Theory of zonal flow growth and propagation in toroidal geometry

Abstract: 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 and geodesic acoustic modes. However, descriptions of the nonlinear zonal flow growth from a turbulent background typically resort to simplified models of the geometry. We present a generalised theory of the secondary instability to model the zonal flow growth from turbulent fluctuations in toroidal geometry, demonstrating that the radial magnetic drift substantially affects the nonlinear zonal flow dynamics. In particular, the toroidicity gives rise to a new branch of propagating zonal flows, the toroidal secondary mode, which is nonlinearly supported by the turbulence. We present a theory of this mode and compare the theory against gyrokinetic simulations of the secondary mode. The connection with other secondary modes - the ion-temperature-gradient and Rogers-Dorland-Kotschenreuther secondary modes - is also examined.