Vertical Momentum (Leading-Order $\mathcal{O}(1/\delta)$):

For the common case where $P \sim \rho_0 f U L$ and assuming $\frac{1}{\delta \mathrm{Ro} \mathrm{Fr}^2} \sim \mathcal{O}(1/\delta)$, the $\mathcal{O}(\delta^2 \mathrm{Ro})$ inertial term is negligible. The largest terms are:

$$0 \approx -\frac{1}{\delta} \partial_z^* p^* + \frac{1}{\delta \mathrm{Ro} \mathrm{Fr}^2} b^*$$

Leading-Order Balance: Hydrostatic Balance (Vertical pressure gradient force balances buoyancy force).