Since the aspect ratio of the motion is small, geometrical considerations of the trajectory of the motion of fluid elements allow us to estimate the ratio of the vertical to horizontal velocities \[ \frac{\underbrace{w}_{\text{vertical velocity}}}{\underbrace{u}_{\text{horizontal velocities}}} = \mathcal{O}\!\left(\frac{w}{\underbrace{v}_{\text{horizontal velocities}}}\right) = \mathcal{O}\!\left(\frac{\underbrace{D}_{\text{thickness of the fluid region}}}{\underbrace{L}_{\text{horizontal scale of variation}}}\right) \ll 1 \Rightarrow p' = \mathcal{O}(\rho \, 2\Omega U L) \]

Vertical pressure gradient \[ \frac{\partial p'}{\partial r} = \mathcal{O}\!\left(\rho \frac{2\Omega U L}{D}\right) \] while the vertical Coriolis acceleration is \[ \rho 2\Omega u \cos\theta = \mathcal{O}(\rho 2\Omega U) \] Their ratio is $$ \frac{\rho 2\Omega u \cos\theta}{\partial p'/\partial r} = \mathcal{O}\!\left(\frac{D}{L}\right) = \delta \ll 1 $$ so the vertical Coriolis term proportional to $2\Omega \cos \theta$ in both the horizontal and vertical equations can be neglected

As long as the Rossby number is small \[ \rho' \leq \mathcal{O}\!\left(\frac{p'}{gD}\right) = \mathcal{O}\!\left(\frac{\rho 2\Omega U L}{gD}\right) \Rightarrow \frac{\rho'}{\rho} = \mathcal{O}\!\left(\frac{U}{2\Omega L}\frac{4\Omega^2 L^2}{gD}\right) = \varepsilon \frac{4\Omega^2 L^2}{gD} \Rightarrow \frac{\rho'}{\rho} \leq \mathcal{O}(\varepsilon) \ll 1 \Rightarrow \rho' \ll \rho_s(r) \]