  A gyrokinetic theory is developed under a set of orderings applicable to the
edge region of tokamaks and other magnetic confinement devices, as well as to
internal transport barriers. The result is a practical set equations that is
valid for large perturbation amplitudes [q{\delta}{\psi}/T = O(1), where
{\delta}{\psi} = {\delta}{\phi} - v_par {\delta}A_par/c], which is
straightforward to implement numerically, and which has straightforward
expressions for its conservation properties. Here, q is the particle charge,
{\delta}{\phi} and {\delta}A_par are the perturbed electrostatic and parallel
magnetic potentials, v_par is the parallel velocity, c is the speed of light,
and T is the temperature. The derivation is based on the quantity
{\epsilon}:=({\rho}/{\lambda})q{\delta}{\psi}/T << 1 as the small expansion
parameter, where {\rho} is the gyroradius and {\lambda} is the perpendicular
wavelength. Physically, this ordering requires that the E\times B velocity and
the component of the parallel velocity perpendicular to the equilibrium
magnetic field are small compared to the thermal velocity. For nonlinear
fluctuations saturated at "mixing-length" levels (i.e., at a level such that
driving gradients in profile quantities are locally flattened), {\epsilon} is
of order {\rho}/L, where L is the equilibrium profile scale length, for all
scales {\lambda} ranging from {\rho} to L. This is true even though
q{\delta}{\psi}/T = O(1) for {\lambda} ~ L. Significant additional
simplifications result from ordering L/R =O({\epsilon}), where R is the spatial
scale of variation of the magnetic field. We argue that these orderings are
well satisfied in strong-gradient regions, such as edge and screapeoff layer
regions and internal transport barriers in tokamaks, and anticipate that our
equations will be useful as a basis for simulation models for these regions.
