what discharges through the bottom coil
The magnetic field and induced current. Charging an inductor via current flow creates a magnetic field, which when collapsing, induces a current flow in the inductor. This is covered in introductory physics textbooks.
Look up "blocking oscillator". This simple circuit is somewhat complicated to analyze because the base-emitter junction will break down due to high reverse voltages, and allow reverse current to flow. To avoid eventual destruction of the transistor, add a diode to protect the base-emitter junction as shown below. L2 current shown, for a guess at the inductance values.