A lot of players find that if they shove in a tourney with 13BBs, as opposed to letting their stack dwindle to teeny and completely unscary, most of the other players at the tourney table will fold, leaving them with the blinds and ante. And the blinds and ante will help them survive a bit longer, during which time they may land a really good hand to play.
Ideally, you wouldn't be down to 13BBs until the blinds are pretty big, and the antes have started. Usually, you are not the only player at the table to have that stack amount. The big stacks will assume that you have waited until you have a premium hand or, at least, a decent PP, and may not have a hand that merits a call, even though they can afford to lose that amount -- if their hand is awful, they may decide not to call. The medium stacks won't want to lose that much -- if they call that raise, it's likely to be a race if you are not already beat. The other small stacks may decide to put off their own shove for another hand. The shove gives a psychological advantage at a certain point in a tourney, like near the bubble. IMO, it's not really about the math -- except the math of how much the other players will have left if they call.
I do not advocate doing this with ATC. Not with 13BB. You have some time. Do it with a hand that feels at least playable.