The earth’s magnetic field: much more complicated than you might think

At the recent NZ Institute of Physics conference, we were treated to a wonderful description of the earth's magnetic proceses, by Gillian Turner.  What makes up the earth's magnetic field? What effect does it have? How is it changing?

At first glance the magnetic field of the earth is pretty straightforward. There's a magnetic north pole and a magnetic south pole. In fact, the earth's magnetic field looks a lot like what you get from a simple bar magnet. 

But look only a little more closely, and it's clear there's a lot more to it than that. For example, the magnetic south pole is not diametrically opposite to the magnetic north pole. In fact,  both are moving about quite substantially – like a lost polar bear / penguin, depending upon your hemisphere. At sixty-something degrees south, the magnetic south pole currently isn't all that far south at all!

Then there are the reversals in magnetic field, that occur from time to time. We're talking hundreds of thousands of years. But they're not regular, suggesting a great deal of randomness is going on. 

It helps in interpreting what we see to remember that we are observing the field at the surface of the earth. It originates deeper within the earth – in the inner and outer cores. The distance from the centre of the earth is rather important. Why? Because as we move away from the centre, the smaller-scale variations get 'ironed-out' more quickly than the larger-scale variations. The effect is that the large-scale behaviour (i.e. the bar-magnet-like shape of the field) is emphasized at the earth's surface, whereas deeper down it is much less like a bar magnet. 

For those more mathematically inclined, one can see this with multipole expansions. This is a mathematical way of breaking up the description of a shape of an object into different components. One starts with a monopole – how much like a sphere an object is. Then we move on to a dipole moment – this is describing how separation of material along an axis there is.  Next are quadrupoles, then octupoles – each describing finer variations in the shape. So, as an example, a perfect sphere has a monopole moment of 1, and no multipoles of any other type. A rugby ball is quite like a sphere, so it has a high monopole moment. While it's got a preferred direction (its axis) both ends of the axis are the same and so there is no dipole moment. The next term, the quadrupole moment, however isn't zero – it's this moment that describes the bulk of the distortion from a sphere.  The link above gives a nice example of a skittle – it has monopole, dipole and quadrupole moments. 

Now, with the earth's magnetic field, there is exactly no monopole moment. Magnetic field isn't like electric charge – while one can have an isolated 'positive' charge, one can't have an isolated 'north' pole. The leading term for the earth's field is the dipole moment – there's an axis and a distinct split of field on the axis – at one end it points away from the centre, at the other towards the centre.  Now, the interesting thing is how the impact of the moments changes with distance away. The n-th order multipole has a strength that varies as the inverse of distance to the power n plus one. So the field due to a monopole varies as 1/r^2 (where r is distance away), the field due to a dipole varies as 1/r^3, that of a quadrupole as 1/r^4 and so on. As r gets large, the effect of the higher-order (higher n) moments diminishes quickly. Consequently, it doesn't matter what mish-mash of magnetic behaviour one has, at large enough distances away, the field from it  will look like that of a dipole. 

At the boundary between the outer core and the mantle, there is such a mish-mash of magnetic behaviour. A picture from an impressive computer simulation of the field by Glatzmaier and others  is here. At the earth's surface, however, it is much smoother and we see it as approximately a dipole – with a clear north pole and south pole, (very) approximately diametrically opposite. 

But the mish-mash of the field in the liquid outer core isn't the whole story. It's tempered by the solid inner core, which isn't going to change its magnetism so easily. It provides a large inertia against any changes, meaning that flipping the field of the inner core required some extreme behaviour in the outer core. It gets extreme enough just occasionally, and indeed the inner core can then be flipped, but it's not often. Our compasses are still likely to work tomorrow.

Glatzmaier, Gary A.; Roberts, Paul H. (1995). "A three-dimensional self-consistent computer simulation of a geomagnetic field reversal". Nature 377 (6546): 203–209.Bibcode:1995Natur.377..203Gdoi:10.1038/377203a0


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