# Must magnetic field lines close upon themselves or go to infinity?

Conventional wisdom tells us that magnetic field lines must either form closed loops, or shoot off to infinity. However, this leaves out the possibility that a magnetic field line, even if it stays confined inside a compact region of space, can in principle come back to close to its starting point but miss it, forming a quasi-loop that never quite closes.

In particular, I suspect it's possible to set this up using a twisted-torus magnetic field like the ones in a tokamak, where by controlling the relative strengths of the toroidal and poloidal components one can engineer situations where magnetic field lines close upon themselves after $N_\mathrm{t}$ toroidal loops around the torus and $N_\mathrm{p}$ poloidal loops about it, but one can also engineer things such that this never quite happens and the magnetic field line occupies a dense subset of a surface (or even a dense subset of space?).

Image source

Alas, in looking for an explicit expression for such a field I got drowned in technical results for tokamaks and magnetic fields in neutron stars, so I'll just leave this here in case someone wants to have a crack at it.

Is this possible? I'm for now mostly interested in results for this specific configuration, but if this configuration won't work (i.e. if some result guarantees that field lines in this setting do close upon themselves) then I'm also interested in examples from further afield.

• "if it stays confined inside a compact region of space, can in principle come back to close to its starting point but miss it, forming a quasi-loop that never quite closes." How? Feb 10, 2017 at 14:56
• @Yashas Like this; it's a standard example in topology. Feb 10, 2017 at 14:58
• Hmmm .... in a vague and handwavey way I am imagining that the problem with confining an infinite length path to a finite subspace is that it will fill the region densely implying infinite field strength. Certainly the approximating cases that confound the search involve high field strengths. One of these decades I have to learn some topology and other bits of usefully abstract math. Feb 10, 2017 at 16:26
• @dmckee Why would this imply infinite field strength? I think you're reading too far into the density-of-field-lines picture. Feb 10, 2017 at 16:37
• @EmilioPisanty - I fear I am misunderstanding something simple here. Does not the lack of magnetic monopoles (i.e., $\nabla \cdot \mathbf{B} = 0$) show that magnetic field lines cannot be "open" and must close except where they go to zero (e.g., as in magnetic reconnection). The reconnection community (and dynamo theory community, I think) has done lots of work on the requirements on/limitations of magnetic field topology. Mar 11, 2018 at 19:05

I had the same question a few years ago, both for magnetic field lines and for current lines in the quasi-static approximation. The following papers from the American Journal of Physics convinced me that, indeed, magnetic field lines are not always closed or extending to infinity.

Nope. Conventional wisdom is wrong about this one. Magnetic field lines can have much more interesting configurations.

Magnetic field lines are trajectories of Hamiltonian systems and, as such, are typically chaotic, which means they never close on themselves (if they did, they'd be periodic, not chaotic). This case is made very clear in this paper by Phil Morrison: Magnetic field lines, Hamiltonian dynamics, and nontwist systems (e-print, without figures).

The dense covering of a toroidal surface you mention is also typical: in fact, these so-called quasiperiodic lines are infinitely more common than their periodic counterparts - essentially because irrational numbers are infinitely more common than rational numbers.

Again, in the same way that a real-life mechanical system is typically chaotic rather than periodic, real life magnetic fields will most often also be chaotic, even for simple configurations and that's actually probably good.

In the image bellow, electric current flows through the black lines (belonging to two perpendicular wire loops), generating the field line showed in red. From a paper that makes a good case that

chaotic magnetic-field lines are ubiquitous in the modern technological world.

• Another good reference for the Hamiltonian perspective on field lines is in Janaki & Ghosh, "Hamiltonian formulation of magnetic field line equations", J. Phys. A 20, 3679 (1987). Aug 23, 2018 at 13:28
• @MichaelSeifert Thanks! I didn't know about this one. Aug 23, 2018 at 13:36