Vertical Lagrangian Remapping notes

documentation/docs/pages/season2.md

@@ -1027,6 +1027,63 @@ You may note that this final equation is similar in form to the approximate equa
 
 ## Vertical Lagrangian remapping
 
+Date: 18/06/2026.
+
+Presenter: Angus Gibson (@angus-g).
+
+With **generalised vertical coordinates (GVC)**, we know that vertical velocities permit
+a range of grid evolution from fully Lagrangian (zero dia-surface transport) to fully
+Eulerian (zero grid velocity). In MOM6, the dynamics are written in the fully Lagrangian sense:
+the GVC $s$ follows fluid elements so that the dia-surface volume flux $w^{(\dot{s})} = 0$.
+
+If the model is using a purely Lagrangian coordinate, how do we have any control
+over it? In general, these coordinates drift to a less useful representation of the
+water column. For example, volume injection at the surface would inflate the
+upper-most layer, losing resolution for the representation of boundary layer
+processes. Even more simply, there is indeed irreversible mixing across surfaces
+that must be captured somehow.
+
+The method has three steps:
+
+1. the Vertical Lagrangian step: evolve the model with a purely Lagrangian
+   generalised coordinate;
+2. the Vertical Regrid step: explicitly set the generalised coordinate;
+3. the Vertical Remap step: ensure consistency between the ocean state
+   and the new coordinate.
+
+![Vertical regrid](../assets/regridding.png)
+
+The *Vertical Regrid* step is where we have the most freedom and flexibility. At
+this stage, we explicitly set the generalised coordinate $s(x,y,z,t)$. The choice
+is arbitrary: it could be predetermined (like a fixed geopotential or
+terrain-following coordinate); state-dependent (following particular isopycnals by
+solving for density levels); or even evolutionary (relaxing the current field
+toward some target value). In fact, you could even change the number of
+vertical levels!
+
+There is a slight issue after the Vertical Regrid step: the underlying ocean
+state is no longer consistent with the GVC. The *Vertical Remap* step fixes
+this by interpolating (or extrapolating) the state onto the new grid. In the
+continuous limit, this doesn't change the ocean state. However, since we have
+limited resolution there is necessarily a spurious change to the state due to
+interpolation error. Ideally, this numerical mixing is reduced while integrated
+quantities are conserved.
+
+![Vertical Lagrangian remapping](../assets/vertical-lagrange-remap.png)
+
+There are two big advantages to using this method:
+
+1. There is no vertical CFL limit! As long as the remapping can handle
+   interpolation over more than a single cell, the target grid can be
+   arbitrary.
+2. Grid evolution can occur on a different timestep to state evolution. You may
+   take several Lagrangian timesteps before performing the regrid/remap steps.
+   Particularly with several tracers (as with BCG), this may give some
+   performance improvements.
+
+A third advantage is that there is a built-in method for remapping the state onto
+an arbitrary grid, which is useful for diagnostics. You may have a lower-resolution
+diagnostic grid, or want density-space diagnostics, etc.
 
 ## Pressure forces
 ## Coriolis term