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 crossscale interactions between small (electron)scale and large (ion)scale turbulence. Since conventional turbulent transport modelling lacks crossscale interactions, it should be clarified whether crossscale interactions are needed to be considered in future experiments on burning plasma, whose high electron temperature is sustained with fusionborn alpha particle heating. Here, we present supercomputer simulations showing that electronscale turbulence in high electron temperature plasma can affect the turbulent transport of not only electrons but also fuels and ash. Electronscale turbulence disturbs the trajectories of resonant electrons responsible for ionscale microinstability and suppresses largescale turbulent fluctuations. Simultaneously, ionscale turbulent eddies also suppress electronscale turbulence. These results indicate a mutually exclusive nature of turbulence with disparate scales. We demonstrate the possibility of reduced heat flux via crossscale interactions.
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Introduction
Plasma is an ionised gas coupled with electromagnetic fields. It is ubiquitously found in nature, laboratories, and industries, for example, in blackhole 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 fusionborn alpha particles in future burning plasma experiments. The magnetic fusion plasma is a nonequilibrium open system, selforganised with transport processes via microinstabilities and associated turbulence driven by the steep density and temperature gradients^{1}. Turbulence in magnetic fusion plasma is regarded as a multiscale problem involving wide temporal and spatial scales, from the radius of electron gyration (~0.1 mm) to that of ion gyration (~1 cm). The electronscale and ionscale turbulence were often analysed separately under the scaleseparation assumption. Since largescale eddies in ionscale turbulence were often dominant, the conventional model was designed to reproduce ionscale turbulent transport. However, recent gyrokinetic simulation studies revealed the existence of crossscale interactions between electronscale and ionscale turbulence^{2,3,4,5,6,7,8}. Comparisons with experiments in Alcator CMod and DIIID tokamaks in the United States suggest that the multiscale interactions are necessary to explain the heat fluxes measured in the experiments^{3} and play an important role in the nearfuture ITER device^{4}. Latest campaigns in TCV, ASDEX Upgrade and JET tokamak devices in Switzerland, Germany and the United Kingdom also report that the electronscale turbulence is responsible for a stiff dependence of electron heat flux against electron temperature gradient (ETG) in ionheated plasma^{8}. Electronscale 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 multiscale problems in fusion plasma, namely, interactions between ionscale turbulence and device scale (~1 m) or mesoscale fluctuations^{14}, which is beyond the scope of this work.
Beyond the existing devices, electron heating is expected to dominate in ITER. In particular for the prefusion power operation 1 (PFPO1) phase, the central ratio of electron to ion temperature T_{e}/T_{i} can be even larger than 3.0 by electron cyclotron heating^{15}. Additionally, fusionborn 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 ionelectron energy exchange, a reactorrelevant temperature ratio is considered around 1 < T_{e}/T_{i} < 2. As the temperature ratio T_{e}/T_{i} increases, ionscale instabilities tend to be destabilised^{16,17}, in contrast to electronscale instabilities that tend to be stabilised^{18}. Therefore, the extrapolation of multiscale interactions toward future burning fusion plasma experiments is nontrivial. To improve the performance prediction of future fusion devices, it should be clarified whether the crossscale interactions are needed to be considered for ITERrelevant electron heated plasma and future burning plasma at T_{e}/T_{i} > 1.
Here, we address this problem by means of numerical simulations of multiscale 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 nonlinear gyrokinetic theory^{19,20,21}, which is widely used for the analyses of magnetic fusion plasma and is also applied to accretiondisc and solarwind turbulence^{22} and auroral arc^{23}. 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 deltaf electromagnetic gyrokinetic equations in a fivedimensional phase space^{26}. See “Methods” for the numerical model and employed plasma parameters.
