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What about vertical motion?
The air over land will rise and the air over water will sink.
The vertical motion will form as a consequence of vertically-directed forces, just as the horizontal motion formed as a consequence of horizontally-directed forces.
There are two ways of looking at. First, we will consider the vertically-directed forces explicitly, then we will see what happens if we make the hydrostatic approximation.
The heating over land has caused the pressure to fall over land. This pressure change will typically be quite small, on the order of a millibar or so, or about 0.1% of the total atmospheric pressure. It may be small, but it's not zero, and it's enough to cause the air to accelerate horizontally toward the area of lowered pressure.
Why wouldn't the air accelerate downward too? After all, the two main forces in the vertical are the vertical pressure gradient force and gravity. So if the vertical pressure gradient force changes, wouldn't the air accelerate downward?
Well, it would, if that were the whole story. But the force of gravity changed too! Think for a moment. The thing that started all this was the air over land heating up and becoming less dense - in effect, lighter. The lighter the air, the less gravitational force it experiences. If the temperature of the air increases by 5 K (about 2% of 300 K), then, according to the ideal gas law, if the pressure stays approximately the same (and it does), the density of the air decreases by about 2%.
Bottom line: the (upward-directed) vertical pressure gradient force decreases by about 0.1%, but the (downward-directed) force of gravity decreases by about 2%, or twenty times as much. With the pressure gradient force ending up larger than the force of gravity, the air will tend to accelerate upward.
The change in the force of gravity caused by changes in density is often referred to as the buoyancy force. The concept of buoyancy is nice and intuitive: air that's warmer or lighter than its surroundings will tend to rise, and air that's colder or denser than its surroundings will tend to sink. The concept applies readily to most thermally-driven circulations, from sea breezes to thunderstorms.
Meanwhile, what happens to the air over water?
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