Turbulent particle pinch in gyrokinetic flux-driven ITG/TEM turbulence

Aiming at a fuel supply through particle pinch effects, turbulent particle transport is studied by gyrokinetic flux-driven Ion-Temperature-Gradient/Trapped-Electron-Mode (ITG/TEM) simulations. It is found that ITG/TEM turbulence can drive ion particle pinch by E × B drift (n ≠ 0) when the ion temperature gradient is steep enough. Electron particle pinch is also driven by E × B drift (n ≠ 0) in the case with the steep electron temperature gradient. Such an electron particle pinch can trigger an ambipolar electric field, leading to additional ion particle pinch by not only magnetic drift but also E × B drift (n = 0). These results suggest that a density peaking of bulk ions due to turbulent fluctuations can be achieved by sufficiently strong both ion and electron heating.


Introduction
Establishment of a refueling method is an important issue for controlling nuclear fusion reactors.But, in DEMO-class high-temperature plasmas, a pellet injection reaches only up to 80%-90% of the minor radius so that the central density peaking depends on particle pinch, making the prediction difficult.It is known that turbulent particle flux consists of a diagonal diffusion term, a non-diagonal thermo-diffusion term, and a convection term.The first term is usually positive for the peaked profile, while the second term can be negative, indicating that it can drive particle pinch.According to the quasi-linear fluid theory [1], the thermo-diffusion coefficient is estimated as D T ∝ − (10L n /3R 0 − ω r /ω * e ) so that Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
the sign depends on the real frequency of dominant mode ω r , the typical scale length of density gradient L n , the major radius R 0 , and the diamagnetic electron frequency ω * e .In the Ion-Temperature-Gradient (ITG) case, D T is always negative because the real frequency is negative, while it can be reversed in the Trapped-Electron-Mode (TEM) case.In fact, the turbulent electron particle flux calculated by gyrokinetic simulations shows the opposite trend for electron temperature gradient between ITG and TEM [2,3].In addition, recent full-f gyrokinetic simulations suggest the importance of global effects in evaluating impurity particle transport [4] such as core helium ash exhaust and edge impurity accumulation.For instance, GYSELA full-f gyrokinetic simulations show that the turbulence-driven Reynolds stress causes (m, n) = (1, 0) mode, leading to an impact on neoclassical impurity transport in a deuterium plasma with helium, neon, or tungsten as tracer impurities [5].GT5D full-f gyrokinetic simulations also show that neoclassical impurity transport is enhanced by a turbulent transport driven ambipolar radial electric field in a deuterium plasma with helium, beryllium, carbon, or argon as tracer impurities [6].These simulations suggest that the interaction between turbulent and neoclassical transport is important for impurity transport, however, the role of the interaction on the particle pinch of bulk ion has been not investigated yet.
Based on this motivation, we perform flux-driven ITG/TEM simulations in the presence of ion/electron heating by means of the full-f electrostatic version of our global gyrokinetic code GKNET [7][8][9][10] with kinetic electron dynamics [11], while there is another version of the GKNET code that solves electromagnetic δf global gyrokinetic equations [12][13][14] including the fast ion effects [15].The full-f electrostatic GKNET enables us to precisely consider the self-consistent mean E r determined by the radial force balance with the pressure and poloidal/toroidal flow profiles controlled by external source and sink.The global gyrokinetic ambipolarity condition can be also precisely treated so that we can investigate the physical mechanism of particle transport caused not only by non-axisymmetric E × B drift but also by axisymmetric E × B and magnetic drifts, which is responsible for the interactions between turbulent and neoclassical transport.
The rest of this paper is organized as follows.In section 2, we present the numerical model of GKNET used in our simulations.In section 3.1, the simulation setting is explained.The effect of ion/electron heating on the ion density and temperature profiles is shown in section 3.2.Then, we separately present turbulent particle fluxes caused by E × B drift (n ̸ = 0), magnetic drift, and E × B drift (n = 0), in sections 3.3-3.5,respectively, and discuss their physical mechanisms.In this paper, we refer all of them as turbulent particle transport by different three drifts because turbulence triggers them, although the second and third components are categorized into the neoclassical particle transport in some papers.The total turbulent ion particle flux caused by these three drifts exhibits a particle pinch, which is summarized in section 3.6.Finally, the summary and some future plans are given in section 4.

