Electromagnetic corrections to the zonal flow residual

The axisymmetric zonal flow residual calculation in tokamak plasmas is generalized to include electromagnetic perturbations. Instead of imposing magnetic perturbations externally, we formulate and solve a description retaining the fully self-consistent temporal and spatial perturbations in the electric and magnetic fields. Simple expressions for the electrostatic, shear and compressional magnetic residual responses derived provide a fully electromagnetic test of the zonal flow residual in gyrokinetic codes. We find that at $\beta\sim\mathcal{O}$(1) the most easily testable quantity is the compressional magnetic perturbation generated by the density perturbation corresponding to the zonal flow potential, while at small values of $\beta$, the electrostatic and shear magnetic responses to an initial compressional magnetic perturbation can also be detectable. Without collisions any initial magnetic perturbation remain completely undamped.