$E\times B$ shear suppression of microtearing based transport in spherical tokamaks

Electromagnetic microtearing modes (MTMs) have been observed in many different spherical tokamak regimes. Understanding how these and other electromagnetic modes nonlinearly saturate is likely critical in understanding the confinement of a high $\beta$ spherical tokamak (ST). Equilibrium $E\times B$ sheared flows have sometimes been found to significantly suppress low $\beta$ ion scale transport in both gyrokinetic simulations and in experiment. This work aims to understand the conditions under which $E\times B$ sheared flow impacts on the saturation of MTM simulations. Two experimental regimes are examined from MAST and NSTX, on surfaces that have unstable MTMs. The MTM driven transport on a local flux surface in MAST is shown to be more resilient to suppression via $E\times B$ shear, compared to the case from NSTX where the MTM transport is found to be significantly suppressed. This difference in the response to flow shear is explained through the impact of magnetic shear, $\hat{s}$ on the MTM linear growth rate dependence on ballooning angle, $\theta_0$. At low $\hat{s}$, the growth rate depends weakly on $\theta_0$, but at higher $\hat{s}$, the MTM growth rate peaks at $\theta_0 = 0$, with regions of stability at higher $\theta_0$. Equilibrium $E\times B$ sheared flows act to advect the $\theta_0$ of a mode in time, providing a mechanism which suppresses the transport from these modes when they become stable. The dependence of $\gamma^{MTM}$ on $\theta_0$ is in qualitative agreement with a recent theory [M.R. Hardman et al (2023)] at low $\beta$ when $q\sim1$, but the agreement worsens at higher $q$ where the theory breaks down. This work highlights the important role of the safety factor profile in determining the impact of equilibrium $E\times B$ shear on the saturation level of MTM turbulence.