Growth of filaments and saturation of the filamentation instability

The filamentation instability of counterstreaming beams is a nonresonant hydrodynamic-type instability whose growth rate is a smooth function of the wavelength (scale). As a result, perturbations with all unstable wavelengths develop, and the growth saturates due to the saturation of available current. For a given scale, the magnetic field at saturation is proportional to the scale. As a result, the instability develops in a nearly linear regime, where the unstable modes stop growing as soon as the saturation of the corresponding wavelength is reached. At each moment there exists a dominant scale of the magnetic field which is the scale that reached saturation at this particular time. The smaller scales do not disappear and can be easily distinguished in the current structure. The overall growth of the instability stops when the loss of the streaming ion energy because of deceleration is comparable to the initial ion energy.