(a) Non-Dimensional Form of the Equations

We introduce the following characteristic scales:

• Horizontal velocity: $\mathbf{u}_h \sim U$; Vertical velocity: $w \sim W = \delta U$
• Horizontal length: $x, y \sim L$; Vertical length: $z \sim H$
• Time scale: $T = L/U$; Coriolis parameter: $f = 2\vert\mathbf{\Omega}\vert$
• Buoyancy: $b \sim B = N^2 \delta L$; Pressure: $p \sim P = \rho_0 f U L$

We define the dimensionless variables (denoted by a star), e.g., $\mathbf{u} = U \mathbf{u}^*$, $w = \delta U w^*$, $p = P p^*$, $b = B b^*$.

Substituting and normalizing the equations (dividing the momentum equations by $f U$ and the stratification equation by $N^2 W$):