Cross-Scale Interactions between Electron and Ion Scale Turbulence in a Tokamak Plasma

Multiscale gyrokinetic turbulence simulations with the real ion-to-electron mass ratio and $\beta$ value are realized for the first time, where the $\beta$ value is given by the ratio of plasma pressure to magnetic pressure and characterizes electromagnetic effects on microinstabilities. Numerical analysis at both the electron scale and the ion scale is used to reveal the mechanism of their cross-scale interactions. Even with the real-mass scale separation, ion-scale turbulence eliminates electron-scale streamers and dominates heat transport, not only of ions but also of electrons. Suppression of electron-scale turbulence by ion-scale eddies, rather than by long-wavelength zonal flows, is also demonstrated by means of direct measurement of nonlinear mode-to-mode coupling. When the ion-scale modes are stabilized by finite-$\beta$ effects, the contribution of the electron-scale dynamics to the turbulent transport becomes non-negligible and turns out to enhance ion-scale turbulent transport. Damping of the ion-scale zonal flows by electron-scale turbulence is responsible for the enhancement of ion-scale transport.

Figure 1

(a) Linear instabilities: Linear growth rate $\gamma$, plotted against the poloidal wave number $k_y$. Square (red) and circular (blue) marks indicate electrostatic ($\beta =0.04\%$) and electromagnetic ($\beta =2.0\%$) cases, respectively. (b) Nonlinear evolution: Time evolution of the field energy $W_{\mathit{k}}$ for the $\beta =2.0\%$ case. The solid (red), long-dashed (green), short-dashed (blue), and dotted (rose) lines correspond to the zonal ($k_y{\rho }_{\mathrm{ti}}=0$), ITG/TEM ($k_y{\rho }_{\mathrm{ti}}\le 1$), intermediate ($1<k_y{\rho }_{\mathrm{ti}}\le 4$, represented as Med.), and ETG ($4<k_y{\rho }_{\mathrm{ti}}$) modes, respectively. The straight line shows the slope of the linear growth rate of the ITG with $\gamma =0.12v_{\mathrm{ti}}/R$.

Figure 2

Poloidal wave number spectrum of the time-averaged electron energy diffusivity ${\chi }_{e\mathit{k}}$ for (a) electrostatic ($\beta =0.04\%$) and (b) electromagnetic ($\beta =2.0\%$) cases. The solid (red), dotted (blue), and dashed (green) lines plot ${\chi }_{e\mathit{k}}$ as obtained from the full-$k$, low-$k$ ($k_y{\rho }_{\mathrm{ti}}<1.3$), and high-$k$ ($k_y{\rho }_{\mathrm{ti}}>1.3$) simulations, respectively.

Figure 3

Wave number $\mathit{q}$ spectrum of the triad transfer $J_{\mathit{k}}^{\mathit{p},\mathit{q}}$ (normalized by $n_0{T}_i{v}_{\mathrm{ti}}{\rho }_{\mathrm{ti}}^2/R^3$) for a typical streamer $\mathit{k}=(0,4.4{\rho }_{\mathrm{ti}})$, averaged over $60<t{v}_{\mathrm{ti}}/R<80$ with $\beta =2.0\%$. Arrows represent dominant triad couplings. Negative and positive values of $J_{\mathit{k}}^{\mathit{p},\mathit{q}}$, respectively, mean that the streamer gives and receives entropy via coupling with resonant modes $\mathit{k}+\mathit{p}+\mathit{q}=0$.

Figure 4

(a) Ratio of zonal-mode entropy transfer to the total entropy drive ${\overline{I}}_{\mathit{k}\mathrm{ZF}}=\sum _{k_y=0}{I}_{\mathit{k}}/X_{\text{drive}}$ and (b) ratio of zonal to nonzonal field energy $\zeta =\sum _{k_y=0}{W}_{\mathit{k}}/\sum _{k_y\ne 0}{W}_{\mathit{k}}$ (where $\beta =2.0\%$). The solid (red) and dashed (blue) lines indicate results from the full-$k$ and low-$k$ ($k_y{\rho }_{\mathrm{ti}}<1.3$) simulations, respectively.