Instabilities in a current sheet with plasma jet

We study the stability problem of a magnetohydrodynamic current sheet with the presence of a plasma jet. The flow direction is perpendicular to the normal of the current sheet and we analyze two cases: (1) The flow is along the anti-parallel component of the magnetic field; (2) The flow is perpendicular to the anti-parallel component of the magnetic field. A generalized equation set with the condition of incompressibility is derived and solved as a boundary-value-problem. For the first case, we show that, the streaming kink mode is stabilized by the magnetic field at $V_0/B_0 \lesssim 2$, where $V_0$ and $B_0$ are the jet speed and upstream Alfv\'en speed, and it is not affected by resistivity significantly. The streaming sausage mode is stabilized at $V_0/B_0 \lesssim 1$, and it can transit to the streaming tearing mode with a finite resistivity. The streaming tearing mode has larger growth rate than the pure tearing mode, though the scaling relation between the maximum growth rate and the Lundquist number remains unchanged. When the jet is perpendicular to the anti-parallel component of the magnetic field, the most unstable sausage mode is usually perpendicular (wave vector along the jet) without a guide field. But with a finite guide field, the most unstable sausage mode can be oblique, depending on the jet speed and guide field strength.