PA-28 Flight Instructor reported engine surging after takeoff at low altitude on a training flight with a student and immediately returned to the departure airport. The aircraft landed safely and maintenance later discovered a leak in a fuel line.

Narrative

At approximately XA:30 pm my student and I were departing from ZZZ to practice commercial maneuvers. After a normal runup and following all procedures as prescribed in the POH; we lined up on Runway XXL. My student was pilot flying at the time. She applied full throttle and we both verbally confirmed 'Engine instruments in the green' meaning and confirming that oil temperature; oil pressure; fuel pressure; and RPM were all within normal range. We were achieving approximately 2300 RPM on our takeoff roll and RPM was steady. After rotation and during our initial climb we experienced RPM surges from 2300 down to 1000 and back up to 2300 continuously. I immediately took control and verbally called out 'my controls' which were confirmed through a three way positive exchange of controls. I called Tower and announced that we were experiencing RPM loss. I made the decision to do a 180 back to the parallel Runway (XYR). I followed all emergency procedures for Engine Power Loss in Flight as prescribed in the POH and confirmed that oil temperature; oil pressure; and fuel pressure were still 'in the green.' I [requested priority handling] with Tower as I turned a right crosswind/base off of XXL for XYR and was cleared to land on XYR. We landed safely on the runway and maintenance found that there was a leak in the fuel line.