(b) Dimensionless group comparing advection and pressure-gradient

Estimate magnitudes. The advective term scales as $U^2/H$. The pressure-gradient term scales as $(\Delta p)/(\rho H)$. If we use a dynamic pressure scale $\Delta p \sim \rho U^2$, then the pressure-gradient magnitude is $\rho U^2/(\rho H)=U^2/H$ and both terms are comparable (this corresponds to a dimensionless pressure $p'=O(1)$). However, in geophysical contexts the relevant pressure scale is often hydrostatic $P\sim\rho g H$. Using that, the pressure gradient magnitude is $P/(\rho H)\sim g$.

The dimensionless group measuring advective term relative to the pressure-gradient term (using a pressure scale $P$) is

$$\frac{\text{advective}}{\text{pressure-gradient}} \sim \frac{U^2/H}{P/(\rho H)} = \frac{\rho U^2}{P}.$$

If we choose $P=\rho U^2$ then this ratio is $1$. If we choose $P=\rho g H$ (hydrostatic pressure), then

$$\frac{\text{advective}}{\text{pressure-gradient}} \sim \frac{U^2}{gH} = \mathrm{Fr}^2.$$

Equivalently, the Euler number is defined as

$$\mathrm{Eu}=\frac{P}{\rho U^2}.$$

So the ratio inertial/pressure is $1/\mathrm{Eu}$. If $P=\rho g H$, then $\mathrm{Eu}=gH/U^2=\mathrm{Fr}^{-2}$ and the advective-to-pressure ratio is $\mathrm{Fr}^2$.