Numerical model of GKNET code
The 5D full-f gyrokinetic code GKNET [7][8][9][10][11] calculates the time evolution of gyro-center distribution function f s of particle species s ≡ i, e governed by the gyrokinetic Vlasov equation ] , (3) where R ≡ (r, θ, ζ) is the position of guiding center, v ∥ is the parallel velocity along the magnetic field line, µ is the magnetic moment, J ≡ rRB * ∥ is the phase space Jacobian.
is the modified magnetic field for particle species s, B ≡ ∇ × A = (0, r/qR, R 0 /R) B 0 is the magnetic field for a circular concentric Tokamak configuration with the safety factor q, R = R 0 + r cos θ is the major radius, and b ≡ B/B, c is the speed of light, m s , e s and Ω s ≡ e s B 0 /m s c are the mass, the charge and the gyro frequency of particle species s, respectively.During the evolution of the distribution function, the phase space volume conservation is satisfied because equations ( 2) and ( 3) are analytically derived from the gyrokinetic Hamiltonian H s = m s v 2 ∥ /2 + µB + e s ⟨ϕ ⟩ s , where ⟨ϕ ⟩ s is the gyro-averaged electrostatic potential.In addition, exact particle conservation is kept including the magnetic axis by utilizing the gauge transformation technique to equations ( 2) and (3).These properties are important for keeping numerical accuracy and stability in full-f gyrokinetic simulations.
The self-collision operator C s,s is modeled by the linear Fokker-Planck one [16] incorporated with the field particle operator [17] to conserve density, momentum and energy at each real space grid.Heat source and energy sink operators for species s are given by where A src (r) and A snk (r) are the deposition profiles, τ src and τ snk are the characteristic time of the heat source and energy sink operators.
is the local Maxwellian distribution function with a density n, and a temperature T. n s0 , v ts0 = √ T s0 /m s , T s0 are the initial density, thermal velocity, and temperature at the half minor radius r = a 0 /2 for species s.Equation ( 5) provides constant heat source, while there is no particle supply.On the other hand, equation ( 6) represents a simple model of energy sink at the plasma boundary by means of a Krook-type operator, which modifies the distribution function towards its initial profile at the outer boundary region [18].
The gyrokinetic Vlasov equation ( 1) is coupled with the gyrokinetic quasi-neutrality conditions based on the hybrid kinetic electron model [11,19,20] given by where 2 , n s and T s are the ion gyro radius, the ion Debye length, the density and temperature for species s, respectively.In equation (9), the Tayler expansion is used for the ion polarization density, which has a stabilization effect on high wavenumber modes.In this study, the most unstable mode is located around k θ ρ ti0 ∼ 0.3 as will be shown in figures 3 and 5, so that we believe it is acceptable to use the Tayler expansion, which is also supported by the mixing length estimate of the turbulent particle and heat transport.Second term of equation ( 11) denotes the adiabatic passing electron response with (m, n) ̸ = (0, 0), where α p is the flux-surface averaged fraction of passing electrons given by in a circular concentric Tokamak case.δf e,t is the perturbed trapped electron distribution function, which satisfies Here, B max is the maximum magnetic field on a magnetic surface.Since both whole the trapped electron and (m, n) = (0, 0) passing electron are treated as kinetic ones in this hybrid kinetic electron model, we can address TEM driven turbulence in addition to kinetic electron effects on neoclassical dynamics.Note that the components with m ̸ = 0, n = 0 are finite so that the radial E × B drift with n = 0 is precisely considered, which is one of the novelties in this study.
Here, we briefly describe numerical methods used in the GKNET code.The spatial derivatives in equation ( 1) are discretized by using the fourth-order Morinishi scheme [21,22] and the time integration is performed using the fourthorder explicit Runge-Kutta method.The magnetic field B is calculated from the vector potential A by using the fourthorder finite difference method to numerically satisfy the phase space conservation given by equation (4).In this version of GKNET, we use 3D MPI decomposition for the (r, θ, µ) domain.Equation ( 8) is 1D Fourier-transformed along the ζ direction and then 1D Fourier-transformed along the θ direction after MPI_ALLtoALL transpose between the θ and ζ directions.Then by using MPI_ALLtoALL transpose between the r and θ directions again, we can solve equation (8) in the (r, k θ , k ζ ) space, which has a tri-diagonal matrix form by applying the fourth-order finite difference method to the r direction.The matrix is not decomposed along the r direction so that LU decomposition can be directly applied without any MPI communications.In order to directly evaluate gyroaveraging for ⟨δf s s ⟩ and ⟨ϕ ⟩ s , we make the 2D local polynomial interpolation on the poloidal plane to calculate the electrostatic potential on a gyro ring and then take 20 sampling points average in real space.

