I used to work at GFDL and I know that they provide some code on GitHub, including at least one climate model (MOM6). Also, the Flexible Modeling System (FMS) is core to the work they do, but you'll have to put your Fortran hat on. You might also look at PCMDI's repo. They provide tooling that helps standardize climate data for the international community for use in the assessment reports. For something more on the distribution and analysis side (which may be more accessible and interesting to people who aren’t climate scientists), take a look at the Earth System Grid Federation (ESGF). Hope this helps!

https://www.gfdl.noaa.gov

https://github.com/NOAA-GFDL

https://github.com/PCMDI

https://github.com/esgf

I'm not involved in climate science, so I'm hardly an expert, but this is something I've heard a bit about. It seems to be pretty popular / widely used:

http://climate.apache.org/

There is this crypto currency called gridcoin where you lend your machine's processing power for researches like climate change,cancer research... Very intersting projects there

I mean its a nice idea, but how does that work - wouldnt the spare cycles use energy and contribute to climate change?

Depends on your locations power source: nuclear, wind, solar, and hydro don't cause co2 emission.

This from a little over a year ago was very informative:

https://news.ycombinator.com/item?id=10622615