{'fraction_non_alphanumeric': 0.03523781902552204, 'fraction_numerical': 0.009667440061871617, 'mean_word_length': 4.756538675570395, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Intrinsic current drive by electromagnetic electron temperature gradient turbulence in tokamak plasmas

The mean parallel current density evolution equation is presented using electromagnetic (EM) gyrokinetic equation. There exist two types of intrinsic current driving mechanisms resulted from EM electron temperature gradient (ETG) turbulence. The first type is the divergence of residual turbulent flux including a residual stress-like term and a kinetic stress-like term. The second type is named as residual turbulent source, which is driven by the correlation between density and parallel electric field fluctuations. The intrinsic current density driven by the residual turbulent source is negligible as compared to that driven by the residual turbulent flux. The ratio of intrinsic current density driven by EM ETG turbulence to the background bootstrap current density is estimated. The local intrinsic current density driven by the residual turbulent flux for mesoscale variation of turbulent flux can reach about 80% of the bootstrap current density in the core region of ITER standard scenario, but there is no net intrinsic current on a global scale. Based on this, the local intrinsic current driven by EM micro-turbulence and its effects on local modification of the profile of safety factor may be needed to be carefully taken into account in the future device with high beta_e which is the ratio between electron pressure to the magnetic pressure.


Introduction
Current density profile is of great importance for tokamak plasmas since it can affect both confinement time and a variety of magnetohydrodynamic (MHD) instabilities, such as the kink mode, and the tearing mode. The current density profile can be strongly affected by the current driving mechanism. Up to now, various current driving mechanisms have been proposed, such as inductive current drive, neutral-beam-injection (NBI) current drive [1], lower hybrid drift instability current drive [2], electron cyclotron current drive [3], the poloidally asymmetric fueling current drive [4], and so on. One of the particular efficient and economical ways is the bootstrap (BS) current driven by the pressure gradient which has been predicted by the neoclassical theory [5]. It is demonstrated to be consistent with the predictions of the neoclassical theory in some experiments [6][7][8]. Nonetheless, deviations from the neoclassical theory are also observed in some cases [9,10]. Inspired by the turbulent-driven intrinsic rotation, naturally, intrinsic current driven by turbulence can be taken into account similarly.
In fact, current driven by drift waves in a slab geometry has been proposed many years ago [11]. It showed that a residual current flux could be caused by the electrostatic (ES) fluctuations in the presence of ∥ (parallel wave number of drift wave) symmetry breaking. A more systematic model for current driven by turbulence has been done by Garbet [12], where two kinds of current source from the gyrokinetic equation were proposed. Both of them require ∥ symmetry breaking and may contribute about 10% of the BS current density in the edge region of tokamak plasmas.
Similarly, an extended Ohm's law modified by magnetic turbulence has also been studied by using the self-consistent action-angel transport theory [13]. In this model, turbulence results in three different kinds of resistances in Ohm's law. The modification of Ohm's law by two kinds of current source from electromagnetic (EM) turbulence has been done [14]. One is the momentum source and the other one is radial flux of the parallel current driven by magnetic flutter. Though the momentum source term can bring a little change of the local current density, it contributes a nonzero total current. In contrast, the flux term will significantly change the local current density without a net total current. It is worth mentioning that a novel anomalous BS current can be caused by turbulent scattering [15][16][17], which is thought to serve as effective collision. The ES trapped electron mode (TEM) turbulence and ion temperature gradient (ITG) turbulence were considered in [17].
Although lacking of experimental evidence, in a slab geometry, the current driven by ES electron temperature gradient (ETG) turbulence has been calculated using a fluid model [18], in which the ratio of current density driven by ETG turbulence to the BS current density can be 10%. Recently, a gyrokinetic simulation result also reports that ES ETG can drive current [19], and it shows that ETG turbulence driven current density reaches about 20% of the BS current density, thus significantly changes the safety factor. On the other hand, experimental evidence found that EM effects may be considerable for intrinsic rotation [20,21], and theoretical work also addressed significant EM effects on intrinsic rotation drive in the H-mode pedestal plasmas [22].
Thus, investigation of intrinsic current driven by EM ETG turbulence in toroidal tokamak plasmas is meaningful.
