Many theories and mechanisms have been proposed to explain the phenomenon of clear-air turbulence (CAT), and some of them have been successful in predicting light, moderate and, in some cases, severe turbulence. It is only recently that skill in the forecasting of the severe form of CAT, which could lead to injuries to passengers and damage to aircraft, has improved. Recent observations and simulations suggest that some severe to extreme turbulence could be caused by horizontal vortex tubes resulting from secondary instabilities of regions of high shear in the atmosphere. We have conducted direct numerical simulations to understand the scale relationship between primary structures (larger-scale structures related to one of the causes mentioned above) and secondary structures (smaller-sized, shear structures of the size of aircraft). From shear layer simulations, we find that the ratio of sizes of primary and secondary vortices is of the right order to generate aircraft-scale vortex tubes from typical atmospheric shear layers. We have also conducted simulations with a mesoscale atmospheric model, to understand possible causes of turbulence experienced by a flight off the west coast of India. Our simulations show the occurrence of primary flow structures related to synoptic conditions around the time of the incident. The evidence presented for this mechanism also has implications for possible methods of detection and avoidance of severe CAT.