Inserting the approximate geostrophic expressions for velocity components $$ u \ \cong\ -\frac{g}{f_0} \frac{\partial \eta}{\partial y} \quad \text{and} \quad v \ \cong\ \frac{g}{f_0} \frac{\partial \eta}{\partial x} $$

From these the vertical vorticity is found as $$ \zeta \ =\ \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} = \frac{g}{f_0} \left( \frac{\partial^2 \eta}{\partial x^2} + \frac{\partial^2 \eta}{\partial y^2} \right) $$

The linearized potential vorticity equation \(\boxed{H\,\frac{\partial\zeta}{\partial t}+H\,\beta v-f_0\,\frac{\partial\eta}{\partial t}=0}\) becomes $$ \frac{gH}{f_0} \ \frac{\partial}{\partial t} \left( \frac{\partial^2 \eta}{\partial x^2} + \frac{\partial^2 \eta}{\partial y^2} \right) + \frac{\beta g H}{f_0} \ \frac{\partial \eta}{\partial x} - f_0 \ \frac{\partial \eta}{\partial t} = 0 $$

$$ \mathrel{\overset{c^2 = gH}{\Rightarrow}}\;\frac{\partial}{\partial t} \left( \frac{\partial^2 \eta}{\partial x^2} + \frac{\partial^2 \eta}{\partial y^2} - \frac{f_0^2}{c^2} \eta \right) + \beta \ \frac{\partial \eta}{\partial x} = 0 $$ which is the quasi-geostrophic form of the linearized vorticity equation for flow fields that span a significant range of latitude