Cold air is more what than warm air?

It is common knowledge that warm air rises. It is normally assumed that is because warm air is lighter than cooler air.

Warm air rises primarily due its lower density as compared to cooler air. As the temperature increases, the density of the air decreases. But even air that is of a lower density will not begin to rise by itself.

Isaac Newton's first law of physics is that the velocity of an object will remain constant unless another force is exerted on that object. The more common way of saying this is 'an object at rest tends to stay at rest and an object in motion tends to stay in motion'.

This is why decreasing the density of air alone is not sufficient enough to cause air to rise. There must be another force exerting on the less dense air for it to begin its upward motion.

That force is 'gravity'. Gravity's role is its pull of cooler, denser air toward the earth's surface. As the denser air reaches the earth's surface it spreads and undercuts the less dense air which, in turn, forces the less dense air into motion causing it to rise.

This is how hot air ballooning works. A flame is used to heat the air inside of the balloon making it less dense. Outside of the balloon, the cooler, denser air is pulled down by gravity. The cooler air undercuts the warmer, less dense air trapped inside the balloon causing it to lift.