indicating the existence of suppression mechanisms of ETG turbulence by cross-scale interactions.

Fig. 4: Poloidal wavenumber spectra of the time-averaged electron energy flux Q_e.

The multi-scale TEM/ETG turbulence simulation result is plotted using a blue line. Results of a single ion-scale TEM turbulence simulation (k_yρ_ti ≤ 1) and single electron-scale ETG turbulence simulation (k_yρ_ti ≥ 0.5) are plotted using orange and green lines, respectively.

Gyrokinetic non-linear entropy transfer analysis

The cross-scale interactions between the TEM and ETG modes can be interpreted in two ways. Moving from the large to small scales, the suppression of ETG turbulence in the appearance of TEMs is due to the distortion of the electron-scale streamers by the ion-scale turbulent eddies, as observed in previous multi-scale simulations^2,33. In contrast to a previous work discussing the suppression of ETG modes by TEM-driven zonal flows (poloidally symmetric flows)³⁴, in this study, there are no strong zonal flows (Fig. 3a). Our results indicate that the shearing of TEM turbulent eddies (not necessarily zonal flows) can suppress ETG turbulence. Moving from small to large scales, the reduction of TEM turbulent transport in the presence of ETG turbulence is in contrast to the multi-scale simulations of ion-temperature-gradient (ITG) modes² but resembles the suppression of micro-tearing modes (MTM) by ETG turbulence⁵. The non-linear cross-scale interactions between electron-scale ETG turbulence and ion-scale TEM turbulence are investigated using the gyrokinetic non-linear entropy transfer analysis³⁵. The perturbed entropy is a measure of amplitude fluctuations of the plasma distribution function, and its non-linear excitation or damping is described by the entropy transfer function. The net entropy transfer to a mode with wave number k is split into contributions of the electron-scale, electron-scale coupling J_k^Ωe,Ωe and the ion-scale, electron-scale coupling 2J_k^Ωi,Ωe using the subspace transfer analysis technique³⁶ and defining the ion and electron scales Ω_i, and Ω_e in the perpendicular wavenumber space. Spectra of the time-averaged transfer function in Fig. 5a shows that the electron-scale, electron-scale coupling has a negative contribution on low-k_y TEM fluctuations (Fig. 5a, J_k^Ωe,Ωe < 0 at k_yρ_ti < 0.5), which directly confirms the damping effect of ETG turbulence on TEMs. From the energy conservation relation among triad subspaces called the detailed balance, J_k∈Ωi^Ωe,Ωe + 2 J_k∈Ωe^Ωi,Ωe = 0, the suppression of TEMs by ETG turbulence (J_k^Ωe,Ωe < 0 at k_yρ_ti < 0.5) indicates entropy transfer from low-k_y TEMs to high-k_y modes (2J_k∈Ωe^Ωi,Ωe = −J_k∈Ωi^Ωe,Ωe > 0). As plotted in Fig. 5b, net entropy gain in high-k_y range via ion-scale coupling is observed at finite k_x (J_k^Ωi,Ωe > 0 at k_yρ_ti ~ 4 and k_xρ_ti > 1) but not at ETG peaks (k_yρ_ti ~ 4 and k_xρ_ti ~ 0, characterised by a peak of energy specrum at high wavenumber, which is close to linearly unstable ETG modes). These high-k_⊥ modes create higher k_⊥ fluctuations (J_k^Ωe,Ωe > 0 at higher k_yρ_ti > 7, not shown), which are eventually damped by collisional dissipation. Additionally, the ion-scale, electron-scale coupling has a negative contribution on ETG (J_k^Ωi,Ωe < 0 at k_yρ_ti ~ 4 and k_xρ_ti ~ 0), confirming the suppression of ETG modes by ion-scale turbulence.

Fig. 5: Perpendicular wavenumber spectra of the time-averaged non-linear electron entropy transfer in the multi-scale turbulence simulation.

a The contribution of the electron-scale, electron-scale coupling J_k^Ωe,Ωe. b The contribution of the ion-scale, electron-scale coupling 2J_k^Ωi,Ωe = J_k^Ωi,Ωe + J_k^Ωe,Ωi, as normalised by n_eT_iv_tiρ_ti²/R₀³.