Flux-driven ITG/TEM simulation for turbulent particle transport
In this section, we first explain simulation settings as well as the linear stability analysis for the initial profile in section 3.1.Second, we present the time evolution of the density and temperature profiles as well as the particle and heat fluxes in section 3.2.Then, we discuss turbulent particle flux by dividing it into three parts: the flux by E × B drift (n ̸ = 0) ) the flux by magnetic drift and the flux by (15) We discuss them in sections 3.3-3.5,respectively.Finally, the total turbulent ion particle flux is summarized in section 3.6.

Simulation condition
In this paper, we consider a circular concentric tokamak configuration with a 0 /R 0 = 0.36 and a 0 /ρ ti0 = 100, where ρ ti0 ≡ v ti0 /Ω i are the ion radius at r = a 0 /2.The initial background profiles are given by where L ns ≡ −n s / (dn s /dr) and L Ts ≡ −T s / (dT s /dr) denote the typical scale lengths of initial density and temperature gradients at r = a 0 /2 and R 0 /L n = 2.22.In order to understand the impact of electron temperature gradient on the particle flux, here we consider two cases.The case (A) uses steep ion and electron temperature profiles given by R 0 /L Ti = R 0 /L Te = 10.In this case, external ion and electron heat sources are introduced near the magnetic axis to sustain the steep temperature profiles (see figure 4).In the case (B), only ion temperature is steep as R 0 /L Ti = 10 while electron temperature profile is set to be R 0 /L Te = 4.Note that only ion heat source is introduced in the case (B), which deposition profile is same as the case (A).The ion-ion and electron-electron collision frequencies are set as ν * ii = ν * ee = 0.025 at r = a 0 /2. Figure 1(a) shows the radial profile of initial density, initial temperature with gentle gradient (R 0 /L Ts = 4) and steep gradient (R 0 /L Ts = 10).Safety factor profile is same as the standard cyclone-base-case as is shown in figure 1(b).From some convergence tests, simulation parameters are chosen as follows; the time step width is ∆t = 5 × 10 −4 R 0 /v t0i , the grid number and the system size are ts0 /B 0 ), respectively.In order to reduce the computational cost, the simulation domain is an 1/4 wedge torus, and the mass ratio between ion and electron is set to be m i /m e = 100.
Figure 2 shows the wave-number dependencies of the linear growth rate obtained from linear δf global simulations with the initial profiles of the case (A) (R 0 /L n = 2.22, R 0 /L Ti = R 0 /L Te = 10) and the case (B) (R 0 /L n = 2.22, R 0 /L Ti = 10, R 0 /L Te = 4).For both cases, n = 12 (k θ ρ ti0 = 0.34) mode is the most unstable, which type is categorized into the ITG mode with positive ballooning angle at the outer board of the torus, as is shown by the color map of linear eigenfunctions on the cross section in figures 3(A-1) and (B-1).In the case (A), the TEM with negative ballooning angle is also excited in the high wavenumber regime, while only ITG is dominant in the case (A) (see figures 2 and 3(A-2), (B-2)).

