Dr Mark Dieckmann
The filamentation instability (FI) of counter-propagating beams of charged particles results in the aperiodic growth of magnetic fields. The FI is driven in its simplest form by two spatially uniform, counter-propagating and unmagnetized beams of electrons that have the same density and temperature. We consider this case here. The growing magnetic fields rearrange the electrons in the plane perpendicular to the beam velocity vector. The electron beams are spatially separated and form current filaments. The strong currents running through these filaments yield powerful magnetic fields perpendicular to the beam velocity vector. It is this magnetic field that makes the FI interesting as a source mechanism for the strong magnetic fields that occur in astrophysical jets, such as those triggering the long GRBs. Being one of the elementary kinetic instabilities, the FI has been widely investigated by many authors, both analytically and numerically. The unprecedented spatio-temporal scales, which are now accessible to particle-in-cell (PIC) simulations through modern parallel computers, imply that the statistical properties of a large ensemble of current filaments can be investigated. We present the results of our recent PIC simulations, addressing some of these statistical properties. We show with a 1D PIC simulation, which resolves one direction orthogonal to the beam velocity vector, that the size distribution of the current filaments can be approximated by a Gumbel distribution, when the FI has just saturated. The saturation is enforced by the electrostatic fields that are driven by the pressure gradient of the magnetic fields, which are tied to the current filaments. The 1D PIC simulation inhibits the merger of the current filaments and the system is frozen-in at the instant of its nonlinear saturation. A 2D PIC simulation, which resolves the plane perpendicular to the beam velocity vector, enables such filament mergers and we demonstrate with two case studies that the characteristic size of the filaments increases linearly with time.