First-order elliptical corrections to the gyroaveraging operation in high E-field-shear regions in XGC

In gyrokinetic Particle-in-Cell codes, a circular charged ring is used in the gyroaveraging operation. However, in a region with large electric field shear, like the edge pedestal and the Chodura sheath, the particle orbit departs from being circular, deformed by the presence of the electric field shear. At those locations, the use of circular rings is no longer a valid assumption, and corrections must be applied. In this work, we analytically solve the Newton-Lorentz equation for a single particle in the presence of a uniform magnetic field and a linear gradient electric field at an arbitrary angle. It is shown that the existence of an electric field gradient will deform the circular gyromotion to an elliptical one. Based on the result obtained, an explicit analytical expression for first-order elliptical corrections to the gyroaveraging operation is derived. The expression obtained is a function of the local field quantities and the 6D Lagrangian coordinates of the particle. To facilitate application in gyrokinetic codes, the analytical expression can be further reduced by exploiting the adiabatic invariant, it can then be expressed as a function of local field quantities and the magnetic moment. Implementation of the correction factor in the edge gyrokinetic code, XGC, will be presented using a DIII-D divertor plasma where the quasi-neutral sheath and H-mode pedestal possess a large electric field gradient.