Time evolutions of density and temperature profiles under ion/electron heating
Here, we present the nonlinear evolutions of density and temperature profiles under the ion/electron heating.Figures 4(A-1) and (B-1) show the radial ion density profiles after the nonlinear saturation in flux-driven simulations which initial temperature gradients are given by (A) R 0 /L Ti = R 0 /L Te = 10 and (B) R 0 /L Ti = 10, R 0 /L Te = 4, respectively.The deposition profiles of applied heat source and energy sink are also shown in each figure.It is found that clear density peaking can be observed in the case (A), while density profile is relaxed in the  case (B).In this study, there is no explicit particle source operator but the Krook-type energy sink operator is applied at the outer boundary region so that the sink operator can work as a particle source in the case (A) and a sink in the case (B).
Figures 4(A-2) and (B-2) show the radial ion temperature profiles in the cases (A) and (B).Initial steep ion temperature gradient is found to be relaxed and keeps globally same constant value around R 0 /L Ti = 6 in both cases.This contrasts with the result that the ion density gradients are not same in both cases and show the non-uniform profiles.This is because the ion temperature gradient is the direct driving force of dominant micro-scale instability and resultant ion turbulent heat transport so that a strong constraint is imposed for global ion profiles.Such a tendency is often observed in some full-f simulations [18,23].
The radial electron temperature profiles in the cases (A) and (B) are also shown in figures 4(A-3) and (B-3), respectively.The initial steep electron temperature gradient is relaxed into globally uniform value around R 0 /L Te = 7 in the case (A).On the other hand, the electron temperature remains almost unchanged in the case (B), indicating that electron turbulent heat transport is sufficiently small in the absence of steep initial electron temperature profile and electron heat source.
Then, we discuss the dominant mode in the quasi-steady state (tR 0 /v ti0 = 800) shown in figure 4. Figure 5 shows the wave-number dependencies of the linear growth rate obtained from linear δf global simulation with the quasi-steady profiles of the case (A) (R 0 /L n = 2.6, R 0 /L Ti = 6.5, R 0 /L Te = 7) and the case (B) (R 0 /L n = 1.8, R 0 /L Ti = 6.5, R 0 /L Te = 4).The corresponding eigenfunctions are also shown in figure 6.It is found that the dominant mode is still ITG even in the case (A) (see figures 5 and 6(A-1)), while the ratio of TEM growth rate to ITG one is 44.0% in the case with initial profile and 61.7% in the case with quasi-steady profile.Hence, the ITG mode dominantly drives the turbulence even after the nonlinear saturation, implying that only the type of dominant mode does not determine the direction of particle transport.It is remarked that the density relaxation observed in the case (B) is different from the prediction by the quasi-linear fluid theory [1].Figures 7 and 8 show the temporal-spatial evolutions of total particle and heat fluxes in the cases (A) and (B).After the nonlinear saturation, large outward heat transport is observed, and then intermittent avalanche-like transport takes places, the direction of which is outward in the positive mean E r shear region and inward in the positive mean E r shear region, as was discussed in the previous works [11,18].It is found that the heat avalanches due to ITG turbulence are accompanied by inward particle transport in the case (A) while outward one in the case (B).Since the heat source provides the freeenergy of ITG turbulence, continuous density peaking/flattening is considered to be observed as shown in the figures 4(A-1) and (B-1).

Turbulent particle flux by
From this sub-section, we separately discuss turbulent particle flux by each radial drift one by one.Figure 9(a) shows the temporal-spatial evolutions of turbulent ion particle flux by E × B drift (n ̸ = 0) Γ E×B(n̸ =0), s given by equation (13) in the case (A).It is found that turbulent particle pinch intermittently takes place even after the saturation, providing the continuous density peaking at the core region.In order to verify the contribution from the diagonal-diffusion term (D n R 0 /L n ) and the non-diagonal thermo-diffusion term (D T R 0 /L T ), the particle fluxes calculated from passive tracer ions [24] with uniform background temperature and density profiles are also shown in figures 9(b) and (c), respectively.The sign of the diagonal diffusion term is positive because the contribution from the non-diagonal thermo-diffusion term is omitted in figure 9(b).On the other hand, the sign of the non-diagonal thermo-diffusion term is considered to be negative from figure 9(c).In fact, by solving the simultaneous equations obtained from the three kinds of simulations shown in figure 9, the diagonal diffusion and non-diagonal thermodiffusion coefficients are estimated as , respectively.This result is consistent with the quasi-linear theory [1] since the thermodiffusion coefficient D T ∝ − (10L n /3R 0 − ω r /ω * e ) is negative for the ITG mode with ω r < 0.
Figure 10 shows the temporal evolution of turbulent particle fluxes by E × B drift (n ̸ = 0) in the cases (A) and (B).In the case (A), strongly negative ion and electron particle fluxes can be observed just after the nonlinear saturation.Even after that, they keep a finite negative level continuously, and the magnitude of electron transport is found to be a little bit larger than that of ion.On the other hand, weakly positive ion particle flux is observed while electron particle flux is strongly positive in the case (B).This originates from the fact that the positive diagonal diffusion term becomes dominant for the electron particle flux in the case (B) because the contribution from the negative non-diagonal thermo-diffusion term is relatively small.It should be noted that the balance between ion and electron particle fluxes breaks, namely As the other candidate to drive turbulent particle pinch, the temperature ratio T i /T e is considered to be the other candidate.According to [1], the particle transport coefficient tends to   more negative once T i /T e smaller.Since T i /T e the case-(A) is smaller than that in the case-(B) (see figures 4(A-3) and (B-3)), our result that particle pinch is observed in the case-(A) is consistent with the theory, implying that T i /T e is considered to be the other origin of turbulent particle pinch.

