The cleansing of saidin was a world-altering event that occurred at the climax of Winter's Heart and was felt throughout the entire world by channelers.
In 1000 NE, Rand al'Thor made an extraordinary claim: he believed he had discovered how to cleanse the Dark One's 3,000-year-old taint on saidin. While Rand was inside the redstone doorway ter'angreal, one of the questions he asked the Aelfinn was how to cleanse saidin. The answer to the question is never revealed to the reader, but it is known to be a riddle that Herid Fel said stated "sound principles, in both high philosophy and natural philosophy."
On Saban 22, Rand al'Thor and Nynaeve al'Meara used the two most powerful sa'angreal ever made, the Choedan Kal, one buried in Cairhien and the other on Tremalking to funnel the taint into Shadar Logoth.
Nynaeve and Rand were linked together for the cleansing. When Nynaeve touched saidin, because she held control of the circle, she openly expressed her shock and fear of the "chaos and rage" of saidin and wondered how Rand was ever able to control it. When Rand "ripped" control of the circle away from her, he was able to touch saidar and, unbelieving of Nynaeve's warning that saidar could not be controlled, only guided, he immediately tried to control saidar like he did saidin, and felt like he was being pulled into a thick, slow-moving stream with eddies and cross-currents pulling every which way. It was not until he could both surrender to saidar and dominate saidin that he was able to weave either half of the Source at all.
Rand's strategy was to create a funnel of saidar from the Source of saidin to Shadar Logoth, and when he guided saidar to that goal, it created something he had not directly intended. Rather than a tunnel as he thought would appear, saidar was actually just an intricate, flowery-looking shape which was a tunnel of infinite and zero distance only relative to the Source and Shadar Logoth; from any other vantage point, it looked nothing like a conduit. Saidin flowed very easily into the conduit, and the taint began to slide off of saidin into Shadar Logoth. Rand's theory was, that after channeling enough of saidin into the conduit, the taint would be drawn in its entirety towards a similar, yet polar opposite sort of taint; the evil residue of Shadar Logoth. After that, he would merely need to keep a steady flow of saidin going through the conduit, and the taint would automatically be pulled into Shadar Logoth so long as he kept saidin touched directly to the evil of the city. Fortunately, the plan worked exactly as predicted.
In Shadar Logoth, a dome-like structure of black darker than black appeared, slowly rising out of the ground as if thick, viscous mud were being piled on the city. Towards the end of the battle, flames even darker than the mass of taint flashed throughout the bubble--Mashadar and the taint destroying each other. This created an immensely powerful whirlwind, sucking Moghedien towards the ruins, and left a gigantic crater sweeping the city.
Six Rebel Aes Sedai who went to investigate the area afterward report that the crater left was "roughly circular hole in the ground... three and a half miles across and one and a half miles deep." The strength of the residues discovered are the impetus for their alliance with the Black Tower.
Near the end of 1000 NE, it is believed that this process was entirely successful, although what it means for the Asha'man and other male channelers, not to mention the Red Ajah, remains to be seen. By the end of the Last Battle, it seems as though the source being cleansed has been accepted by most, and the fear of male channelers by the general populace is beginning to fade.
See also the Battle near Shadar Logoth