= Solution
Substitute the stated scales and divide the momentum equation by $2\Omega V$. The ratio of inertial to Coriolis acceleration is the <Rossby number>
$$
\operatorname{Ro}=\frac{V}{2\Omega L},
$$
while the dimensionless buoyancy coefficient is
$$
\frac{\alpha g\Delta\theta}{2\Omega V}
=\frac{B}{\operatorname{Ro}},
\qquad
B=\frac{\alpha g\Delta\theta}{4\Omega^2L}.
$$
Thus
$$
\boxed{
\operatorname{Ro}(\mathbf u_t+\mathbf u\mathbin\cdot\nabla\mathbf u)
+\widehat{\mathbf z}\times\mathbf u
=-\nabla p+\frac{B}{\operatorname{Ro}}\theta\widehat{\mathbf z}},
$$
$$
\boxed{\theta_t+\mathbf u\mathbin\cdot\nabla\theta=0,
\qquad\nabla\mathbin\cdot\mathbf u=0}.
$$
The dimensionless $B$ is the <Burgers number> for this rotating stratified flow.
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