(From the questions topic)
Does anyone know of a paper or discussion on the heat energy of falling water in the cryosphere?
The power of falling water is given as P = ηpQgh where
η = efficiency factor, percentage efficiency expressed as a decimal
P = power in Watts
p = density of the water
Q = flow rate in cubic meters/second
g = gravitational acceleration
h = head in meters (difference between inlet and outlet levels)
For ice, as for turbines, an efficiency factor is needed. When water flows over ice or falls through it, heat is generated at each disturbance in the flow. This heat is less than would be generated by water falling unimpeded.
My point is that water flowing over ice or through it into the sea must be heated to at least some degree on its downward journey. This will likely promote melting of glacier ice and / or promote local sea temperature rise.
So, has this been discussed anywhere?
This bit makes perfect sense to meteorologists (and many others with a grounding in the right part of physics) but is counter intuitive for many others.
Among other things, the atmosphere is a giant heat engine, accepting heat from the sun and converting it into mechanical energy of various sorts. Not very efficient, but that matters little - the sun provides lots of energy to work with.
To turn your question around, every drop of rain and flake of snow that falls does so only because the heat engine raised it from ground level to whatever height it fell from. Every scrap of mechanical energy lost in falling was originally added, with the sun being the original source of the lift.
The tricky part is, our senses tell us that warm moist air feels 'close', it feels heavy to us. The reality is that (because a water molecule is much lighter than nitrogen and oxygen molecules) moist air is not dense at all. Hot moist air tends to rise, cold dry air tends to sink. Once water at the surface evaporates, it will tend to rise until conditions prevent the air from holding it - a cloud results. Under appropriate conditions, the water/ice suspended in the cloud becomes chunks large enough to fall, as rain or snow. Air resistance and impact processes turn all the mechanical potential energy the water once had back into heat. Evaporation is the key, everything else naturally (if counter-intuitively) follows.
You can plot the process on something like a Mollier diagram and see exactly how and why it works, but I'm not sure it is that exciting. Others have pointed out that (depending on the height the precipitation forms) the mechanical energy involved amounts to enough to heat the precipitate something less than 10 degrees, or melt falling ice at a constant temperature something less than 10%.
The real significance of Arctic precipitation is that the water (for the most part) gained enough energy to evaporate it somewhere more temperate, traveled to the Arctic and then transferred enough energy to the atmosphere in order to allow the water to leave the vapor phase. This is the biggest significance of moist air intrusions into the Arctic (which, at one time, were not an expected wintertime occurrence).
One centimeter of precipitation falling, say, two thousand meters turns about 50 kcal per square meter of mechanical potential (and then kinetic) energy into heat. Condensing that same precipitation left about 900 kcal of heat behind in the Arctic atmosphere.
One cm per hour is a respectable (but far from torrential) rainfall. From a (swag - wild guess) 2 km height, this is very roughly 60 watts per square meter of mechanical energy (most absorbed by the air, in braking the rain/snow to a comparatively sedate terminal velocity). Condensing the same precipitation, however, added almost 1100 watts per square meter to the atmosphere (considerably more than the maximum summer insolation at 80N). Of course it rarely rains in the far north, but this puts it in everyday context. The precipitable water that has found its way past the Arctic Circle in the last couple of years represents a major thumb on the scales balanced by sea ice.
(This is back of the envelope math from parameters I gathered quickly and informally - if someone notices I've slipped a decimal point or whatever, let me know. This is well outside my expertise, so if I've boned this entirely, let me know that as well.)