Saturation of Gyrokinetic Turbulence through Damped Eigenmodes

In the context of toroidal gyrokinetic simulations, it is shown that a hierarchy of damped modes is excited in the nonlinear turbulent state. These modes exist at the same spatial scales as the unstable eigenmodes that drive the turbulence. The larger amplitude subdominant modes are weakly damped and exhibit smooth, large-scale structure in velocity space and in the direction parallel to the magnetic field. Modes with increasingly fine-scale structure are excited to decreasing amplitudes. In aggregate, damped modes define a potent energy sink. This leads to an overlap of the spatial scales of energy injection and peak dissipation, a feature that is in contrast with more traditional turbulent systems.

Figure 1

Energy drive $Q_k$ and dissipation $C_k$ time averaged over the nonlinear state and averaged over the $z$ direction, as a function of (a) $k_x$ summed over $k_y$ and (b) $k_y$ summed over $k_x$.

Figure 2

The spectrum of singular values (a), energy drive $Q_k$ for each normalized POD mode (b), and dissipation $C_k$ for each normalized mode (c). For $k_y{\rho }_i=0.2$, $k_x{\rho }_i=0.0$.

Figure 3

Mode structures for a selection of POD modes at the peak of the nonlinear spectrum, $k_y{\rho }_i=0.2$, $k_x{\rho }_i=0.0$. The mode structures of the electrostatic potential are shown in the top row for modes 1, 2, 10, and 100, and the $v_{\parallel }$ dependence is shown in the bottom row for the same POD modes at $\mu =0.18$ and $z=0$. Fine-scale structure develops in both coordinates as $n$ increases.