You are correct in saying that the pole near the Arctic circle is really the south pole of the Earth's magnetic field because the north pole of a bar-magnet points towards it. (Wiki)
That said, it is purely convention what you define on a bar magnet to be north and south - we've named the side that points towards the Arctic circle to be the north pole of the bar magnet, and since people don't immediately identify this seeking behavior as opposite poles attracting, we tend to think the earth's field at the Arctic circle is also a north pole when it is strictly speaking, a south magnetic pole.
Also, I would add that the magnetic poles don't coincide with the geographic poles - the latter define the points of intersection of the Earth's rotational axis with the surface of the Earth. We know for a fact that the magnetic poles have shifted (even reversed) with respect to the geographic poles during the history of the Earth. (Wiki)
This question was cross-posted on Electronics Stack Exchange. Here is my answer from over there:
Exactly what charged particles are flowing outside (and inside) a permanent magnet that create the magnetic "lines"?
The magnetic field of a permanent magnet is not caused by flowing particles.
The electrons within a ferromagnetic material, even if they aren't flowing, have quantum mechanical spin. If the spin vectors of many of the electrons within the material are aligned, they produce a net magnetic dipole moment, producing the macroscopic magnetic field lines associated with a permanent magnet.
(This is just another way of saying, even when electrons aren't moving, they produce a magnetic field. We don't really know "why" that is, but we have a mathematical model of how much field they produce and how it interacts with other objects, and we call that model the "spin" of the electron).
You can read more about this in the Wikipedia article on Ferromagnetism.
Do those particles come from something inside the magnet or does the magnet do something outside of it to affect unknown particles to make the lines?
It comes from the electrons in the magnetic material.
If there is a current (i.e. a continuous flow of charged particles), then why don't we harness that current like a water wheel
Since the magnetic field doesn't derive from the flow of particles, we can't harvest it as if it were a flow of particles.
We measure B in terms of Newtons/meter/Ampere ... Consequently, those "magnetic lines" are currents (or flows) of charges
The B-field has amperes in its units because it produces a force on a moving charge according to the Lorentz law:
$$\vec{F}=q\vec{v}\times{}\vec{B}$$
Since it is multiplied by a charge and a velocity to produce a force, it must have units $\dfrac{[\mathrm{N}][\mathrm{s}]}{[\mathrm{C}][\mathrm{m}]}$
in order for the equation to balance.
Just as a force itself has $[\mathrm{kg}]$ in its units because it has an effect on something with mass, although a force does not have mass itself; a B-field must have charge in its units because it effects charges, not because it is composed of charge or contains charge.
Best Answer
The field would disappear completely.
I think the simplest explanation is in terms of the surface currents that account for the field (assuming constant magnetization, which is reasonable for thin slices). For the initial torus magnet (your second image) the magnetic field is generated, in practice, by surface currents on the planar ends of the torus. One runs clockwise as seen on the image - the one nearest the viewer - while the rear planar face has an anticlockwise current on it.
If you now stack two of those together, the surface currents on the faces in contact will cancel out, giving a bigger version of the same thing. The field will then be created just by the surface currents at the ends of the tube.
You then propose adding more magnets in this fashion until the ring is closed. That will bring the two remaining planar surfaces with surface currents into contact, cancelling out those currents. The resulting magnet will have zero surface current and therefore zero magnetic field.