Why does wake turbulence sink? There has been a remarkable story this week of a Challenger 604 business jet being rolled after encountering the wake turbulence of an A380.

A Challenger 604 at FL340 operating from Male-Abu Dhabi passed an A380 opposite direction at FL350, one thousand feet above, about 630nm southeast of Muscat, Oman, over the Arabian Sea.
A short time later (1-2 minutes) the aircraft encountered wake turbulence sending the aircraft into an uncontrolled roll, turning the aircraft around at least 3 times (possibly even 5 times), both engines flamed out, the aircraft lost about 10,000 feet until the crew was able to recover the aircraft, restart the engines and divert to Muscat. The aircraft received damage beyond repair due to the G-forces, and was written off.

I subsequently discovered that wake turbulence sinks; according to Wikipedia:

At altitude, vortices sink at a rate of 90 to 150 metres per minute and stabilize about 150 to 270 metres below the flight level of the generating aircraft.

Well, now I know where not to fly my business jet...
Incorrect assumption
(See accepted answer below)
Presumably the vortices sink because they are denser than the air they were whipped up out of, but it's not clear to me why they are denser.
Lift is created by accelerating air downwards. The same goes for wake turbulence: The vortices form at the boundary between the downward-moving strip of air (wake) from surrounding air that tries to fill the void left by the wake.

Friction will eventually dissipate the kinetic energy of the wake, but in case of heavy aircraft the motion will continue for several minutes. By that time it will sink down hundreds of feet or hit the ground. This is driven by inertia; density differences do not exist unless the atmospheric lapse rate is above the adiabatic rate and the wake warms less quickly from increasing atmospheric pressure than the surrounding air.

In the end, the weight of any heavier-than-air vehicle is supported by an area of higher pressure on the ground. Some mechanism has to transfer the weight force from the flying aircraft to the ground, and that is the downward-moving wake. When it slowly dissipates, it will leave a slightly higher air pressure below it. When that happens at supersonic speed, the more focussed pressure change is high enough to be audible on the ground.