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They keep saying that spinning molten iron generates Earth's magnetic field. But, even if that iron is somewhat ionized, those two lost electrons are still nearby in the fluid. So the spin of the ++ and the spin of the -- should cancel out any magnetic field they'd generate.

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The Earth's magnetic field is not generated by spin alignment, because you can't make a ferromagnet out of molten iron. The Earth's core is at a temperature of 5700 K, far higher than iron's Curie temperature (1043 K), so any domains you would try to form would be broken by thermodynamic fluctuations.

What actually generates the Earth's magnetic field is large-scale electric currents flowing in the molten iron. Convection in the core causes the molten iron to flow, creating powerful currents that generate the Earth's magnetic field in the same way that running a current through a loop of wire does.

As to why you can get large internal currents by moving a neutral conductor around, we need to understand two things. First, "neutral" is only a macroscopic property. At any finite temperature, thermodynamic fluctuations will produce microscopic pockets of positive and negative charges in any conductor, whether it's a copper wire or a ball of molten iron. If you allow these microscopic pockets to move relative to each other, they can generate magnetic fields which will move oppositely-charged pockets in the opposite direction. In turn, those oppositely-charged pockets will produce magnetic fields that move still more charges around, each time reinforcing the initial motion. At some point, this self-reinforcing dynamic will produce large numbers of positive charges moving in one direction and large numbers of negative charges moving in the opposite direction, giving you a large-scale current in a neutral conductor, all because you allowed charge fluctuations to move relative to each other. This phenomenon is closely related to turbulence, which also gives macroscopic flow from microscopic fluctuations.

The key to having this self-reinforcing current generation is allowing fluctuations to move relative to each other inside the conductor. In other words, you must have the ability to support internal mechanical flow inside your conductor. A copper wire, for example, does not have this property. As such, you won't get current from waving a copper wire around.

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    $\begingroup$ How does moving a neutral conductor (molten iron in this case) cause a magnetic field? Just waving a copper wire in the air doesn't generate a magnetic field. $\endgroup$ – Samuel Weir Feb 9 '18 at 19:34
  • $\begingroup$ @SamuelWeir See edited answer. Essentially, what matters is the motion of different internal components of the conductor relative to each other (i.e. internal flow), which you can't get by waving a copper wire around. $\endgroup$ – probably_someone Feb 9 '18 at 20:24
  • $\begingroup$ Shouldn't we consider the solar magnetic field acting on rotating and convecting iron melt as a current inductor? From that the process could evolve.... I mean even without fluctuations there will be an induced current generating a magnetic field and so on till earth reached a kind of equilibrium as we know (mysteries and poles reversal omitted, for the purpose of this discussion). $\endgroup$ – Alchimista Feb 9 '18 at 21:20
  • $\begingroup$ @Alchimista We certainly could do that. I wanted to demonstrate that it could happen even in the absence of something like that, though, because 1.) we don't know exactly how much the solar field actually played a role, and 2.) I wanted to demonstrate the process's link to turbulence, which is easier when you can start from a microscopic picture. The fact is, this is still an active research area, so there are of course lots of details that aren't yet ironed out. $\endgroup$ – probably_someone Feb 9 '18 at 21:30
  • $\begingroup$ Ok. Thanks. I think there are even experiments with rotating molten metals inexorably finding that magnetic fields arise. It is just me having difficulties with the fluctuations picture. But I am not saying I don't believe it. ... $\endgroup$ – Alchimista Feb 9 '18 at 21:35
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Larmor proposed a geodynamo mechanism back in 1919, but the details are still being worked out. The general idea: Convective heat transport drives fluid flows, the (preexistent) magnetic field induces electrical currents in the conductive fluid moving though it, and these current “regenerate” the magnetic field.

The chicken-and-egg riddle: Which came first, the field or the currents? As another answer explained, there are always seed fields from thermodynamic and quantum fluctuations, and also from the Sun. A missing piece of the theory: How do we know that convection currents promote exponential growth of a seed field?

So-called kinematic dynamo theory identified flow patterns that promote runaway exponential growth, but they are hydrodynamically unrealistic. It has been proven that simple axisymmetric flow patterns don’t hack it, and none of the viable patterns are time-independent, so the modern question is: How does turbulent convection sustain the magnetic field?

Modern theories have begun to clarify the role of rotation in regenerating and orienting the magnetic field. Differential rotation (visible on the Sun, but merely presumed in the Earth) bends the dipole field lines into hairpins, thereby creating a strong toroidal field. (What drives differential rotation? Some suspect hydrodynamic $(\mathbf{v}\cdot \nabla )\mathbf{v}$ forces, others magnetic $\mathbf{J}\times \mathbf{B}$ forces.)

In Parker’s loop mechanism, Coriolis forces twist rising convective plumes, which in turn twist the toroidal field lines, thereby regenerating the dipole field. This picture is geometric, but there are algebraic theories that arrive at similar conclusions. (I happened to be true believer in Parker’s picture, after developing an algebraic theory consistent with his.)

Detailed simulations on supercomputers may someday validate an algebraic theory that predicts correlations between certain significant flow patterns. At present, these simulations are producing too much indigestible data. It's hard to find a needle in a haystack, and harder yet if you don't know what a needle looks like.

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