The Kelvin-Hemholtz (K-H) instability in the corona EUV jet is studied via 2.5 MHD numerical simulations. The jet results from magnetic reconnection due to the interation of the new emerging magnetic field and the pre-existing magnetic field in the corona. Our results show that the Alfvén Mach number along the jet is about 5-14 just before the instability occurs, and it is even higher than 14 at some local areas. During the K-H instability process, several vortex- like plasma blobs of high temperature and high density appear along the jet, and magnetic fields have also been rolled up and the magnetic configuration including anti-parallel magnetic fields forms, which leads to magnetic reconnection at many X-points and current sheet fragments inside the vortex-like blob. After magnetic islands appear inside the main current sheet, the total kinetic energy of the reconnection outflows decreases, and cannot support the formation of the vortex-like blob along the jet any longer, then the K-H instability eventually disappears. We also present the results about how the guide field and the flux emerging speed affect the K-H instability. We find that the strong guide field inhibits the shock formation in the reconnection upward outflow regions but helps secondary magnetic islands appear earlier in the main current sheet, and then apparently suppresses the K-H instability. As the speed of the emerging magnetic field decreases, the K-H instability appears later, the highest temperature inside the vortex blob gets lower and the vortex structure gets smaller.
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