Our recent numerical studies have led to a better understanding of aircraft wake vortices and have contributed to the development of the AVOSS. The primary objective of these numerical studies has been to quantify vortex transport and decay in relation to atmospheric variables. Prior to this program, many controversies about aircraft wake vortices existed, such as: 1) are the character of aircraft wakes dependent upon either ambient meteorology or the aircraft and its configuration; 2) do wake vortices decay from atmospheric turbulence or do they decay very little until three-dimensional vortex instabilities cause a sudden demise; 3) can wake vortices rise, and if so, why; 4) what is the behavior of wake vortices in strongly-stratified flow; and 5) do the vortex cores remain tight as the vortices decay, or do the vortices decay from the inside with an expanding core diameter. These controversies are addressed in this paper. Our LES studies have shown that the rate of vortex decay increases with increasing levels of ambient turbulence (as quantified by the eddy dissipation rate). However, near the ground, the decay rate is independent of the ambient turbulence level. Here, vortex decay is dominated by frictional interaction with the ground, and transport is affected the production of countersign vorticity. Our results also have shown that the time of vortex linking (and the onset of three-dimensional instabilities) was primarily influenced by ambient turbulence. This was true even for cases of vortex linking with the ground. Vortex descent was found to be influenced by crosswind shear, large atmospheric eddies, and thermal stratification. In addition, results have show that unexpected stalls during vortex descent can be attributed to either crosswind shear or vertical transport due to large planetary boundary layer eddies. In strongly stable environments, our LES studies show enhanced decay from baroclinic vorticity generation; with the wake vortices dissipating once their descent is stalled by stable stratification.