Hysteresis in a swirling jet as a model tornado

Abstract

A swirling jet, emerging normal to a plane, serves as a model of a tornado and is characterized by its flow force and outer circulation. This model is examined here using the full Navier–Stokes equations. Three branches of solutions are found which form a hysteresis loop and a cusp catastrophe that means jump transitions between flow regimes. One of the jumps relates to vortex breakdown and the other relates to a new (opposite) effect: abrupt vortex consolidation. These results are compared with those of Long [J. Fluid Mech. 11, 611 (1961)], who considered a near-axis jet in the boundary layer approximation. More detailed analysis made here for high circulation values allows discovery of two new types of asymptotic solutions corresponding to a near-plane fan jet and a two-cell flow. It was also found that the boundary layer approach for the near-axis jet fails to accurately yield the total flow force because the outer flow contributes a share of the momentum flux of comparable magnitude to that of the inner flow. The prediction of the jump transitions between one- and two-cell flow patterns agrees with observations of abrupt changes in tornado patterns in nature.