Electron-scale Kelvin-Helmholtz instabilities (ESKHIs) are found in several astrophysical scenarios, ranging from the magnetopause [1] to active galactic nuclei (AGNs) and gamma-ray bursts (GRBs). Different from the classical KHI, the ESKHI is a collisionless electromagnetic instability which is able to take place in relativistic shear flows and generate magnetic fields [2], thus having a potential to explain the mechanism of field generation, particle acceleration and non-thermal radiations in relativistic jets.
In scenarios mentioned above, ESKHIs are naturally subject to a background magnetic field. However, an analytical dispersion relation and an accurate growth rate of ESKHI under such a circumstance are long absent. Former works concerning this problem rely on MHD assumptions which are not applicable in the relativistic regime [3,4], leading to obvious discrepancies from accurate theories (in absence of magnetic field) [5] and simulation results [4].
To address this problem, we avoid MHD assumptions and start with the two-fluid equations to derive a generalized dispersion relation of ESKHI in magnetized shear flows. The dispersion relation is generic to equilibrium profiles and numerically solvable for certain cases. From the numerical results, we conclude that the external magnetic field stabilizes ESKHI for small shear velocities but can destabilize it for large velocities. The external magnetic field also broadens the unstable band and increases the wavenumber of the most unstable mode. Moreover, in some cases we find activation of long-wavelength modes coupled with electromagnetic waves, which is both travelling and growing. These discoveries are way beyond the MHD theories of ESKHI.
In order to verify our analytical results, PIC simulations are also carried out with a fully kinetic code LAPINS. Most of our analytical conclusions are confirmed by the simulations. Especially, the wavenumbers of the most unstable modes in simulations reach a good agreement with our theoretical predictions. In simulations, we also find a suppressing of the kinetic DC magnetic field generation. We point out that it is related with electron gyration across the shear interface induced by the external magnetic field.
Our work shed light on the problem of ESKHI in magnetized shear flows. For the first time, we derived the dispersion relation of magnetized ESKHI in the relativistic regime, and analyzed the effect of the external magnetic field. These results may contribute to explanation of the field generation mechanism in relativistic shear flows.

instability,Kelvin-Helmholtz不稳定波