Salt water and ice, vs. plain water and ice A famous puzzle is whether an ice cube melts to nothing more quickly in salt water or fresh water. The answer is fresh water, because the water melting off the ice cube sinks in the plain water and rises in the denser salt water. The sinking causes convection to play a large role, and the time difference is very large. See Ice melts slower in salt water? and many videos, using dye, that are easily found.
But what happens in the first second, before convection has an effect? In a "pure" situation where it is just salt water melting ice vs plain water melting ice, which melts the ice faster? In other words, which has the faster instantaneous rate of melting right at the start?
I and  a friend have done several experiments. 


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*In a 35-degree refrigerator, the overall melting story is reversed. This is expected because now the melt water floats on plain water (and on salt water), and so stays at the top. But why exactly does the salt water melt the cube a lot faster than the fresh water? This seems to indicate that in the pure situation, salt is faster.

*At room temperature, with regular stirring of the water, the cubes melt at the same time. This is puzzling and seemingly contradictory to (1).
We have seen several possible explanations, but at this point we do not know the true story. There are likely several effects going on, but is there one dominant effect that drives the melting in the "pure" case?   
Edited to respond to Sam Gerbil comment: 
Classic case: Room temperature. The meltwater floats in saline; so saline is slower by a lot (loses).
In refrigerator: meltwater floats in both cases. But saline wins by a lot. Not clear why. Maybe because the meltwater rising rate is different because of different density. Or maybe saline melts ice faster because of freezing point depression (but I am not clear as to why exactly this affects overall energy transfer).
In refrigerator, with ice forced to bottom of container. My attempt failed. Jim just finished an experiment with nails frozen into the ice for weight. This worked very well and, again, salt is a big winner. 
Stirring: Jim tried it at room temperature and states that the melt rates are the same. This is a bit of a mystery, as it contradicts the stirring experiment. Perhaps the stirring removes the refreezing effect.
One suggested answer:  I've always assumed as the ice melts in saltwater the dissolved sodium and chlorine ions interfere with the strength of the H-bonds between water molecules and the process just becomes a runaway train. I love the idea of doing it in a refrigerator to eliminate the convection.
One difficulty is that convection is never entirely eliminated, so it is hard to be certain that the salt naturally melts the ice faster. But right now that is what it looks like. The two suggested solutions are: the refreezing that occurs in plain water slows the process down; the H-bonds weaken b/c of sodium and chlorine ions. 
 A: We have done more experiments and, to our surprise, it appears that salt water and plain water melt ice at essentially the same speed, when the effect of convection is eliminated. Jim Tilley carried out two experiments: (1) the classic room temperature situation, but stirring the water with a spoon every 30 seconds; (2) still at room temperature, but using wire mesh to place the cubes in the center of the glass. In (2) the meltwater will either fall or rise away from the cube. In both cases the melting speeds were very close, within experimental error. Another interesting context is to do the classic experiment in a refrigerator. Then the meltwater rises in both the plain water and the saline. But there is no refreezing in the saline. In this case the salt water melted the ice in 4 hours, but plain water took more than 10 hours; perhaps 11 hours. So, to our surprise, the answer to the question about a "natural" melting speed appears to be that the two solutions melt the ice at the same rate.
Back to the refrigerator case. For plain water a certain amount of refreezing occurs at the interface. That does not happen in salt water because of the lower freezing point. The extra ice takes time to melt, and we believe that that is the reason for the vast difference in melt times when the ambient water is just under 2 degrees C. (35 F).   
What would happen in 0 gravity? The meltwater would hang around the ice cube and there might be some refreezing, though not a lot because of the room temperature ambient water. So possibly salt would win in this case. But really it is not clear. Is there an earth-bound experiment that would indicate the zero-gravity story? Not sure.
A: In the 35-degree test, the convection currents are lower than at room temperature in both cases, but still bigger with pure water. The absolute difference may not be as much, but the proportional difference is still significant. That is still the explanation. It’d be true even at 33-deg, because even a little convection helps, and there will be less (especially proportionally as just mentioned) in salt water. I don’t believe either of the two solutions mentioned will have an effect. There is no refreezing; the temperature for some distance into the surface is at the melting point as heat transfer must occur at the melting point for a little while for phase transition between solid at freezing and liquid at freezing. And the thing being melted is the same in both cases, so the properties are also.
With stirring however, you truly have eliminated that factor. So they should be almost exactly equal, depending almost entirely on your ability to provide equivalent mixing. The viscosity and density (therefore kinematic viscosity as well) of salt water is quite close to that of water.
Finally, there is another way that one could eliminate convection differences between the cases. That is to eliminate convection. If you had an object that didn’t melt and accurately measured its temperature in very still water, and in salt water, the results would basically return. On good object would be the thermocouple itself, but anything where temperature could be very accurately tabulated through time. In that case, the coefficients of thermal conductivity would matter. And salt water does not conduct heat as well as regular water. (Note: I’m not saying this is much of a factor in the experiments done so far; it wasn’t. Convection dominates conduction at even extremely low flows.)
Part of the reasoning for adding this last situation was your “first second” thinking. Because of its lower coefficient of thermal conductivity, the temperature gradients in salt water are higher for the same amount of heat transfer, so it’s less effective. It seems reasonable to imagine even the first moment (although less than a second) would be faster for regular water.
And the difference is not small, $6-9$% worse at conducting heat due to the salt: https://thermtest.com/application/thermal-conductivity-of-salty-water-mixtures