Results
Microinstabilities at ion and electron scales
In this simulation, two types of microinstabilities 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 mode^{28}. 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}/R_{0} and linear growth rate γ = 0.145 v_{ti}/R_{0}, where ρ_{ti}, v_{ti} and R_{0} 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 electronscale k_{y} = 4.5 ρ_{ti}^{−1}, ω_{r} = 24.3 v_{ti}/R_{0} and γ = 1.16 v_{ti}/R_{0}. 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_{ti}ρ_{ti}/R_{0} and (ω_{r}/k_{tor})_{ETG} = 45.1 v_{ti}ρ_{ti}/R_{0}, with the toroidal wavenumber k_{tor} = ε_{r}k_{y}/q, where the ε_{r} = r/R_{0} 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_{e}v^{2}q/(2erB_{0}), where m_{e}, e, r, and B_{0} 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 nonlinear analytic theory^{29,30}. Recent gyrokinetic simulation studies have also attracted researchers’ attention because of the impact on the tokamak edge transport^{31} and the longstanding mystery of the hydrogen isotope effect^{32}. Nevertheless, the crossscale interactions between TEM and ETG have not been fully investigated yet.
Tracer particle trajectory analysis
The multiscale turbulence simulations show that largescale TEM and smallscale ETG fluctuations can coexist in nonlinearly 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_{ti}ρ_{ti}/R_{0}) over a toroidal path length yq/(ε_{r}ρ_{ti}) < 400, which causes radial motion by the E × B drift (Fig. 2b). Further, an offresonant particle (v = 2.5 v_{te}, v_{pre} = 73.9 v_{ti}ρ_{ti}/R_{0}) 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, smallscale ETG electric fields are also averaged out for the offresonant 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 smallscale turbulence (Fig. 2b). A secular displacement is observed (the red line in Fig. 2b after yq/(ε_{r}ρ_{ti}) > 700) because the tracer particle trajectory is modified by smallscale turbulence and moved in an oppositely rotating largescale eddy. Since the statistical correlation between turbulent flows and perturbations of plasma distribution functions determines the resultant turbulent transport, the impacts of small electronscale 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, socalled 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, hightemperature fluctuations go outward and lowtemperature 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, smallscale ETG turbulent eddies are observed to coexist with largescale TEM fluctuations. This means that the smallscale 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 TEMdriven turbulent transport, multiscale simulation results are compared with an ionscale simulation resolving only TEM scales and an electronscale 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 multiscale spectrum shows two peaks attributed to lowk_{y} TEM and highk_{y} ETG turbulence. Lowk_{y} TEM components in the multiscale simulation are reduced compared with those in the single ionscale simulation, suggesting the suppression of TEM by ETG turbulence. The crossscale interactions modify ionscale 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 multiscale 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 singlescale 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_{e}T_{e}c_{a}ρ_{a}^{2}/R_{0}^{2} is the gyroBohm unit with the speed of acoustic wave c_{a} = √(T_{e}/m_{i}) and the acoustic gyroradius ρ_{a}. The agreement of highk_{y} ETG peaks of turbulent flux spectra in multiscale and single electronscale simulations seems coincident. Although ETG turbulence initially saturates at a higher transport level, ETGdriven zonal flows suppress turbulent transport after a long time tv_{ti}/R_{0} ≥ 50 in the single electronscale simulation (not shown), which is consistent with a recent study on singlescale ETG turbulence^{10}. Therefore, the suppression of ETG turbulence by ETGdriven zonal flows dominates the single electronscale simulation result, whereas no significant zonal flow is observed in the multiscale simulation, indicating the existence of suppression mechanisms of ETG turbulence by crossscale interactions.
