Dusty debris in tornadoes modelled by high Reynolds number two cells vortices

Exact solutions of Navier-Stokes equations enable to describe nice features of atmospherical flows as tornadoes. For this very particular and singular kind of fluid motions, the flows are very often modelled by conical solutions. Famous authors as Burgers (1948) [J.M. Burgers, A mathematical model illustrating the theory of turbulence, Advan. Appl. Mech. 1 (1948) 197-199], Rott (1958) [N. Rott, On the viscous core of a line vortex, Z. Angew. Math. Phys. 96 (1958) 543-553], Serrin (1972) [J. Serrin, The swirling vortex, Phil. Trans. Roy. Soc. Lond. A 271 (1972) 320-360; J. Serrin, The swirling vortex, Phil. Trans. Roy. Soc. Lond. A 271 (1972) 357-358] and Goldshtik and Shtern (1990) [M. Goldshtik, V. Shtern, J. Fluid Mech. 218 (1990) 483-508], improved greatly research. Most of the time, except Serrin's model, these models applied to tornadoes, rather produced fields of velocity, with orders of magnitudes which are way too low to be appropriate for describing thunderstorms observed in meteorology. Moreover, these models do not include any explicit mechanisms which take into account the potential actions of dust particles often present in tornadoes. Here, we do suggest a new way of modelling the mature phase of a tornado by considering two swirling cells separated by a intermediate cone, which position results by the equilibrium of normal stresses exerted on the two sides of the cone. This choice of modelling is motivated by considering two kinds of flows, inside and outside the cone, to get significant and different characteristics of the associated resulting vortices. This equilibrium condition, completed by the discontinuity of the tangential stresses on the cone, leads to realistic magnitude of the velocity field, i.e., about 102 m s−1. This discontinuity is motivated by several observations which show the ejection of dust particles along a specific cone direction. At our modelling scale, we do want to model the integral of the dragging forces, resulting from the motion of those particles inside the fluid, by a discontinuity of the tangential stresses on the cone which split the flows into two separate cells.