Extrapolation of multi-scale turbulent interactions toward high T _e/T _i regime

Finally, the dependence of turbulent energy flux Q_e on the temperature ratio T_e/T_i is examined in Fig. 6. Corresponding linear dispersion relations (as in Fig. 1) and poloidal wavenumber spectra of electron energy flux (as in Fig. 4) for each T_e/T_i are respectively shown in Supplementary Figs. 1 and 2. Since ETGs are highly unstable at T_e/T_i = 1, turbulent transport in the multi-scale simulation agrees with that in the single electron-scale simulation. As the electron temperature increases, ETGs are stabilised (green line), whereas TEMs are destabilised (orange line). At T_e/T_i = 4, TEM dominates turbulent transport even in the multi-scale simulation. The T_e/T_i = 3 case is thoroughly investigated in this study, where TEMs are close to marginal stability and significantly affected via the cross-scale interactions with ETG turbulence. From the ITER profile modelling and a reference value of a high T_e discharge of the DIII-D tokamak³⁷, a reactor-relevant temperature ratio is considered around 1 < T_e/T_i < 2 due to the ion–electron energy exchange. Our survey of dependence of Q_e on T_e/T_i covers this range. An important consequence is that the ETG contribution survives in a wider parameter range 1 < T_e/T_i < 3, although it may have been considered that ETG could have non-negligible contribution when T_e ~ T_i and sufficiently large electron temperature⁸. Additionally, the multi-scale turbulence simulation shows the existence of an appropriate T_e/T_i range where the cross-scale interactions suppress turbulent transport. The suppression of near-marginal TEM by ETG turbulence leads the upshift of the critical temperature ratio for an increase of TEM-dominated energy flux (from T_e/T_i ~ 2 in the ion-scale simulation to T_e/T_i ~ 3 in the multi-scale simulation), analogous to the Dimits upshift of critical ion temperature gradient where zonal flows suppress near marginal ITG modes³⁸.

Fig. 6: Electron energy flux Q_e as a function of the temperature ratio T_e/T_i.

The multi-scale TEM/ETG turbulence simulation result is plotted using a blue line with standard deviation error bars. The results of a single ion-scale TEM turbulence simulation (k_yρ_ti ≤ 1) and single electron-scale ETG turbulence simulation (k_yρ_ti ≥ 0.5) are plotted using orange and green lines, respectively. Note that an electron-scale simulation at T_e/T_i = 4 and ion-scale simulations at T_e/T_i = 1 and 2 contains no unstable modes (see Supplementary Fig. 1) and decays in time, and therefore cannot define finite amplitudes. The corresponding points are plotted on x-axis. A large standard deviation bar for the case of the electron-scale simulation at T_e/T_i = 1 (268 Q_gB) is omitted for visibility.

Discussion

A series of our works on multi-scale turbulence in magnetised plasma indicates some common features: large-scale turbulence tends to suppress small-scale turbulence (ITG/ETG^2,3,4, MTM/ETG¹³, and TEM/ETG in this study), whereas small-scale turbulence tends to destroy large-scale structures (damping of short-wave-length zonal flows by ETG³⁶, destruction of radially localised current sheet of MTM by ETG⁵ and disturbance of drift resonance between trapped electrons and TEM by ETG in this study). These findings suggest the mutual conjunction between disparate-scale turbulence as a generic nature of cross-scale interactions beyond a conventional turbulence theory described using single-scale energy injection/cascade/dissipation processes.

Our results answer the question of whether the cross-scale interactions are needed to be taken into account for future burning plasma experiments. Even beyond the existing ion-heated devices T_e ~ T_i, electron-scale turbulence can have impacts in electron heated plasma with high electron temperature T_e > T_i. This study demonstrates the possibility of reducing total electron heat flux via cross-scale interactions. Considering the opposite dependence of TEM and ETG instabilities on the temperature ratio T_e/T_i, our results reveal the existence of an appropriate T_e/T_i range where the cross-scale interactions reduce turbulent transport, which will be advantageous for optimum tokamak operation. This finding is particularly important in burning plasma, because electron heating by the fusion-born alpha particles keeps electron temperature high T_e > T_i. It also has impacts on understanding electron-heated plasma at the PFPO-1 phase in the ITER research plan, contributing the early success of fusion energy development.

