Detecting Shearless Phase-Space Transport Barriers in Global Gyrokinetic Turbulence Simulations with Test Particle Map Models

TL;DR

This work addresses how weakly sheared, non-degenerate regions inside zonal $E\times B$ jets can form robust transport barriers in gyrokinetic turbulence. By constructing a single-mode, rigidly rotating test-particle model around the zonal jet and reducing gyrokinetic dynamics to a planar map, the authors identify shearless invariant tori via the kinetic rotation number $q_{kin}$ and demonstrate their persistence as shearless phase-space barriers. In direct comparison with self-consistent XGC simulations, these barriers manifest as reduced cross-barrier particle transport (transmissivity $\eta_t$, notably for trapped particles) and as organized phase-space structures in $F_i$ that resist radial mixing, with avalanches terminating through barrier reconnection and blob spin analogous to oceanic eddy detachment. The findings suggest that shearless transport barriers, arising from zonal-flow curvature and tertiary-instability dynamics, may be generic across turbulence regimes and influence density peaking and turbulence spreading in reactor-relevant plasmas. They also outline how extending these map-based insights to collisional and broadband regimes could inform confinement strategies and turbulence control in fusion devices.

Abstract

In magnetically confined fusion plasmas, the role played by zonal E$\times$B flow shear layers in the suppression of turbulent transport is relatively well-understood. However, less is understood about the role played by the weak shear regions that arise in the non-monotonic radial electric field profiles often associated with these shear layers. In electrostatic simulations from the global total-f gyrokinetic particle-in-cell code XGC, we demonstrate how shearless regions with non-zero flow curvature form zonal "jets" that, in conjunction with neighboring regions of shear, can act as robust barriers to particle transport and turbulence spreading. By isolating quasi-coherent fluctuations radially localized to the zonal jets, we construct a map model for the Lagrangian dynamics of gyrokinetic test particles in the presence of drift waves. We identify the presence of shearless invariant tori in this model and verify that these tori act as partial phase-space transport barriers in the simulations. We also demonstrate how avalanches impinging on these shearless tori cause reconnection events that form "cold/warm core ring" structures analogous to those found in oceanic jets, facilitating transport across the barriers without destroying them completely. We discuss how shearless tori may generically arise from tertiary instabilities or other types of discrete eigenmodes, suggesting their potential relevance to broader classes of turbulent fluctuations.

Figures (11)

• Figure 1: (a-b) Equilibrium profiles used to initialize the simulations. The region of interest is marked between the vertical gray lines. (c) Poloidal cross-section showing the non-zonal component of the electrostatic potential, as as $v_{E}$ on the outboard midplane. (d-h) Temporally-averaged quantities after several turbulence times as a function of radial coordinate, including the zonal $E \times B$ rotation rate $\Omega_E$, the Waltz-Miller shearing rate $\gamma_E$, ion gyrocenter density gradient scale length $a/L_{N_i}$, ion temperature gradient $a/L_{T_i}$, and density skewness $\langle \tilde{n}_e^3 \rangle / \langle \tilde{n}_e^2 \rangle^{3/2}$. The shearless region $\psi=\psi^*$ is marked with a dashed vertical line.
• Figure 2: Sequence of color plots showing the evolution of various flux-surface-averaged quantities over time and radial coordinate. The shearless region is overplotted with a gray line on all of the plots.
• Figure 3: (a) Toroidal mode spectra and (b) angular phase velocities, both measured at one instant in time. $\langle \phi_n^2 \rangle$ is the flux surface averaged squared electrostatic potential, and $\hat{\Omega}_n$ is the toroidally-directed angular phase velocity. The quantity $\Gamma_0 k_\theta^2 \langle \phi_n^2 \rangle$ mimics the gyroaveraged $\langle\mathcal{J}[E_\perp]^2\rangle$ spectrum. Different toroidal mode numbers are shown with different colors and linestyles. $\Omega_E$ is shown on both plots for reference, and a dashed black line is also plotted over the angular phase velocities show the rotation rate $\Omega$ used for the model fluctuations.
• Figure 4: Poloidal slices of (a) The electrostatic potential $\phi_n$ for the $n=39$ Fourier mode, and (b) the single-mode model electrostatic potential $\delta \hat{\psi}$ at a single instant in time. The fields are plotted against the radial coordinate $\psi_n$ and the perpendicular field line label $\alpha = \varphi - q \theta$. The shearless region $\psi=\psi^*$ is demarcated with a dashed gray line. (c) Space-time plot showing the value of the electrostatic potential fluctuations $\delta \phi$, evaluated at the shearless region $\psi = \psi^*$ plotted against $\alpha$. A line with $\varphi \propto \Omega t$ is shown demonstrating the fixed phase velocity of the fluctuations.
• Figure 5: Plots of $q_{kin}(P_\varphi)$ for (a) passing $\mathcal{E}_\perp/\mathcal{E} = 1/3$ and (b) trapped particles $\mathcal{E}_\perp/\mathcal{E} = 2/3$ in the vicinity of the zonal jet, with the kinetic energy $\mathcal{E}$ equal to the ion temperature in the shearless region. The ratio of perpendicular to total kinetic energy $\mathcal{E}_\perp/\mathcal{E}$ is taken at the outboard midplane. The magnetic safety factor $q(\psi)$ and $E \times B$ rotation $\Omega_E(\psi)$ divided by the bounce frequency $\omega_b$ are shown for comparison.

Theorems & Definitions (2)

• Proposition 1
• proof