r cos 0
In local Cartesian coordinates, it is reduced to d u d v d w — + — + — = 0 dx dy d z
For global-scale motions, L = a = 6,400 km is the radius of the Earth, f = 2«, the horizontal velocity scale is U = 0.1 m/s, and the depth scale of the motion is H = 800 m. The corresponding nondimensional numbers are Ro — 10—4 and F — 100. Thus, errors in replacing the mass conservation with the volume conservation for global-scale motions are approximately 1%.
On the other hand, if RoF is not very small, the errors introduced by the volume conservation approximation may not be entirely negligible. Since RoF = fUL/gH, we expect that the errors introduced by neglecting the density change terms can be relatively large for large-scale heating/cooling of the thin layer in the upper ocean.
It is very important to emphasize that the volume conservation is only an approximation to the continuity equation; it does not mean that the density of the fluid is constant. In fact, both temperature and salinity of water parcels can change with time due to surface forcing and diffusion. The common practice in dynamical oceanography is that the density of seawater is calculated from the equation of state, after both temperature and salinity of a given water parcel are determined. Therefore, the following set of equations is used:
where QT and QS are the sources of heat and salt due to diffusion and surface fluxes.
Since density is approximately constant, we will neglect density variation in the horizontal momentum equations, keeping the buoyancy contribution due to density variation only in the vertical momentum equation, giving dM -> 1 , p'
dt po po where p' = p - p(z), p' = p - p (z), p (z) and p (z) are the mean density and pressure profiles, and p0 is the mean reference density (p0 ~ 1,035 kg/m3 for the world's oceans). Using Taylor series to expand the right-hand-side of Eqn. (2.148):
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