  We present two dimensional (2D) particle-in-cell (PIC) simulations of 2D
Bernstein-Greene-Kruskal (BGK) modes, which are exact nonlinear steady-state
solutions of the Vlasov-Poisson equations, on a 2D plane perpendicular to a
background magnetic field, with a cylindrically symmetric electric potential
localized on the plane. PIC simulations are initialized using analytic electron
distributions and electric potentials from the theory. We confirm the validity
of such solutions using high-resolutions up to a 2048^2 grid. We show that the
solutions are dynamically stable for a stronger background magnetic field,
while keeping other parameters of the model fixed, but become unstable when the
field strength is weaker than a certain value. When a mode becomes unstable, we
observe that the instability begins with the excitation of azimuthal
electrostatic waves that ends with a spiral pattern.
