This doesn't look good for my theory:
50 x 50 km = 2 500 km^2
tidal fluctuation 1.5m/6hrs => 25cm/hr
625 billion cubic metres per hour
~170 million cubic metres per second
boundary influenced: 50 km wide (edge of box) by 1km high (nominal) = 50 billion square metres
speed through this area: 0.0034 m/s => 0.02 knots.

That would be the average velocity change if the effect was confined to the first 1000m of the atmosphere, but how do we work that out? It's already known that the windspeed at sea level is less than 30 or 100m up anyway, though this piston effect may be strongest at sea level anyway, cancelling that out.
Fluid dynamics wasn't one of my stronger subjects.