From a thematic perspective, it makes perfect sense, but from a gameplay perspective, you have to figure out a reward/incentive structure that would make players want to participate in such an event. To that end, I have a few ideas:
1. If we take your tornado idea as an example, you could make the tornado itself targetable, and perhaps attacking it can lessen its size or otherwise weaken the repel or other effects it may generate. You could also have people trapped under rubble and allow players to destroy the debris to free them, with each such successful rescue rewarding XP. Further, you could have some villains taking advantage of the disaster to loot stores, so you could defeat them as well.
2. We could do a similar implementation with an earthquake or landslide, where the epicenter of the disaster can be attacked to disburse or weaken it, while others can rescue trapped victims and you fight off looters.
3. You could have a flood, perhaps accompanied by waves of Coralax, and maybe the Leviathan makes an appearance that you have to repel. There could be civilians trapped on pieces of debris in the water, and you have to defeat some surrounding spirit sharks to free them.