Link from Water Pumps.

There are two possibilities:

Consider a vertical pipe of, say, 60 feet. The bottom end is immersed in the water to be pumped and has a non return valve close to the base. At a point 25 feet above the base a 3 foot long side arm joins the vertical pipe and then bends up in parallel to the main pipe to a point 28 feet above the base. The side arm is engineered to withstand modest explosive forces. The height of 28 foot is chosen as being less than the 30 foot height of a water barometric leg. The end of this stub is closed off but has a valve at the top to allow gas to be fed in. The main vertical pipe has a non return valve just above the side arm. The assembly is pre-primed with water to a height of 30 feet.

Simple DC electrolysis of an alkaline solution is used to produce hydrogen and oxygen at modest pressure in a stoichiometric 2:1 ratio and in this case will, ideally, be mixed in a closed electrolysis vessel. There is evidence that pressure can be built up without drastically affecting efficiency and there is no pressure component in the Nernst equation. For the purposes of this idea, the pressure will be limited to a very modest one atmosphere.

To start the cycle, the pressurised hydrogen/oxygen mix is fed in to the top of the side arm which will force water up through the top non-return valve. When a suitable volume of gas mixture has been fed in, (this to be determined in practice,) the gas is turned off and the gas mixture in the side arm is ignited with a spark and the instantaneous pressure from the explosion will drive more water up through the top non-return valve. The gases will then cool at the interface with the water and, being water vapour, collapse to something approaching a vacuum which will allow atmospheric pressure to force water up from the bottom via the bottom non-return valve which replacing the water pumped to the top in the first two parts of the cycle. The cycle can now recommence.

This uses only the atmospheric pump part of the above idea.

This must operate below the 28 foot barometric leg for water. A vertical pipe 25 feet long has the bottom immersed in the water to be pumped. At the top there is a 3 foot tube with a weighted piston, this part of the tube being engineered to withstand modest explosive forces. At the bottom of this top tube there is a non-return valve. Pressurised hydrogen/oxygen gas is fed in to the bottom of the top tube raising the piston an amount to be determined. The flow of pressurised gas is cut off and the gas in the tube ignited with a spark which will force the piston upwards a distance to be determined in practice. At the limit of its upward travel, the piston will lock in position and the water vapour resulting from the explosion start to cool and condense causing a near vacuum which will allow atmospheric pressure to force water up from the bottom through the non-return valve. The water thus pumped into the top tube can be drained out at the 25 foot level and the cycle started again.

Notes:

In my youth many years ago it was a given that the majority of all laboratory explosions involved hydrogen in some form. There are dangers which would have to be overcome with proper design and engineering. Maybe the two component gases could be kept separate until mixed in the reaction vessel. Perhaps any feed tube transporting the mixed gases could be packed with a stainless steel wire wadding that might act like the gauze in the Davy Safety Lamp. Certainly any design should allow any inadvertent explosion to dissipate safely.

The mention of hydrogen and oxygen together tends to lead into the area of fringe science. Please note that I have in mind simple DC electrolysis of alkaline solutions with an efficiency of 50% upwards and note that the power could very conveniently come from solar cells. Needless to say, any genuine advances in electrolysis would be of great benefit to ideas like this.

Imbalances in the oxygen/hydrogen ratio or dissolved gases coming out of solution might result in gases accumulating in the reaction vessel and impairing the vacuum and it may be that regular bleeding might be necessary to remove these, certainly in the first example. The weighted piston should purge away any accumulations in the second. The idea depends on hot gases being cooled by the water being pumped and it may be necessary to route some of the water being sucked into the reaction vessel via spray mechanisms to speed this up.