Ian, I'm sorry to have to correct you water,
most definatly can be compressed, i'll give you an example, it the method they use to test diving cylinders, they pump water into the cylinder way past the normal working pressure of the cylinder, and measure the stretch of the metal, it would be to dangerous using air to perform this test,as if the cylinders wall failed an explosion could occur, however i know what you are getting at, as this can be done because the water has nowhere to go so the desity will increase as more water is pumped in.
Air pressure in open air Does exert approx 1kg of pressure at sea level, i think i chose the word density badly, in the sea the variances in air pressure will be negated by the water being able to move lateraly , and in any case air pressure variances are local and are not uniform across the globe, BUT! in an enclosed body of water, the water has nowhere to move , it can't go up unless the air pressure decreases, if air pressure increases so must the pressure it exerts on the waters surface, and so will alter the pressure sub surface as well. to avoid complicating the issue a standard figure of 1 bar of surface pressure is added to that of the ambient water pressure at a given depth in order that a decompression profile can be worked out for a diver at a given depth (all done by computer worn by the diver nowadays) a safety margin is built in to the calculations anyway , so the slight variances of air pressure are of no significance to these calculations, but they are there, So e.g lets assume that on a given day the air pressure over a lake is
900mb - .9 bar at 10 metres the water will exert a pressure of 1 bar = a total of 1.9 bar, however next day the air pressure has altered to 1000mb adding a further .1 of a bar to surface pressure at 10 metres - 30 foot -the pressure would have increased to 2
bar, this increase and decrease in pressure
will affect all compressible objects within a body of water including air spaces within living creatures, also the gases in dissolved solution within the bloodstreams of any living creature, however this will vary in a fish as it alters it's depth anyway, in conclusion Ian you are right that water is not compressible where it is free to move in 3 dimensions ,but where a body of water is confined it's desity will alter according to the surface pressure applied to it, Phew!