## 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 turbulence2,3,4,5,6,7,8. 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 tokamaks9,10,11 and tokamak edge plasma12,13. 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 Te/Ti 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 Te higher than that of ions Ti. Due to the relaxation by ion-electron energy exchange, a reactor-relevant temperature ratio is considered around 1 < Te/Ti < 2. As the temperature ratio Te/Ti increases, ion-scale instabilities tend to be destabilised16,17, 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 Te/Ti > 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 theory19,20,21, 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 GKV24,25 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 ky = 0.25 ρti−1 with the real frequency ωr = 1.50 vti/R0 and linear growth rate γ = 0.145 vti/R0, where ρti, vti 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 ky = 4.5 ρti−1, ωr = 24.3 vti/R0 and γ = 1.16 vti/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, (ωr/ktor)TEM = 42.7 vtiρti/R0 and (ωr/ktor)ETG = 45.1 vtiρti/R0, with the toroidal wavenumber ktor = εrky/q, where the εr = 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 vpre ~ mev2q/(2erB0), where me, e, r, and B0 are the electron mass, electric charge, tokamak minor radius, and equilibrium magnetic field strength, the resonant condition of TEM (vpre = ωr/ktor) is satisfied when the particle velocity v = 1.9 vte. TEM has been considered relevant for experimental density scaling or electron temperature profile based on non-linear analytic theory29,30. 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.

### 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 vte, vpre = 47.3 vtiρti/R0) over a toroidal path length yq/(εrρti) < 400, which causes radial motion by the E × B drift (Fig. 2b). Further, an off-resonant particle (v = 2.5 vte, vpre = 73.9 vtiρti/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/(εrρti) > 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.

### 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.

### 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 (ky) spectra of electron energy flux are shown in Fig. 4. The multi-scale spectrum shows two peaks attributed to low-ky TEM and high-ky ETG turbulence. Low-ky 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 [Qe(TEM/ETG) = 5.66 QgB, QD + QT(TEM/ETG) = 0.24 QgB, QHe(TEM/ETG) = 0.02 QgB] are reduced compared with those in single-scale TEM turbulence [Qe(TEM) = 33.63 QgB, QD + QT(TEM) = 1.11 QgB, QHe(TEM) = 0.09 QgB] because of the suppression of TEM turbulence by ETG turbulence, where QgB = neTecaρa2/R02 is the gyro-Bohm unit with the speed of acoustic wave ca = √(Te/mi) and the acoustic gyroradius ρa. The agreement of high-ky 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 tvti/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.

### 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 simulations2,33. 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 JkΩe,Ωe and the ion-scale, electron-scale coupling 2JkΩi,Ωe using the subspace transfer analysis technique36 and defining the ion and electron scales Ωi, and Ωe 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-ky TEM fluctuations (Fig. 5a, JkΩe,Ωe < 0 at kyρti < 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, JkΩiΩe,Ωe + 2 JkΩeΩi,Ωe = 0, the suppression of TEMs by ETG turbulence (JkΩe,Ωe < 0 at kyρti < 0.5) indicates entropy transfer from low-ky TEMs to high-ky modes (2JkΩeΩi,Ωe = −JkΩiΩe,Ωe > 0). As plotted in Fig. 5b, net entropy gain in high-ky range via ion-scale coupling is observed at finite kx (JkΩi,Ωe > 0 at kyρti ~ 4 and kxρti > 1) but not at ETG peaks (kyρti ~ 4 and kxρti ~ 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 (JkΩe,Ωe > 0 at higher kyρti > 7, not shown), which are eventually damped by collisional dissipation. Additionally, the ion-scale, electron-scale coupling has a negative contribution on ETG (JkΩi,Ωe < 0 at kyρti ~ 4 and kxρti ~ 0), confirming the suppression of ETG modes by ion-scale turbulence.

### Extrapolation of multi-scale turbulent interactions toward high Te/Ti regime

Finally, the dependence of turbulent energy flux Qe on the temperature ratio Te/Ti 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 Te/Ti are respectively shown in Supplementary Figs. 1 and 2. Since ETGs are highly unstable at Te/Ti = 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 Te/Ti = 4, TEM dominates turbulent transport even in the multi-scale simulation. The Te/Ti = 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 Te discharge of the DIII-D tokamak37, a reactor-relevant temperature ratio is considered around 1 < Te/Ti < 2 due to the ion–electron energy exchange. Our survey of dependence of Qe on Te/Ti covers this range. An important consequence is that the ETG contribution survives in a wider parameter range 1 < Te/Ti < 3, although it may have been considered that ETG could have non-negligible contribution when Te ~ Ti and sufficiently large electron temperature8. Additionally, the multi-scale turbulence simulation shows the existence of an appropriate Te/Ti 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 Te/Ti ~ 2 in the ion-scale simulation to Te/Ti ~ 3 in the multi-scale simulation), analogous to the Dimits upshift of critical ion temperature gradient where zonal flows suppress near marginal ITG modes38.

