Vertical Momentum (Normalized by $f U$):

$$\underbrace{\frac{\delta U}{fL}}_{\delta \mathrm{Ro}} \frac{D w^*}{Dt^*} = -\frac{P/\rho_0 H}{fU} \partial_z^* p^* + \frac{B}{fU} b^*$$

Using $\frac{P/\rho_0 H}{fU} = \frac{\rho_0 f U L / \rho_0 H}{fU} = \frac{L}{H} = \frac{1}{\delta}$, and $\frac{B}{fU} = \frac{N^2 \delta L}{fU} = \frac{N^2 H}{fU} = \frac{N^2 H^2}{U^2} \frac{U}{fH} = \frac{1}{\mathrm{Fr}^2} \frac{1}{\mathrm{Ro} \delta}$. Multiplying the entire vertical momentum equation by $\frac{1}{\delta}$:

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

(A simplified, standard form for the vertical equation, using the factor $\frac{\delta}{\mathrm{Fr}^2}$):

$$\delta^2 \mathrm{Ro} \frac{D w^*}{Dt^*} = -\frac{1}{\delta} \partial_z^* p^* + \frac{1}{\delta \mathrm{Ro} \mathrm{Fr}^2} b^*$$