Unfortunately your analogy isn't correct. Hot and cold water don't mix unless you force them to mix or the density differences pushed them against each other. The hot will attempt to rise to the top, the cold to the bottom. It's a physical property exploited by tank water heaters to get more heating power out than gets put in by heating different areas of the water tank differently at different times knowing that the fresh cold water added at the bottom won't mix with the hot. Your bucket example upside down would result in a bucket that is half hot, and half cold. Air on the other hand does mix a bit more chaotically, it's a function of gas's density to be more chaotic.
By the way have you ever swam in the ocean and noticed some spots are hotter than others? It's not because you're swimming through fresh fish pee.
The atmosphere part of your post is correct through. But this "mixing" in water is not an example of this in action. The mixing of air in the atmosphere is the result of air currents created both locally due to localised temperature differences and globally due to the earth's rotation.
A good analogy of how the polar vortex works are the air curtain airconditioning units you find at entrances into buildings. When you turn them off, drafts and currents as well as people moving in and out cause the hot air inside and the cold air outside to be exchanged. When the polar votex breaks down, it is localised temperature effects that causes the colder parts of the atmosphere to be drawn towards them creating mixing.
