6.4.2 The barotropic vorticity equation

In order to eliminate the lid and atmospheric pressures, it is sufficient to form the time tendency of the barotropic vorticity

 

$$\overline{\zeta} \hat{z} \cdot \nabla \wedge \overline{{\bf\omega}}$$ =

$$\left(\frac{1}{a \, \cos\phi} \right) [ \overline{v}_{\lambda} - (\overline{u} \, \cos\phi )_{\phi} ]$$ =

$$\frac{1}{a \, \cos\phi} \left[ \frac{1}{\cos\phi} \left( \frac{\p... ...ht)_{\lambda} + \left( \frac{\psi_{\phi} \, \cos\phi}{H} \right)_{\phi} \right]$$ =

$$\nabla_{h} \cdot \left( \frac{1}{H} \, \nabla_{h} \psi \right).$$ = (6.30)

A few lines of manipulations renders

 

$${ \overline{\zeta}_{t} + \beta \, \overline{v} = }$$

$$f \, \nabla_{h} \cdot \overline{{\bf u}_{h}} - \left( \frac{1}{a^... ...wedge \left( \frac{\Delta {\bf\tau}}{\rho_{o} \, H} + \Gamma^{{\bf u}} \right),$$ - (6.31)

where

$$\beta \frac{1}{a} \, \frac{\partial f}{\partial \phi}$$ =

$$(2 \, \Omega/a) \, \cos\phi$$ = (6.32)

is the planetary vorticity gradient. The forcing for tendencies in $\overline{\zeta}$ consists of the following terms:

1.

Meridional advection of planetary vorticity: $-\beta \, \overline{v}$.

2.

Curl of the Coriolis force, which takes the form of the convergence of the barotropic velocity weighted by the Coriolis parameter: $-f \, \nabla_{h} \cdot \overline{{\bf u}_{h}}$. For a flat bottom, this term vanishes with the rigid lid. Combined with the $\beta \, \overline{v}$ term, these provide the forcing $-\nabla_{h} \cdot (f \, {\bf u}_{h})$ to the barotropic vorticity.

3.

The antisymmetric term proportional to $\partial_{\lambda} (\overline{\partial_{\phi} p_{b} } ) - \partial_{\phi} (\overline{\partial_{\lambda} p_{b} } )$. This term vanishes in a barotropic model, or in a baroclinic model with a flat bottom.

4.

Curl of the depth weighted surface minus bottom stresses: $\hat{z} \cdot \nabla \wedge (\Delta {\bf\tau}/\rho_{o} \, H)$.

5.

Curl of the nonlinear lateral friction terms and advection terms embodied by $\Gamma$.

To touch bases with familiar textbook dynamics, note that a steady state ocean with $\Gamma = 0$ and a flat bottom will result in the familiar barotropic Sverdrup balance

 

$$\beta \, \overline{v} = \hat{z} \cdot \nabla \wedge \left( \frac{\Delta {\bf\tau}}{\rho_{o} \, H} \right).$$ (6.33)