In this work, the mean parallel current density evolution equation is derived using the EM gyrokinetic equation. It is shown that there are two types of intrinsic current driving mechanisms. Similar to the turbulent-driven intrinsic rotation, the first type is the divergence of residual turbulent flux, which comes from a residual stress-like term and a kinetic stress-like term. The second type is named as residual turbulent source, which is driven by the correlation between density and parallel electric field fluctuations. The quasilinear estimation of the divergence of the residual turbulent flux and the residual turbulent source of current density driven by EM ETG turbulence are presented. Both of the driving mechanisms need ∥ symmetry breaking. By taking the typical core parameters in International Thermonuclear Experimental Reactor (ITER) standard scenario, we compare the intrinsic current density driven by EM ETG turbulence to the background BS current density. The results show that the EM effects could be important for relatively high which is the ratio between electron pressure to the magnetic pressure. Although the contribution to the total net current is neglected, the local modification of current density profile due to EM ETG turbulence could play a significant role in future device.
The remainder of this paper is organized as follows. In section 2, the derivation of the mean parallel current density evolution equation is presented. The quasi-linear estimation of intrinsic current density driven by EM ETG turbulence and the comparison between the intrinsic current density and the BS current density are also presented. Finally, conclusions and some discussions are given in section 3.

Quasilinear estimate for intrinsic parallel current drive
We start from the nonlinear EM gyrokinetic equation [23], with the electron gyro-center equations of motion in the symplectic formulation, i.e., ∥ representation and Here, = ( , , ∥ , ) is the electron gyro-center distribution function where is the magnetic moment of electron, ̂= ⁄ is the unit vector of equilibrium magnetic line, ∥ * =̂⋅ * is the Jacobian of the transformation from the particle phase space to the gyro-center phase space with * = + ⊥ + − ∥ ∇ ×̂, c is the speed of light, is the electron mass, is the elementary charge. is the ES potential fluctuation. δ ⊥ ≈ −̂× ∇ ∥ is the perturbed magnetic field. In this paper, the index ∥ refers to the components parallel to the equilibrium magnetic field, and the index ⊥ refers to the components perpendicular to the equilibrium magnetic field. Only the shear component of magnetic perturbation, i.e, ∥ , is considered, and ∥ is not included in this work. This simplified model was also widely adopted in previous works on investigation of EM ETG [24][25][26]. However, for spherical tokamak where can reach 10% or even higher, neglecting δ ∥ will lead to an under estimation of the growth rates of ETG as well as ITG [27].
Multiplying Eq. (1) by − 2 ∥ and integrating in the velocity space, the evolution equation of parallel current density can be obtained, Here, we assume the current is mainly carried by electron. ∥ = − ∫ ∥ 3 = − ∥ is the parallel current density with being the density of electron and ∥ being the parallel speed of the electron fluid. ∥ = ∫ ( ∥ − ∥ ) 2 3 is the parallel electron pressure. ⊥ = ∫ 3 is the perpendicular electron pressure.
is the magnetic gradient drift velocity and = ∥ −̂× (̂• ∇)̂ is the magnetic curvature drift velocity. The magnetic drift velocity will be neglected in the mean parallel current density equation, since they are higher order terms ( ⁄ ) as compared to × drift velocity [28]. For the same reason, all the terms in Eq. (4) related to magnetic gradient will be neglected. The terms related to the equilibrium electron fluid velocity 0 will also be neglected because of 0 2 ≪ ℎ 2 , where ℎ = √ is the electron thermal velocity with being the electron temperature. Taking flux-average of Eq. (4), the evolution of mean parallel current density can be obtained, Here, ∥ = − ∫ with the intrinsic rotation drive in [22], the first one is the Reynolds stress-like term, and the second one is the kinetic stress-like term denoting dynamo effects. The terms on the right hand side (RHS) are turbulent source terms, which is analogous to the momentum source for ion [29]. The source terms are driven by the correlation between density and parallel electric field fluctuations including inductive electric field. Eq. (5) seems to be similar with Eq. (3) derived from fluid model in [18].
