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Existence and repeatability of tornadoes could be straightforwardly explained if there existed instability,  responsible for their formation.However, it is well known that convection is the only instability in initially stable air, and the usual convective instability is not applicable for these phenomena. In the present paper we describe an instability in the atmosphere, which can be responsible for intence vortices. This instability appears in a fluid with Coriolis force and dissipation and has oscillatory behaviour, where the amplitude growth is accompanied by oscillations with freguency comparable to the growth rate of the instability. In the paper, both analytycal analysis of the linear phase of the instability and nonlinear simulation of the developed stage of the air motion are addressed. This work was supported by the RFBR grant no. 09-05-00374-a.