Transient growth in stable linearized Vlasov–Maxwell plasmas

Large amplitude transient growth of kinetic scale perturbations in stable collisionless magnetized plasmas has recently been demonstrated using a linearized Landau fluid model. Initial perturbations with lengthscales of the order of the ion gyroradius were shown to have transient timescales that in some cases were long compared to the ion gyroperiod, Ωit⪢1. Moreover, it was suggested that such perturbations are not rare but instead form a large class within the set of all possible initial conditions. For collisionless plasmas, the Vlasov–Maxwell equations provide a more complete description of kinetic physics and the existence of transient growth of solutions for the linearized Vlasov–Maxwell system is an interesting question. The existence of transient growth of solutions is demonstrated here for a special case of the Vlasov–Maxwell equations, namely, the one dimensional Vlasov–Poisson system. The analysis is different from the standard approach of nonmodal analysis since the initial value problem is described by a Volterra integral equation of the second kind, reflecting the fact that the time evolution of the system depends on the memory of the state from time zero through time t. For the case of a thermal equilibrium plasma, it is shown how initial conditions may be constructed to obtain solutions that grow linearly in time; the duration of this growth is the time required for a thermal electron to traverse the wavelength of the initial perturbation, a timescale that can last for many plasma periods 2π/ωpe, thus demonstrating the existence of transient growth of solutions for the linearized Vlasov–Poisson system. The results suggest that the phenomenon of transient growth may be a common feature of the linearized Vlasov–Maxwell system as well as for Landau fluid models.