ALE algorithm

The semi-discrete, vertically integrated, Boussinesq hydrostatic equations of motion in general-coordinate \(r\) are

\[\begin{split}D_t \boldsymbol{u} + f \widehat{\boldsymbol{k}} \wedge \boldsymbol{u} + \frac{\rho}{\rho_o}\boldsymbol{\nabla}_z \Phi + \frac{1}{\rho_o} \boldsymbol{\nabla}_z p &= \boldsymbol{\nabla} \cdot \boldsymbol{\underline{\tau}} ,\\ \rho \delta_k \Phi + \delta_k p &= 0 ,\\ \partial_t h + \nabla_r \cdot ( h \boldsymbol{u} ) + \delta_k ( z_r \dot{r} ) &= 0 ,\\ \partial_t (h \theta) + \nabla_r \cdot ( h \boldsymbol{u} \theta ) + \delta_k ( z_r \dot{r} \theta ) &= \boldsymbol{\nabla} \cdot \boldsymbol{Q}_\theta ,\\ \partial_t (h S) + \nabla_r \cdot ( h \boldsymbol{u} S ) + \delta_k ( z_r \dot{r} S ) &= \boldsymbol{\nabla} \cdot \boldsymbol{Q}_S ,\\ \rho &= \rho(S, \theta, z) .\end{split}\]

The Arbitrary-Lagrangian-Eulerian algorithm we use is quasi-Lagrangian in that in the first (Lagrangian) phase, regardless of the current mesh (or coordinate \(r\)) we integrate the equations forward with \(\dot{r}=0\), i.e.:

\[\begin{split}D_t \boldsymbol{u} + f \widehat{\boldsymbol{k}} \wedge \boldsymbol{u} + \frac{\rho}{\rho_o}\boldsymbol{\nabla}_z \Phi + \frac{1}{\rho_o} \boldsymbol{\nabla}_z p &= \boldsymbol{\nabla} \cdot \boldsymbol{\underline{\tau}} ,\\ \rho \delta_k \Phi + \delta_k p &= 0 ,\\ \partial_t h + \nabla_r \cdot ( h \boldsymbol{u} ) &= 0 ,\\ \partial_t (h \theta) + \nabla_r \cdot ( h \boldsymbol{u} \theta ) &= \boldsymbol{\nabla} \cdot \boldsymbol{Q}_\theta ,\\ \partial_t (h S) + \nabla_r \cdot ( h \boldsymbol{u} S ) &= \boldsymbol{\nabla} \cdot \boldsymbol{Q}_S ,\\ \rho &= \rho(S, \theta, z) .\end{split}\]

Notice that by setting \(\dot{r}=0\) all the terms with the metric \(z_r\) disappeared.

After a finite amount of time, the mesh (\(h\)) may become very distorted or unrelated to the intended mesh. At any point in time, we can simply define a new mesh and remap from the current mesh to the new mesh without an explicit change in the physical state.