What appears to be a curved rainbow in the sky is actually what’s called a circumzenithal arc and is quite different from the more familiar rainbows that grace the sky after a rainy day. The thin, wispy cirrus clouds in this picture occur at high levels in the atmosphere where temperatures are way below freezing. At these cold temperatures, these clouds are therefore made up of ice crystals. Ice crystals can come in many shapes and sizes, as seen in this diagram.
What controls the type of ice crystal that forms? Well, that depends on the temperature and moisture content of the environment it forms and grows in. For the circumzenithal arc to form, the ice crystals need to be plate-like. These plates have horizontal faces and shorter vertical side faces. Sunlight enters these faces and is bent within the crystal; this is called refraction. A simple diagram will help to visualize this concept:
The separation of the lines in this picture represents the sunlight being separated into the colors of the visible spectrum (ROYGBIV), such as what you see when light passes through a prism. For the circumzenithal arc to form, the sun’s rays enter the uppermost horizontal face and exists through one of the vertical side faces. For the light to enter these crystals at these angles (nearly parallel), the sun has to be lower than about 32 degrees above the horizon.
So why is it called a circumzenithal arc? Well, the word “zenith” refers to directly overhead. If this arc were a complete circle, the center would be directly above you. The arc is seen directly above the sun.
How does it differ from a circumhorizon arc, which is typically seen below the sun? Circumhorizon arcs also form as light is refracted through plate-like ice crystals, but the sun’s rays first enter the vertical side face of the crystals and exit out of the bottommost horizontal face. For this to happen, the sun has to be at a much higher angle in the sky (~58 degrees above the horizon).