R.H .S. ~ — Vp' - kg p p\d&Jp,S p\d SJp,0 p\dp) &,S

where © is potential temperature. Note that ^dpf^ = 1 ~ (2 x 103) 2; hence in the vertical momentum equation we have the following estimation for the ratio between the vertical pressure gradient and the buoyancy contribution from the pressure perturbation (after subtracting the standard pressure!):

where we assume that H = 5,000 m. Thus the last term on the right-hand side of Eqn. (2.149) can be neglected. Physically, when the Boussinesq approximations apply, the fluid is regarded as incompressible with infinite sound speed; hence the above treatment is self-consistent.

The Boussinesq approximations Combining the analysis above, the oceanic circulation problem can be solved in terms of a basic state consisting of an adiabatic state of no motion and a dynamical perturbation part to the basic state. Thus, the dynamical structure of the oceanic circulation can be solved as follows.

1. The adiabatic basic state

In the basic state, density and pressure are in hydrostatic equilibrium:

dt p0 p0

where

Both a = - = ( M ) and P = = ( ^ ) are assumed to be constant.

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