Scientists thought strong upwelling of colder deep waters spared the region from the warming seen in other parts of the Pacific, she said. (sciencedaily.com)
Consider, as you say, that «If the warm water heated by the sun is driven by the wind into the deeper layers, and also bringing colder deep waters to the surface, the surface stays cool, the heat is transferred down...» If we agree that oceanic warming (especially at depth) means more heat in the overall system without a corresponding increase in its radiative efficacy, then from a systems point of view, La Nina warms, not cools. (tamino.wordpress.com)
The Atlantic Meridional Overturning Circulation (AMOC)-- characterized by warm surface waters flowing northward and cold deep waters flowing southward throughout the Atlantic basin — is defined as the zonal integral of the northward mass flux at a particular latitude. (nap.edu)