Turbulent particle flux by magnetic drift
In section 3.3, it is confirmed that turbulent particle transport by E × B drift (n ̸ = 0) breaks the balance between ion and electron particle fluxes.Such a break can then induce the other turbulent particle transport to satisfy the total balance of particle fluxes.Figure 11 shows the temporal evolution of the sum of turbulent ion and electron particle fluxes by E × B drift (n ̸ = 0) and turbulent ion particle flux by magnetic drift Γ B, s given by equation (14).When the sum of turbulent ion and electron particle fluxes by E × B drift (n ̸ = 0) is non-zero, that by magnetic drift is enhanced (see the green line in figure 11), which cancels with the E × B drift (n ̸ = 0) driven particle transport although its level depends on the case.Such a trend is observed in impurity transport in GT5D full-f gyrokinetic simulation, too [6].The physical mechanism why the magnetic drift driven particle transport is enhanced can be explained as follows.When the sum of turbulent ion and electron particle fluxes by E × B drift (n ̸ = 0) becomes positive as in the case (A), it yields a negative mean radial electric field E r with (m, n) = (0, 0).Consequently, the poloidal  has a poloidal up-down asymmetry arising from the magnetic field B, which leads to ion density perturbations with (m, n) = (1, 0) through the continuity equation given by ∂ t δn i ∼ −n 0 /r∂ θ (v E, θ ) = −n 0 E r sin θ/R 0 .Such density perturbations are in phase with the magnetic drift so that it can provide negative particle flux by magnetic drift as ∂ t ⟨v B,r δn⟩ ∝ E r ⟨ sin 2 θ ⟩ < 0. The fact that its sign is opposite to that of Γ i, E(n̸ =0) − Γ i, E(n̸ =0) is consistent with the observation in figure 11.

Turbulent particle flux by
The other candidate to satisfy the total balance is turbulent particle flux by E × B drift (n = 0) Γ E×B(n=0), s given by equation (15), which was not precisely considered in the other simulations.Figure 12 show the temporal evolutions of the sum of turbulent ion and electron particle fluxes by E × B drift (n ̸ = 0), turbulent ion particle flux by magnetic drift, and turbulent ion particle flux by E × B drift (n = 0).It is newly found that particle flux by not only magnetic drift but also E × B drift (n = 0) are enhanced, which also cancels with the E × B drift (n ̸ = 0) driven particle transport.It should be noted that while the magnetic drift driven ion particle transport quickly reaches to zero after the nonlinear saturation in the case (A) (see the green line in figure 12(A)), the E × B drift (n = 0) driven ion particle transport keeps a finite level after the saturation (see the yellow line in figure 12(A)), which can drive continuous density peaking.Note that ion particle flux by E × B drift (n = 0) is not driven by (m, n) = (0, 0) radial electric field but (m, n) = (1, 0) one.This physical mechanism is similar to that of particle flux by magnetic drift.Namely, the poloidal E × B compression by (m, n) = (0, 0) radial electric field leads to ion density perturbations with (m, n) = (1, 0).It can also cause (m, n) = (1, 0) electric field through the quasi-neutrality condition, which is considered to enhance ion particle flux by E × B drift (n = 0).E × B drift (n = 0) also plays an important role on electron particle transport, too. Figure 13 shows the time evolution of electron particle fluxes by magnetic drift and E × B drift (n = 0) in the case (A).Because of the fast-scale electron motion along the magnetic field line, which is equivalent to the return current, electron density perturbations along the poloidal direction quickly disappears, leading to the perfect balance between electron particle transport by magnetic drift and E × B drift (n = 0).On the other hand, these two terms do not completely balance with each other in the case of ions because the parallel convection is relatively slow.Since the species with larger mass has slower-scale parallel convection, total net particle flux given by magnetic drift and E × B (n = 0) drift is expected to become larger for heavier bulk ion and impurities.

