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The spontaneous filamentation of a dielectric barrier glow discharge plasma (Townsend, not streamer breakdown), i.e., an instability of the homogeneous state has been simulated and understood with the help of a self-consistent two-dimensional fluid model of the discharge. The formation of self-organized or solitary filaments observed experimentally and described in previous papers can be explained in terms of electron and ion transport coefficients only, without including gas heating, plasma chemistry or surface effects. The conditions favoring the plasma filamentation are discussed. © 1999 American Institute of Physics.