Description
The toroidal geometry of tokamaks and stellarators has long been known to play a crucial role in the linear physics of zonal flows (ZFs), leading to the Rosenbluth-Hinton residual [1] for stationary ZFs and the rapidly oscillating geodesic acoustic modes [2]. However, descriptions of the nonlinear ZF dynamics due to the interaction with drift-wave turbulence typically resort to simplified models of the geometry which do not include toroidicity. We show that the toroidal geometry in fact substantially modifies the nonlinear ZF physics of both short- and long-wavelength ZFs. First, we develop a theory of short-wavelength ZFs by generalising the secondary instability theory of [3] to toroidal geometries, revealing a new nonlinear branch of propagating ZFs, the toroidal secondary mode (TSM) [4]. Second, we develop a gyrokinetic theory of the nonlinear stresses governing the evolution of stationary long-wavelength ZFs. We show that the saturation of these ZFs is governed by a balance between the ZF drive through perpendicular stresses and the ZF damping through parallel stresses, generalising the fluid theory of [5] to a gyrokinetic setting. The theory is verified through extensive gyrokinetic simulations using the code stella [6], and we consider the implications of our findings for the saturation of turbulence in the regimes of strongly-driven [7] and near-marginal (Dimits [8]) turbulence.
[1] Rosenbluth, M.N., Hinton, F.L., (1998) PRL 80
[2] Winsor, N et al. (1968). PoF 11
[3] Rogers, BN et al. (2000). PRL 85
[4] Nies, R., Parra, F. (2025). arXiv:2504.09785 (to be published in PPCF)
[5] Hallatschek, K. (2004) PRL 93
[6] Barnes, M. et al. (2019) JCP 391
[7] Nies R. et al. (2024) arXiv:2409.02283 (to be published in PRR)
[8] Dimits, A. et al. (2000) PoP 7
Supported by DOE DE-AC02-09CH11466 and a Leverhulme Trust International Professorship Grant to S. L. Sondhi (No. LIP-2020-014).