Saturation of magnetised plasma turbulence by propagating zonal flows

Abstract

Strongly driven ion-scale turbulence in tokamak plasmas is shown to be regulated by a new propagating zonal flow mode, the toroidal secondary mode, which is nonlinearly supported by the turbulence. The mode grows and propagates due to the combined effects of zonal flow shearing and advection by the magnetic drift. Above a threshold in the turbulence level, small-scale toroidal secondary modes become unstable and shear apart turbulent eddies, forcing the turbulence level to remain near the threshold. This threshold condition is used to derive scaling laws for the turbulent heat flux, fluctuation spectra, and zonal flow amplitude, which are validated in nonlinear gyrokinetic simulations and explain previous experimental observations.

Figures (9)

• Figure 1: Normalised zonal flow velocity $\langle v_E^\mathrm{Z}/v_{Ti} \rangle_{\theta} R/\rho_i$ from saturated ITG turbulence far from marginality ($q=2.8, R/L_T = 10.4$) in real space (left) and the absolute value of its Fourier amplitude (right). Remarkably, the oscillation frequency of the secondary mode due to a primary streamer (dashed line) reproduces the frequency in the nonlinear turbulence simulation. The streamer has amplitude $v_{Ex}^P = \langle v_{Ex}^2\rangle_{y\theta}^{1/2} = \rho_i v_{Ti}/R$, is sinusoidal in the binormal direction with wavenumber $k_y^P \rho_i = 0.01$, is constant in $\theta$, and is bi-Maxwellian in velocity space with parallel temperature $T_\parallel^P = 0$ and a perpendicular temperature to density ratio of $T_\perp^P/T_i = 3 n^P/n_i$. We note that the details of the primary do not substantially affect the secondary mode frequency (see nies_theory_2025-1).
• Figure 2: Oscillation frequency (left) and growth rate (right) of secondary modes with radial wavenumber $k_x^S$ due to a primary streamer mode with varying amplitude, measured by $v_{Ex}^P = \langle v_{Ex}^2\rangle_{y\theta}^{1/2}$. The primary is a linear ITG mode with binormal wavenumber $k_y \rho_i = 0.05$, at $q=1.4$ and $R/L_T=13.9$.
• Figure 3: Generation of $(P+e_i n_i \varphi)^\mathrm{Z}$ asymmetry by nonlinear heat flux, illustrated above the tokamak midplane where $\tilde{v}_{Mx}>0$: starting from a streamer mode (left), the $v_E^\mathrm{Z}$ perturbation shears the $\mathop{}\!\boldsymbol{v}_E^\mathrm{NZ}$ and $P^\mathrm{NZ}$ contours equally, leaving the heat flux $\langle Q_\parallel + Q_\perp/2 \rangle_y \propto \langle v_{Ex}^\mathrm{NZ} P^\mathrm{NZ} \rangle_y$\ref{['eq:heat_flux']} unchanged (middle). Due to the advection by the velocity-dependent $\tilde{v}_{Mx}$\ref{['eq:vMx']}, a relative displacement (phase shift) between $\mathop{}\!\boldsymbol{v}_E^\mathrm{NZ}$ and $P^\mathrm{NZ}$ ensues, leading to a radial modulation of $\langle Q_\parallel+Q_\perp/2 \rangle_y$ (right). This mechanism is reversed below the midplane where $\tilde{v}_{Mx}<0$, causing an up-down asymmetry in the heat flux and thence in $(P+e_i n_i \varphi)^\mathrm{Z}$.
• Figure 4: Energy transfer rate $\mathcal{T}_{K_x}$\ref{['eq:energyflux']} normalised by the total injection rate $\mathcal{I}=\lim_{K_x\rightarrow \infty} \mathcal{I}_{K_x}=\langle Q \rangle_{xy\theta}/L_T$ in turbulence simulations. The contributions to the transfer rate (solid) are obtained by decomposing $\mathop{}\!\boldsymbol{v}_E$ in \ref{['eq:energyflux']} into its nonzonal (dashed), small-scale $\abs{k_x \rho_i} > 0.3$ zonal (dash-dotted), and large-scale $\abs{k_x \rho_i} \leq 0.3$ zonal (dotted) components. Different colours correspond to different simulations.
• Figure 5: Numerical experiments rescaling the nonzonal distribution function. The rescaling occurs at $t=t_\mathrm{scale}$ (vertical dotted line), during the saturated phase of the stella turbulence simulation with $R/L_T=13.9$ and $q=1.4$. The plots show the ZF velocity (color coded) as a function of $x$ and $t$, and the temporal evolution of the nonlinear heat flux $Q$ (green solid) and the ZF energy $E^\mathrm{ZF}$ (green dashed) \ref{['eq:E_ZF']}. The values of $Q$ and $E^\mathrm{ZF}$ are given on the right vertical axis.