I've got livestock panels over the top of the runs. Horse panels with 2X4 squares. Then a tarp over that. The livestock panels give plenty of support to the tarp, but I still go out and rake the snow load off the top.
My son, who does all my construction, is doing a lot of thinking about how to put a permanent cover over the runs that won't be too heavy for the run to support. I think he is considering sinking some posts on both sides to support a roof and doing a roof of corrugated metal.
Some expense involved there, but a tarp to cover is getting up around $100 and it disintegrates in the sun and only lasts about a year. I found a good surplus canvas tarp that would cover the top, but it was over $200. A permanent roof wouldn't be too much more than that.
With a permanent roof, I'd still need tarps down the sides to keep blowing weather out.