Turbulent ion particle pinch by all the drifts
From figure 13, since electron particle transport by the E × B drift (n = 0) and the magnetic drift balance with each other, the gyrokinetic ambipolarity condition in the quasi-steady state is given by Equation ( 18) indicates that the electron particle pinch Γ e,E(n̸ =0) (< 0) can indirectly trigger the ion particle pinch.Figure 14 shows the temporal evolutions of turbulent ion particle flux by E × B drift (n ̸ = 0), magnetic drift, and E × B  drift (n = 0) in the cases (A) and (B).First, n ̸ = 0 E × B drift driven ion particle pinch is triggered by steep ion temperature gradient in the case (A) (see the red line in figure 14(A)).E × B drift (n ̸ = 0) driven electron particle pinch is also triggered by steep electron temperature gradient (see the blue line in figure 10(A)).Then, ion particle flux by magnetic drift and E × B drift (n = 0) is triggered, satisfying the total particle flux balance given by equation (18) (see the green and yellow lines in figure 14(A)), leading to an additional ion particle pinch in the case (A).This is the schematic process of turbulent ion particle pinch observed in the case (A).On the other hand, turbulent electron particle flux becomes positive in the case (B) (see the blue line in figure 10(B)) because the diagonal diffusion term is dominant.As the result, turbulent ion particle flux by magnetic drift and E × B drift (n = 0) becomes clearly positive, which is the origin of density flattening observed in figure 4(B-1).

Summary
We have performed flux-driven ITG/TEM simulations for studying turbulent ion particle transport in the presence of ion/electron heating by means of the full-f gyrokineitc code GKNET with hybrid electron model.It is found that steep ion temperature gradient can drive turbulent ion particle pinch by E × B drift (n ̸ = 0) because the negative thermo-diffusion term becomes dominant.Turbulent electron particle pinch is also driven in the case with steep electron temperature gradient.Such an electron particle pinch can trigger an ambipolar field, leading to up-down asymmetric density perturbations and resultant ion particle pinch by not only magnetic drift but also E × B drift (n = 0).This means that neoclassical transport (n = 0) transiently changes on a turbulent time scale.In other words, when the macroscopic structure changes on a turbulence time scale shorter than a collision time scale, the scale separation is not satisfied and the interaction between neoclassical and turbulent transport becomes essential.Note that it is not easy to separate the transport into neoclassical and turbulent components in such a case.A more precise definition other than axial symmetry should be also considered [25].
As future plans, larger plasma-size simulations will be performed to confirm the proposed ion particle pinch mechanism can be utilized for the fuel supply in future DEMO reactors.By introducing helium as the other species, the optimal condition for refueling and helium ash exhaust by the selective external heating will be also addressed.To study edge impurity accumulation, we will also perform full-f simulation with SOL and diverter region by means of the new version of GKNET with field aligned coordinate [26].

Figure 7 .
Figure 7.The temporal-spatial evolutions of (A) total particle flux and (B) total heat flux in the case (A) R 0 /L Ti = R 0 /L Te = 10.

Figure 8 .
Figure 8.The temporal-spatial evolutions of (a) total particle flux and (b) total heat flux in the case (B) R 0 /L Ti = 10, R 0 /L Te = 4.

Figure 9 .
Figure 9.The temporal-spatial evolutions of turbulent ion particle flux by E × B drift (n ̸ = 0) in the case (A).(A) Original ion, (B) tracer ion with uniform T i0 and (C) tracer ion with uniform n i0 are used to calculate each flux.

Figure 11 .
Figure 11.Temporal evolutions the sum of turbulent ion and electron particle by E × B drift (n ̸ = 0) (purple) and turbulent ion particle flux by magnetic drift (green) in the cases (A) and (B).The particle fluxes are spatially averaged among 0.4a 0 < r < 0.6a 0 .

Figure 12 .
Figure 12.Temporal evolutions of the sum of turbulent ion and electron particle fluxes by E × B drift (n ̸ = 0) (purple), turbulent ion particle flux by magnetic drift (green), and turbulent ion particle flux by E × B drift (n = 0) (yellow) in the cases (A) and (B).The particle fluxes are spatially averaged among 0.4a 0 < r < 0.6a 0 .

Figure 13 .
Figure 13.Temporal evolutions of electron particle flux by magnetic drift (green) and electron particle flux by E × B drift (n = 0) (yellow) in the case (A).The particle fluxes are spatially averaged among 0.4a 0 < r < 0.6a 0 .