Andy's notes on Generalised vertical coordinates

documentation/docs/pages/season2.md

@@ -975,7 +975,59 @@ Although $w$ varies with $z$ in each layer, we don't need it in either the momen
 Note that we are still using $z$ as a vertical coordinate, in that our horizontal derivatives are being taken by infinitesimal differences at constant $z$ within each layer. However, because $\rho$ is a montonic function of $z$ (assuming a stable stratification), another approach is to use $\rho$ instead of $z$ as a "vertical" coordinate in the continuously-stratified primitive equations. This is called using _isopycnal coordinates_. In this case the horizontal derivatives are along isopycnals (surfaces of constant $\rho$), rather than surfaces of constant $z$, but the equations can be made as tidy as the primitive equations we've derived here if the horizontal pressure gradient is replaced by the gradient of Montgomery potential $M=\frac{p+\rho gz}{\rho_0}$ on isopycnal surfaces. The $\rho$ coordinate can then be discretised to arrive at layered shallow-water equations with isopycnal coordinates. See [these notes](https://decoding-access-om3.readthedocs.io/AOMSS_Lecture_Notes/) or [Vallis](https://www.vallisbook.org/) section 3.9 for further details.
 
 ## Generalised vertical coordinates
+
+Date: 11/06/2026.
+
+Presenter: Andy Hogg (@AndyHoggANU). 
+
+In a z-level model, we have vertical velocities, $w$, which passes through coordinate interfaces.
+
+![RGLayer](../assets/z-level-andy.png)
+
+In this case, we can write $w$ from
+
+$$ \frac{\partial w}{\partial z} = - \nabla \cdot \mathbf{u} $$
+
+Conversely, in the stratified shallow water equations the vertical velocity is entirely associated with the vertical motion of the interface itself. We can call this $w_{\mathrm{grid}}$.
+
+![RGLayer](../assets/ssw-wgrid-andy.png)
+
+Note that $w_{\mathrm{grid}}$ has a contribution from both the local rate of change of the interface and an advective component:
+
+$$ w = w_{\mathrm{grid}} = \frac{\partial \eta_k}{\partial t} + \mathbf{u} \cdot \nabla \eta_k $$
+
+In a nutshell, **generalised vertical coordinates** aims to enable both a cross-interface velocity and the movement of the interface itself.
+We would take a scalar field $s(x,y,z,t)$, where $s$ is monotonic in $z$ (actually, its derivative cannot be zero).
+Then, to first order, we can calculate the diasurface flux across a surface of constant $s$ to be
+
+$$ w^{(\dot{s})} \approx w - w_{\mathrm{grid}} $$
+
+That is, the diasurface velocity, $w^{(\dot{s})}$, and the velocity of the interface, $w_{\mathrm{grid}}$, combine to make up the actual vertical velocity.
+
+**However,** this is a caveat here, and that is the ``$\approx$'' in the equation above. In fact, the situation is made more complex by geometric effects (basically, the coordinate interface not being flat). For a full treatment of this derivation, I refer you to [Appendix D of the Griffies et al. (2020)](https://doi.org/10.1029/2019MS001954) paper on ALE. There, they show that we can write
+
+$$ w^{(\dot{s})} \, dA = \mathbf{\hat{n}} \cdot (\mathbf{v} - \mathbf{v_{\mathrm{grid}}}) \, dS, $$
+
+where $dS$ is the area of the surface over the grid cell, $dA$ is the projection of that area onto a flat plane, $\mathbf{\hat{n}}$ is the unit normal and $\mathbf{v}$ is the 3D velocity field.
+
+I won't reproduce the full derivation here, but to summarise, a little trigonometry allows the above equation to be written as
+
+$$ w^{(\dot{s})}  = \frac{dz}{ds} \frac{Ds}{Dt} $$
+
+$$ w^{(\dot{s})}  = \frac{dz}{ds} (\frac{\partial s}{\partial t} + \mathbf{v} \cdot \nabla s) $$
+
+...
+
+$$ w^{(\dot{s})}  = w - (\frac{\partial z}{\partial t} + \mathbf{u} \cdot \nabla_s z) $$
+
+You may note that this final equation is similar in form to the approximate equation above, except that there is an additional advection term at the end.
+
+**In summary,** generalised vertical coordinates are mathematically complicated, but the principle is relatively simple: both vertical Lagrangian motion of the grid and dia-surface flux through the coordinate interface is permitted. Importantly, GVCs are a nice framework to help understand the logic of vertical Lagrangian remapping, which Angus will talk about next week ...
+
+
 ## Vertical Lagrangian remapping
+
+
 ## Pressure forces
 ## Coriolis term
 ## Pressure solver - barotropic / baroclinic split