Recently I have been considering the fundamental nature of a range of analytical models of tornado like vortices, and have written up my musings as an extended essay that can be read and downloaded below. In the essay I look at the class of tornado models that are solutions of the Navier-Stokes or Euler equations. It is clear that they all share a common analytical basis based on the assumption, either implicit or explicit, that the three velocity components (radial, vertical and circumferential) can each be specified by the multiple of two functions – one a function of radius only, and one a function of height only. Assumptions are made concerning the nature of one particular velocity component, and this assumption then allows the other components to be calculated from the continuity and momentum equations via the method of separating the variables. The recognition of this commonality allows a common analytical formulation to be developed that underlies all the models.
Those models that are solutions of the full Navier-Stokes equations (the Burgers-Rott, Sullivan and Vasistas et al models) derive velocity component formulae that are functions of Reynolds number. In the context of a full-scale tornado, this is a Reynolds number based on turbulence eddy viscosity rather than molecular viscosity. The assumptions required to obtain analytical solutions result in vertical velocities that are unbound with height and in some cases radial velocities that are unbound with distance from the vortex centre.
Those models that are solutions of the Euler equations (two by Baker and Sterling and two new models A and B) have, on the whole, rather more realistic formulations of the velocity components and, with one exception, all components for these models are bound in the vertical and radial directions. Instead of the Reynolds number, the velocity components are functions of constants of integration that relate to the Swirl ratio – the ratio of the maximum circumferential to radial velocities. As the circumferential velocity profiles in these models fall to zero at ground level in a reasonably realistic way, the boundary layer at the bottom of the tornado is modeled to some extent. The common analytical framework of these models allows, in principle, the derivation of a large number of different models, provided that they are of a form that allows the solutions to be obtained through simple integrations. However the drawback of such models is that the pressure is zero at the ground for all distances from the vortex core and thus the dip in pressure at the centre of tornadoes is not modeled. This is broadly a consequence of viscous effects not being properly modeled near the ground.
Whilst most of the models represent single cell tornado vortices, two of them – those of Sullivan and new model B – give solutions for two cell vortices. The essay shows that the Sullivan model, based on the Navier-Stokes equations, has a more general form than that given in the original paper and can model one-cell and two-cell vortices and the transition between them. New model B, based on the Euler equations is also able to model both sorts of vortex.
The essay concludes that further work is required in two areas. Firstly there is a need to develop methods that do not rely on the assumption that the velocity components are multiples of two functions – one of radius and one of height – as recent experimental data suggests that the vortex radius can vary significantly with height. Secondly, the tornado boundary layer needs to be modeled in a more satisfactory way than at present, and the essay suggest that this might be done through matching a viscous solution of the Navier-Stokes equation near the ground, with an inviscid solution from the Euler solution away from the ground. I may have more to say on this in the future.