Observations, modeling and understanding of atmospheric tornadoes have known large developments these last ten years. In spite of the current power of computers, simplified models through exact solutions of the fluids dynamic equations are still an important tool to determine the singular conditions that start up these whirlwinds.
For that reason, we present in this work different contributions to the modeling of atmospheric tornadoes by using exact and similar solutions of the Navier-Stokes equations (called conical solutions). These solutions represent the interaction between a swirling half line vortex and the ground described by a no-slip horizontal plane.
By using these solutions, we took into account the thermomechanical effects in order to model a realistic phenomenology of the tornadoes. We first allowed the presence of a mass flux from the axis of the vortex, which is able to describe new phenomena near the axis. Secondly, we implemented thermal effects by new exact and similar solutions of the Boussinesq equations.
Then we considered a lost of revolution symmetry into the vortex in order to model situations illustrated by meteorological observations, and we presented a special case describing a vortex whose axis won't be perpendicular to the ground.
We finally developed a new model that allow to access to realistic orders of magnitude for the velocities. As observed in atmospheric tornadoes, it describes a severe swirling downdraft near the vertical axis and a slower flow whirling up from the ground.
With these contributions, we can now propose a model of the end lifetime of tornadoes.