Debuggin potential vorticity in E3SM

Choosing actual functions for PV formulation:

We use the folliwng formulation of potential vorticity:

$$\mathrm{PV} = - \frac{g}{p_0 \partial_\eta a + p_s \partial_\eta b } \left[ \left(\zeta_\eta + f \right) \partial_\eta \theta - \frac{1}{\overline{r} \cos \varphi} \left(\partial_\eta v \right) \left(^\eta\partial_\lambda \theta\right) + \frac{1}{\overline{r}} (\partial_\eta u)\left(^\eta\partial_\varphi \theta\right) \right]$$

with $\zeta_\eta = \frac{1}{\overline{r}\cos\varphi} \left((^\eta \partial_\lambda v) - (^\eta \partial_\varphi u\cos \phi ) \right)$

Pitfalls of comparing against lat-lon grids:

Assuming I calculate the above quantity correctly, then $\partial u$ and $\partial v$ are are calculated to machine precision. That is, assuming we have an internal (numerical) state $\mathbf{u, v}_{mijk}$ where $m$ indexes element ID, $i,j$ index the tensored GLL points within an element, and $k$ indexes level.