 Taylor identity for quasi-geostrophic flux

For zonal averages, $u^{\prime }=-{\psi }_y^{\prime }$, $v^{\prime }={\psi }_x^{\prime }$, and $Q^{\prime }={\nabla }_H^2{\psi }^{\prime }+{\partial }_z(S{\psi }_z^{\prime })$, with $S=f_0^2/N^2$ depending only on height, define $F=-\overline{u^{\prime }{v}^{\prime }}$ and $G=S\overline{{\psi }_x^{\prime }{\psi }_z^{\prime }}$. Integrating zonal derivatives by parts gives $\overline{v^{\prime }{Q}^{\prime }}={\partial }_y\overline{{\psi }_x^{\prime }{\psi }_y^{\prime }}+{\partial }_z(S\overline{{\psi }_x^{\prime }{\psi }_z^{\prime }})$. This identity is exact within quasi-geostrophic potential vorticity, without assuming small disturbance amplitudes. Periodicity or appropriate vanishing zonal boundary terms is essential.