Reference-buoyancy convention in quasi-geostrophic potential vorticity

The hydrostatic convention $b=f_0{\psi }_z$ may refer to full buoyancy or to its anomaly relative to $N^{2}z$. Subtracting the uniform reference profile changes $\psi$ by $-N^2{z}^2/(2f_0)$ and changes $q=f_0+{\nabla }_h^2\psi +(f_0^2/N^2){\psi }_{zz}$ by the uniform constant $-f_0$. Horizontal velocity and potential-vorticity gradients are unchanged, so the advection and stability dynamics are identical. Stating the convention avoids mistaking this reference constant for a dynamical gradient.