The first is that if the water gets to its source without any work being done on it, then this planet is a perpetual motion machine. The second is that as the water falls, its potential energy is being converted into kinetic energy. Kinetic energy is basically temperature, so the water is getting REALLY hot. I do not know what happens to water when it gets hot enough to boil while free falling in a confined space and I will bet money that it is not good.
That I have figured out so far:
My second solution is an Archimedes' screw goes from the South Pole to the North Pole. This solution limits the speed the water can travel, which puts a limit on the efficiency of the system. Another problem is that because the water is losing altitude, its potential energy must be decreasing; I am not sure what happens to this energy though. My guess is that it becomes mechanical force that makes the Archimedes' screw turn (which would be an advantage) or that the temperature of the water increases (which could be a disadvantage). If the potential energy becomes mechanical force, then it will do all the work required in order to make the the screw turn until the water reaches the height of the South Pole. This in turn [pun unintended] lowers the energy input of this solution to that of the first solution. This will also solve the problem of the water boiling. If the potential energy is converted to thermal energy, then the water may reach its boiling point while in a more confined space than in the first solution.
Thanks for your help.