It is shown through scaling arguments that for flow at small aspect ratio delta the three-dimensional (3D) inversion of vorticity is completely determined by the two horizontal vorticity components--that is, the vertical vorticity does not induce any velocity in the limit of small delta. A corresponding small-delta vorticity inversion is then derived and shown to accurately reproduce the results of the full vorticity inversion for flows past topographic obstacles of sufficient width. These results together suggest an approximate formulation of small-delta fluid mechanics in which the three governing prognostic variables are the two horizontal vorticity components and the potential temperature. We then proceed to explore the process of orographic wake formation within the context of this small-delta vorticity dynamics framework.
We first consider wake formation in the two-dimensional (2D) case through the use of high-resolution 2D-3D calculations of flows past infinitely long barriers. Consistent with previous work, the 2D-3D simulations suggest two distinct physical mechanisms associated with wake formation: gravity wave breaking and upstream blocking. The two mechanisms are shown to produce wakes of distinctly different vertical structure, and we attempt to interpret this difference in terms of vorticity arguments applied to each type of flow. We then consider the upstream blocking case in greater detail and show that in some essential respects the flow evolution resembles a simple hydraulic dam-break problem. As the obstacle is set in motion the cold low-level fluid upstream of the ridge becomes blocked and warmer air from aloft descends along the lee slope to take its place. The cold surface air downstream of the ridge then propagates into the lee-slope warm anomaly and thus effectively acts as a as a gravity current that follows the obstacle as it moves through the fluid.
The analysis of the blocked case is then extended to 3D flows past an isolated mountain and the mechanisms leading to flow reversal are found to be similar to that described for the 2D case. However, in three dimensions the wake air spreads laterally as it ascends the lee slope and thus ultimately reconnects with the low-level blocked flow passing around the barrier from upstream.