Geostrophic thermal wind with separable buoyancy ($\bar{u}=-z{A}^{\prime }(y)/f_0$)

For $b=A(y)+B(z)$, constant nonzero $f_0$, and zero basic velocity at $z=0$, thermal-wind balance givesWith full buoyancy represented by $f_0{\bar{\psi }}_z$, the constant-coefficient three-dimensional quasi-geostrophic potential vorticity is $\bar{q}=f_0+z{A}^{\prime \prime }/f_0+f_0{B}^{\prime }/N^2$. The reference-buoyancy convention in quasi-geostrophic potential vorticity changes only the uniform reference contribution.