We have this rule in the Player's Handbook, p. 204:
Spells such as burning hands and cone of cold cover an area, allowing them to affect multiple creatures at once. A spell's description specifies its area of effect, which typically has one of five different shapes: cone, cube, cylinder, line, or sphere. Every area of effect has a point of origin, a location from which the spell's energy erupts. The rules for each shape specify how you position its point of origin. Typically, a point of origin is a point in space, but some spells have an area whose origin is a creature or an object.
However, not all Area of Effect spells fit into one of those five shapes. For example, Tiny Hut creates a dome, and Wall of Force creates a wall.
But I'm specifically interested in Control Weather:
Range: Self (5-mile radius)
Spell entry: You take control of the weather within 5 miles of you for the duration. [...]
My guess is that the area is defined by a circle with a radius of 5 miles, without any limitation in terms of height. This would make sense to me because you're controlling the local area's precipitation, temperature, and wind within the atmosphere.