What's the relationship between the Carnot Cycle and Hurricanes? (and more)

Question

I'm trying to understand the relationship between the Carnot Cycle and Hurricanes. My understanding thus far is that the Carnot Cycle follows 4 stages: 2 Isothermic (same temperature) and 2 Adiabatic (no heat transfer).

Put simply, there's a reservoir (hot/cold) and gas. In the first stage, the hot reservoir is in contact with the gas. The gas expands and does work and its temperature does not change. Everything's great.

Now you take out the reservoir ... why? And the gas still expands! Why?

Now you put the gas in contact with a cold reservoir and the piston presses on the gas. The volume decreases, but the temperature remains the same because the gas transfers its energy to the cold reservoir. Great, but why exactly do we need this stage?

And then we take the cold reservoir away from the gas and the gas keeps contracting ... ? Why?

And perhaps the question I'm struggling most to answer ... what's the relationship between the Carnot Cycle and Hurricanes?

I know there are many other questions regarding the Carnot Cycle, but having looked at all of them, I see that everyone learns it a bit differently. I would be very grateful if anyone could help me answer these questions.

Answer 1

Note that the real-world effects happening are all mixed together, not completely separated the way an ideal engine would be.

The gas expands and does work and its temperature does not change.

The air is being warmed by ocean waters. This expansion reduces the density and it will tend to rise in the atmosphere.

Now you take out the reservoir ... why?

Because it has transferred all the heat necessary. The air can leave the vicinity of the hot reservoir and move the energy to another location.

And the gas still expands! Why?

Because the pressure has reduced. In the case of an engine, this might be because a piston is moving allowing expansion. In the case of the storm, it's because the air parcel is moving upward. As it moves upward, the pressure drops. The parcel expands in size as this happens.

So the phase 1 expansion is because the temperature increases. The phase 2 is because the pressure is reduced.

Now you put the gas in contact with a cold reservoir and the piston presses on the gas.

The piston is always pressing on the gas. It's not pressing any harder at this point. But because of cooling the (same) pressure will contract the gas.

Great, but why exactly do we need this stage?

Without it, you couldn't pick up much heat in stage 1. The gas would be much warmer than if the heat were rejected.

And then we take the cold reservoir away from the gas and the gas keeps contracting ... ? Why?

Now because the pressure is increased. In the storm, this is because the now denser air falls. As it descends, the increased pressure further compresses the gas.