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 $mathbf{E} + mathbf{v}_e imes mathbf{B}/c = 0$ 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., $