This work has extensively investigated the cross-scale interactions between TEM and ETG turbulence by assuming electron temperature and its gradients exceed those of ions, though the dominant instabilities at ion scale (e.g, ITG, TEM, and MTM) depend on magnetic configuration and plasma parameters. This work is also limited to simulations with electrons, deuterium and tritium fuels, and helium ash, excluding energetic alpha particle dynamics. It has been reported that TEM drives low-level transport of alpha particles³⁹. The resonant interactions between energetic particles and micro-instability will be significant for ITG modes^40,41 but not for TEM and ETG modes because the propagation directions are opposite⁴². Therefore, interactions between energetic particles and TEM/ETG turbulence are expected to be weak. When we consider interplay from electron to ion-scale turbulence and macroscopic fluctuations, magnetohydrodynamic instability driven by energetic particles will become a candidate for the third player⁴³, which is a theoretically and numerically challenging subject.

Methods

Simulation details

Micro-instabilities and turbulent transport in magnetised plasma are simulated using the gyrokinetic Vlasov simulation code GKV^24,25. The code is parallelised by a hybrid of message-passing interface and open multi-processing. Optimisation techniques for high-performance computing are implemented, such as the pipelined computation-communication overlap, segmented process mapping on the three-dimensional torus interconnect²⁵, and communication-avoiding iterative solver for the implicit collision operator⁴⁴. This implementation ensures that the GKV code achieves a good scalability up to 12,288 nodes on the Fugaku supercomputer, 3.1 Peta-FLOPs (floating-point operation per second), which comprises 7.5% of the theoretical peak performance of the architecture and parallel efficiency of 83.7%. The highly optimised code and plentiful computational resources of Fugaku enable us to investigate the unexplored multi-scale nature of turbulent transport in burning fusion plasma.