Gyrokinetic nonlinear entropy transfer analysis
The crossscale 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 electronscale streamers by the ionscale turbulent eddies, as observed in previous multiscale simulations^{2,33}. In contrast to a previous work discussing the suppression of ETG modes by TEMdriven 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 multiscale simulations of iontemperaturegradient (ITG) modes^{2} but resembles the suppression of microtearing modes (MTM) by ETG turbulence^{5}. The nonlinear crossscale interactions between electronscale ETG turbulence and ionscale TEM turbulence are investigated using the gyrokinetic nonlinear entropy transfer analysis^{35}. The perturbed entropy is a measure of amplitude fluctuations of the plasma distribution function, and its nonlinear 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 electronscale, electronscale coupling J_{k}^{Ωe,Ωe} and the ionscale, electronscale coupling 2J_{k}^{Ωi,Ωe} using the subspace transfer analysis technique^{36} and defining the ion and electron scales Ω_{i}, and Ω_{e} in the perpendicular wavenumber space. Spectra of the timeaveraged transfer function in Fig. 5a shows that the electronscale, electronscale coupling has a negative contribution on lowk_{y} TEM fluctuations (Fig. 5a, J_{k}^{Ωe,Ωe} < 0 at k_{y}ρ_{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_{y}ρ_{ti} < 0.5) indicates entropy transfer from lowk_{y} TEMs to highk_{y} modes (2J_{k∈Ωe}^{Ωi,Ωe} = −J_{k∈Ωi}^{Ωe,Ωe} > 0). As plotted in Fig. 5b, net entropy gain in highk_{y} range via ionscale coupling is observed at finite k_{x} (J_{k}^{Ωi,Ωe} > 0 at k_{y}ρ_{ti} ~ 4 and k_{x}ρ_{ti} > 1) but not at ETG peaks (k_{y}ρ_{ti} ~ 4 and k_{x}ρ_{ti} ~ 0, characterised by a peak of energy specrum at high wavenumber, which is close to linearly unstable ETG modes). These highk_{⊥} modes create higher k_{⊥} fluctuations (J_{k}^{Ωe,Ωe} > 0 at higher k_{y}ρ_{ti} > 7, not shown), which are eventually damped by collisional dissipation. Additionally, the ionscale, electronscale coupling has a negative contribution on ETG (J_{k}^{Ωi,Ωe} < 0 at k_{y}ρ_{ti} ~ 4 and k_{x}ρ_{ti} ~ 0), confirming the suppression of ETG modes by ionscale turbulence.
Extrapolation of multiscale 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 multiscale simulation agrees with that in the single electronscale 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 multiscale 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 crossscale interactions with ETG turbulence. From the ITER profile modelling and a reference value of a high T_{e} discharge of the DIIID tokamak^{37}, a reactorrelevant 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 nonnegligible contribution when T_{e} ~ T_{i} and sufficiently large electron temperature^{8}. Additionally, the multiscale turbulence simulation shows the existence of an appropriate T_{e}/T_{i} range where the crossscale interactions suppress turbulent transport. The suppression of nearmarginal TEM by ETG turbulence leads the upshift of the critical temperature ratio for an increase of TEMdominated energy flux (from T_{e}/T_{i} ~ 2 in the ionscale simulation to T_{e}/T_{i} ~ 3 in the multiscale simulation), analogous to the Dimits upshift of critical ion temperature gradient where zonal flows suppress near marginal ITG modes^{38}.
Discussion
A series of our works on multiscale turbulence in magnetised plasma indicates some common features: largescale turbulence tends to suppress smallscale turbulence (ITG/ETG^{2,3,4}, MTM/ETG^{13}, and TEM/ETG in this study), whereas smallscale turbulence tends to destroy largescale structures (damping of shortwavelength zonal flows by ETG^{36}, destruction of radially localised current sheet of MTM by ETG^{5} and disturbance of drift resonance between trapped electrons and TEM by ETG in this study). These findings suggest the mutual conjunction between disparatescale turbulence as a generic nature of crossscale interactions beyond a conventional turbulence theory described using singlescale energy injection/cascade/dissipation processes.
Our results answer the question of whether the crossscale interactions are needed to be taken into account for future burning plasma experiments. Even beyond the existing ionheated devices T_{e} ~ T_{i}, electronscale 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 crossscale 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 crossscale 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 fusionborn alpha particles keeps electron temperature high T_{e} > T_{i}. It also has impacts on understanding electronheated plasma at the PFPO1 phase in the ITER research plan, contributing the early success of fusion energy development.
This work has extensively investigated the crossscale 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 lowlevel transport of alpha particles^{39}. The resonant interactions between energetic particles and microinstability will be significant for ITG modes^{40,41} but not for TEM and ETG modes because the propagation directions are opposite^{42}. Therefore, interactions between energetic particles and TEM/ETG turbulence are expected to be weak. When we consider interplay from electron to ionscale turbulence and macroscopic fluctuations, magnetohydrodynamic instability driven by energetic particles will become a candidate for the third player^{43}, which is a theoretically and numerically challenging subject.