However, the ES limit was adopted in quasilinear calculation of intrinsic current drive in [18]. Both the EM effects and toroidal effects are kept in our quasilinear calculation which will be shown later. Then, Eq. (5) can be rewritten as where Γ is the turbulent flux of current density, and represents the turbulent source. The turbulent flux can be usually divided into diffusive term, convective term and residual term, i.e., where ∥ is the diffusion coefficient of the parallel current density, is the convective velocity of the parallel current density and Γ is the residual turbulent flux. The residual turbulent flux is independent of the parallel current density or its gradient, so it can provide intrinsic current drive. This is analogous to the intrinsic rotation drive by residual stress [29,30]. The turbulent source term can be divided into residual turbulent source and non-residual turbulent source terms, similarly. The residual turbulent source is also independent of the parallel current density or its gradient, and so can provide intrinsic current drive, too. This is also analogous to the intrinsic rotation drive by momentum source [29].
Next, we make quasi-linear estimation for the turbulent flux and the residual turbulent source. Therefore, the linear calculations of δ , δ ∥ , and δ ∥ are required.
The electron density fluctuation can be obtained using the quasi-neutrality condition δ = and adiabatic ion approximation = − 0 for EM ETG mode. The adiabatic ion model is often employed in studying ETG turbulence [25,26].
Especially, the validity of adiabatic ion model used for ETG turbulence at low magnetic shear was demonstrated via parameter scan in magnetic shear in [31]. The model is used in [18,24], which is analogous to the model for electrons used in ITG turbulence. Assuming the equilibrium electron distribution function to be shifted Maxwellion where 0 is the equilibrium electron density, the linearized EM perturbed electron distribution function in Fourier space can be written as Here  2 )] + . (9) The lowest order of non-resonant part will be kept.
Taking the first order moment of Eq. (8), i.e., ∥ = − ∫ ∥ 3 we can obtain the perturbed parallel current density where = is the ratio of electron temperature to ion temperature. Here, the assumption 2̂∥ 2 = ⊥ 2 has been used for calculation of the resonant part. The first two terms in Eq. (10) result from non-resonant ES contribution, the third term comes from non-resonant EM effects, and the last two terms are caused by resonant effects.
Different from [12], the non-resonant parts are also calculated here. Moreover, we will consider the relation between ̂∥ and ̂ in more detail to investigate the explicit EM effects. In the following, we will focus on the residual turbulent flux and the residual source, which contribute to the intrinsic current drive. Combing the Ampere's law with Eq. (10) and neglecting terms related to 0 , the general relation between ̂ and ̂∥ can be obtained, ( 1 − 2 ) 2 +( 3 + 4 ) 2̂.
Here, 1 , 2 , 3 , 4 and 1 , 2 , 3 are all dimensionless and independent of parallel current density or gradient of parallel current density. Eq. (14) is important for explicit estimate of the EM effects. In [12], this relation is not mentioned and the EM effects are not the focus. Besides, the spatial scale of turbulence is about ion gyroradius in [12]. The details of the calculation of Eq. (14) can be found in the Appendix.
In the following, we assume ≃ * which is appropriate for ETG mode.
The residual turbulent flux can be written as For turbulent source, = which is also independent of parallel current density or its gradient due to Eq. (14), and so can provide an intrinsic current drive. The first term on the RHS of Eq. (15) represents the non-resonant ES contribution, the second term and the third term represent the non-resonant EM contribution, the forth term represents resonant ES contribution, and the last term represents resonant EM contribution. We will use * ≃ later. The terms related to come from toroidal effects. Eq. (13) shows that the residual turbulent source is only caused by EM effects, since the turbulent source driven by the correlation between density and ES field fluctuations vanishes for the adiabatic ion response. Both the residual turbulent flux and the residual turbulent source require parallel symmetry breaking. Theoretical works have proposed various symmetry breaking mechanisms, such as E × B shear [32,33], charge separation from polarization drift [34], intensity gradient [35], geometrical up-down asymmetries [36], etc. In [19], it is found that the contribution to the ETG turbulence driven current from symmetry breaking induced by the turbulence intensity gradient is more important than that by zonal flow shear from ES gyrokinetic simulations. We also take the symmetry breaking caused by the turbulence intensity gradient in this work. This is reasonable for core region with flat pressure profile where the × shear is not strong. The mean parallel wave number is ̅ ∥ ≃ 2 with ̅ ∥ = ∑ ∥ |̂| Here, the length scale of variation of the residual turbulent flux is taken as mesoscale, i.e., √ . There does not exist experimental evidence for the length scale. The sign of ∓ corresponds to positive (negative) gradient of flux. In [18], only the ES part, namely the first term on the RHS of Eq. (15) is considered. While the toroidal effects, EM effects and resonant effects are neglected in [18]. Similarly, intrinsic current density driven by residual turbulent source can be calculated The residual turbulent source is caused by momentum exchange between ions and electrons. To illustrate how important the intrinsic current density driven by EM ETG turbulence as compared to the BS current density, we also estimate the BS current density as following [12], where is the length scale of pressure gradient, and q is the safety factor.
