Nonlinear entropy transfer via zonal flows in gyrokinetic plasma turbulence

Abstract

Nonlinear entropy transfer processes in toroidal ion temperature gradient (ITG) and electron temperature gradient (ETG) driven turbulence are investigated based on the gyrokinetic entropy balance relations for zonal and non-zonal modes, which are coupled through the entropy transfer function regarded as a kinetic extension of the zonal-flow production due to the Reynolds stress. Spectral analyses of the "triad" entropy transfer function introduced in this study reveal not only the nonlinear interactions among the zonal and non-zonal modes, but also their effects on the turbulent transport level. Different types of the entropy transfer processes between the ITG and ETG turbulence are found: The entropy transfer from non-zonal to zonal modes is substantial in the saturation phase of the ITG instability, while, once the strong zonal flow is generated, the entropy transfer to the zonal modes becomes quite weak in the steady turbulence state. Instead, the zonal flows mediate the entropy transfer from non-zonal modes with low radial-wavenumbers (with contribution to the heat flux) to the other non-zonal modes with higher radial-wavenumbers (but with less contribution to the heat flux) through the triad interaction. The successive entropy transfer processes to the higher radial-wavenumber modes are associated with transport regulation in the steady turbulence state. In contrast, in both the instability-saturation and steady phases of the ETG turbulence, the entropy transfer processes among low-wavenumber non-zonal modes are dominant rather than the transfer via zonal modes.

Figures (11)

• Figure 1: (Color online) Wavenumber spectrum of the linear growth rate $\gamma_{\mathrm{L} }$ of the toroidal ITG instability. The spectrum for the toroidal ETG instability is the same as that in the ITG case except for the normalizations with $\varv _{\mathrm{te} }$ and $\rho_{\mathrm{te} }$.
• Figure 2: (Color online) Time evolutions of each term in the entropy balance relation of the turbulence part, Eq. (21), for toroidal (a)ITG ($\mathrm{s} \! =\! \mathrm{i}$) and (b)ETG ($\mathrm{s} \! =\! \mathrm{e}$) turbulence. The deviation from the exact balance is also plotted by the dashed line.
• Figure 3: (Color online) Time evolutions of each term in the entropy balance relation of the zonal-flow part, Eq. (22), for toroidal (a)ITG ($\mathrm{s} \! =\! \mathrm{i}$) and (b)ETG ($\mathrm{s} \! =\! \mathrm{e}$) turbulence. The deviation from the exact balance is also plotted by the dashed line. (c)The time-histories of the entropy transfer function normalized by the time-averaged heat flux $\mathcal{T}_{\mathrm{s} }^{\mathrm{(zf)} } /\eta_{\mathrm{s} }\overline{Q}_{\mathrm{s} }$.
• Figure 4: (Color online) Contours of the potential fluctuations $\delta \phi(x,y,z)$ at $t \! = \! 315(L_{n_{0}}/\varv _{\mathrm{ts} })$ for toroidal (a)ITG and (b)ETG turbulence, where the unit is $(T_{\mathrm{s} }\rho_{\mathrm{ts} }/eL_{n_{0}})$. The box size is $L_{x} \times L_{y} \times L_{z} \! = \! 266 \rho_{\mathrm{ts} } \times 168\rho_{\mathrm{ts} } \times \pi$. The ($x,\,y$)-cross section shown here is the perpendicular plane in the outboard side of the torus, where $z \! = \! 0$.
• Figure 5: (Color online) Wavenumber spectra of the turbulent heat flux $\eta_{\mathrm{s} }Q_{\mathrm{s} \space\bm{k}_{\perp}}$ [(a) and (b)] and the potential fluctuation $\langle |\delta \phi_{\space\bm{k}_{\perp}} | \rangle$ [(c) and (d)] in the steady states of the ITG (upper row) and ETG (lower row) turbulence, where the amplitudes are averaged over $220 \! \leqslant \! t \! \leqslant 320$. The amplitudes of potential fluctuations are normalized by the maximum value in the non-zonal components.