Air taxi First Officer reported encountering severe up and down drafts with the lose and gain of 1000 feet of altitude.

Narrative

While flying at FL400 we were advised of a line of storms ahead of us. We turned on our on board weather radar and clearly saw two cells; one to the left of us and one to the right; but there was no weather detected directly ahead of us on our route.We then looked outside and saw the cells to the left and right and saw clouds below us in front of us but it was clear we would be higher than them. As we flew over the clouds we could clearly see past them and see the ground on the other side. At some point over the top of the clouds we started to experience turbulence; then a very strong downdraft that forced the airplane down to FL390 regardless of any action we took. I reported severe turbulence in accordance with AIM 7-1-21 definitions and asked for a block altitude FL390-410 which was granted. Within several seconds we had climbed back to 400 and seemed stable. Then we experienced an updraft and the aircraft climbed to 410. We regained control and were then on the other side of the storm and had an uneventful rest of the flight.