\hypertarget{Baroclinic_Momentum_Equations_section_BC_momentum}{}\doxysection{Baroclinic Momentum Equations}\label{Baroclinic_Momentum_Equations_section_BC_momentum}
The baroclinic momentum equations are the stacked shallow water equations\+:

\[ \frac{\partial \vec{u}_k}{\partial t} + (f + \nabla_s \times \vec{u}_k) \hat{z} \times \vec{u}_k = - \frac{\nabla_s p_k}{\rho} - \nabla_s (\phi_k + \frac{1}{2} || \vec{u}_k ||^2 ) + \frac{\nabla \cdot \tilde{\tau}_k}{\rho} \] \[ \frac{\partial h_k}{\partial t} + \nabla_s \cdot (\vec{u}h_k) = 0 \]

The timestepping for these equations is a (quasi?) second-\/order Runge-\/\+Kutta step for the inertial oscillations and a forward-\/backward Euler step for the pressure (gravity) waves. Using the graphical notation from \cite{shchepetkin2005}, it looks like\+:

 
\begin{DoxyImage}
\includegraphics[width=\textwidth,height=\textheight/2,keepaspectratio=true]{timestep_MOM6.png}
\doxyfigcaption{Graphical notation for timestepping schemes in which the black line represents the ideal solution and the red line shows the actual solution. Phase errors are represented by the grey shapes between the bars normal to the circle.}
\end{DoxyImage}


The timestep used in R\+O\+MS looks instead like\+:

 
\begin{DoxyImage}
\includegraphics[width=\textwidth,height=\textheight/2,keepaspectratio=true]{timestep_ROMS.png}
\doxyfigcaption{Graphical notation for the Adams-\/\+Bashforth technique used in the R\+O\+MS model.}
\end{DoxyImage}


The R\+O\+MS timestepping has smaller phase errors, strong damping at high frequency. The M\+O\+M6 use as a global climate model has made the phase errors of lower priority. However, the phase errors may become more problematic for future uses of M\+O\+M6. While the M\+O\+M6 use of the A\+LE remapping makes an Adams-\/\+Bashforth scheme impractical, there may be a better timestepping scheme out there for M\+O\+M6. Please let the M\+O\+M6 developers know if you would like to work on this problem. 