Why does my tea periodically alternate its rotational speed after stirring? (Link to video below) I noticed that after stirring, a bubble in the centre of my mug of tea changed the speed it was rotating at periodically. Speeding up, then slowing down, then speeding up again, etc. Almost like when a ballerina pulls in her arms to increase her speed.
Tea after stirring
Edit:
I've repeated this with room temperature water to try and rule out any temperature-related effects and the same effect is present.
 A: When a liquid spins, there is often a vortex that spins quickly. Velocity drops away from the center. Usually the vortex spins quietly in the center of the vessel. 
In this case, there may be a vortex that is off center orbiting around the center of the cup. When the vortex passes under the bubbles, they spin quickly. When it passes on, they slow down. 
This sounds plausible at first, but I am not sure if it is realistic. The vortex affects the shape of the surface and draws the initially separated bubbles together. It looks like the center is depressed, and the largest component of the circular motion is around the center of the cup. 
When the spoon swooshes through the tea, two counter-rotating vortices are created. The spoon travels in a circle, and this is what gives the tea its overall circular motion. 
Perhaps one or more counter rotating vortices survive. It is clear that the bubbles are pulled off and on center in sync with their rotational speed changes. Perhaps a counter rotating vortex orbits the center, and slows the bubbles when it passes underneath. Perhaps it attracts or repels the bubbles. 
This is all speculative, but there are a couple points. First I suspect that rapidly spinning vortices are involved somehow. And second while period doubling might explain the motion, it might also be explained by two separate causes where one of them comes and goes in a regular way. 
Really cool video. I would like to know what is really going on. 
A: Just looking at the video, it appears that the shape of the surface is varying quasi-periodically, as if the liquid is moving outward (and upward) toward the cup walls, then moving inward and rising in the center of the cup.  This can be expected, if in the beginning the shape is not a perfect equilibrium shape (e.g., like a parabolic surface in a rotating cup).  But when the the liquid moves toward the center, the rotation necessarily speeds up due to conservation of angular momentum; and when it moves outward the opposite happens.
A crude analogy:  If you rolled a marble in a large wok with smooth spherical curvature, in such a way that it looped near the center/bottom, then out near the edge, you would see that its angular velocity increases when it approaches the center/bottom, and decreases when it recedes from the center/bottom.  You can think of the volume of liquid doing the same thing as the surface shape changes from a shallow curve to a deep curve.
A: You have created a Lissajous spiral from the tea: the raising of the mixing utensil creates a partial secondary flow which deviates some of the concentric flow up and down. The continuous rotational flow dominates and it is interrupted by a group that flows up and down as well as concentrically.
 
The effect slightly follows a Lissajous trajectory, although it is a complex turbulent flow and would necessitate some direct observation to understand precisely.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
As others have said, it's very likely that the lissajous travels downwards through the center of the mug and back upwards trhough the sides, so that it actually becomes a spiralling torus.
You can use oat flour or glitter in a transparent mug to see the effect. You can glue a syringe with milk to a spoon to see if you can observe a Lissajous spiral and film it. It probably necessitates a big jar and a video because there are a lot of small turbulent vortices that spiral and mix the boundary of the concentric vortices, especially when the Lissajous flow hits upper and lower boundaries. 
A: The liquid in your mug does not rotate as a solid body due to viscosity; it also goes up at the center and down at the mug walls ("rotates in a vertical cross section") due to heat losses and thus stratification effects. So the 3D motions are not stationary anyway - they will stop in a finite time. Their interplay (roughly saying, transport time and rotation time) lead to some surface effects that you observe.
A: The variable rotation speed is due to (semi-)turbulent motion. 
You can see this more clearly in this YouTube video that doesn't show the rotation by a few bubbles but by thermal imaging (which also shows the convection).

https://www.youtube.com/watch?v=Va8xP2Q6sgo#t=2m00s
It is not the entire flow that changes uniformly, but instead it looks more chaotic. Medium size rapidly rotating edies occur (randomly) and dissipate their energy and dissapear after which new eddies form.
The pattern seems to occur rhythmically because it only appears like that (but it is a bit random). Although the eddies may have a narrow distribution in lifetime because the Rayleigh-Bénard convection on top of the rotation causes fluid to move inward (where liquid is cooler and sinks) which creates a consistent pattern of acceleration in the angular speed and eventual collapse of the eddie. 
Note that these eddies keep occurring also when the stirring was a long time ago. They are powered by the 


