Off-peak saturation effects of beam-plasma instability

For beam-plasma instability in the absence of a magnetic field, hybrid simulations produce almost identical results to those of particle simulations (both agree with linear and nonlinear analyses), but with much reduced computing cost and noise level. With the verification by simulations, it is shown that the saturation level is very strongly influenced by the discrete wave number spectrum, sometimes giving different, by an order of magnitude, results by choosing slightly different off-peak (in growth rate curve) parameters, which must be considered in most simulations (whether hybrid or particle or whether magnetized or unmagnetized)