[Gravityloss]

We can look at it this way too: assume an oiler has maybe 10,000 t of oil and takes ten days to travel: 1E7 kg with energy density of 1E7 J/kg, in 1E6 seconds means an average power supplied by one oiler to be in the order of 100 megawatts. Or 10 kg/second. If that synfuel generation rate can be sustained, then the oiler is not needed.

The Nimitz class carriers have two 100 MW reactors: http://en.wikipedia.org/wiki/A4W_reactor

Of course, there's efficiencies, probably the oilers don't always go full speed, there's different distances etc etc...

If we assume 1% efficiency and one dedicated reactor, then we get 0.1 kg/second, 8 tons per day. You could load about two Hornets' internal fuel tanks with that.

In the sea you could probably also try other things than nuclear reactors, like farm algae and harvest it to produce biofuels (biofuel has already been tested in a B-52), or put solar or wind plants out in the ocean where there's space. No energy storage problems if the fuel is generated in situ.