*

*energy/motion from thermal convection (or only stirring if there is no strong temperature difference)

*turned into eddies/rotation due to turbulence (and regular instabilities that can already be explained by laminar flow)

*and a tornado effect because the liquid is moving to the inside (convection) which accelerates the angular speed (like the ballerina making the pirouette and explained by conservation of angular momentum)



The exact type of instabilities that cause the oscillations might be difficult to pinpoint because there are multiple effects (initial chaotic state due to stirring, temperature/gravity gradients, flow instabilities in rotating fluids, deceleration of the rotation when stirring stops). There are many visualisations of such effect (with ink or particles in liquid) an example is this YouTube video

https://m.youtube.com/watch?v=emWThWDNjsE
related to a MITopencourseware course and http://weathertank.mit.edu
A: To corroborate on the observation of @Vladimir Kalitvianski, this hypothesis might be understood by simple Newtonian physics. 
There are two timescales in the system - $\tau_J$ timescale of the dissipation of angular momentum and $\tau_I$timescale of (quasiperiodic) change in tensor of inertia of the liquid. It seems that $\tau_J$ large enough in comparison to $\tau_I$ such that tensor of inertial changes significantly without significant change in angular momentum at small finite times. The rest is the consequence of the $\vec{J}=I\vec{\omega}$ relation.
To experimentally test it, it would be nice to measure the time dependence of the circulation at the center (image processing the orientation of the bubble at the center) and measure the time dependent local height profile of the fluid, from which it is possible to obtain the time dependent component of tensor of inertia. These two independent measurement should correlate strongly in Fourier space if the hypothesis is correct.
A: As pointed out by Vladimir Kalitviansky: as the fluid slows down (due to friction) an internal circulation commences: along the walls fluid is descending and hence in the center fluid is rising.
This circulation can also be inferred from the following observation: when stirred tea is slowing down the tea leaves that are on bottom of the glass accumulate in the center.
This circulation would not form if the glass itself would be co-rotating with the fluid. Then you get solid body rotation of the entire body of fluid.
In the case of solid body rotation the surface of the fluid redistributes to a shape with a parabolic cross section. Thus at every distance to the center of rotation the required centripetal force is provided by the inclination of the surface.
Here the fluid that is touching the wall is being slowed down, so it doesn't have the angular velocity that is required at that distance to the center of rotation. As a consequence the fluid that is touching the wall descends, and in turn that pushes up fluid in the center.
At the top the fluid must spread out again. That is: when the swirling fluid is in the process of slowing down the top layer consists of fluid that is flowing outward from the center. 
As the top layer flows outward it loses angular velocity. Friction with the fluid underneath it brings the top layer up to speed. By the looks of it: this transfer of momentum happens in bursts, rather than continuously.
That, I seems to me, is the explanation for the changes in angular velocity of the top layer of fluid that you are observing
A: Welcome Luke, and congratulations on such a great question.
If the fluid just rotated uniformly, we'd expect to see a low spot in the middle (a vortex). due to the inward acceleration of the fluid by the cup. This is almost all we need to know to see why the bubble rotates more quickly in the low spot than when it jumps out. In the center of the vortex, the fluid rotates around the bubble uniformly and the little surrounding bubbles rotate with the fluid at the radius of the big bubble.
When the bubble group jumps out of the vortex, it keeps rotating in the same direction but slowly. Here, we see the part of the bubble group closer to the vortex moving more slowly than the part closer to the cup's edge. Crudely, $V = r\omega$. Since the outer edge of the group is further from the vertex center than the inner edge, the little bubbles spin around the big bubble slowly.
This is a remarkably complicated system. Your spoon accelerates the fluid in unpredictable ways, surface tension makes for a little equilibrium for the group at the center, there may be different density fluids present (if you recently added cold milk, for example), bubbles don't behave intuitively (check out the corks in the video I linked) and the friction between the fluid and the mug cause turbulence (depending on Reynolds number). The perturbations from these factors help explain why the vortex isn't in the center of the cup, why the bubbles "jump" in and out of the vortex and why the behavior gets slower and more uniform as the experiment progresses.
