Usually when a tornado is starting, it happens during a storm. So there's lots of air moving around already. Now comes the part that we don't understand quite as well. So far the spinning air is still really really wide. No energy has to be added to do this, just a reduction in the width and angular momentum conservation.
Hurricanes work by similar principles.
Wind patterns near the ocean surface spiral inward around a low pressure area. Bands of thunderstorms form, allowing the air to warm further and rise higher into the atmosphere. If the winds at these higher levels are relatively light, this structure can remain intact and allow for additional strengthening. Astrophysics is also replete with examples of angular momentum conservation as well as a few mysteries related to high energy processes involving rotation. Find the period of rotation of the neutron star.