Description
Zonal flows (ZFs) are radially-sheared $\mathbf{E}\times\mathbf{B}$ poloidal flows that are believed to play a role in the L-H transition in tokamaks. The zonal potential $\phi_\mathrm{ZF}=\phi(k_x, k_y=0)$ is constant on a flux surface, so it does not drive radial transport; on the contrary, the variation of $\phi_\mathrm{ZF}$ across flux surfaces causes differential rotation that shears eddies apart, causing transport to saturate at lower levels. ZFs are driven nonlinearly through the Reynolds-Maxwell stress arising from drift wave turbulence. Energy transfer into ZFs from turbulence then naturally leads to a reduction in energy in non-zonal ($k_y \neq 0$) turbulence. Previous attempts to measure this transfer during the L-H transition have been limited by the poloidal extent of turbulence diagnostics. To facilitate comparisons with experimental data, and to probe ZF dynamics numerically, a diagnostic has recently been added to the gyrokinetic code GS2 that resolves the nonlinear free energy transfer to ZFs as a function of $(t, \theta, k_{xs}, k_{ys}, k_{xt})$; that is, time, poloidal angle, source $k_x$ and $k_y$, and target $k_x$. This is distinct from an existing nonlinear kinetic energy transfer diagnostic that adopts a fluid-based approach [T. M. Schuett et al, Plasma Phys. Control. Fusion 67 115022 (2025)]. The new diagnostic adopts a kinetic approach (determining $\partial_t |h_s (k_y = 0)|^2$ rather than $\partial_t |\mathbf{u}_{\mathbf{E}\times\mathbf{B}}(k_y = 0)|^2$) and is fully electromagnetic ($\delta A_\parallel$ and $\delta B_\parallel$) and species-resolved. It has been observed in nonlinear electromagnetic gyrokinetic simulations that above a critical $\beta$, zonal flow formation is suppressed and transport saturates at much higher levels [M. J. Pueschel et al, Phys. Rev. Lett. 110, 155005 (2013)]. Recent work has shown that it's possible to access saturated states above this critical $\beta$, provided persistent mesoscale ZF patterns can develop [F. Rath & A. G. Peeters, Phys. Plasmas 29, 042305 (2022)]. This work probes this further and also provides insights into the effects of shaping on ZF drive at finite $\beta$.