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Tornado vortex is believed to be essentially nonlinear phenomenon; and the puzzle to choose the nonlinear term(s) responsible for its formation is still unresolved. In the present work we consider the nonlinear term associated with atmosphere humidity, by introducing variable temperature gradient depending on the vertical velocity of the fluid. Such term is able to yield energy to the system and is very suitable for such a problem. Other nonlinear terms are neglected, assuming slow rotation, or in other words a "weak" tornado approximation. We consider one-dimensional radial boundary problem, and use a modification of shooting method to satisfy boundary conditions at large radii. Obtained numerical solutions of the nonlinear differential equation qualitatively agree with the observed atmosphere vortices  (tornados, tropical cyclones). The obtained results show general possibility of existence of unstable motion even in convectively stable atmosphere stratification.