You already know abut the North American Conveyor current. Briefly: The major ocean currents happen because the equatorial ocean is warmer, and since water (unlike land) can move (though not as fast as air) the dissipation of this heat across the surface of the Earth results in warm water moving, at the surface, north or south away from the Equator, where it loses its heat and finds it way back to the equatorial regions, usually as deeper, cooler water.
Conveniently, this process also involves increasing the salinity of the water far from the equator, as evaporating water becomes saltier. This saltier water is therefore both cold and dense, so it sinks, drawing the warm surface water into the evaporation regions. Something like this is happening at a small scale around all the oceans, but the density driven conveyor is the biggest driver of ocean currents, most significant with respect to weather, and most famous, in the North Atlantic.
With global warming, the fresh water budget and distribution in the northern latitudes, in the Atlantic, changes, with more fresh water coming out of the Arctic and off of Greenland. This freshens up the hypersaline engine of the Atlantic Conveyor, also known as the Atlantic Meridional Overturning Circulation (AMOC). When that engine slows or shuts down, the currents in the entire North Atlantic, and beyond, change.
Here is the number one reason this is important (though number two may be more important, I’ll get to that in a moment). You know how England is warm and Maine is cold, though they are both really far north? London, Saskatoon, and Adak are all at about the same latitude. Paris, Quebec City, and Thunder Bay. Northern Europe is warmish, and habitable, even in Scandinavia, because of the heat that the AMOC transfers from ocean to land.
If the AMOC shuts down or moves really far south, Scandinavia, which is at the same latitude as Hudson Bay, will act more like Central Canada, which it does not do today.
Visiting London from Minneapolis in the Winter now means going to a warmer (if dreary and foggy) place. Without the AMOC, it will be more like going to central Canada.
The above strip maps make it look like there is an equivalence across different longitudes at a given latitude. This is not true. The ocean, even without the AMOC, will still warm Western Europe. But now, there is a gradient of warmth from eastern North America over to Europe, where a mostly non freezing winter shifts north to a degree that is nothing short of spectacular. Without the AMOC, that shift will be modest. And, interior areas in Eurasia, such as Moscow, will also cool down (though relatively not as much).
Newly published research tells us something new and troubling about AMOC deterioration. Current climate models suggest that this may happen, but it is unclear to what degree and when. Physical evidence shows the actual real life weakening of AMOC in recent years. So, reality seems to be outpacing the models. Some have suggested that this means that AMOC varies a lot, and will likely swing partly out and back in. Others are not so sure.
The recent research identifies a bias in the generally used climate models that causes AMOC to be more stable and long lasting, under global warming, than it might in real life. When the model is run with and without the bias corrected, you get very different results (see graph above).
This is a preliminary finding. The model has not been run enough times, and a few other things that are usually done have not yet been done. But the results are interesting enough that it is getting some serious attention.
One of the world’s experts on this topic, Stefan Rahmstorf, has written this up on RealClimate: The underestimated danger of a breakdown of the Gulf Stream System. The original research is here, but you may need a subscription.
I should mention that the collapse of the AMOC that happens when this model is run occurs in the somewhat distant future. That makes it worse, of course, because even more people will be living in, and depending on, the affected region than today. But it also allows us to ignore the problem because, hey, who cares about what happens to our children anyway, right?
Oh, and on that other thing that could happen if AMOC shuts down. This is speculative, but we do know that in the past large areas of ancient versions of the Atlantic Ocean and other seas have essentially died, become anoxic over large areas, so they become sources of dead matter rather than edible fish and stuff. This is how many of the major oil supplies we exploit today formed. I would imagine that shutting off the relatively restricted North Atlantic basin from much of the global circulation would be a first step in killing the ocean. So, there goes that food supply, and possibly, that source of oxygen. You know, for eating and breathing and stuff.