How does the Earth's core affect Earth's atmosphere?
It's composition an movement affects the "tilt" and "wobble" of the earth, and the length of the day.
The degree of tilt and the frequency of a "wobble" in the earth's orientation towards the sun have a significant effect on the amount of solar radiation reaching the planet. Solar radiation is the primary energy source that drives all the rest of the planet's weather through atmospheric movements.
Really Technical Paper here:
Easier explanation here: