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 expression =
  
  Rewrite Ideal Gas Law and take derivative as a function of T remembering that p is constant so the derivative of that term is zero.
  
  ln(V) + ln(P) = ln(T)
  
  α =
  
  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:
  
  (because its an ideal gas)
  
  For the Oceans c_p ~ 4000 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. In fact, this is the fundamental physical difference between the way these two fluids behave.
  
  To formulate an expression for the Density Scale Height (H) we use the speed of sound and hydrostatic equilibrium:
  
  Speed of sound = c_s² = (P/ρ)*γ ; γ is the adiabatic constant or 5/3 for air. Note, this directly tells you that a sound wave is a pressure wave moving through a dense medium. This means the the speed of sound is the change in pressure divided by the change in density (since either of those along the moving fluid will change the fluid velocity).
  
  
  
  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 cannot 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 that can cause local regions of denser water to exist (recall your homework problem related to this). Note that local here means a scale of about 1000 km.
  
  Ocean salinity is largerly determined by (E-P) variations.
  
  
  
  Incompressible vs Compressible Flow:
  
  Incompressible flow means that flow-induced pressure (ρv²) and/or temperature changes DO NOT cause significant density changes.
  
  Liquid flow are usually incompressible. Gas flows can also be incompressible if speeds much less than the speed of sound (< 0.3 Mach number). This is because the flow pressure (ρv²) is not "big" enough to produce density variations.
  
  As discussed, in the oceans it is salinity variations that produces density variations. IN the atmosphere it is temperature. (what varies fractionally the most in the atmosphere, density, pressure of Temperature from say 0 to 1 km)
  
  Density variations ultimately lead to buoyancy forces in the fluid, even when that fluid is magma beneath the Earth's crust. Buoyancy arises from density differences (large ones) in the Mantle of the Earth that leads to convective upwelling of material that drives the surface lithospheric plates.
  
  
  
  Finally when density variations are not flow-induced then operationally that fluid can be treated as incompressible. That doesn't necessarily mean the fluid is uniform density.