Roof edge vents and ice dams?

Usualy the cause is warm air rising from poor insulation or air infiltration from lower heated spaces is melting snow and wasting your $$. If you want to do it right, a blower door test and an evaluation of your insulation, needs to be done, few have insulation to 2008 standards which is usualy double of code.

ridge vents should definitely help. But you need plenty of intake vents to supply them. Normally these are soffits vents, but .....

Thanks. I had been thinking "How can I get the heat out of the attic without heating the roof?". I now realize I should also/instead be thinking "How can I keep the heat from getting out of the heated area (and in to the attic) instead?".

Still, I think it needs more ventilation, for moisture if not for heat. (Moisture is another problem in the attic.)

No soffits plays a part here, I think (parent's house has similar feature/similar problem). Keep the heated air from getting in contact with any part of the roof. Rafter baffles are used in houses with soffit vents, this allows the insulation to be packed in against the low part of the roof yet allow air from the soffit vents to enter.

Having a ridge vent without soffit vents is a waste of time & money. You have half of the system in place, now it's time for soffit vents, and lots of them. DAGS for "ridge vent" or "soffit vents" or similar. The ridge vent system needs soffit vents to provide natural convection; cool air comes in at the bottom & goes out thru the ridge vent.