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Two counter-propagating cool and equally dense electron beams are modelledwith particle-in-cell (PIC) simulations. The electron beam filamentationinstability is examined in one spatial dimension, which is an approximation fora quasi-planar filament boundary. It is confirmed, that the force on theelectrons imposed by the electrostatic field, which develops during thenonlinear stage of the instability, oscillates around a mean value that equalsthe magnetic pressure gradient force. The forces acting on the electrons due tothe electrostatic and the magnetic field have a similar strength. Theelectrostatic field reduces the confining force close to the stable equilibriumof each filament and increases it farther away, limiting the peak density. Theconfining time-averaged total potential permits an overlap of current filamentswith an opposite flow direction.

One-dimensional particle simulation of the filamentation instability: electrostatic field driven by the magnetic pressure gradient force