### Fluid Dynamics

#### 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 10^{2} 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.

#### Appearance of a source/sink line into a swirling vortex.

Several mathematical models applied to tornadoes consist of exact and axisymmetric solutions of the steady and incompressible Navier{Stokes equations. These models studied by Serrin,^{9} Goldshtik and Shtern^{8} describe families of fluid motions vanishing at the ground and are restricted not to develop a source nor a sink near the vortex line. Therefore, Serrin showed that the flow patterns of the resulting velocity field may have some realistic characteristics to model the mature phase of the lifetime of a tornado, in comparison with atmospheric observations. On the other hand, no reason has been given to motivate the restriction of the absence of a source/sink vortex line. Therefore, we present here the construction and the analysis of a fluid motion driven by the vertical shear near the ground, the rate of the azimuthal rotation and by the intensity of a central source/sink line. We prove the local existence and uniqueness of a family of fluid motions, leading to the genesis of such source/sink lines inside a non-rotating updraft which does not develop, before perturbation, a source nor a sink.

#### Non-axisymmetric boundary layer induced by a swirling downdraft.

Most mathematical models of conical swirling flows are exact axisymmetrical solutions of the steady and incompressible Navier–Stokes equations. Some of these models take into account the coupling mechanisms (Chaskalovic and Chauvière 1999, Shtern et al. 1998), but very few mathematical studies try to model non-axisymmetrical conical flows. We propose here a generalisation of the formulation used by Aristov (1998) to study a swirling vertical downdraft limited by a nearly horizontal plane. The use of asymptotic analysis for high Reynolds numbers allows to find non-axisymmetrical analytical solutions for the whole flow.

#### Thermal convection into a swirling source/sink line vortex.

These last ten years, observations of atmospheric tornadoes knew a very

large development. The main goal of this kind of exploration is to determine singular conditions that are responsible of the genesis of this variety of thunderstorms. These investigations made it possible to conclude that one of the essential mechanisms of this phenomenon is due to a thermomechanical process, resulting of a strong gradient of temperature, driving the fluid between the ground and the atmosphere. In this paper, we present exact solutions of Navier-Stokes equations linked with the energy equation, which modeled this type of thermomechanical process. Our model consists of stationary and axisymmetrical flows of a viscous fluid in a half space, limited by a horizontal plane, in the Boussinesq approximation. These solutions, which satisfy the no-slip condition at the ground, belong to a family of fields of velocity and temperature that depends on five parameters. The mathematical problem we consider is derived by the streamfunction-vorticity method in order to substitute the classical formulation of Navier-Stokes equations.

#### Genesis of a swirling source/sink line into an updraft.

Several mathematical models applied to tornadoes consist of exact and axisymmetric solutions of the steady and incompressible Navier-Stokes equations. These models studied by J. Serrin [5] and M.A. Gol’dshtik and V.N. Sthern (1995) describe families of fluid motions vanishing at the ground and are restricted for not developing neither a source nor a sink near the vortex line. Therefore, J. Serrin showed that the flow patterns of the resulting velocity field may have some realistic characteristics to model the mature phase of the lifetime of a tornado, in comparison with atmospheric observations. On another hand, no reason has been explicated to motivate the absence of a source/sink vortex line. We present here a result of local existence and uniqueness of a family of fluid motions, leading to the genesis of such lines inside a nonrotating updraft which do not develops neither a source nor a sink.