Shearing rate of time-dependent E×B flow

Theory of ${E\times B}$ shear suppression of turbulence in toroidal geometry [Phys. Plasmas 2, 1648 (1995)] is extended to include fast time variations of the ${E\times B}$ flows often observed in nonlinear simulations of tokamak turbulence. It is shown that the quickly time varying components of the ${E\times B}$ flows, while they typically contribute significantly to the instantaneous ${E\times B}$ shearing rate, are less effective than the slowly time varying components in suppressing turbulence. This is because the shear flow pattern changes before eddies get distorted enough. The effective ${E\times B}$ shearing rate capturing this important physics is analytically derived and estimated from zonal flow statistics of gyrofluid simulation. This provides new insights into understanding recent gyrofluid and gyrokinetic simulations that yield a reduced, but not completely quenched, level of turbulence in the presence of turbulence-driven zonal flows.