I see no evidence of liquid moving en mass to the center of the cup, nor of a higher center - it looks like your tea is like every other spinning liquid with the low point in the center. Further, I would be shocked if the mug cooling the fluid caused a down-current at the mug's edge during the experiment that was bigger than the factors I've listed so far. 
If you wanted to test this hypothesis, place bubbles at varying distances from the vortex. As they move further from the center, they should spin at roughly the same rate until you place them on the edge where they may stop spinning or spin backwards due to the surface friction. There's an integral you could do to see how they vary, but these measurements are crude, and the differences will be small.
A: That's a good question for a new contributor! Let me try to give you an answer.
When you pour the coffee you give the body of coffee a rotation (clockward) in the cup. You can put a velocity vector field on the surface of the coffee. Initially, all these vectors have the same magnitude and directions parallel to the tangent vector on the side of the cup.
So in the middle of the cup, the rotation velocity has the highest value, simply because there is the least distance to travel before a complete rotation is completed. 
The bubble goes along with this rotation (also clockward; just try pouring the coffee in the opposite direction; it would be very strange if this wasn't the case!). When the bubble is exactly in the middle (as well as the middle point of the rotation) it should stay there and will rotate along with the coffee (though in the video it doesn't seem the middle of the coffee surface has the highest rotation velocity, but let's assume the highest rotations velocities find themselves around the middle). The surface, in this case, must be symmetrical wrt the middle point of the round glass. The water molecules whirling around the vertical axis going through the middle point don't form a vortex. 
But this is theoretically. Just a tiny displacement from the center is enough to let the effect occur, and this will surely happen after pouring. So in reality, when the bubble moves outward a bit, its angular velocity is diminished because of the interaction with the different (smaller) angular velocities of the coffee surface and because the angular momentum has to be conserved. This also drives the bubble back to the middle. You will never see the bubble moving too much away from the center of rotation. All angular velocities are reduced because of friction, tending to make all angular velocities equal. The bubble is doing a rotation dance!
Finally, the (angular) velocities all tend to zero because of friction and the effect is gone, obviously. 
You made a great video. The temperature has to do with it insofar the viscosity of coffee becomes less if the temperature is increased. Maybe you can try to put the bubble in the middle and see what happens when trying to let the coffee rotate in such a way that the center of rotation will find itself in the middle, though this is very difficult to do! 
The bubble spends more time in the middle when the temperature is higher (where it spins faster) because it's easier pulled to it than driven away from it. Obviously. Why? Maybe you can think about that for yourself (viscosity).
Here is another, new video.
A: Possibly the Dzhanibekov effect in a fluid:
https://www.youtube.com/watch?v=L2o9eBl_Gzw
Perhaps a more controlled experiment where motions outside the surface plane are minimized could be created to prevent the occurrence of the effect, but even then, I suspect the effect is unavoidable due to irregularities created in a liquid by the spinning motion itself that would cause the moment to move away from the exact center.
A: My best guess is that the whirlpool that carries the mass of the water downwards has only a limited pathlength to do so. 
At some point you start stirring, and at that point the whirlpool starts accelerating mass downwards. So it carries the mass down, it bounces back up and from the surface back down, periodically. 
If you now imagine a conical whirlpool, the lowest point, where the fewest mass is displaced, rotates the fastest, the highest point rotates the slowest. These points periodically interchange.
I think this is the same thing that @Alexander tried to express, but unfortunately unnecessarily complicated.
A: https://en.wikipedia.org/wiki/Rayleigh%E2%80%93B%C3%A9nard_convection
https://www.youtube.com/watch?v=ovJcsL7vyrk&t=660s
Your system is more complicated due to the rotation, but the oscillations are related to that general bifurcation effect.  Technically, the Bénard cells may not be necessary (or even present) in your cup, but the bifurcation is.
