= Solution
The <quasi-geostrophic approximation> requires small <Rossby number>, nearly horizontal geostrophic balance, hydrostatic vertical balance, small interface or density displacements, stable background stratification, and horizontal scales much larger than the vertical scale. The <Boussinesq approximation> and a <beta plane> are used, ageostrophic motion enters only at the order needed to evolve potential vorticity, and here the <buoyancy frequency> $N$ is constant. Under these assumptions the materially conserved <three-dimensional quasi-geostrophic potential vorticity> is
$$
q=\nabla_h^2\psi+\frac{f_0^2}{N^2}\psi_{zz}+\beta y.
$$
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