Magnetic forces within a closed ferrous core

[Dorian Black]

If a toroidal ferrous core has a number of windings round it and a DC current going through them, would this magnetic core be subjected to a 'shrinking force'? To phrase it more clearly, if the toroidal core was compressible, would it actually get compressed?

On the one hand, since the magnetic field (and hence magnetic energy density) is equal everywhere inside the core, it seems there should be no force. But then if you take a certain volume within the core, calculate the magnetic energy associated with it, then differentiate it with respect to any dimension, it wouldn't yield a zero. So what is exactly happening inside the core?

 

[Baluncore]

Analyse one turn of the coil with the circle of slightly wedge shaped magnetic material inside. That wedge will be magnetised with North and South magnetic poles. All these wedges will attract and so slightly reduce the magnetic path. That is independent of the current direction in the coil.
https://en.wikipedia.org/wiki/Magnetostriction#Explanation