[Curious Cat (OP)]

.

[Bored chemist]

https://en.wikipedia.org/wiki/Coriolis_force

[alancalverd]

Nothing is going the wrong way. For some reason I always imagine myself spinning anticlockwise viewed from above, so every off-axis atom is moving anticlockwise. Now I alter my moment of inertia by extending or contracting my arms, angular momentum is conserved, various  atoms move faster or slower, but still anticlockwise.

[Halc]

Suppose she starts out with her arms straight out, brings them in by
bending them at the elbows, horizontally, and speeds up, and yet those
atoms in her lower arms are now "pointing/going" the wrong way,
to the direction of motion, if U get/know what I mean!?

The direction of spin never changes, as Alan points out. Nothing 'goes the wrong way'. Just at a higher angular velocity.

Relative to the rotating frame of the skater, yes, it is Coriolis force that causes the spin rate to increase.

Angular momentum decreases (due to friction) only slowly if you're on ice, so the reduced angular moment (the rotating version of inertia) must be accompanied by a greater spin rate to preserve the angular momentum.

[Halc]

Ah, yes, but/t, the atoms in her lower arms have a forward momentum.

All parts of her have angular momentum in the same direction, which I suppose by definition must be 'forward'. Assuming she's spinning counterclockwise as viewed from above, her angular momentum including her arms and other parts) have an angular momentum vector pointing up, not some other way. Fold the arms inward, crossing to the other side and their angular momentum is still up, but less, having transferred some of it to the torso. The direction of the momentum never changes no matter what contortion she puts herself into, including inverting, legs up. Angular momentum remains up the whole time.

Her leading edge is now her trailing.

There is no leading edge to rotation. That's a linear momentum trait, which is also conserved, but the spinning skater doesn't have much of that in the frame of the ice into which she's eroding a hole.

What's the Coriolis force got to do with it? Never thought about that.

It's what makes hurricanes do the same thing: spin faster when the air draws inward like the skaters arms, due to the low pressure center at the middle. It's what makes you pee in a curved path while standing on a rotating space station. It happens on Earth too since it rotates, but few pee far enough to notice the curvature.

[alancalverd]

I think I can see the confusion.

Spinning anticlockwise with arms outstretched, if you cross your arms forwards and all the way across your body, your right arm temporarily rotates a little quicker and your left arm a little slower, but there is no net change in angular momentum and the whole lot continues to rotate anticlockwise.

Not to be confused with spinning an airplane, where the outer wing speeds up and the inner wing slows down with respect to the airflow and becomes stalled. If you really screw up, you might even get part of the inner wing to move backwards, with truly  disastrous results.

[alancalverd]

t's what makes you pee in a curved path while standing on a rotating space station.

Ah yes, we've all observed that. Bloody nuisance, having to review rotating differential geometry after a few beers.

[Halc]

No, not fully across, just at the elbow.
Suddenly the leading edge of Ur lower arm is facing backwards.

Linearly, yes, it is moving in the opposite direction as the elbow (relative to the ice). But that's linear momentum, also conserved. Her total linear momentum was zero and stays that way regardless of where she puts her arms.
The hand and elbows are all still spinning the same direction (up) so bending them that way didn't change the direction of the angular velocity or momentum of anything. If she's really talented, she can impart angular momentum in the downward direction to her long braided hair while she's spinning, but here total angular momentum is still unaffected by that.

It appears to be a paradox if U look at it from the point of view
of individual atoms, having momentum, and taking time to realign.

The conservation laws are actually derived from the laws governing those tiny particles. In a closed system of any number (small or large) of atoms, it is impossible to alter the linear or angular momentum of the system. These are old laws of motion first stated by Newton. No new fancy physics going on here.

I thought the Coriolis was all about stuff going N-S appearing to curve.

Naw, it's stuff going towards or away from the spin axis that does it. So there's no Coriolis force at the surface of Earth at the equator because motion north and south do not change your distance from the axis there. Toilets flushing don't spin (unless they utilize angled jets). No hurricanes at the equator.

[Petrochemicals]

How does a spinning skater really work?

Suppose she starts out with her arms straight out, brings them in by
bending them at the elbows, horizontally, and speeds up, and yet those
atoms in her lower arms are now "pointing/going" the wrong way,
to the direction of motion, if U get/know what I mean!?

Yes but the skater is one cohesive bundle,

[Halc]

What if she folded her arms very quickly? How would the atoms react, exactly?

Her hand atoms would move faster from doing that, gaining kinetic energy, but the total momentum and the total angular momentum* of all her atoms would be unchanged.

*Total angular momentum actually slowly goes down, regardless of what she does with the arms, due to friction with the ice. So what I say is only true if she's spinning in space or some frictionless environment like that.

It's drawing in her one leg that really gets things spinning. They all start with one leg extended because that's how to give yourself maximum angular moment and still be on one skate.

[Colin2B]

Yes but/t is still made up of individual atoms, with a momentum, of their own.
What I'm really saying is: What if she folded her arms very quickly? How would the atoms react, exactly?

The atoms are her arms, it’s the atoms doing the moving.