Kinetic instability in inductively oscillatory plasma equilibrium

A uniform in space, oscillatory in time plasma equilibrium sustained by a time-dependent current density is analytically and numerically studied resorting to particle-in-cell simulations. The dispersion relation is derived from the Vlasov equation for oscillating equilibrium distribution functions, and used to demonstrate that the plasma has an infinite number of unstable kinetic modes. This instability represents a kinetic mechanism for the decay of the initial mode of infinite wavelength (or equivalently null wave number), for which no classical wave breaking or Landau damping exists. The relativistic generalization of the instability is discussed. In this regime, the growth rate of the fastest growing unstable modes scales with ${\gamma }_T^{-1/2}$, where ${\gamma }_T$ is the largest Lorentz factor of the plasma distribution. This result hints that this instability is not as severely suppressed for large Lorentz factor flows as purely streaming instabilities. The relevance of this instability in inductive electric field oscillations driven in pulsar magnetospheres is discussed.

Figure 1

Theoretical prediction of real and imaginary components of wave frequencies excited with $\delta v/c=0.14$ and $\mathrm{\Delta }v/c=0.1$: (a) shows the purely real solutions obtained from individual branches of the full dispersion relation in Eq. (4). Lines labeled with $n^{\pm }$ correspond to solutions where only term $n$ was kept in the series and where $\partial {\omega }_r/\partial k\gtrless 0$ for $k\gg 1$, respectively; (b) illustrates the growth rate of unstable modes obtained by combining the symmetric branches of Eq. (4).

Figure 2

Temporal evolution of electron phase space (in color) and electric field (in black lines) for a simulation with $E_0/(m_{e}c{\omega }_0/e)=\delta v/c=0.14$ and $\mathrm{\Delta }v/c=0.1$: (a) Initial plasma configuration, (b) linear stage of the instability, showing electric field perturbations grown on top of its uniform oscillatory component, and (c) final plasma state, after the instability has saturated.

Figure 3

Fourier analysis of unstable modes in the simulation illustrated in Fig. 2: (a) and (b) show respectively the time evolution of the electric field Fourier transform and of the energy in $k$ bands corresponding to the main oscillatory mode (R1) and to the three unstable modes with lowest $k$ (R2–R4); (c) illustrates the numerical plasma dispersion relation, obtained by taking the Fourier transform of (a) in time during the linear stage of the instability.