Time evolution of the perturbed distribution functions f_s of plasma species s, the electrostatic potential ϕ, and the magnetic vector potential parallel to the equilibrium magnetic field A_|| is solved based on the delta-f gyrokinetic Vlasov–Poisson–Ampère equations²⁶. The configuration space is represented by field-aligned coordinates x = r − r₀, y = [q(r)θ − ζ]r₀/q(r₀), z = θ, where r, θ, and ζ are the tokamak minor radius, and the poloidal and toroidal angles, respectively. The velocity–space coordinates are the parallel velocity v_|| and magnetic moment μ. The local flux-tube model⁴⁵ resolves perturbed quantities in a long and thin simulation box along a field line, whereas the equilibrium quantities are approximated at the flux-tube centre r = r₀, consistent with delta-f gyrokinetic ordering. Then, periodicities in perpendicular directions x and y are assumed along with homogeneous turbulence in fluids. The last boundary condition is the torus periodicity f(r, θ + 2π, ζ, v_||, μ) = f(r, θ, ζ, v_||, μ). In this study, the equilibrium magnetic field is assumed to have a concentric circular torus geometry (the so-called s-α model with geometric α = 0), which is characterised by the tokamak inverse aspect ratio ε_r, safety factor q, and magnetic shear ŝ. We selected ε_r = 0.18, q = 1.42 and ŝ = 0.8. The following plasma species were included in simulations: electron, deuterium, tritium, and helium (s = e, D, T, and He). The employed plasma parameters are e = e_D = e_T = −e_e = e_He/2, m_i = 1837 m_e = m_D/2 = m_T/3 = m_He/4, T_i = T_D = T_T = T_He, n_D = n_T = 0.45 n_e, n_He = 0.05 n_e, R₀/L_ne = R₀/L_nD = R₀/L_nT = R₀/L_nHe = 3, R₀/L_Te = 9.342 and R₀/L_TD = R₀/L_TT = R₀/L_THe = 1, where L_ns = −den n_s/dx and L_Ts = −den T_s/dx are the density and temperature gradient scale lengths. The elementary electric charge e, hydrogen mass m_i, and tokamak major radius R₀ are used as references. The charge density satisfies the quasi-neutrality condition ∑_s e_sn_s = 0 and ∑_s e_sn_s/L_ns = 0. The plasma beta value is β = μ₀n_eT_D/B₀² = 0.05%, normalised Debye length is λ² = ε₀B₀²/(m_in_e) = 10⁻³ and normalised collision frequency is ν^* = qR₀τ_ee⁻¹/(√2ε_r^3/2v_te) = 0.05 with electron–electron collision time τ_ee. The above plasma parameters are chosen from a previous study³² but the number of plasma species is increased. Most parameters are comparable to the existing tokamak device (e.g., a high T_e discharge of DIII-D #173147³⁷), whereas the electron temperature and its gradient are slightly enhanced to mimic the electron-heated burning fusion plasma. Although electron heating by the alpha particles is the main heat source in the burning plasma, the ion–electron energy exchange process also heats ion species. When the ITG increases due to the energy exchange, the ITG modes are destabilised. For the multi-scale ITG/ETG turbulence at T_e = T_i and R₀/L_Te = R₀/L_Ti, detailed mechanisms of cross-scale interactions between ITG and ETG modes have been reported previously^2,36. In this study, we focus on the parameter regime at T_e > T_i and R₀/L_Te > R₀/L_Ti which has not yet been analysed in existing devices. We examined the electron temperature dependence in the range 1 ≤ T_e/T_i ≤ 4, and detailed analyses are presented for the case of T_e/T_i = 3, where ETG turbulence significantly affects near-marginal TEMs. These parameters are also related to the electron-heated plasma at the PFPO-1 phase in the ITER research plan which is an important step for early success of ITER. The prediction of the integrated modelling of PFPO-1 phase H-mode plasma reported high central temperature ratio T_e/T_i > 3 and R₀/L_Te > R₀/L_Ti¹⁵. For the multi-scale turbulence simulations, we employed simulation box sizes 0 ≤ x/ρ_ti < 125, 0 ≤ y/ρ_ti < 40π,–π ≤ z < π,−4.5 ≤ v_||/v_ts ≤ 4.5, 0 ≤ μB₀/T_s ≤ 12.5, and grid points in each dimension (N_x, N_y, N_z, N_v||, N_μ) = (2048, 2048, 40, 64, 16) for the T_e/T_i = 1 case and (N_x, N_y, N_z, N_v||, N_μ) = (1024, 1024, 40, 64, 16) for the 2 ≤ T_e/T_i cases. Since the GKV code treats perpendicular x and y space using the Fourier spectral method with 2/3 de-aliasing rule, f(x, y, z, v_||, μ) = ∑_kx∑_ky f_k(z, v_||, μ) exp(ik_xx + ik_yy), the corresponding perpendicular wavenumber resolutions are k_x,min = k_y,min = 0.05ρ_ti⁻¹, k_x,max = k_y,max = 33.9ρ_ti⁻¹ for the T_e/T_i = 1 case and k_x,max = k_y,max = 16.95ρ_ti⁻¹ for the 2 ≤ T_e/T_i cases. For single ion-scale turbulence simulations, we employed reduced perpendicular wavenumber space k_x,max = 4ρ_ti⁻¹, k_y,max = 1ρ_ti⁻¹, which well resolves large ion-scale micro-instabilities and associated turbulence but excludes small electron-scale dynamics. Further, for single electron-scale turbulence simulations, we employed smaller perpendicular box sizes k_x,min = k_y,min = 0.5ρ_ti⁻¹, which covers only electron-scale micro-instabilities but excludes ion-scale ones.

Tracer particle analysis

In Fig. 2, tracer particle trajectories are calculated to explain behaviour of resonant and off-resonant particles and to examine the effects of small electron-scale turbulence on the trajectories. Because of the low β value, magnetic perturbations are neglected in diagnostics. The equation of motion of a gyrokinetic electron is solved up to the lowest order of ρ_te/R₀,

$$\frac{{{{{{\rm{d}}}}}}{{{{{\bf{x}}}}}}}{{{{{{{\rm{d}}}}}}t}}={v}_{\parallel }{{{{{\bf{b}}}}}}+{{{{{{\bf{v}}}}}}}_{{{{{{\rm{ed}}}}}}}+{{{{{{\bf{v}}}}}}}_{{{\rm{E}}}},\,\frac{{{{{{\rm{d}}}}}}{v}_{\parallel }}{{{{{{{\rm{d}}}}}}t}}=-\frac{\mu {\nabla }_{\parallel }{B}_{0}}{{m}_{{{\rm{e}}}}},\,\frac{{{{{{\rm{d}}}}}}\mu }{{{{{{{\rm{d}}}}}}t}}=0,$$

