I am an amateur physics enthusiast, so admittedly this idea needs a lot of polishing. But it occured to me there must be a way to exploit the temperature difference between the base of a space elevator and its orbital top for power purposes. Here's what I thought up.

Taking a page of meteorology, the reason wind exists is that cold air sinks and hot air rises. As air moves away from the equator and higher into the atmosphere, it becomes colder and heavier. Similarly, as air moves toward the equator and toward the Earth's surface, it becomes hotter and lighter. This causes changes in pressure that creates the eternal loop of our planet's wind cycle.

Now, let us assume we have a space elevator. Take two large tubes (let's say about 10 feet in diameter) and run them side by side along most of the length of the elevator. Then connect them at either end and make them airtight/vacuumtight. Fill them with a stable and plentiful gas, like Nitrogen. A single element being involved will probably make the physics easier to work out.

So, we have one end of the tube sticking out into space, and the other end down near the surface of the Earth. That's a huge temperature difference. The air at the top of the tube will be extremely cold, and will want to move downward, while the hot air near the bottom will want to move upward. And as hot air move upwards, the ambient temperature will chill it, and the reverse with the cold air moving downwards.

Exaggerate this effect by implementing a valve system that only lets air pass one way through either of the tubes (one down, one up). This should create a fairly powerful wind force by manipulating the temperature differences.

Stick in wind turbines modified to work in the tube. And you have a endless power source that uses no resource other than the temperature differences created by the sun, and has no toxic side effects.

Tell me why this wouldn't work, or a way to use the same idea but better. There has to be a way to harness the temperature differences on a space ribbon for power.

Temperature of atmosphere as a function of height

The pressure drops off as you increase height, but the temperature does not.
Firstly, building the tubes is not an easy task, your talking about tubes that are of lengths around 100Km, and it has to be airtight, so its goto be strong enough to not collaspe due to its own weight (needs to be strong in the vertical direction) and not collaspe due to pressure differences (strong in the other horizantal directions).

Pressure drops off as a function of height, so even though you may have a high temperature in space (you dont) then you dont have hardly any gas either, thats the reason its called space in the first place!. Putting a gas in a tube still means it obeys the laws of gravity, which ends up having a huge pressure at the bottom ( you said one atmosphere) and very little at the top.

The main problem with your idea is density of the gas, which will be almost none at the top. To solve that problem you would need to pump in alot of gas, making the pressure at the bottom huge!

There is a big difference in the intensity of the suns light as you go higher up, so instead of your idea just attach nice big solarpanels all along the ribbon, it will act as a nice shade for pods travelling up and down and produce a ton of power with no moving parts too.

The solar method is obvious. But this is mainly a mental exercise.

Maybe the pressure problem could be solved by using a liquid rather than gas to create the whirlwind effect. Like an inert oil.

The pressure problem would be worse with a dense liquid. Pressure is a function only of depth and density (which might be a little counter-intuitive, since you might expect the total volume of liquid above you to count for something, but it doesn't). If you have tubes of oil that are hundreds of kilometers high, the pressure at the bottom will be many times the pressure at the deepest point in the ocean. The only way to avoid that would be if the tube were small enough to get some sort of capillary effect.

I don't mean to ruin your fun - it's a good question, and i know you described this as a mental exercise. But also, trying to hang giant oil filled tubes on the ribbon is not gonna work. As of yet, we still haven't made a single CNT that's strong enough to be a part of the ribbon. Theoretically the strength of a single CNT is significantly higher than what we've developed, but we've not been able to make any.

But if we do develop the materials, we won't want to be hanging heavy objects from it. Sure, a few climbers that are each carrying 20 tons is to be expected. But once you start talking about tubes that are many kilometers long, and full of oil - now you're talking a LOT of weight. The pressure at the bottom would be huge, and you'd need a lot of structural reinforcement to keep it contained. But even then, another major objection would be the danger it poses if it falls. A 36,000km ribbon of CNT would float to the ground like an open newspaper. But big solar panels or oil filled tubes are going to make that a lot more dangerous.

If we're gonna use the Elevator for power, we'll use it by making incredibly fragile, ultra-thin solar panels. We'll cart them up to orbit, and unfurl them to form many square kilometers worth of solar panel. The energy could then be beamed back to earth using microwaves.