That link doesn't tell us how wind systems are able to maintain their distinct disk shapes without falling apart. It just tells us that the Coriolis Effect compels the system to rotate in one direction rather than another.
So again, how do wind systems maintain their disk shapes without falling apart?
Not quite sure what you mean by "maintain their disk shapes without falling apart" but I'm going to guess you mean that it doesn't provide an explanation for why the coriolis effect causes a circular motion of the winds that propel the currents in gyres? Rather than myself giving an explanation on this, I would suggest you read some of the sections in "Applied to Earth", particularly "Rotating sphere", "Meteorology" (Which speaks about the Rosbby number and your confusion on how the winds caused by the coriolis effect actually work), "Flow around a low-pressure area" and "Inertial circles".
I would also like to site an experiment by Ascher Shapiro (in the wikipedia article btw), in which the coriolis effect is even shown to exist in small scale water systems.
"Coriolis rotation can conceivably play a role on scales as small as a bathtub. It is a commonly held myth that the every-day rotation of a bathtub or toilet vortex is due to whether one is in the northern or southern hemisphere. An article in Nature, by Ascher Shapiro, describes an experiment in which all other forces to the system are removed by filling a 6 ft. tank with water and allowing it to settle for 24 hrs (to remove any internal velocity), in a room where the temperature has stabilized (temperature differences in the room can introduce forces inside the fluid). The drain plug is then very slowly removed, and tiny pieces of floating wood are used to observe rotation. During the first 12 to 15 mins, no rotation is observed. Then, a vortex appears and consistently begins to rotate in a counter-clockwise direction (the experiment was performed in the Northern hemisphere, in Boston, MA). This is repeated and the results averaged to make sure the effect is real. The Coriolis effect does indeed play a role in vortex rotation for draining liquids that have come to rest for a long time. ["Bath-Tub Vortex", Nature. Dec 15th, 1962. Vol 195, No. 4859, p. 1080-1081]"
---------------------Can you provide a better explanation for the existence of this?