Let's start with the ideal gas law: PV = T and also introduce coefficients of thermal expansion defined as:

• α = isobaric thermal expansion =
  
  
• β = isobaric density express =
  
  Rewrite Ideal Gas Law and take derivative as a function of T remembering the p is constant so the derivate of that term is zero.
  
  ln(V) + ln(P) = ln(T)
  
  
  
  Since the adiabatic lapse rate is about 10 K per km, then over the 10 km height of the troposhere this is a 100 K temperature change which is large and would obviously lead to thermal expansion (contraction) and hence significant density changes over this length scale.
  
  For water, α depends strongly on Temperature and is expermintally determined. At around 300K α = 1.7 x 10^-4.
  
  The general expression for an adiabatic temperature change does include α and T:
  
  
  
  For the Pacific Ocean which has mean depth of 4000 m (=dz) we get:
  
  
  
  This low value reflects the near-incompressibility of water. In addtion, this low temperature gradient in the ocean means that thermal processes can not induce density variations in that fluid, unlike that which happens in the atmosphere.
  
  To formulate an expression for the Density Scale Height (H) we use the speed of sound and hydrostatic equilibrium:
  
  
  
  Hence large sound speed makes the fluid less compressible which will in turn induce a large value for H.
  
  
  • For air, c_s = 300 m/s H = 9km
    
    
  • For water, c_s = 1500 m/s H = 200km
  
  Thus over the 4 km average depth of the Pacific Ocean, the density variations can only be 4/200 = 2%.
  
  So density variations can also not be caused simply by the thickness of the fluid (the sound speed is too high, precisely because the fluid is nearly incompresssible - this gives no time for individual molecules to relax (as they do in air) and gives rise to a high sound speed.
  
  Since neither temperature or depth can give rise to significant density variations in the oceans, then the only way that density driven ocean currents can exist is due to salinity variations.