WARNING - What you are about to read DOES NOT take the place of qualified training!!
The circle search is nothing more than a variation of the arc search pattern or visa-versa. Instead of conducting an arc you simply continue until you've made a complete circle. The other difference is that the tender is underwater with you. This search pattern is mainly used when you're away from shore and is best performed with a length of rope (50 to 75 feet) placed in some form of stuff bag.
The circle search can be quick to set up and performed but, on the other hand, it can be rather troublesome if you don't know what you're doing down below.
This search is primarily used for medium to large items like bodies, safes, cars, etc., but can be used for smaller items depending on the conditions, your environment and the searchers abilities and technique along the bottom.
One of the keys to a successful circle search is controlling the rope that's used in the search. I prefer to use a medium size goody bag (red) with a large metal mouth. This helps to control the rope as it is being fed out and makes it easy to re-stuff underwater after the search has ended.
Again, I prefer to use 5/16 nylon rope. It is negatively buoyant so it stays in the bag and does not want to float away when I open it, and its size makes it easy to handle and control. A larger diameter rope would limit the length of rope I can search with and is not needed. Using a too small of a rope can make it hard to control and find if dropped or lost.
Some teams prefer to use rescue throw bags or wreck reels when doing an arc or circle search pattern. I find that rescue throw bags are tough to re-stuff underwater after you've searched. Wreck reels are awkward to use in zero visibility and the line is too thin to feel and control. If you lose the line underwater, it's almost impossible to relocate by feel in zero visibility, especially with gloved hands. So, just like the rescue throw bag, it does not work well as a search rope. I prefer to use wreck reels and rescue throw bags for what they were designed for.
With this in mind, it's important that the tender use some form of a stuff bag to control the rope because underwater, while sitting on the bottom, visibility will usually drop to zero and the rope can become knotted or entangled. When using some kind of a "stuff bag," like a medium size goody bag, you simply take one end of your search rope and start stuffing it into the bag. As you get to the other end, you tie a loop in this end for the diver to hold. If you stuffed it right, starting from one end and stuffing until you get to the other end, the rope will easily feed itself out making it safe to control. When it's time to "close up shop," the tender can quickly and easily re-stuff the bag prior to making his ascent. This way there is no loss rope hanging around and entangling divers.
To control the goody bag and to have my hands free for other diving needs, I dive with large snap hooks on my weight belt. I believe these hooks are a must for a search and recovery diver.
Like I have said before on other patterns on this site, is there more to say about the circle search? You be there is! Unfortunately, I don't have the time or the space here to write it all down. Good luck in your search and recovery diving!