Description
It is widely known that cross field $E\times B$ shear stabilises ion temperature gradient (ITG) turbulence in magnetised plasma via deforming eddies to dissipation scales. However, in the presence of a sheared magnetic field, this effect can be negated for turbulence moving at an angular frequency of $\Omega_E = \gamma_E/\hat{s}$, where $\gamma_E$ is shearing rate and $\hat{s}$ is magnetic shear. Due to variations in curvature along a given magnetic field in a tokamak, our instantaneous growth rates for these modes oscillate periodically in time. However our growth rates averaged over the period $T = 2\pi/\Omega_E$ are constant complex numbers. This behaviour is characteristic of Floquet modes, which have been heavily studied and are seen in an a range of physically important systems.
We perform the transform as seen in Maeyama et al. (2025), before collapsing to two dimensions and integrating to produce fluid equations by taking the approximations of Ivanov et al. (2020). We then impose the geometry of large-aspect ratio circular flux surfaces. We then numerically study these equations linearly and non-linearly. At sufficiently high $\Omega_E$, we find our linear modes and non-linear turbulence are completely suppressed. However, we find that maximum linear growth rates and non-linear heat flux do not decrease monotonically with increasing $\Omega_E$, as they would for flow shear alone. We use qualitative models to explore this effect and predict the point at which complete suppression of turbulence happens.