  A plasma rotating spoke in a crossed field discharge is studied using 2D
radial-azimuthal fully kinetic Particle-In-Cell Monte Carlo Collision (PIC/MCC)
simulations. The kinetic model reveals the whole perturbation spectrum of the
gradient drift instability in the linear stage: Simon-Hoh, lower-hybrid and ion
sound modes, providing direct evidence of the spoke of the gradient drift
instability nature. The two-fluid dispersion relation of the gradient drift
instability was utilized to analyze the linear development of instabilities in
the simulations. The charge separation effect was incorporated in the fluid
linear theory and a super-resolution signal processing method (multiple signal
classification) was applied to obtain the numerical frequency spectrum. The
simulated spectrum and growth rate show excellent agreement with the
theoretical dispersion relation (real frequency and imaginary frequency) over
investigated cases. The most linearly unstable mode was found to be the lower
hybrid instability and the mode transition into the m=1 macroscopic rotating
structure after saturation of the linear phase is accompanied by an inverse
energy cascade. In the nonlinear stage, the pronounced spoke phenomena can
occur when the heating of $\mathbf{E_{\theta}\times B}$ electron flow channeled
in the spoke front passage suffices to provide the enhanced ionization.