Methods
Simulation details
Microinstabilities 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 messagepassing interface and open multiprocessing. Optimisation techniques for highperformance computing are implemented, such as the pipelined computationcommunication overlap, segmented process mapping on the threedimensional torus interconnect^{25}, and communicationavoiding iterative solver for the implicit collision operator^{44}. This implementation ensures that the GKV code achieves a good scalability up to 12,288 nodes on the Fugaku supercomputer, 3.1 PetaFLOPs (floatingpoint 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 multiscale 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 deltaf gyrokinetic Vlasov–Poisson–Ampère equations^{26}. The configuration space is represented by fieldaligned coordinates x = r − r_{0}, y = [q(r)θ − ζ]r_{0}/q(r_{0}), 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 fluxtube model^{45} resolves perturbed quantities in a long and thin simulation box along a field line, whereas the equilibrium quantities are approximated at the fluxtube centre r = r_{0}, consistent with deltaf 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 socalled sα 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}, R_{0}/L_{ne} = R_{0}/L_{nD} = R_{0}/L_{nT} = R_{0}/L_{nHe} = 3, R_{0}/L_{Te} = 9.342 and R_{0}/L_{TD} = R_{0}/L_{TT} = R_{0}/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 R_{0} are used as references. The charge density satisfies the quasineutrality condition ∑_{s} e_{s}n_{s} = 0 and ∑_{s} e_{s}n_{s}/L_{ns} = 0. The plasma beta value is β = μ_{0}n_{e}T_{D}/B_{0}^{2} = 0.05%, normalised Debye length is λ^{2} = ε_{0}B_{0}^{2}/(m_{i}n_{e}) = 10^{−3} and normalised collision frequency is ν^{*} = qR_{0}τ_{ee}^{−1}/(√2ε_{r}^{3/2}v_{te}) = 0.05 with electron–electron collision time τ_{ee}. The above plasma parameters are chosen from a previous study^{32} 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 DIIID #173147^{37}), whereas the electron temperature and its gradient are slightly enhanced to mimic the electronheated 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 multiscale ITG/ETG turbulence at T_{e} = T_{i} and R_{0}/L_{Te} = R_{0}/L_{Ti}, detailed mechanisms of crossscale 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 R_{0}/L_{Te} > R_{0}/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 nearmarginal TEMs. These parameters are also related to the electronheated plasma at the PFPO1 phase in the ITER research plan which is an important step for early success of ITER. The prediction of the integrated modelling of PFPO1 phase Hmode plasma reported high central temperature ratio T_{e}/T_{i} > 3 and R_{0}/L_{Te} > R_{0}/L_{Ti}^{15}. For the multiscale 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 ≤ μB_{0}/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 dealiasing rule, f(x, y, z, v_{}, μ) = ∑_{kx}∑_{ky} f_{k}(z, v_{}, μ) exp(ik_{x}x + ik_{y}y), 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 ionscale turbulence simulations, we employed reduced perpendicular wavenumber space k_{x,max} = 4ρ_{ti}^{−1}, k_{y,max} = 1ρ_{ti}^{−1}, which well resolves large ionscale microinstabilities and associated turbulence but excludes small electronscale dynamics. Further, for single electronscale turbulence simulations, we employed smaller perpendicular box sizes k_{x,min} = k_{y,min} = 0.5ρ_{ti}^{−1}, which covers only electronscale microinstabilities but excludes ionscale ones.
