General coordinate equations

Transforming to a vertical coordinate \(r(z,x,y,t)\), with \(\dot{r} = \frac{\partial r}{\partial t}\)

The Boussinesq hydrostatic equations of motion in general-coordinate \(r\) are:

(1)\[\begin{split}\begin{eqnarray} \rho_0 \left( \frac{\partial \mathbf{u}}{\partial t} + ( f + \zeta ) \, \hat{\mathbf{z}} \times \mathbf{u} + \dot{r} \, \frac{\partial \mathbf{u}}{\partial r} + \nabla_r \, K \right) &= -\nabla_r \, p - \rho \nabla_r \, \Phi + \boldsymbol{\mathcal{F}} &\mbox{momentum} \label{eq:r-horz-momentum} \\ \rho \, \frac{\partial \Phi}{\partial r} + \frac{\partial p}{\partial r} &= 0 &\mbox{hydrostatic} \label{eq:r-hydrostatic-equation} \\ \frac{\partial z_r }{\partial t} + \nabla_r \cdot \, \left( z_r \, \mathbf{u} \right) + \frac{\partial ( z_r \, \dot{r} ) }{\partial r} &= 0 &\mbox{thickness} \label{eq:r-non-divergence} \\ \frac{\partial ( \theta \, z_r ) }{\partial t} + \nabla_r \cdot \left( \theta z_r \, \mathbf{u} \right) + \frac{\partial ( \theta \, z_r \, \dot{r} )}{\partial r} &= z_r \boldsymbol{\mathcal{N}}_\theta^\gamma - \frac{\partial J_\theta^{(z)}}{\partial r} &\mbox{potential temp} \label{eq:r-temperature-equation} \\ \frac{\partial ( S \, z_r) }{\partial t} + \nabla_r \cdot \left( S \, z_r \, \mathbf{u} \right) + \frac{\partial ( S \, z_r \, \dot{r} )}{\partial r} &= z_r \boldsymbol{\mathcal{N}}_S^\gamma - \frac{\partial J_S^{(z)}}{\partial r} &\mbox{salinity} \label{eq:r-salinity-equation} \\ \rho &= \rho\left( S, \theta, -g \rho_0 z(r) \right) &\mbox{equation of state.} \end{eqnarray}\end{split}\]

The time derivatives are now computed with the generalized vertical coordinate fixed rather than the geopotential. We introduced the specific thickness, \(z_r = \partial z/\partial r\), which measures the inverse vertical stratification of the vertical coordinate surfaces.

Similar to [7], MOM6 is discretized in the vertical by integrating between surfaces of \(r\) to yield layer equations where the layer thickness is \(h = \int z_r dr\) and variables are treated as finite volume averages over each layer:

(2)\[\begin{split}\begin{eqnarray} \rho_0 \left( \frac{\partial \mathbf{u}}{\partial t} + \frac{( f + \zeta )}{h} \, \hat{\mathbf{z}} \times h \, \mathbf{u} + \underbrace{ \dot{r} \, \frac{\partial \mathbf{u}}{\partial r} } + \nabla_r K \right) &= -\nabla_r \, p - \rho \nabla_r \, \Phi + \boldsymbol{\mathcal{F}} &\mbox{momentum} \label{eq:h-horz-momentum} \\ \rho \, \delta_r \Phi + \delta_r p &= 0 &\mbox{hydrostatic} \label{eq:h-hydrostatic-equation} \\ \frac{\partial h}{\partial t} + \nabla_r \cdot \left( h \, \mathbf{u} \right) + \underbrace{ \delta_r ( z_r \dot{r} ) } &= 0 &\mbox{thickness} \label{eq:h-thickness-equation} \\ \frac{\partial ( \theta \, h )}{\partial t} + \nabla_r \cdot \left( \theta h \, \mathbf{u} \right) + \underbrace{ \delta_r ( \theta \, z_r \dot{r} ) } &= h \boldsymbol{\mathcal{N}}_\theta^\gamma - \delta_r J_\theta^{(z)} &\mbox{potential temp} \label{eq:h-temperature-equation} \\ \frac{\partial ( S \, h )}{\partial t} + \nabla_r \cdot \left( S \, h \, \mathbf{u} \right) + \underbrace{ \delta_r ( S \, z_r \dot{r} ) } &= h \boldsymbol{\mathcal{N}}_S^\gamma - \delta_r J_S^{(z)} &\mbox{salinity} \label{eq:h-salinity-equation} \\ \rho &= \rho\left( S, \theta, -g \rho_0 z(r) \right) &\mbox{equation of state,} \label{eq:h-equation-of-state} \end{eqnarray}\end{split}\]

where \(\delta_{r} = \mathrm{d}r \, (\partial/\partial r)\) is the discrete vertical difference operator. The pressure gradient accelerations in the momentum equation (2) - momentum are written in continuous-in-the-vertical form for brevity; the exact discretization is detailed in [3]. The MOM6 time-stepping algorithm integrates the above layer-averaged equations forward allowing the vertical grid to follow the motion, i.e. \(\dot{r}=0\), so that the underbraced terms are dropped. This approach is generally known as the Lagrangian method but here the Lagrangian method is used only in the vertical direction. After each Lagrangian step, a remap step is applied that generates a new vertical grid of the user’s choosing. The ocean state is then mapped from the old to the new grid. The physical state is not meant to change during the remap step, yet truncation errors make remapping imperfect. We employ high-order accurate reconstructions to minimize errors introduced during the remap step ([67], [68]). The connection between time-stepping and remapping is described in section ALE Timestep.