The ocean currents circulate along consistent paths known as the global conveyor belt, or thermohaline circulation. Bottom-water in the earth's oceans is formed at the poles (chiefly south of Greenland, but also off Antarctica) where saltier ocean water is cooled, and being more dense, sinks to the bottom. There are two major implications for deep ocean currents from global warming.
The most important is the threat of increased melt-water at the poles diluting the salty water at Greenland and Antarctica. Not only will that water become less salty (and therefore, less dense), if the ice shrinks enough, it is hypothesised that it will no longer be cool enough to force that water to sink to the bottom. This would, in theory, shut down or severely impact the global conveyor belt, by shutting down normal thermohaline circulation.
Another implication is the effect the El Nino Southern Oscillation (ENSO) cycle has on upwelling off the coast of South America, and continental downwelling off the continental shelf of Australia, and at the Indonesian Archipelago. During an El Nino event, weakening winds and currents along the South American coast cause the weakening, then total stalling of upwelling of cool, nutrient-rich water off that coast. The result is a collapse in local fisheries. As the globe has warmed, El Nino events seem to be getting stronger and longer. Climate science has not reached a concensus on what the long-term implications for the ENSO cycle are from a warmer earth - some hypothesise longer El Nino, some say stronger, some say more often, some say more variable, some say that El Nino could become a permanent condition.
To summarise, a warming earth has the potential to reduce the strength of the thermohaline circulation that drives the ocean conveyor belt, by inhibiting the formation of cold, dense deep water at Greenland and Antarctica. It is also changing the nature of the ENSO cycle which affects upwelling of cold water off South America in a way we cannot yet predict.
