The morphology of electrostatic tripolar regions

Electrostatic tripolar regions in plasmas develop a skewness of their own electric potential waveform as a peculiar morphological property, which distinguishes them from symmetric electrostatic solitary waves. Within the collision-less, kinetic treatment developed here, this property holds if the velocity distributions of electrons and ions are singular in value, irrespective of their smoothness at the region’s boundary and of the smoothness of the potential waveform and of the electron and ion density distributions. These singularities are integrable, and are of the logarithmic and jump type: the former occur at isolated points in phase space; the latter occur on the left branch of the electron separatrix and on the left branch of the ion sub-separatrix. The distributions are non-negative if, at its local extrema, the potential waveform is skewed to the left, in agreement with observations, and if the skewness is smaller than a given bound: a sufficient condition for such skewness to be small about the minimum of the potential waveform is that a sufficiently fast electron beam exists on the high-potential boundary of the tripolar region. In those special cases in which the particle distributions are continuous in value, the above mentioned singularities affect their space and velocity derivatives. These results could be extracted from very general considerations on the degree of smoothness of the spatial distribution of the electric potential and on the non-negativity of the electron and ion distributions, without the assistance of any specific models.