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One of the surprises for me in working out the answer to this question: Why is the Earth so fat? , is that the core is more elliptical than the surface, the extra ellipticity builds up gradually to about 10% (this is false, I miscalculated the ellipticity, it is actually less elliptical by a factor of 1.47, so 47 percent less ellipticity. This doesn't change the question, except the imagined effects become smaller). As the Earth slows down, the interior has to relax to the new shape, and the inner core is solid. This means that the solid core is the wrong shape for a slower rotating Earth, and it must relieve the gravitational stress by a "core-quake" to move material to the poles from the equator. Unlike the crust, I don't see how it can do this by slow flows, because it isn't floating on anything, it's solid.

I was wondering if anything is known about these core-stresses caused by rotation slowing. Is the core ellipticity wrong for the current rotation of the Earth? By how much? How does the Earth's core relax to the new stable shape when the Earth slows down? What's the time scale? Does this mechanism have any effect on the Earth's magnetic field?

If all of this is unknown, or uncertain, this is a fine answer too.

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Could you please be more specific and explain why you think that "stresses must be relieved" when the Earth is slowing down? What stresses? Formulae? The slowdown itself (billions of years for a dozen of percent change) is so incredibly slow that it surely doesn't exert any detectable forces by itself, does it? Geology has pretty much established that there is a solid inner core inside the liquid outer core, so the inner core is just swimming and whatever the pressures from the outside are, the core doesn't have a problem to adjust, does it? – Luboš Motl Aug 20 '12 at 8:53
Over billions of years, the shape of the Earth may change so that the different ellipsoid radii increase or decrease by those 10 km or so from the fat-Earth question. It a negligible percentage, isn't it? The Earth is like a tomato - filled by the liquid outer core, among other things, and rather flexible other layers - that may get 0.1% more squeezed from the top down over billions of years. Why is it a problem? Of course that if the motion of the interior slows down or changes, the magnetic field also does. But why would you study the "stresses" during the slowdown to quantify it? – Luboš Motl Aug 20 '12 at 8:57
If the Earth stopped spinning, it would be more energetically favored to have concentric spherical layers. As long as the layers may get mixed and transported from a place or another, the motion will favor the ordering according to the new potential surfaces. But as long as the layers are solid enough, the remixing will be low. But the stresses/forces from the change itself will always be negligible relatively to other drivers. Just look at continental drift. It moves continents by thousands of km in 10 mil years. It's not hard to deform some layers by 10 km in 5 bil years, is it? – Luboš Motl Aug 20 '12 at 9:01
I also want to say that the motion of every volume of liquid or stone within the Earth is determined by the total force. If pressure acts from all sides, it's the gradient of the pressure that matters for the motion. But when it's nonzero and is accelerating a piece of matter, it will immediately reshuffle the pressures so that an equilibrium is more or less instantly restored. Tiny changes of density are enough to change the pressure by a lot (near incompressibility). – Luboš Motl Aug 20 '12 at 9:03
@LubošMotl: The issue I had is that the inner rotation rate is controlled with feedback to the magnetic field (as my limited understanding suggested), I don't know whether the (solid) core can relieve ellipticity stresses caused by a 5% slowdown, since unlike the surface, it isn't floating on liquid. – Ron Maimon Aug 20 '12 at 17:12

While googling around, I found a simple answer--- the Earth's solid core is believed to be freezing out of the molten layer that surrounds it, gradually as the Earth cools. The core started freezing around, say, 4 billion years ago, and has reached 1000 km, so the rate of growth over the last 2 billion years only needs to be altered by a percent at the poles vs. the equator to fix the ellipticity.

There are no rotational stresses on the Earth's core if this is correct, the core is always at the proper ellipticity for the rotation.

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