Hot air rises. When it does, it can’t just leave a vacuum of nothingness behind in the place where it was. Instead, other cooler air gets sucked in from the sides. By definition that air is colder, otherwise that would be the air that’s rising. The result is air on the surface moving from areas of relatively high to low pressure. At an ELI5 level [it looks like this.](https://www.e-education.psu.edu/earth111/sites/www.e-education.psu.edu.earth111/files/Module2/Earth111Mod2GWImage5.png)
So the energy to move the cooler air still came from heat, just in a more indirect way than you were considering.
TLDR: the wind doesn’t blow, it sucks.
Wind is caused by a difference in heat in a body of air. The easiest example is a flame because it’s a strong heat difference– the fire is very hot and the surrounding air is much cooler.
Heat is added to the air around the fire, which causes it to expand. The less dense air is squeezed upward by the cooler air– this is why fire and hot smoke tend to rise. That creates a momentary low-pressure area though, so cooler air moves in to replace it, and itself gets heated by the fire. This phenomenon is called convection.
That happens on a global scale because the sun doesn’t heat the earth evenly. Some places are hotter than others, which creates all sorts of high and low pressure zones in the atmosphere, so air is always blowing every which way as it moves to equalize the pressure. But of course it won’t achieve that because there will always be some sort of heat difference until maximum entropy, aka heat death.
There’s also the Coriolis force, which is the earth rotating and not completely taking the atmosphere with it. IIRC this is what causes the jet stream.
You are right that heat is energy, more specifically it is the kinetic energy of the particles. Cold air has slower moving particles than hot air. This means cold air is denser than walm. However importantly the cold air is still moving and still has energy.
Wind comes from cooler, denser air filling in the gap left by walm air as it rises. Think of a mushroom cloud, very hot air risers and the cooler air fills up the gap left.
Cold water particles bounce around slowly compared to hot water particles, but if you put a straw into a cold drink and suck, it will move just as quickly, because the atmosphere is pushing the drink up into your mouth.
Wind is the result of massive pressure differentials moving millions of tons of air around. That movement will occur no matter how cold or hot the air being moved is.
Cold is relative. Anything above 0 Kelvin has energy via vibrating particles and has warmth relative to absolute zero. So, even liquid helium which is around 4.15K (-269C or -452.2F) technically has some ‘warmth’ to it.
And what you define as ‘cold’ air is blisteringly hot compared to liquid helium.
Cold isn’t an energy, it’s the absence of heat energy. Cold is also generally goes hand in hand with a high pressure area of weather. Think the coldest days you’ve ever seen, it’s not cloudy and snowy. It’s clear and crisp and so cold trees explode.
So when you have a high pressure Cold area in one place and a lower pressure warmer area next to it the atmosphere wants to equalize the pressure. So high pressure flows to low pressure area, bringing that cold with it.
This is to the best of my understanding, I am not a meteorologist.
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