transformer – Unwanted saw tooth waveform on DC rails

I am somewhat new to electronics and am building up a dual rail power supply. Under no load the DC rails sit at about +15V and -15V. Please refer to the schematic and oscilloscope screenshot (below) for the following discussion.

I discovered a saw tooth waveform sitting on top of both DC rails. This saw tooth waveform is a problem as it grows in amplitude with increased load (22mA = 50mVpp, 0.6A = 700mVpp). At a 3A load the amplitude of the saw tooth waveform is greater than 4Vpp.

I do not understand the origins of this saw tooth waveform. Why is it there? What is generating it?

The saw tooth waveform appears on either rail when a load is attached to that rail.

The saw tooth waveform is not discernible on the oscilloscope when no load is attached.

If I replace the transformer and the rectifier with a commercial DC power supply, but keep a load in the circuit, the saw tooth waveform is not present.

It seems that the combination of a rectifier and a transformer and a circuit load generate this issue but I do not have a clue why.

I tried full bridge rectifiers, transformer, and capacitors from different manufacturers but got the same result.

I thought maybe this could be the result of inductive coupling of the AC wires to the DC wires. To that end I isolated as best I could all wires and the transformer but there was no change in the presence or amplitude of this saw tooth waveform (for a given load).

Any help in understanding why this issue exists would be greatly appreciated. If this issue is inherent in this design, I would very much appreciate any ideas on how to filter it out or even design changes to avoid this issue in the first place.

Thanks.