_ALE_timestep.dox

/*! \page ALE_Timestep Vertical Lagrangian method in pictures

\section section_ALE_remap Graphical explanation of vertical Lagrangian method

Vertical Lagrangian regridding/remapping is not a timestep method in

the traditional sense. Rather, it is a sequence of operations

performed to bring the vertical grid back to a target specification

(the regrid step), and then to remap the ocean state onto this new

grid (the remap step). This regrid/remap process can be chosen to be

less frequent than the momentum or thermodynamic timesteps.  We are

motivated to choose less frequent regrid/remap steps to save

computational time and to reduce spurious mixing that occurs due to

truncation errors in the remap step. However, there is a downside to

delaying the regrid/remap.  Namely, if delayed too long then the layer

interfaces can become entangled (i.e., no longer monotonic in the

vertical), which is a common problem with purely Lagrangian methods.

On this page we illustrate the regrid/remap steps by making use of

Figure 3 from Griffies, Adcroft, and Hallberg (2020)

\cite Griffies_Adcroft_Hallberg2020.

For purposes of this example, assume that the target vertical grid is

comprised of geopotential \f$z\f$-surfaces, with the initial ocean

state (e.g., the temperature field) shown on the left in the following

figure.

\image html remapping1.png "Initial state with level surface (left) and perturbed state after a wave has come through (right)"  width=60%

\image latex remapping1.png "Initial state with level surface (left) and perturbed state after a wave has come through (right)" width=0.6\textwidth

Some time later, assume a wave has perturbed the ocean state. During

the Lagrangian portion of the algorithm, the coordinate surfaces move

vertically with the ocean fluid according to \f$\dot{r}=0\f$.  Assume

now that the algorithm has determined that a regrid step is needed,

with the target vertical grid still geopotential \f$z\f$-surfaces, so

this new target grid is shown overlaid on the left as a regrid.

\image html remapping2.png "The regrid operation (left) and the remap operation (right)" width=60%

\image latex remapping2.png "The regrid operation (left) and the remap operation (right)" width=0.6\textwidth

The most complex part of the method involves remapping the wavy ocean

field onto the new grid.  This step also incurs truncation errors that

are a function of the vertical grid spacing and the numerical method

used to perform the remapping.  We illustrate this remap step in the

figure above, as well as in the frame below shown after the old

deformed coordinate grid has been deleted:

\image html remapping3.png "The final state after regriddinig and remapping" width=30%

\image latex remapping3.png "The final state after regridding and remapping" width=0.3\textwidth

The new layer thicknesses, \f$h_k\f$, are computed and then the layers

are populated with the new velocities and tracers

\f{align}

  \sum h_k^{\scriptstyle{\mathrm{new}}} &= \sum h_k^{\scriptstyle{\mathrm{old}}}

\\

  \mathbf{u}_k^{\scriptstyle{\mathrm{new}}}

  &= \frac{1}{h_k}

  \int_{z_{k + 1/2}}^{z_{k + 1/2} + h_k} \mathbf{u}^{\scriptstyle{\mathrm{old}}}(z') \, \mathrm{d}z'

\\

  \theta_k^{\scriptstyle{\mathrm{new}}}  &= \frac{1}{h_k}

  \int_{z_{k + 1/2}}^{z_{k + 1/2} + h_k} \theta^{\scriptstyle{\mathrm{old}}}(z') \, \mathrm{d}z'

\f}

*/