Supernumerary arcs are the narrow coloured fringes sometimes seen close inside a primary rainbow. To see one fringe is fairly common to see 4-5 is rare.
Supernumeraries are seen when raindrops are small (1mm dia. or less) and of uniform size. The fringe spacing widens with decreasing drop size and uniform sized raindrops show most fringes because there is less blurring from overlap of different fringe spacings.
Supernumeraries are diffraction effects associated with a light caustic.
Light rays reflected once inside a rain drop form the primary rainbow. The once reflected rays emerge in a number of directions and as they do so they fold over and intersect each other.
The surface separating the region of space occupied by two rays intersecting and that where there are no rays is a caustic sheet. The sheet (a fold caustic) marks the intensely bright rainbow rim.
Close to the caustic sheet, each intersecting ray/wave pair coalesces and interferes. Coincident wave crests give brightness, out of phase crests give darkness.
The result is a set of light and dark diffraction fringes parallel to the caustic sheet – The supernumerary arcs.