Theory of zonal flow growth and propagation in toroidal geometry*

The toroidal geometry of tokamaks and stellarators is known to play a crucial role in the linear physics of zonal flows, leading to e.g. the Rosenbluth-Hinton residual [1] and geodesic acoustic modes [2]. However, descriptions of the nonlinear zonal flow growth from a turbulent background typically resort to simplified models of the geometry. We present a generalized theory of the secondary instability that explains the zonal flow growth from turbulent fluctuations [3], demonstrating that the parallel streaming and radial magnetic drift substantially affect the nonlinear zonal flow dynamics. The classical secondary of [3] is shown to be altered at small turbulence drives, as neoclassical effects become dominant. Furthermore, we find a new branch of propagating zonal flows, the toroidal secondary. This new mode results from the Stringer spin-up [4] of zonal flows, where the required up-down asymmetric flux is caused by the magnetic drift advecting turbulent eddies sheared by the zonal flow. The generalized theory is validated by stella [5] gyrokinetic simulations of the secondary, and also explains the zonal flow behavior in fully nonlinear simulations.

[1] Rosenbluth, M.N., Hinton, F.L., (1998) Phys. Rev. Lett. 80, 724–727

[2] Winsor, N et al. (1968). The Physics of Fluids 11, 2448–2450

[3] Rogers, BN et al. (2000). Phys. Rev. Lett. 85, 5336–5339

[4] Stringer, TE. (1969). Phys. Rev. Lett. 22, 770–774

[5] Barnes, M. et al. (2019) Journal of Computational Physics 391, 365–380