Magnetic field generation and open-quotes Weibel resonanceclose quotes in inhomogeneous plasmas

The Weibel instability is an effective general mechanism of magnetic field generation in anisotropic plasmas. Recently, it has been invoked in order to explain the magnetic field wake observed in PIC simulations of the interaction of ultrashort ultraintense laser pulses with an underdense plasma. In this case the Weibel instability is thought to arise because of the formation in the plasma of two opposite streams of electrons. When the electric currents carried by these streams are displaced one respect to the other by a transversal disturbance, a magnetic field is produced that grows because the repulsion of the oppositely directed currents reinforces the initial displacement. The evolution of this instability can be described, before wavebreaking takes place, within the framework of the (relativistic) EMHD equations where the ions axe considered to be at rest and the electrons are described as a (relativistic) fluid. (1) If the plasma is inhomogeneous and/or if the two electron streams are spatially localized, the EMHD equations show the presence of a singularity which is reminiscent of the Alfven resonance, found for Shear-Alfven waves. (2) In the simple case of two symmetric streams, this open-quotes Weibel resonanceclose quotes occurs where γ2= 2v2 where γ is the normalized growth rate and v the local value of the normalized electron velocity. As a result, a resonant mode localized around the singularity develops from an initial perturbation at a rate which depends on the characteristic gradient of the electron velocities. (3) The growth rate of this mode is nearly independent of the initial wavelength of the perturbation and is comparable to the growth rate that would be obtained with homogeneous electron velocities equal to their maximum value in the inhomogeneous case