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The spatial-temporal evolution of the purely transverse current filamentationinstability is analyzed by deriving a single partial differential equation forthe instability and obtaining the analytical solutions for the spatially andtemporally growing current filament mode. When the beam front always encountersfresh plasma, our analysis shows that the instability grows spatially from thebeam front to the back up to a certain critical beam length; then theinstability acquires a purely temporal growth. This critical beam lengthincreases linearly with time and in the non-relativistic regime it isproportional to the beam velocity. In the relativistic regime the criticallength is inversely proportional to the cube of the beam Lorentz factor$\gamma_{0b}$. Thus, in the ultra-relativistic regime the instabilityimmediately acquires a purely temporal growth all over the beam. The analyticalresults are in good agreement with multidimensional particle-in-cellsimulations performed with OSIRIS. Relevance of current study to recent andfuture experiments on fireball beams is also addressed.

Spatial-temporal evolution of the current filamentation instability