  The guiding center and gyrokinetic theory of magnetized particle motion is
extended to the regime of large electric field gradients perpendicular to the
magnetic field. A gradient in the electric field directly modifies the
oscillation frequency and causes the Larmor orbits to deform from circular to
elliptical trajectories. In order to retain a good adiabatic invariant, there
can only be strong dependence on a single coordinate at lowest order, so that
resonances do not generate chaotic motion that destroys the invariant. When the
gradient across magnetic flux surfaces is dominant, the guiding center drift
velocity becomes anisotropic in response to external forces and additional
curvature drifts must be included. The electric polarization density remains
gyrotropic, but both the polarization and magnetization are modified by the
change in gyrofrequency. The theory can be applied to shear flows that are even
stronger than those observed in the edge transport barrier of a
high-performance tokamak (H-mode) pedestal, even if the toroidal field is as
small as or even smaller than the poloidal field. Yet, the theory retains a
mathematical form that is similar to the standard case and can readily be
implemented within existing simulation tools.
