How fast does the Earth's core rotate currently, and what would happen if its speed of rotation were to increase dramatically?
Earth's core currently rotates at about the same rate as the rest of the planet, 360 degrees every 24 hours. There is a hypothesis that it may rotate very slightly faster, an extra 0.3 to 0.5 degree every year, but this is unproven, either way.
If Alien Space Bats somehow made it spin a lot faster, things would get exciting. It would change shape from a slightly flattened sphere into a somewhat more flattened sphere. It would also start to transfer angular momentum to the rest of Earth, because of friction between the liquid outer core and the inner layers of the mantle, and the same friction would start to convert the energy of the faster rotation into heat.
All of this means earthquakes and volcanoes. The change of shape and the transfer of angular momentum would be noticeable first, up here on the surface, first in earthquakes, and them probably in volcanoes triggered by the 'quakes.
How powerful are the 'quakes? That depends on how much the core has been sped up. Doing that hard enough will tear the planet apart, doing it rather less would just melt the whole thing, and doing it more gently would just wreck everything we've built and cause an apocalypse. It's fortunate that there's no obvious way of doing this, and that it's going to be far easier for the ASBs to just nuke us, or hit us with extinction-level asteroids.
Increasing the rotation of the core of the earth will increase the magnetic field around the earth which might have further ramifications. This increase will occur because of the existence of electric charge in the liquid core which will rotate at higher speed thus producing high electric current and magnetic field.
If the Earth's core were to speed up then the liquid outer core would heat up. As you have mentioned only the core is spinning faster not the whole Earth. This could result in massive friction from the outer core and the mantle. This means more heat and more volcanoes. In the short term it would ruin everything, however in the long term faster Earth's core means protection from the sun. Eventually however the core would slow down as all the rotational energy is converted into thermal energy/heat. However during the time the Earth's core does rotate we would get stronger. This could be good and bad. Magnetic field protects from solar radiation, but around the Earth there is a layer of radiation around the Earth called the Van Allen's belt. So if the Earth's magnetic field got any stronger going to space would still be a bad idea even in the Earth's magnetic field as the Van Allen's belt would accumulate more charged particles which could affect electronics in space.
All of this is the positive side of things and considering that the core did not speed up very fast very quickly. If Earth's core sped up very fast it would quickly melt the mantle and the crust. It would probably make Earth the hottest planet now. Not because of the greenhouse effect(Molten surface means more carbon dioxide means more greenhouse effect) but because the whole surface is molten and hot, from the heat of the Earth's core.
You might look into the Dinosaurs on the moon hypothesis. It roughly goes like this:
- In the beginning or our solar system, the sun ignites, the remaining accretion disk gets hot.
- Big hunks of accretion are drawn into fast spinning molten metal spheres.
- The sphere cools to make a crust. Life evolves. The world has no oceans and is a huge hunk of swamp with low gravity on the surface from the rapid spin. Go dinosaurs!
- Internal friction causes the Earth crust to spin a bit faster than the interior. An instability occurs and slowly magnifies. A large blister forms lifting some portions of the crust high on pockets on magma.
- The system is unstable. A cataclysmic break in the crust causes the ripping of large sections of the crust as the high rotational blisters rip off. Serious mass departs from the earth, taking huge swaths of crust. The remaining crust spins slower. Water accumulates into the low sections. The earth develops a tilt. Some life survives, but its not the big swampy paradise any more. The big blister looks suspiciously like a meteor strike.
- The ejecta exits the atmosphere but is captured by the planetary pull. We call it the moon. With enough exploration, we will eventually find dinosaur bones on the moon.
The theory, while ridiculous, has a surprising number of correlations with reality: fluid simulations; timing; composition; rotational velocity; etc.
So changing the rotation speed of the core might make things unhappy on the surface. It should take a while.