  A gyrokinetic model is presented that can properly describe strong flows,
large and small amplitude electromagnetic fluctuations occurring on scale
lengths ranging from the electron Larmor radius to the equilibrium
perpendicular pressure gradient scale length, and large deviations from thermal
equilibrium. The formulation of the gyrokinetic model is based on a second
order description of the single charged particle dynamics, derived from Lie
perturbation theory, where the fast particle gyromotion is decoupled from the
slow drifts, assuming that the ratio of the ion sound Larmor radius to the
perpendicular equilibrium pressure scale length is small. The collective
behavior of the plasma is obtained by a gyrokinetic Boltzmann equation that
describes the evolution of the gyroaveraged distribution function and includes
a non-linear gyrokinetic Dougherty collision operator. The gyrokinetic model is
then developed into a set of coupled fluid equations referred to as the
gyrokinetic moment hierarchy. To obtain this hierarchy, the gyroaveraged
distribution function is expanded onto a velocity-space Hermite-Laguerre
polynomial basis and the gyrokinetic equation is projected onto the same basis,
obtaining the spatial and temporal evolution of the Hermite-Laguerre expansion
coefficients. The Hermite-Laguerre projection is performed accurately at
arbitrary perpendicular wavenumber values. Finally, the self-consistent
evolution of the electromagnetic fields is described by a set of gyrokinetic
Maxwell's equations derived from a variational principle, with the velocity
integrals of the gyroaveraged distribution function explicitly evaluated.
