Three-dimensional centrifugal instability and the Rayleigh critertion
 

Lord Rayliegh postulated that for a two-dimensional symmetric swirling flow, a three-dimensional instability may occur if the angular momentum magnitude decreases in the radial direction with respect to the centre of rotation. This is known as the Rayleigh criterion. This happens because the balance (for a fluid volume) between the centrifugal force and the radial pressure force is disrupted and the fluid volume is further displaced from its original position if perturbed.
My question is: why when this instability happens, the flow creates three-dimensional structures instead of just breaking the steady swirling flow in a two-dimensional manner. Why is there fluid being ejected into the normal direction of the plane?
Thanks.

A quick answer to that would be to look at the Biot-Savart law.

Thank you for your answer. Could you be a but more specific?
The Biot-Savart law just recovers de velocity field from a vortex line, but I can't see how for a vortex line of a 2D velocity field (straight line) a 3D velocity field can be recovered.

I think that a useful way is to look also to the vorticity vector field and its equation. In general, instability would onset the stretching and the vorticity vector does not remain normal to the velocity plane