Put an index card on top of the bottle filled with cold water. While holding the index card in place, slowly and carefully turn the bottle upside down and sit it on top of the very warm water bottle. The index card should now be in between the two bottles. Slowly remove the index card from in between the two bottles. The reason why the warm water rises and cold water sinks is that the warm water is less dense than cold water.
Density is something that makes things float.
So, that means that warm water actually floats in cold water. So, we think the temperature of the water makes the water rise and sink, not the red food coloring that we added to the warm water.