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Strongly inspired by the current thread Are there necessarily always at least two points where the Earth's magnetic field is vertical?, this is the opposite question:

Is there necessarily always at most one point (on Earth's surface) where the Earth's magnetic field is directed vertically downwards?

The English Wikipedia page North Magnetic Pole currently says:

The North Magnetic Pole is the point on the surface of Earth's Northern Hemisphere at which the planet's magnetic field points vertically downwards (in other words, if a magnetic compass needle is allowed to rotate about a horizontal axis, it will point straight down). There is only one location where this occurs, [...]

(emphasis mine).

Of course the Earth's true $B$ field is far from an exact dipole field, see e.g. these diagrams.

Can it not occur that there are two distinct magnetic poles in the Arctic, say separated by 100 kilometers, where the inclination is exactly +90 degrees?

Are there any geometric, electromagnetic or geophysical "laws" preventing that? I cannot think of any.

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If some fluctuation in the currents in Earth's core was to originate a second magnetic axis, the magnetic moments relative to the two fields (the "normal one" and the "anomalous one") would start to exert forces on each other until they merged in a single field, according to the equation for the torque acting on a magnetic dipole in an external field:

$$\vec \tau = \vec m \times \vec B$$

Therefore any fluctuation would be quickly suppressed by the forces exerted by the magnetic field itself. The only possible equilibrium situation is one in which there is only one magnetic axis.

In the picture below (source) you can see a simulation of the magnetic field of Earth in an equilibrium situation and during a geomagnetic reversal; blue lines represent outgoing field, and orange lines ingoing field.

As you can see, during the reversal there are multiple "poles", but eventually the field returns to a dipolar one (but with a differently oriented magnetic axis).

enter image description here

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  • $\begingroup$ At least this shows (image on the right) that in some geological periods there will be many "north poles", even if these periods are extremely short. But even when I look at the picture on the left where, clearly, the general dipole trend of the field is evident, the blue "hair" on the top does not appear to be completely uniform. I cannot tell, from the picture, that there are not two points near the word "north", maybe separated by 100 kilometers, where the blue lines are exactly "vertical"/radial. Evidently, the force you mention has not made the dipole a perfect one. $\endgroup$ – Jeppe Stig Nielsen Jul 21 '17 at 9:07
  • $\begingroup$ @JeppeStigNielsen Well, Earth's magnetic field is not perfectly dipolar, like you said, so you shouldn't expect it to look like a perfect dipole. This doesn't change the fact that two separated poles wouldn't last long, for the mechanism that I have explained above: they would quickly merge into a single one. $\endgroup$ – valerio Jul 21 '17 at 9:37
  • $\begingroup$ Just to be the devil's advocate, you say there cannot be two north poles, because if we had a minor field competing with the main field, the torque formula $\vec \tau = \vec m \times \vec B$ shows that the minor field would "quickly" be turned to align with the main field anyway? To the devil, that argument can also prove that the entire field will "quickly" become an exact dipole field. Yet we agree it is not a perfect dipole; there are quadupole, octupole, etc. contributions. With those higher order terms, is anything preventing the north pole from splitting into two north poles? $\endgroup$ – Jeppe Stig Nielsen Jul 21 '17 at 12:05
  • $\begingroup$ @JeppeStigNielsen Those terms are much weaker and can generally be neglected. My argument is a very rough one, to make it more precise you should take the equations of magnetohydrodynamics for Earth's core and show that the equilibrium solution is an approximately dipolar field. But that would be terribly complicated (and overkill). $\endgroup$ – valerio Jul 21 '17 at 13:45
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Jeppe, as the images above show, there can be multiple "north" poles at the same time. However, I disagree with the stability analysis by valerio92. There is nothing about magnetic torques that favors a dipolar structure; just look at magnetic domains in ferromagnets.

There are two main reasons the Earth's field generally has two poles that are nearly aligned with the geographic poles: the Earth's rotation tends to organize the flow, and the solid inner core tends to slow down any changes, giving the rotation time to assert its influence. For more information, see this summary by the scientist who generated the images.

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