 Solution

At leading order as $\mathrm{Ro}\to 0$, horizontal geostrophic balance and vertical hydrostatic balance giveThe temperature equation givesMeanwhile ${\zeta }^{\prime }=p_{xx}^{\prime }+p_{yy}^{\prime }$. The leading vertical-vorticity equation is $D{\zeta }^{\prime }=w_z^{\prime }$. Since derivatives in $z$ do not commute with $D$,The $p_{xz}^{\prime }$ terms therefore cancel, leaving the conserved three-dimensional quasi-geostrophic potential vorticityNo normal flow at $y=0,1$ means $v^{\prime }=p_x^{\prime }=0$. At $z=0,1$, $w^{\prime }=0$, so