As far as I know, the only reasons to have circulation of water are:

1) To distribute chlorine (and other chemicals) evenly to prevent low concentrations in parts of the pool (usually caused by chlorine getting used up by algae trying to grow in such regions) so that algae growth is prevented.

2) To move large objects (e.g. leaves) to the skimmer for easy collection.

3) To filter the water to remove suspended particles (i.e. to prevent cloudiness).

4) To prevent freezing of the water, mostly in pipes (only during freezing weather, of course).

With extremely low temperatures and a lack of strong sunlight, the risk of algae growth is very, very low. I don't know if there is an easy way to tell how much circulation is good enough except to see if algae starts to develop. Even if it does, its growth would be so slow as to be easy to see and remove with more chlorine. It seems to me that it's just an experiment for your pool, but that the risk is low.

My own pool only gets down to around 50F this time of year so I still have my pump running every day, though only for 2 hours. This is probably overkill, especially since the pool has an opaque cover, but it's what I'm comfortable with.

Richard