In the planetary-geostrophic formulation, the horizontal velocity is divergence-free only because we have vertically integrated. If the scales of motion are not too large the horizontal flow at every level is divergence-free for another reason: because it is in geostrophic balance

In reality, over a single oceanic gyre (say from 15° to 40° latitude), variations in Coriolis parameter are not large, and this prompts us to formulate the model in terms of the quasi-geostrophic equations. An advantage of the quasi-geostrophic equations is that they readily allow for the inclusion of both nonlinearity and stratification

Such a model would be restricted to length scales, $L$, of no more than $\mathcal{O}(Ro^{-1})$ larger than the deformation radius, and for gyre scales this criterion is marginally satisfied if $L_d = 100 \, \mathrm{km}$

The quasi-geostrophic vorticity equation for a Boussinesq system is \[ \frac{D\zeta}{Dt} + \beta v = f_0 \frac{\partial w}{\partial z} + \text{curl}_z \frac{\partial \boldsymbol{\tau}}{\partial z} \]