Subsequently, the ratios of the intrinsic current density driven by residual turbulent flux and residual turbulent source to the BS current density can be written as and Then, we take the typical parameters of ITER [37]  Based on these parameters, we can obtain ( ∥ ℎ ) ⁄ ∼⁄~4 3 > 1, which is consistent with previous assumption. Of course, it would be more accurate to deal with transit and drift resonance simultaneously. However, we have to adopt simplified model and keep some key physics to be able to derive an analytical expression. Note that, the finite Larmor radius effects are neglected in this work, which may quantitatively modify the results and should be carefully considered in future.
Numerical results showed that the linear growth rate of EM ETG increases with [25]. Then, the ratios can be written as Γ = ∓(17.3% + 9.9% + 59.6% − 3.7%), On the RHS of Eq. of the BS current density, but does not contribute to the total net current. On the other hand, the residual turbulent source can drive a nonzero total current, but the order of magnitude is less than 1% as compared to the BS current. Moreover, a rough estimation for ratio of the ETG turbulence driven current density to BS current density by taking similar parameters to those from [19] is about 10% which is comparable to the ratio in [19]. In this paper, the contribution from magnetic drift resonance is calculated since a kinetic model is used. While in [18], only the non-resonant ES contribution was considered due to a fluid model in pedestal region, and the toroidal effects were not taken into account, either. From Eq. (21), it is shown that the resonant contribution is very important as compared to the non-resonant one.
This is because the density profile is flat in the core region, making the characteristic frequency of ETG turbulence comparable to the magnetic drift frequency.
The current density profile may be locally modified on the length scale of √ (about hundreds of electron gyroradii or several ion gyroradii) by the EM ETG turbulence driven intrinsic current in the core region of tokamak H-mode plasmas.
This may lead to the local modification of q profile, and hence affect the MHD behaviors. However, if the length scale of the residual turbulent flux is at macroscale, e.g., , the ratio of intrinsic current density driven by turbulent flux to the BS current density will reduce to 1 277 ⁄ of that estimated for mesoscale turbulent flux.
Then, the ratio becomes less than 1%, and the intrinsic current density driven by ETG turbulence is thus negligible for this case. Therefore, the selection of the length scale of the residual turbulent flux is important and deserved to be verified in experiments.
A brief summary of the result is given in Table 1.

Summary
In this work, an evolution equation of mean parallel current density has been derived using the EM gyrokinetic equations. There exist two intrinsic current driving mechanisms. One is the residual turbulent flux and the other is the residual turbulent source. Both of them can provide intrinsic current drive and need ∥ symmetry breaking, and the symmetry breaking caused by the turbulence intensity gradient is taken in the present work.
Although the net intrinsic current driven by the EM ETG turbulence can be neglected, the local current density profile can be significantly affected. Quasi-linear estimations show that while the residual turbulent source may not contribute a lot to the current density as compared to the local BS current density, the residual turbulent flux can locally drive about 80% of the local BS current density by using the core parameters of standard scenario ITER. Therefore, we conclude that in the high beta fusion devices like ITER, the EM ETG turbulence driven intrinsic current density may significantly change the local current density profile, thus may change the q profile in core region. This modification of current density profile may be important for MHD instabilities. Particularly, the local modification of current density profile in the narrow pedestal region might be important for edge localized modes, which will be considered in future.
Until now, there is no direct experimental observations for turbulence driven current. However, we expect that in the future fusion reactor with high beta the EM ETG turbulence driven intrinsic current density could be observed. We should point out that the diffusive coefficient and convective speed of current flux are not calculated in the present work. The diffusion and convection may be important for accurate prediction of current density profile in the pedestal region. Moreover, the modification of current density profile in pedestal could affect the edge localized modes control. Therefore, extending this work from core region to pedestal region and considering the effects of diffusion and convection induced by EM turbulence on current density profile may be investigated in future. In addition, collisionless TEM turbulence is another prominent candidate for the electron heat transport in high temperature plasmas. Now, we are also working on the intrinsic current driven by EM collisionless TEM turbulence.