Electromagnetic Electron Kelvin-Helmholtz Instability

On electron kinetic scales, ions and electrons decouple, and electron velocity shear on electron inertial length $\sim d_e$ can trigger electromagnetic (EM) electron Kelvin-Helmholtz instability (EKHI). In this paper, we present an analytic study of EM EKHI in an inviscid collisionless plasma with a step-function electron shear flow. We show that in incompressible collisionless plasma the ideal electron frozen-in condition $\mathbf{E} + \mathbf{v}_e \times \mathbf{B}/c = 0$ must be broken for the EM EKHI to occur. In a step-function electron shear flow, the ideal electron frozen-in condition is replaced by magnetic flux conservation, i.e., $\nabla \times (\mathbf{E} + \mathbf{v}_e\times \mathbf{B}/c) = 0$, resulting in a dispersion relation similar to that of the standard ideal and incompressible magnetohydrodynamics KHI. The magnetic field parallel to the electron streaming suppresses the EM EKHI due to magnetic tension. The threshold for the EM mode of the EKHI is $(\mathbf{k}\cdot\Delta\mathbf{U}_e)^2>\frac{n_{e1}+n_{e2}}{n_{e1} n_{e2}}[n_{e1}(\mathbf{v}_{Ae1}\cdot\mathbf{k})^2+n_{e2}(\mathbf{v}_{Ae2}\cdot\mathbf{k})^2]$, where $\mathbf{v}_{Ae} =\mathbf{B}/(4\pi m_e n_e)^{1/2}$, $\Delta\mathbf{U}_e$ and $n_e$ are the electron streaming velocity shear and densities, respectively. The growth rate of the EM mode is $\gamma_{em} \sim \Omega_{ce}$, the electron gyro-frequency.