And having thought about it, I'm happy to challenge your physics.
At a constant rate of turn at a constant altitude, there is a constant acceleration towards the the axis of the turn, and a constant acceleration upwards to match gravity (whose vector is hardly likely to change over this scale). There is no energy change and no requirement for power change.
Now as three blades said, though L2 egregiously misquoted him, many pilots DO record a change in power requirement when turning downwind. In that case the physics must be a little more complex than L2 implies.