(a) Non-dimensionalization

Introduce nondimensional variables (primed)

$$\vec x = H\vec x',\qquad t = \frac{H}{U}t',\qquad \vec u = U\vec u',\qquad p = \rho U^2 p',$$

and $z = H z'$ for the vertical coordinate. The gradient operator scales as $\nabla = \tfrac{1}{H}\nabla'$ and time derivative $\partial_t = \tfrac{U}{H}\partial_{t'}$.

Substitute into the momentum equation:

$$\frac{U}{H}{\vec u'}_{t'} + U\left(U\frac{1}{H}\right)(\vec u'\cd... ...')\vec u' = -\frac{1}{\rho}\left(\frac{\rho U^2}{H}\nabla' p'\right) + \vec g.$$

Divide through by $U^2/H$:

$${\vec u'}_{t'} + (\vec u'\cdot\nabla')\vec u' = -\nabla' p' + \frac{H}{U^2}\vec g.$$

Write $\vec g = -g\hat{\vec z}$ (downwards). Then the nondimensional Euler equations are

$$\boxed{{\vec u'}_{t'} + (\vec u'\cdot\nabla')\vec u' = -\nabla' p' - \frac{gH}{U^2}\,\hat{\vec z}\; }, \qquad \nabla'\cdot\vec u' = 0.$$

The dimensionless parameter multiplying $\hat{\vec z}$ is $gH/U^2$, which is the inverse-square of the Froude number:

$$\frac{gH}{U^2} = \mathrm{Fr}^{-2},\qquad \mathrm{Fr}=\frac{U}{\sqrt{gH}}.$$