Identifying the linear phase of the relativistic Kelvin-Helmholtz instability and measuring its growth rate via radiation

For the relativistic Kelvin-Helmholtz instability (KHI), which occurs at shear interfaces between two plasma streams, we report results on the polarized radiation over all observation directions and frequencies emitted by the plasma electrons from ab initio kinetic simulations. We find the polarization of the radiation to provide a clear signature for distinguishing the linear phase of the KHI from its other phases. During the linear phase, we predict the growth rate of the KHI radiation power to match the growth rate of the KHI to a high degree. Our predictions are based on a model of the vortex dynamics, which describes the electron motion in the vicinity of the shear interface between the two streams. Albeit the complex and turbulent dynamics happening in the shear region, we find excellent agreement between our model and large-scale particle-in-cell simulations. Our findings pave the way for identifying the KHI linear regime and for measuring its growth rate in astrophysical jets observable on earth as well as in laboratory plasmas.

Figure 1

(Left) Illustration of the Kelvin-Helmholtz instability (KHI) in the relative velocity frame with two counterpropagating plasma streams, showing the magnetic field strength and electron density from the simulation [14] at $t=31{\omega }_{pe}^{-1}$. (Right) Illustrates vortex trajectories based on a microscopic model [Eq. (2)] and the emitted radiation with its characteristic polarization $P_x$ and $P_y$.

Figure 2

The polarization $〈P_j〉$ extracted from simulation, integrated over all observation directions, frequencies and particles [Eq. (4)]. The polarizations expected during the initial phase ($〈P_y^{\mathrm{syn}}〉=〈P_z^{\mathrm{syn}}〉=\frac{7}{16},〈P_x^{\mathrm{syn}}〉=\frac{2}{16}$) and the linear phase $〈{\tilde{P}}_j〉$ are highlighted with dashed lines. The growth rate ${\mathrm{\Gamma }}_{Bx}$ of the magnetic field energy is plotted on the right axis (dashed line).

Figure 3

(Left) Temporal evolution between the radiation power ${\dot{\epsilon }}_{\mathrm{rad}}$ from the entire stream and the magnetic field energy (${\epsilon }_{Bx},{\epsilon }_{Bz}$) on the shear surface are equal. The deviation between ${\epsilon }_{Bz}$ from the entire plasma and ${\dot{\epsilon }}_{\mathrm{rad}}$ shows that radiation is solely emitted from the KHI. (Right) Illustrates the equality of the growth rate ${\mathrm{\Gamma }}_{\mathrm{rad}}$ of the radiation power and the growth rate ${\mathrm{\Gamma }}_{Bx}$ of the entire magnetic field energy stored in the $B_x$ component, which best represents the evolution of $B_z$ on the shear during the linear phase.

Figure 4

The ratio between the polarization $〈P_y^{\prime }〉$ and $〈P_z^{\prime }〉$ for several boosts in $x$ direction with ${\gamma }_{\mathrm{boost}}$ being the respective Lorentz factor. The peaking anisotropy between the polarizations is a hallmark of the KHI linear phase.