For this example, I will use a wizard character with 5 Conviction and 5 Discipline.
The way casting an evocation spell works, is you cast the spell with a number of shifts up to your conviction (5), then try to control (and in the case of attacking something, aim) it with your Discipline (5). Assuming you're successful in controlling all the power, you take a 1 shift mental stress hit. You can call on more power than your conviction would allow, but for every shift above, you take an additional stress in backlash. Not only that, but every shift of power you can't control with your Discipline roll, if you want those shifts to stay in your spell, you have to take another shift of stress in backlash. On average, attempting to cast a 7 shift spell, would result in a 5 stress hit to yourself.
What I think calling on a storm would do, is it would give you a massive boost to your power (Conviction), but not your ability to control it (Discipline).
For a working example, our theoretical wizard needs to blast a toad demon into oblivion, right then and there, but also knows that the base average damage (10 shift hit - 5 power from conviction, and 5 to hit from Discipline) isn't going to do it. He calls call on the power of the storm.
The power drawn in gets a boost of 9 shifts, but it's an all or nothing thing. He can choose how much power he brings to the table, but if he wants to use any of the power of the storm, he has to use all of it. So, the cast would be done with a minimum of 9 shifts, and a maximum of 14. For that spell, in game terms it would work the same way as getting a 9 shift boost to conviction skill. The wizard would receive no backlash from calling more power than he's used to, but he'd most likely get a ton from trying to control it all.
Casting the spell- a lightning evocation attack- the wizard draws in 3 shifts of power on his own, with the added 9 from the storm, for a total of 12 shifts that need to be controlled. As stated, Discipline is 5 and the roll is all blanks (0). So the wizard takes 7 stress in backlash from the uncontrolled power, and manages to pull off a weapons:12 attack with 5 to hit.