Our atmosphere does have tides, and the gravity of the moon is part of why they exist, but it turns out that the moon’s influence on our atmosphere is absolutely swamped by the influence the sun has on our atmosphere. However, it’s not the gravity of the sun that’s making such a large impact.
You can guess that the sun’s gravitational impact on atmospheric tides must be small by looking at the ocean tides, which are largely driven by simple gravity. The ocean tides follow the moon as it circles us - if the sun were a major player in the ocean tides, then high tide would happen at local noon every day, instead of varying based on the moon’s orbit. The sun is still a player; the highest tides occur when the sun and the moon align so both of them are pulling in the same direction, but the difference between a normal high tide and high tide when all forces align isn’t as large as the difference between high tide and low tide; the moon wins this round.
Unlike our liquid oceans, the atmosphere is a gas, and that introduces some other possibilities. Liquids are fairly straightforward under normal conditions; they fill their vessels, and are relatively constant in density - it’s hard to compress or expand a liquid by very much. Gases, on the other hand, are highly variable in density - it’s fairly easy to compress a gas - we do this simply by talking. Gases are also incredibly sensitive to temperature; one of the easiest way to make a gas more dense is to simply cool it down. Conversely, if you want to make a gas puff up and take up more space, heat is an easy way. Each particle of gas gains a little more energy than it had before, and it bounces around a little faster, and so the whole thing winds up slightly more puffed up than it used to be. So the sun, as an enormous source of heat, has a pretty major influence on the gas by simply heating it up.
When the sun is shining on the atmosphere (anytime it’s daytime), it heats up the gas that makes up our atmosphere, which makes the whole atmosphere expand outwards towards space. This expansion can be measured at sea level as a very slight reduction in atmospheric pressure, but is a lot more dramatic at higher altitudes, where the density of gas is really low to start with – as the atmosphere underneath it expands, suddenly there’s an upwelling of denser gas from below.
This isn’t a tide in the way that we’re used to thinking about it – usually if we say “tide” we mean a gravitational tide, but if by a tide we mean a regular, cyclical change, then these are definitely tides, and indeed they are officially termed atmospheric tides.
Our moon does have a gravitational pull on the atmosphere as well, but like the sun’s impact on our ocean tides, it’s a much weaker effect than the heating provided by the sun. If the moon were the main cause behind this atmospheric stretching, it would work the same way as the ocean tides; high tide would mean that you also had the most atmosphere above you, instead of what we see; a 24 hour cycle of our atmosphere heating and cooling under the sun’s rays.
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