How can magnetic field lines form non closed loops? I recently came across this paper
"Topology of Steady Current Magnetic Fields", Am.J.Phys (1953)
The author points out the erroneous implications of representing magnetic field lines as closed loops.
He has used the below example to illustrate the problem.

Let us examine the field generated by a fixed current $I_1$, flowing in a ring solenoid and an adjustable current $I_2$ flowing along its axis.  A line of force, produced by the ring solenoid alone ($I_2=0$), which originates at a point $P$ will link the circuit of $I_1$, and return to $P$, the line always remaining in the plane through the axis and point $P$. Now if we insert the circuit $I_2$, we see, from the right-hand rule, that the field of $I_2$ produces the following effects, depending on the magnitude of $I_2$: The lines of force originating at P will link both I1, and $I_2$, and (a) return to P after an integral number of linkages $n$ of $I_1$, and $m$ of $I_2$, or (b) will never return to $P$ (incommensurable case). Thus, if we start an assemblage of lines from a two-dimensional region $R$ and follow it continuously around the wire, we find that the tube of this assemblage does not return to the individual points where it originated.

From the above text can someone explain why the field lines do not reach the same point where we started from in the incommensurable case? what does the integral number of linkages mean? and what do they(n,m) depend on?
 A: Honestly the entire concept of "magnetic field lines"  is a little problematic.  The magnetic field is a vector field, associated with a direction at every point in space.  Choosing a starting point and following the vector field in some direction is a useful visualization technique, but it leads new students to think that the magnetic field is somehow large "on" the field lines and small "off" of them.
The "field line" visualization is helpful in magnetism because, in a number of simple systems, choosing a point and following it through space will eventually lead you back to your starting point.  Two example systems with this property are the long straight wire (where the "field lines" are loops) and the current loop (where the "field lines" form a toroid).
What McDonald has done in your quoted figure is to add the field from a straight wire to the field from a loop.  The field on the axis of a loop varies in a particular way with distance but is always directed along the loop axis.  This transforms the "field lines" close to the wire from closed circles into a helix.
Likewise, the the field near the loop now has a component parallel to the loop, caused by the current in the wire.  This circumferential component is larger inside the loop and smaller outside the loop, so if you choose a point near the loop and follow the magnetic field, you'll no longer walk back to your starting point.
The business about integers is related to the frequency ratios that determine whether a Lissajous curve is sparse (for "nice" ratios) or space-filling (for real-number ratios).
Note that, in a system consisting of only stationary electric charges and constant fields, following a field line in either direction will always land you on a charge.  This is represented in Maxwell's equations as
$$
\vec\nabla\cdot\vec E = \rho \times\text{constants}
$$
which in words is "the electric field diverges from positive charges and converges towards negative charges."  The corresponding magnetic equation is
$$
\vec\nabla\cdot\vec B = 0
$$
or "the magnetic field never diverges or converges" (or even "there are no magnetic charges").  This is frequently a justification for "magnetic field lines form closed loops."  McDonald's point is that "magnetic field lines don't have beginnings or ends" is a more correct statement for non-trivial geometries.
A: All the author is saying here is that if we follow the magnetic field line starting at $P$, it will do one of two things:

*

*Eventually return to $P$ after making $n$ circuits of the ring and $m$ circuits of the wire.  This is what is meant by "an integral number of linkages":  the number of circuits (linkages) made is an integer (it's integral.)

*Never return to $P$.  This means that the "period" of the loops around the ring and the corresponding "period" of the loops around the wire are incommensurable, i.e., their ratio is not a rational number.  So by definition, the loops are not closed in the incommensurable case.

The values of $n$ and $m$ would depend in a complicated, non-continuous way on the magnitudes of the currents.
