I may be embarrassing myself here (not for the first time) - but it occurs to me that I don’t know how to resolve the following dilemma.

It is a very simple and intuitively understandable fact that hydrostatic pressure (P) for any ideal liquid is determined by: change in pressure P = pgh [fluid density (rho), times gravity (g), times depth (h) ]. So assuming constant g and constant rho (incompressibility), the pressure is merely a linear relationship with depth under the surface. So far so good - have taught and used that for many years.

Here is what I can’t resolve. The internal energy of any fluid is dependent upon the sum of the kinetic energies of all particles present, right? … which in turn is a function of temperature - and *only* of temperature. I assume this would be true for liquids just as it is for gases - very basic kinetic theory of matter. This means that no matter how deep underwater I go, the speeds of the molecules do not necessarily increase - but might even *decrease* if water temps drop with depth as they often would in real life. Given the near-incompressibility of water, there are approximately then the same number of water molecules colliding with any given cm^2 of wall area at speeds no faster than (and possibly even slower than) near the surface. Given all that, *what then accounts for the greater force on that given area?*

This is easy for a gas, where temperature (molecule kinetic energies) can be held constant, but due to their linear compressibility there are more molecules colliding with the given wall area - hence the greater pressure. But this doesn’t work for incompressible liquids! I know - no liquid is perfectly ideal, and water is ever-so-slightly compressible. So it occurred to me that, perhaps that miniscule compressibility makes water act like a super powerful spring (really high spring-constant). You push a powerful spring down just a tiny amount, and it pushes back with tremendous force. But this would imply that the liquid is then having kinetic energy (temperature) added to it due to compression work (something that wouldn’t even be possible to the extent that it truly approached being an ideal fluid). It also implies that the internal energy of the moving water molecules ought to increase with depth - thus giving greater temperatures as one descends (as actually does happen with gases). But none of that is sounding quite right.

What simple thing am I forgetting? Which two of these things are accounting for the increased pressure at depth under an ideal liquid: 1> increased speed of molecules there? or 2> higher density of molecules striking the wall?