Charybdis' Mouth is the 10th Cyclone Stage. This stage introduces the Relic-trait Cyclone, Primeval Cyclone. Very few other enemies appear in this stage, although they are reasonably powerful. Upon completion, the player has a 30% chance to unlock Primordial Cat. It has no known schedule, and it is only available after clearing Stories of Legend.
Prof. Cat Jobs, along with a Weaken UP (M) combo (e.g. Power of Darkness), can completely nullify the boss as he outranges it. Simply send out some meatshields with the Professor in the back and weaken it. Then finish it off with some damage dealers.
Note that in close range, the Primeval Cyclone can easily destroy most of your cats, especially with its enormous strength boost of 200% at half health, so it is important to keep this enemy at range with any kind of stalling possible.