The ocean contains many large-scale, long-lived vortices, called mesoscale eddies, that are believed to have a role in the transport and redistribution of salt, heat, and nutrients throughout the ocean. Determining this role, however, has proven to be a challenge, since the mechanics of eddies are only partly understood; a standard definition for these ocean eddies does not exist and, therefore, scientifically meaningful, robust methods for eddy extraction, characterization, tracking and visualization remain a challenge. In order to shed light on the nature and potential roles of eddies, we have combined our previous research on eddy identification and tracking, and have used those approaches as the basis for analysis-driven computational experiments on the nature of eddies. Based on the resulting visualizations of eddy behavior, we have devised a new metric to characterize the transfer of water into and out of eddies across their boundary, and have developed visualization methods for this new metric to provide clues about the role eddies play in the global ocean and, potentially, climate change.