(1)

where B = B₀b, v_ed, and v_E are the equilibrium magnetic field, electron magnetic drift, and E × B drift velocities, respectively. Spatio-temporal data of gyrophase-averaged electrostatic potential from multi-scale turbulence simulations are used for evaluating the E × B drift. Since the pitch angle at the poloidal angle θ = 0 is set as 0.45π, the parallel velocity and magnetic moment for a resonant (v = 2v_te) particle are v_|| = 0.31v_te, μ = 2.38T_e/B₀ and v_|| = 0.39v_te, μ = 3.72T_e/B₀ for an off-resonant (v = 2.5v_te) particle. Without E × B flows, the conservation of the canonical angular momentum ensures that there is no radial displacement of a particle. Even when E × B flows exist, there will still be no net radial transport if the electrostatic potential is time-independent because particles move only along the static electrostatic potential contour. Therefore, net radial displacement of a collisionless particle is induced by fluctuating E × B flows. For the analysis of turbulent transport, a statistical correlation between the fluctuating E × B flows and perturbations of the plasma distribution functions is necessary to be evaluated.

Definition of the velocity–space-dependent turbulent energy flux

Instead of calculating trajectories of large numbers of particles, we have solved the time evolution of plasma distribution functions. Then, the correlation between a turbulent radial E × B flow and a perturbed distribution function is obtained by taking the average in homogeneous directions x and y in a simulation box L_x × L_y and over a range of time t₀ < t < t₀ + T. When the particle kinetic energy is multiplied, the velocity–space-dependent turbulent energy flux in Fig. 3c is expressed as

$${Q}_{{{{{{\rm{e}}}}}}}^{{{{{{\rm{v}}}}}}}\left(z,{v}_{\parallel },\mu \right)=\left(\frac{{m}_{{{{{{\rm{e}}}}}}}{v}_{\parallel }^{2}}{2}+\mu {B}_{0}\right)\int _{{t}_{0}}^{{t}_{0}+T}\frac{{{{{{{\rm{d}}}}}}t}}{T}\int_{0}^{{L}_{x}}\frac{{{{{{{\rm{d}}}}}}x}}{{L}_{x}}\int_{0}^{{L}_{y}}\frac{{{{{{{\rm{d}}}}}}y}}{{L}_{y}}{{{{{{\bf{v}}}}}}}_{{{{{{\rm{E}}}}}}}\cdot {{{{{\boldsymbol{\nabla }}}}}}x{f}_{{{{{{\rm{e}}}}}}},$$

(2)

where we retain the poloidal angle θ = z dependence to analyse the trapped-passing boundary, which is regarded as a microscopic heat flux per unit of velocity–space volume⁴⁶. Taking velocity–space and poloidal integrals, one obtains the time-averaged turbulent energy flux Q_e = 〈∫ dv³ Q_e^v〉_θ, where angle brackets 〈⋯〉_θ denote the flux-surface average.

Definition of the turbulent energy flux spectrum

Because the turbulent transport is a convolution of a turbulent radial E × B flow and a perturbed distribution function, the time-averaged perpendicular wavenumber spectrum of the turbulent energy flux is given by

$${Q}_{{{\rm{e}}}{{{{{\bf{k}}}}}}}=\int_{{t}_{0}}^{{t}_{0}+T}\frac{{{{{{{\rm{d}}}}}}t}}{T}{{{{{\rm{Re}}}}}}\left[{\left\langle -\frac{{{{{{\bf{b}}}}}}.{{{{{\boldsymbol{\cdot }}}}}}{{{{{\boldsymbol{\nabla }}}}}}x\times {{{{{\boldsymbol{\nabla }}}}}}y}{B}{{{{{\rm{i}}}}}}{k}_{y}{\phi }_{{{{{{\bf{k}}}}}}}{p}_{{{{{{\rm{e}}}}}}{{{{{\bf{k}}}}}}}^{* }\right\rangle }_{\theta }\right].$$

(3)

The gyrophase-averaged perturbed electron pressure is denoted by p_ek = ∫ dv³ (m_ev_||²/2+μB₀) J₀(k_⊥ρ_e) f_ek with the zeroth-order Bessel function J₀ and the perpendicular wavenumber k_⊥. The poloidal wavenumber spectra are calculated by taking the summation of Q_ek over k_xρ_ti, Q_{eky =} ∑_kx Q_ek. In Fig. 4, the plot is normalised to compare the simulations with different minimum wavenumber Δk_y = k_y,min on an equal footing, Q_e = ∑_ky Q_eky = ∫dk_y (Q_eky/Δk_y).

