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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.

Use $(\bar{u},\bar{v})=(-{\bar{\psi }}_y,{\bar{\psi }}_x)$ and the hydrostatic thermal-wind balance convention $\bar{b}=f_0{\bar{\psi }}_z$. Because the basic state is independent of $x$, $\bar{v}=0$. The thermal-wind relation isIntegration from the resting lower boundary givesA compatible quasi-geostrophic streamfunction iswhere $C$ is a spatial constant. Its horizontal Laplacian is $z{A}^{\prime \prime }(y)/f_0$ and its second vertical derivative is $B^{\prime }(z)/f_0$. Substituting into the given three-dimensional quasi-geostrophic potential vorticity yieldsIn particular, the horizontal-curvature term has a plus sign: $\bar{\zeta }=-{\bar{u}}_y=z{A}^{\prime \prime }/f_0$.

The reference-buoyancy convention in quasi-geostrophic potential vorticity should be stated. If $\bar{b}$ includes the full reference profile $N^{2}z$, the formula above includes its constant contribution $f_0$. If instead the quasi-geostrophic streamfunction represents pressure relative to that reference hydrostatic state, use $\bar{b}-N^{2}z=f_0{\bar{\psi }}_z$. Then the quoted $\bar{q}$ is reduced by the constant $f_0$. Both conventions give exactly the same velocity and potential-vorticity gradients, and therefore the same subsequent dynamics. The coefficient $N$ in the stated linear model is a prescribed reference buoyancy frequency; if it is also required to equal the full physical buoyancy frequency everywhere, then $B^{\prime }(z)=N^2$, rather than an arbitrary function. The formal expression above retains the requested arbitrary $B$ with the stated constant coefficient.