Microstructure formation in radially counterstreaming electron flows

A theoretical model is developed to describe the formation of microstructures due plasma streaming instabilities in radially convergent geometries. Microstructures form radial spokes are found experimentally laser wakefield accelerators. The eigenvalues a set coupled linear ordinary differential equations obtained and complex wavenumbers calculated give local growth rates. predictions confirmed using particle-in-cell (PIC) simulations carried out for two counter-propagating converging/diverging annuli. consistently demonstrate unstable interactions between counterpropagating annuli with different number densities. rates from PIC agree well semi-analytical model. theory presented this paper can provide powerful insight into converging beams space laboratory scale plasma.