Definition of the non-linear triad transfer function

The fluctuation intensity of distribution functions is measured by the perturbed entropy variable⁴⁷. The entropy balance equation describes the entropy production due to transport fluxes under thermodynamic gradient forces, which balances the collisional dissipation in a steady state⁴⁸. In the perpendicular wavenumber space, the quadratic non-linearity of the E × B advection term characterises the triad interactions^35,49. The gyrokinetic triad transfer function for the electron entropy balance is given by

$${J}_{{{{{{\bf{k}}}}}}}^{{{{{{\bf{p}}}}}},{{{{{\bf{q}}}}}}}\left(t\right)={\delta }_{{{{{{\bf{k}}}}}}+{{{{{\bf{p}}}}}}+{{{{{\bf{q}}}}}}={{{{{\bf{0}}}}}}}\frac{{{{{{\bf{b}}}}}}{{\cdot }}{{{{{\bf{p}}}}}}\times {{{{{\bf{q}}}}}}}{2{B}_{0}}{{{{{\rm{Re}}}}}}\left[{\left\langle {{\int }}{{{{{\rm{d}}}}}}{v}^{3}\left({\chi }_{{{{{{\bf{p}}}}}}}{g}_{{{{{{\rm{e}}}}}}{{{{{\bf{q}}}}}}}-{\chi }_{{{{{{\bf{q}}}}}}}{g}_{{{{{{\rm{e}}}}}}{{{{{\bf{p}}}}}}}\right)\frac{{T}_{{{{{{\rm{e}}}}}}}{g}_{{{{{{\rm{e}}}}}}{{{{{\bf{k}}}}}}}}{{F}_{{{{{{\rm{eM}}}}}}}}\right\rangle }_{\theta }\right],$$

(4)

where χ_k = 〈ϕ_k − v_|| A_||k〉 and g_ek = f_ek + e_e〈ϕ_k〉F_eM/T_e denote the gyrophase-averaged generalised potential and the non-adiabatic part of the perturbed electron distribution function for a mode k. F_eM is the equilibrium Maxwell distribution. The triad transfer function J_k^p,q describes the interaction of entropy variables among the three wavenumber modes {k, p, q} satisfying the non-linear coupling condition k + p + q = 0. Namely, positive or negative J_k^p,q means entropy gain or loss of mode k via the coupling with p and q. The detailed balance relation J_k^p,q + J_p^q,k + J_q^k,p = 0 ensures entropy conservation.

To extract the large ion-scale and small electron-scale contributions separately, the subspace transfer function is defined as^36,50

$${J}_{{{{{{\bf{k}}}}}}}^{{\varOmega }_{{{{{{\rm{s}}}}}}},{\varOmega }_{{{{{{{\rm{s}}}}}}}^{{\prime} }}}\left(t\right)=\mathop{\sum} \limits_{{{{{{\bf{p}}}}}}\in {\varOmega }_{{{{{{\rm{s}}}}}}}}\mathop{\sum }\limits_{{{{{{\bf{q}}}}}}\in {\varOmega }_{{{{{{{\rm{s}}}}}}}^{{\prime} }}}{J}_{{{{{{\bf{k}}}}}}}^{{{{{{\bf{p}}}}}},{{{{{\bf{q}}}}}}},$$

(5)

where Ω_i and Ω_e are subspaces in wavenumber space corresponding to ion and electron scales. The subspace transfer function represents the energy gain or loss of the analysed mode via the coupling with subspaces. In Fig. 5, we split the wavenumber space into the ion-scale Ω_i = {k |−4 < k_xρ_ti < 4, −1 < k_yρ_ti < 1} and electron-scale Ω_i = {k | the others}. Although there is the arbitrariness of the boundary choice between ion and electron scales, doubling the |k_yρ_ti | boundary from 1 to 2 gives no qualitative difference to the analysis because the peaks of the ion and electron scales are well separated as shown in Fig. 1. The subspace transfer also satisfies the symmetric property J_k^Ωp,Ωq = J_k^Ωq,Ωp, and the detailed balance among three subspaces J_k∈Ωk^Ωp,Ωq + J_k∈Ωp^Ωq,Ωk + J_k∈Ωq^Ωk,Ωp = 0.