What causes the warm air ahead of a cold front to rise

The correct answer is - difference in density.

The warm and the cold air have different densities. The reasons for that is the way the molecules behave when exposed to higher or lower temperature. On higher temperature, the molecules spread out, thus making the air mass less dense, thus having lower air pressure. On lower temperatures, the molecules are coming closer together, thus making the air mass denser, having higher air pressure.

When a cold front has warm air on its path, it pushes it upwards. As explained, the cold front is denser, heavier, so it manages to push away the less dense, lighter, warm air. The cold front is a moving very low, thus it pushes the warm air up, making it rise into the atmosphere.

river is shown in this type of map