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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. The box length resolves onepair of current filaments. A small, a medium-sized and a large filament areconsidered and compared. The magnetic field amplitude at the saturation time ofthe filamentation instability is proportional to the filament size. It isdemonstrated, that the force on the electrons imposed by the electrostaticfield, which develops during the nonlinear stage of the instability, oscillatesaround a mean value that equals the magnetic pressure gradient force. Theforces acting on the electrons due to the electrostatic and the magnetic fieldhave a similar strength. The electrostatic field reduces the confining forceclose to the stable equilibrium of each filament and increases it farther away.The confining potential is not sinusoidal, as assumed by the magnetic trappingmodel, and it permits an overlap of current filaments (plasmons) with anopposite flow direction. The scaling of the saturation amplitude of themagnetic field with the filament size observed here thus differs from thatexpected from the magnetic trapping model. The latter nevertheless gives a goodestimate for the magnetic saturation amplitude. The increase of the peakelectrostatic and magnetic field amplitudes with the filament size implies,that the electrons heat up more and that the spatial modulation of their meanspeed along the beam flow direction increases with the filament size.

The saturation of the electron beam filamentation instability by the  self-generated magnetic field and magnetic pressure gradient-driven electric  field