6.3 Hydrostatic pressure with the rigid lid

The hydrostatic equation $p_{z} = -\rho \, g$ can be integrated from the surface z=0 to some position z < 0 to yield

$$p(\lambda,\phi,z,t) p_{a}(\lambda,\phi,t) + p_{l}(\lambda,\phi,t) + g \int^{0}_{z} dz \; \rho$$ =

$$p_{a}(\lambda,\phi,t) + p_{l}(\lambda,\phi,t) + p_{b}(\lambda,\phi,z,t),$$ = (6.10)

where p_a is the atmospheric pressure, p_l is the surface lid pressure, and p_b is the hydrostatic pressure arising from the ocean's density field. The surface lid pressure is the pressure which would be exerted by an imaginary rigid lid placed on top of the ocean. An alternative interpretation is that it is the pressure exerted by undulations of a free surface. The latter interpretation is further disscussed in Section 7.1. In summary, the horizontal pressure gradient is given by

$$\nabla_{h} p = \nabla_{h} (p_{a} + p_{l}) + g \int^{0}_{z} dz \; \nabla_{h} \rho.$$ (6.11)

The horizontal pressure gradient at some depth z has a contribution from gradients in the atmospheric and lid pressure, gradients which act the same for all depths, and the vertical integral of horizontal gradients in the interior density. These latter gradients are due to baroclinicity in the density field, which prompts the often used name baroclinic pressure gradient, and which motivates the ``b'' subscript. The vertically integrated horizontal pressure gradient is needed for the development of the barotropic vorticity equation. This gradient is given by

$$\int^{0}_{-H} dz \; \nabla_{h} p \int^{0}_{-H} dz \; \nabla_{h} (p_{a}+p_{l}) + \int^{0}_{-H} dz \; \left( g \int^{0}_{z} dz' \; \nabla_{h} \rho \right).$$ = (6.12)

Since the horizontal gradient of the atmospheric and lid pressures are independent of depth,

$$\int^{0}_{-H} dz \; \nabla_{h} (p_{a}+p_{l}) = H \, \nabla_{h} (p_{a}+p_{l}),$$ (6.13)

which renders

 

$$\int^{0}_{-H} dz \; \nabla_{h} p H \biggl(\nabla_{h} (p_{a}+p_{l}) + \overline{\nabla_{h} p_{b}} \biggr).$$ = (6.14)