Thermohaline Circulation in the Oceans
Salinity and temperature both affects the density of sea water. When water with a fixed salinity cools down, it becomes heavier and sinks. In the same way, when vapor or ice removes water from sea water, the remains is more saline and heavier. 
Thermohaline circulation can work as you describe. Surface water in the tropics is saline, due to evaporation, but warm due to high temperature in the atmosphere and therefor low density. As it reach colder climate (less solar energy per area), it cools down and the high salinity makes it sink. Surface water in polar regions also get heavier as ice is formed of water and leave the salt behind in the sea. 

This is very simplified model, you can read more on the topic here and here.