Branching lemon drop "smoke rings" This might be hard to ask, but here goes nothing.
I recently poured a cup of water into a black coffee cup.  There was a light source--not very bright--above the cup.  Anyways, I was squeezing a lemon into my water mindlessly, and to make sure I got every last drop of lemon juice into the water, I watched the lemon juice hit the water.  Then I looked closer as something pretty neat was happening.
When a single lemon drop hit the water, it dissipated into a shape that many would describe as a "smoke ring."  Knowing some physics/fluids, I understood what was happening here was nothing out of the ordinary.  But then I kept watching.  As the ring dissipated, it eventually broke off into about 5 other smaller "smoke rings." I couldn't see further down, so who knows if it continued; but I would assume that the 5 smaller rings would turn into 5 + X amount more, etc.
At what point would they stop breaking up into smaller rings? Whats causing this to happen?  Does this movement/shape have a scientific term? Does anything else do this that can be easily seen?
I'm more interested in what the movement/pattern that is happening here, not the chemistry aspect.
 A: Edit: 
The original explanation focused on the vortex production. For a description of what's happening when the vortex breaks off into child vortices, I've added this new section. 
The breakup of a single vortex ring into many smaller rings looks sounds like it should be driven by an azimuthal instability. Indeed, under certain conditions, azimuthal waves can grow around a ring's circumference, with instabilities showing as many as 20 peaks around a circumference [1A].
With this in mind, consider what happens when these instabilities drive two vortex filaments closer together. In an inviscid fluid, the filaments could pass through each other and continue developing. But in a viscous fluid, diffusion can cause the overlapping filaments to reconnect into different topologies, spawning child vortices. Such behavior is explained in [2A, 3A] for an elliptical vortex ring splitting into two vortices, but the same mechanism could drive the breakup for higher-frequency instabilities.
(Note, the initial instability in the vortex ring would likely be initiated by the drop-splashing dynamics, and would thus be governed by factors like surface tension, impact speed, etc.)
References
1A. Green, Sheldon. Fluid Vortices. 1995. p124-129
2A. Ibid. p134-140
3A. Dhanank and Bernadinis. J. Fluid Mech., 109. p189-216. (1981) DOI: 10.1017/S0022112081001006

Original Post:
Cool observation! Seems like it would make an interesting research project. It's hard to explain exactly what you saw without pics/video, but here are some initial ideas/background on the problem based on some research in the area of droplet splashing:
Why does this happen?


*

*A lot of researchers dating back to the 1800s have observed regimes of droplet impact can generate subsurface vortex rings (see all refs. below). The driving mechanism is a balance of drop inertia, water viscosity, and surface tension. Somehow, these dominant mechanisms should be able to explain the vortex ring dynamics you see.

*Morton et al. [1] use computer simulations to show that droplet impact on a surface can generate multiple vortex rings under certain conditions. Furthermore, if a small jet or other drops from the lemon impacted the water after the initial drop does, you could have additional vortices formed. 


*

*Interaction of any of these multiple vortices could lead to complex break-up events.


*A lot of the surface tension effects have been studied by many researchers including [2] and [3] below. The branching of the vortex rings sounds reminiscent of common surface-tension driven instabilities. Going off of some of the above research, one hypothesis could be that the combination of surface waves and collapsing unstable jets drives vortex breakup to follow the patterns you see.
How deep would the vortex rings go?


*

*This should depend on both the viscosity of the water, the speed of impact, and the surface tension of the water.

*An earlier study [4] (confirmed by others later, eg [1]) demonstrates that the vortex penetration distance is affected by the droplet shape when it hits the water surface (since the drop oscillates around a spherical shape). This could potentially also set the stage for a directional vortex breakup like you saw.

*Given these mechanisms, I would be surprised if the vortices broke up more than once or twice in the cup.
References


*

*Morton et al., Phys. Fluids 12, 747 (2000) DOI: 10.1063/1.870332

*Cresswell and Morton, Phys. Fluids 7, 1363 (1995) DOI: 10.1063/1.868524

*Thoraval et al., Phys. Rev. Lett. 108, 264506 (2012) DOI: 10.1103/PhysRevLett.108.264506

*Rodriguez and Mesler (1987) DOI: 10.1016/0021-9797(88)90414-6

A: The smokering is the epitomy of perfect turbulence as is the 'smokering' described in this question. The fluid through which the smokering is passing flows laminarly against the circumference of the rotating ring as the smoke ring travels. The energy imparted to the molecules of the ring is stored in the circular rotation of the ring. A drop falling into the water is molecularly identical to a huff of air forming a smoke ring.
Part dieux. Surface tension will eventually 'gather' the molecules of the ring into 'pieces' and each piece will be 'required' to preserve the rotational energy of the original ring. This results in dAughters of the mother ring with the same rotatioal characteristics as the mother.
A: This is a wild-ass guess.
You understand why the lemon drops form into a "smoke ring".  OK so far.  My guess is that after a while, this coherent structure falls apart, just like nice laminar smoke going up from a candle does after some distance.  This is based on the Reynolds number.  Roughly when that distance is exceeded, the smoke ring structure disintegrates into a number of smaller droplets that are individually still coherent.
From there the phenomenon repeats.  These individual droplets now form smoke rings of their own, for the same reason the original drop formed its smoke ring.  These break apart due to turbulance after a while, forming another level of smaller droplets, which cause their own smoke rings, etc, etc.
Eventually there will be enough mixing of the droplets with the bulk of the liquid in the cup that properties of the droplets are no longer distinct enough from the bulk liquid to do much of anything other than diffuse apart even further.
If this guess is correct, then how far the rings go and how many droplets they disintegrate into should be largely a function of the Reynolds number.  How many levels down this can be sustained would have to do with how fast the drop liquid can diffuse into the bulk liquid.