## 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/ETG2,3,4, 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 Te ~ Ti, electron-scale turbulence can have impacts in electron heated plasma with high electron temperature Te > Ti. 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 Te/Ti, our results reveal the existence of an appropriate Te/Ti 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 Te > Ti. 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 modes40,41 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 GKV24,25. 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 fs 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 εr, safety factor q, and magnetic shear ŝ. We selected εr = 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 = eD = eT = −ee = eHe/2, mi = 1837 me = mD/2 = mT/3 = mHe/4, Ti = TD = TT = THe, nD = nT = 0.45 ne, nHe = 0.05 ne, R0/Lne = R0/LnD = R0/LnT = R0/LnHe = 3, R0/LTe = 9.342 and R0/LTD = R0/LTT = R0/LTHe = 1, where Lns = −den ns/dx and LTs = −den Ts/dx are the density and temperature gradient scale lengths. The elementary electric charge e, hydrogen mass mi, and tokamak major radius R0 are used as references. The charge density satisfies the quasi-neutrality condition ∑s esns = 0 and ∑s esns/Lns = 0. The plasma beta value is β = μ0neTD/B02 = 0.05%, normalised Debye length is λ2 = ε0B02/(mine) = 10−3 and normalised collision frequency is ν* = qR0τee−1/(√2εr3/2vte) = 0.05 with electron–electron collision time τee. 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 Te 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 Te = Ti and R0/LTe = R0/LTi, detailed mechanisms of cross-scale interactions between ITG and ETG modes have been reported previously2,36. In this study, we focus on the parameter regime at Te > Ti and R0/LTe > R0/LTi which has not yet been analysed in existing devices. We examined the electron temperature dependence in the range 1 ≤ Te/Ti ≤ 4, and detailed analyses are presented for the case of Te/Ti = 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 Te/Ti > 3 and R0/LTe > R0/LTi15. For the multi-scale turbulence simulations, we employed simulation box sizes 0 ≤ x/ρti < 125, 0 ≤ y/ρti < 40π,–π ≤ z < π,−4.5 ≤ v||/vts ≤ 4.5, 0 ≤ μB0/Ts ≤ 12.5, and grid points in each dimension (Nx, Ny, Nz, Nv||, Nμ) = (2048, 2048, 40, 64, 16) for the Te/Ti = 1 case and (Nx, Ny, Nz, Nv||, Nμ) = (1024, 1024, 40, 64, 16) for the 2 ≤ Te/Ti 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) = ∑kxky fk(z, v||, μ) exp(ikxx + ikyy), the corresponding perpendicular wavenumber resolutions are kx,min = ky,min = 0.05ρti−1, kx,max = ky,max = 33.9ρti−1 for the Te/Ti = 1 case and kx,max = ky,max = 16.95ρti−1 for the 2 ≤ Te/Ti cases. For single ion-scale turbulence simulations, we employed reduced perpendicular wavenumber space kx,max = 4ρti−1, ky,max = 1ρti−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 kx,min = ky,min = 0.5ρti−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 ρte/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, ved, and vE 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 = 2vte) particle are v|| = 0.31vte, μ = 2.38Te/B0 and v|| = 0.39vte, μ = 3.72Te/B0 for an off-resonant (v = 2.5vte) 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 Lx × Ly 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 Qe = 〈∫ dv3 Qevθ, 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 pek = ∫ dv3 (mev||2/2+μB0) J0(kρe) fek with the zeroth-order Bessel function J0 and the perpendicular wavenumber k. The poloidal wavenumber spectra are calculated by taking the summation of Qek over kxρti, Qeky = kx Qek. In Fig. 4, the plot is normalised to compare the simulations with different minimum wavenumber Δky = ky,min on an equal footing, Qe = ∑ky Qeky = ∫dky (Qekyky).

### 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 interactions35,49. 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 χk = 〈ϕk − v|| A||k〉 and gek = fek + eeϕkFeM/Te denote the gyrophase-averaged generalised potential and the non-adiabatic part of the perturbed electron distribution function for a mode k. FeM is the equilibrium Maxwell distribution. The triad transfer function Jkp,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 Jkp,q means entropy gain or loss of mode k via the coupling with p and q. The detailed balance relation Jkp,q + Jpq,k + Jqk,p = 0 ensures entropy conservation.

To extract the large ion-scale and small electron-scale contributions separately, the subspace transfer function is defined as36,50

$${J}_{{{{{{\bf{k}}}}}}}^{{\varOmega }_{{{{{{\rm{s}}}}}}},{\varOmega }_{{{{{{{\rm{s}}}}}}}^{{\prime} }}}\left(t\right)=\mathop{\sum} \limits_{{{{{{\bf{p}}}}}}\in {\varOmega }_{{{{{{\rm{s}}}}}}}}\mathop{\sum }\limits_{{{{{{\bf{q}}}}}}\in {\varOmega }_{{{{{{{\rm{s}}}}}}}^{{\prime} }}}{J}_{{{{{{\bf{k}}}}}}}^{{{{{{\bf{p}}}}}},{{{{{\bf{q}}}}}}},$$
(5)

where Ωi and Ωe 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 Ωi = {k |−4 < kxρti < 4, −1 < kyρti < 1} and electron-scale Ωi = {k | the others}. Although there is the arbitrariness of the boundary choice between ion and electron scales, doubling the |kyρti | 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 JkΩpq = JkΩqp, and the detailed balance among three subspaces JkΩkΩpq + JkΩpΩqk + JkΩqΩkp = 0.