While there are approaches to make a habitable Dyson sphere that are technically reasonable, I am curious about what the climate would be like. For simplicity, consider a scenario where a thick shell is constructed at about 3.6 million kilometres from the sun, where surface gravity would be one g. It is supported by dynamic stabilization from the inside. Assume excess heat is radiated by some additional cooling system, like a cooling tower connected at some point of the shell or whatever. Put a star on an equatorial orbit to provide illumination, and make sure it's sufficiently far away to avoid it's gravitational pull to have more than negligible impact on the surface conditions. Assume earth-like atmosphere and surface. Can actual climate models be used here, and what would they predict? Has climate modelling been done for any megastructure of vaguely similar kind? Scientifically informed speculation, perhaps?
Edit: I chose to not assume anything about rotation of the sphere. This is of course highly significant but I can't come up with a default value, so discussion of that issue is welcome. Also, my first idea of the world is to have a lot of open water, like on earth, so there would possibly be a huge world ocean where most bodies of water are connected.
Edit2: a star in orbit provides a day-night cycle of years, so I guess a different solution is needed for what I have in mind, although extreme day-night cycles could also be interesting. I think we should assume 24 hour cycles with the same spectrum and power as on earth for the simplest scenario.