Maybe. Wake vortices tend to sink down and out. But, they are affected by prevailing winds, too.
It would take a more sophisticated computer model than I could do in a reasonable amount of time. I hope that someone is doing so, though.
Actually, they ARE the wind -- so to speak. That is, wake turbulence is a local disturbance in the mass of air that is moving with respect to the ground and gives rise to the concept of wind.
In addition, to keep a plane up, air has to move downward. Of course, this creates a low pressure above, which then sucks in air, hence you get a circulating flow -- but it must drift downward until it is supported by the surface of the earth, which of course is non-fluid.
But yeah, downward, outward, and carried with the wind.