Explanation of A\+LE remapping

The Arbitrary Lagrangian-\/\+Eulerian (A\+LE) remapping is not a timestep in the traditional sense, but rather an operation performed to bring the vertical coordinate back to the target specification. This remapping can be less frequent than the momentum or thermodynamic timesteps, but must be done before the layer interfaces become entangled with each other.

Assuming the target vertical grid is level $z$-\/surfaces, the initial state is shown on the left in the following figure\+:


\begin{DoxyImage}
\includegraphics[width=\textwidth,height=\textheight/2,keepaspectratio=true]{remapping1.png}
\doxyfigcaption{The initial state with level surface (left) and the perturbed state after a wave has come through (right).}
\end{DoxyImage}
 Some time later, a wave has perturbed the surfaces which move with the fluid and it has been determined that a remapping operation is needed. The target vertical grid is still level $z$-\/surfaces, so this new target grid is shown overlaid on the left as regrid\+:


\begin{DoxyImage}
\includegraphics[width=\textwidth,height=\textheight/2,keepaspectratio=true]{remapping2.png}
\doxyfigcaption{The regrid operation (left) and the remap operation (right).}
\end{DoxyImage}
 The complex part of the operation is remapping the wavy field onto the new grid as shown on the right and again in the final frame after the old deformed coordinate system has been deleted\+:


\begin{DoxyImage}
\includegraphics[width=\textwidth,height=\textheight/2,keepaspectratio=true]{remapping3.png}
\doxyfigcaption{The final state after remapping.}
\end{DoxyImage}
 Mathematically, the new layer thicknesses, $h_k$, are computed and then populated with the new velocities and tracers\+:

\[ h_k^{\mbox{new}} = \nabla_k z_{\mbox{coord}} \] \[ \sum h_k^{\mbox{new}} = \sum h_k^{\mbox{old}} \] \[ \vec{u}_k^{\mbox{new}} = \frac{1}{h_k} \int_{z_{k + \frac{1}{2}}}^{z_{k + \frac{1}{2}} + h_k} \vec{u}^{\mbox{old}}(z')dz' \] \[ \theta^{\mbox{new}} = \frac{1}{h_k} \int_{z_{k + \frac{1}{2}}}^{z_{k + \frac{1}{2}} + h_k} \theta^{\mbox{old}}(z')dz' \] 