Three-Dimensional Bernstein-Greene-Kruskal Modes with Finite Field Magnitude

Previously we presented analytic forms of three-dimensional (3D) Bernstein-Greene-Kruskal (BGK) modes in a magnetized plasma with a finite background uniform magnetic field. The existence of solutions was shown analytically and by numerical solutions in the limit of small field magnitude of the modes. We have now developed an iteration scheme that can solve for converged solutions with moderately large field magnitudes. These modes are exact nonlinear solutions of the steady-state Vlasov equation with an electric potential localized in all three spatial dimensions that satisfies the Poisson equation self-consistently, as well as a localized self-consistent magnetic field perturbation satisfying the Ampère Law. Dynamics of both ions and electrons are included in the formulation. These solutions have cylindrical symmetry with distribution functions of ion and electron depending on particle energy, and a disk species with distribution depending also on the axial component of the canonical angular momentum. Example solutions will be presented, showing that the magnetic field of the mode is strong enough to perturb the background magnetic field significantly.