2.3 Ocean

We adopt quasi-geostrophic dynamics in a ``1 1/2-layer'' ocean, with a moving upper layer and a very deep lower layer which remains at rest; there is a a rigid lid at the surface (Pedlosky, 1987). Neglecting thermal PV sources (Q_o=0 in ( 5)), the potential vorticity in the upper layer of ocean evolves according to (see figure 1)

$$\frac{D}{D{t}}q_o=\nabla \times \frac{\tau }{\rho_{o0}h}$$

where

$$q_o=\nabla ^{2}\psi_o-\frac{1}{L_o^2}\psi_o +\beta y$$

Here $\psi _{o}$ is the oceanic streamfunction in the upper layer, $L_o^2 \equiv \frac{gh\Delta \rho /\rho_{o0}}{f^2}$ is the square of the oceanic baroclinic Rossby radius of deformation, with $\rho _{o0}$ a constant reference value of density and $\Delta \rho$ the density difference between the two layers. Linearizing about a state of rest we have:

$$\frac{\partial}{\partial t}\left(\nabla^2\psi_o-\frac{1}{L_o^... ...tial x}\psi_o=\frac{1}{\rho _{o0}}\nabla \times \frac{\tau}{h}$$

We are interested in motions with spatial extents (L) of thousands of km. The Rossby radius in the ocean (L_o) is $\sim$50 km, so we may make the long-wave approximation and neglect the relative vorticity contribution to the PV, giving our final equation for the dynamic ocean:

$$-\frac{1}{L_o^2}\frac{\partial}{\partial t}\psi_o +\beta \fra... ...tial x}\psi_o=\frac{1}{\rho _{o0}}\nabla \times \frac{\tau}{h}$$ (16)