Tracer particle analysis
In Fig. 2, tracer particle trajectories are calculated to explain behaviour of resonant and offresonant particles and to examine the effects of small electronscale 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}/R_{0},
where B = B_{0}b, v_{ed}, and v_{E} are the equilibrium magnetic field, electron magnetic drift, and E × B drift velocities, respectively. Spatiotemporal data of gyrophaseaveraged electrostatic potential from multiscale 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}/B_{0} and v_{} = 0.39v_{te}, μ = 3.72T_{e}/B_{0} for an offresonant (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 timeindependent 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–spacedependent 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 t_{0} < t < t_{0} + T. When the particle kinetic energy is multiplied, the velocity–spacedependent turbulent energy flux in Fig. 3c is expressed as
where we retain the poloidal angle θ = z dependence to analyse the trappedpassing boundary, which is regarded as a microscopic heat flux per unit of velocity–space volume^{46}. Taking velocity–space and poloidal integrals, one obtains the timeaveraged turbulent energy flux Q_{e} = 〈∫ dv^{3} Q_{e}^{v}〉_{θ}, where angle brackets 〈⋯〉_{θ} denote the fluxsurface 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 timeaveraged perpendicular wavenumber spectrum of the turbulent energy flux is given by
The gyrophaseaveraged perturbed electron pressure is denoted by p_{ek} = ∫ dv^{3} (m_{e}v_{}^{2}/2+μB_{0}) J_{0}(k_{⊥}ρ_{e}) f_{ek} with the zerothorder Bessel function J_{0} and the perpendicular wavenumber k_{⊥}. The poloidal wavenumber spectra are calculated by taking the summation of Q_{ek} over k_{x}ρ_{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 nonlinear triad transfer function
The fluctuation intensity of distribution functions is measured by the perturbed entropy variable^{47}. The entropy balance equation describes the entropy production due to transport fluxes under thermodynamic gradient forces, which balances the collisional dissipation in a steady state^{48}. In the perpendicular wavenumber space, the quadratic nonlinearity 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
where χ_{k} = 〈ϕ_{k} − v_{} A_{k}〉 and g_{ek} = f_{ek} + e_{e}〈ϕ_{k}〉F_{eM}/T_{e} denote the gyrophaseaveraged generalised potential and the nonadiabatic 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 nonlinear 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 ionscale and small electronscale contributions separately, the subspace transfer function is defined as^{36,50}
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 ionscale Ω_{i} = {k −4 < k_{x}ρ_{ti} < 4, −1 < k_{y}ρ_{ti} < 1} and electronscale Ω_{i} = {k  the others}. Although there is the arbitrariness of the boundary choice between ion and electron scales, doubling the k_{y}ρ_{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.
Data availability
The data depicted in the plots of this paper will be made available at the following https://github.com/smaeyama/maeyama_ncomm_2022 upon publication^{51}.
Code availability
The GKV code is an opensource project available from GitHub: https://github.com/GKVdevelopers/gkvp. The version of the GKV code used in this paper is gkvp_f0.59. The source code with relevant physical/numerical parameter settings are available^{51}.
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Acknowledgements
One of the authors (S.M.) acknowledges helpful discussion with Dr. M. Honda. All the authors were supported by MEXT as ‘Programme for Promoting Researches on the Supercomputer Fugaku’ (Exploration of burning plasma confinement physics, JPMXP1020200103). S.M. was supported by JSPS KAKENHI Grant Number JP20K03892. S.M. and Y.A. were supported by ‘Joint Usage/Research Centre for Interdisciplinary Largescale Information Infrastructures’ and ‘HighPerformance Computing Infrastructure’. S.M. was supported by QST Research Collaboration for Fusion DEMO. S.M. and T.H.W. acknowledge the NIFS Collaboration Research Programme (NIFS20KNST162, NIFS21KNST181) in Japan. Numerical analyses were performed on the Fugaku supercomputer at RIKEN RCCS (Project ID: hp200127, hp210178), the JFRS1 at Computational Simulation Centre of International Fusion Energy Research Centre (IFERCCSC), and the plasma simulator at National Institute for Fusion Science.
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S.M. performed numerical simulations and analysed the data. T.H.W. led the project of exploration of burning plasma confinement physics. M. Nakata, M. Nunami, and A.I. contributed to the physical interpretation. Y.A. contributed to the optimisation of the simulation code on the Fugaku supercomputer. All authors contributed to the development of the simulation code and the preparation of the manuscript.
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Maeyama, S., Watanabe, TH., Nakata, M. et al. Multiscale turbulence simulation suggesting improvement of electron heated plasma confinement. Nat Commun 13, 3166 (2022). https://doi.org/10.1038/s41467022308520
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DOI: https://doi.org/10.1038/s41467022308520
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