A lee-side trough (low) is commonly taught as caused by the stretching of a column of air as it descends on the lee-ward side of highly elevated terrain such as the Rocky Mountains. This stretching causes a spin-up of vorticity and a resulting upper level divergence. Lee-side cyclones can intensify over time as they propagate with the upper level flow and can initiate cyclogenesis. A downslope flow will be most intense if the windflow is perpendicular to the mountain range and the winds have a high velocity. Understanding how this produces a low can be complicated by other processes that occur with a downsloping flow. First, a downsloping flow produces adiabatic warming. Air warms at the dry adiabatic lapse rate as it sinks. This should cause temperatures to warm and heights to RISE (thermal expansion). Second, a downsloping wind is associated with stability (RH decreases, downward vertical velocities). How does a trough develop where air is sinking and stabilizing?
Theory 1: Mid-level winds which come in contact with elevated terrain suddenly experiences friction. This causes a directional change in the wind speed (ageostrophic flow). Once the air moves on the lee-ward side of the mountains it no longer experiences the friction. The friction over the mountains creates a shortwave that advects with the upper level flow. This creates a lee-side low.
Theory 2: Although the sinking air is warming at the dry adiabatic lapse rate on the lee-side, the air is still colder than the air it is replacing (katabatic wind). This creates a lee-side low since a layer of colder air is related to lower heights.
Theory 3: Flow over a mountain produces a spin-up of vorticity due to a stretching of air as the lower levels of the atmosphere fall on the lee-side (most common argument). What is troubling, is that I have often noticed lee-side lows with no vorticity showing on the 500-mb model progs and strong downward UVV's.
Theory 4: A combination of 2 or more of the previous theories. The most impressive lee-side lows occur when a pre-existing shortwave moves perpendicular over a mountain range.