\hypertarget{ALE_section_ALE}{}\section{Basics of the Vertical Lagrangian-\/\+Remap Method in M\+O\+M6}\label{ALE_section_ALE}
As discussed by \cite{adcroft2006}, there are two general classes of algorithms that frame how hydrostatic ocean models are formulated. The two classes differ in how they treat the vertical direction. Quasi-\/\+Eulerian methods follow the approach traditionally used in geopotential coordinate models, whereby vertical motion is diagnosed via the continuity equation. Quasi-\/\+Lagrangian methods are traditionally used by layered isopycnal models, with the Lagrangian approach specifying motion that crosses coordinate surfaces. Indeed, such dia-\/surface flow can be set to zero using Lagrangian methods for studies of adiabatic dynamics. M\+O\+M6 makes use of the vertical Lagrangian remap method, as pioneered for ocean modeling by \cite{bleck2002}, which is a limit case of the Arbitrary-\/\+Lagrangian-\/\+Eulerian method (\cite{hirt1997}). Dia-\/surface transport is implemented via a remapping so that the method can be summarized as the Lagrangian plus remap approach and is essentially a one-\/dimensional version of the incremental remapping of \cite{dukowicz2000}.

The M\+O\+M6 implementation of the vertical Lagrangian-\/remap method makes use of two general steps. The first evolves the ocean state forward in time according to a vertical Lagrangian limit with $\dot{r}=0$. Hence, the horizontal momentum, thickness, and tracers are time stepped with the red terms removed in equations \ref{eq:h-horz-momentum}, \ref{eq:h-thickness-equation}, \ref{eq:h-temperature-equation}, and \ref{eq:h-salinity-equation}. All advective transport thus occurs within a layer as defined by constant $r$-\/surfaces so that the volume within each layer is fixed. All other terms are retained in their full form, including subgrid scale terms that contribute to the transfer of tracer and momentum into distinct $r$ layers (e.\+g., dia-\/surface diffusion of tracer and velocity). Maintaining constant volume within a layer yet allowing for tracers to move between layers engenders no inconsistency between tracer and thickness evolution. The reason is that tracer diffusion, even dia-\/surface diffusion, does not transfer volume.

The second step in the algorithm comprises the generation of a new vertical grid following a prescription, such as whether the grid should align with isopcynals or constant $z^{*}$ or a combination. The ocean state is then vertically remapped to the newly generated vertical grid. The remapping step incorporates dia-\/surface transfer of properties, with such transfer depending on the prescription given for the vertical grid generation. To minimize discretization errors and the associated spurious mixing, the remapping step makes use of the high order accurate methods developed by \cite{white2008} and \cite{white2009}.

The underlying algorithm for treatment of the vertical can be related to operator-\/splitting of the red terms in equations \ref{eq:h-thickness-equation}--\ref{eq:h-temperature-equation}. If we consider, for simplicity, an Euler-\/forward update for a time-\/step $\Delta t$, the time-\/stepping for the continuity and temperature equation can be summarized as

\textbackslash{}f\{eqnarray\} \textbackslash{}label\{html\+:ale-\/equations\}\textbackslash{}notag \textbackslash{} h$^\wedge$\textbackslash{}dagger \&= h$^\wedge$\{(n)\} -\/ \textbackslash{}\+Delta t \textbackslash{}left\mbox{[} \textbackslash{}nabla\+\_\+r \textbackslash{}cdot \textbackslash{}left( h \textbackslash{}, \textbackslash{}mathbf\{u\} \textbackslash{}right) \textbackslash{}right\mbox{]} \&\textbackslash{}mbox\{thickness\} \textbackslash{}label\{eq\+:ale-\/thickness-\/equation\} \textbackslash{} \textbackslash{}theta$^\wedge$\textbackslash{}dagger \textbackslash{}, h$^\wedge$\textbackslash{}dagger \&= \textbackslash{}theta$^\wedge$\{(n)\} \textbackslash{}, h$^\wedge$\{(n)\} -\/ \textbackslash{}\+Delta t \textbackslash{}left\mbox{[} \textbackslash{}nabla\+\_\+r \textbackslash{}cdot \textbackslash{}left( \textbackslash{}theta h \textbackslash{}, \textbackslash{}mathbf\{u\} \textbackslash{}right) -\/ h \textbackslash{}boldsymbol\{\textbackslash{}mathcal\{N\}\}\+\_\+\textbackslash{}theta$^\wedge$\textbackslash{}gamma + \textbackslash{}delta\+\_\+r J\+\_\+\textbackslash{}theta$^\wedge$\{(z)\} \textbackslash{}right\mbox{]} \&\textbackslash{};\textbackslash{};\textbackslash{};\textbackslash{};\textbackslash{}mbox\{potential temp\} \textbackslash{}label\{eq\+:ale-\/temperature-\/equation\} \textbackslash{} h$^\wedge$\{(n+1)\} \&= h$^\wedge$\textbackslash{}dagger -\/ \textbackslash{}\+Delta t \textbackslash{}, \textbackslash{}delta\+\_\+r \textbackslash{}left( z\+\_\+r
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\doxyfigcaption{\{r\} \textbackslash{}right)}
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