\hypertarget{Discrete_Coriolis_Coriolis}{}\section{Coriolis Term}\label{Discrete_Coriolis_Coriolis}
In general, the discrete equations are written as simple difference equations based on the Arakawa C-\/grid as described in section \hyperlink{Discrete_Grids_horizontal_grids}{Horizontal grids}. One of the more interesting exceptions is the Coriolis term. It is computed in the form shown in \ref{eq:h-horz-momentum}, or\+:

\[ \frac{( f + \zeta )}{h} \, \hat{\mathbf{z}} \times h \, \mathbf{u} \]

This term needs to be evaluated at $u$ points for the $v$ equation and vice versa, plus we need to keep the thickness, $h$, positive definite. M\+O\+M6 contains a number of options for how to compute this term.

\begin{DoxyItemize}
\item S\+A\+D\+O\+U\+R\+N\+Y75\+\_\+\+E\+N\+E\+R\+GY Sadourny \cite{sadourny1975} figured out how to conserve energy or enstrophy but not both. This option is energy conserving. The term in the $u$ equation becomes\+: \[ \frac{1}{4 dx} \left( q_{i,j} (vh_{i+1,j} + vh_{i,j}) + q_{i,j-1} (vh_{i+1,j-1} + vh_{i,j-1}) \right) \] where $q = \frac{f + \zeta}{h}$ and $h$ is an area-\/weighted average of the four thicknesses surrounding the $q$ point, such that it is guaranteed to be positive definite.\end{DoxyItemize}
There is a variant on this scheme with the C\+O\+R\+I\+O\+L\+I\+S\+\_\+\+E\+N\+\_\+\+D\+IS option. If true, two estimates of the thickness fluxes $vh$ are used to estimate the Coriolis term, and the one that dissipates energy relative to the other one is used.

\begin{DoxyItemize}
\item S\+A\+D\+O\+U\+R\+N\+Y75\+\_\+\+E\+N\+S\+T\+RO Also from \cite{sadourny1975}, this option is enstrophy conserving. \[ \frac{1}{8 dx} ( q_{i,j} + q_{i,j-1} ) ((vh_{i+1,j} + vh_{i,j}) + (vh_{i+1,j-1} + vh_{i,j-1}) ) \]\end{DoxyItemize}
\begin{DoxyItemize}
\item A\+R\+A\+K\+A\+W\+A\+\_\+\+L\+A\+M\+B81 From \cite{arakawa1981} is a scheme which is both energy and enstrophy conserving. Its weaknesses are a large stencil and differing thickness stencils in the numerator and denominator. This scheme and several others (with differing values of $a, b, c, d$ and $ep$) are implemented as\+: \begin{eqnarray} \frac{1}{dx} (a_{i,j} vh_{i+1,j} &+ b_{i,j} vh_{i,j} + d_{i,j} vh_{i+1,j-1} + c_{i,j} vh_{i,j-1} \\ &+ ep_{i,j}*uh_{i-1,j} - ep_{i+1,j}*uh_{i+1,j}) \label{eq:Coriolis_abcd} \end{eqnarray} with \begin{eqnarray} a_{i,j} &= \frac{1}{24} (2.0*(q_{i+1,j} + q_{i,j-1}) + (q_{i,j} + q_{i+1,j-1})) \\ b_{i,j} &= \frac{1}{24} ((q_{i,j} + q_{i-1,j-1}) + 2.0*(q_{i-1,j} + q_{i,j-1})) \\ c_{i,j} &= \frac{1}{24} (2.0*(q_{i,j} + q_{i-1,j-1}) + (q_{i-1,j} + q_{i,j-1})) \\ d_{i,j} &= \frac{1}{24} ((q_{i+1,j} + q_{i,j-1}) + 2.0*(q_{i,j} + q_{i+1,j-1})) \\ ep_{i,j} &= \frac{1}{24}((q_{i,j} - q_{i-1,j-1}) + (q_{i-1,j} - q_{i,j-1})) \end{eqnarray}\end{DoxyItemize}
\begin{DoxyItemize}
\item A\+R\+A\+K\+A\+W\+A\+\_\+\+H\+S\+U90 From \cite{arakawa1990} is a scheme which always conserves energy and conserves enstrophy in the limit of non-\/divergent flow. This one has a larger stencil than Sadourny\textquotesingle{}s energy scheme, but it\textquotesingle{}s much better behaved in terms of handling vanishing layers than Arakawa and Lamb. This scheme is implemented with\+: \[ \frac{1}{dx} (a_{i,j} vh_{i+1,j} + b_{i,j} vh_{i,j} + d_{i,j} vh_{i+1,j-1} + c_{i,j} vh_{i,j-1}) \] and \begin{eqnarray} a_{i,j} &= \frac{1}{12} (q_{i,j} + (q_{i+1,j} + q_{i,j-1})) \\ b_{i,j} &= \frac{1}{12} (q_{i,j} + (q_{i-1,j} + q_{i,j-1})) \\ c_{i,j} &= \frac{1}{12} (q_{i,j} + (q_{i-1,j-1} + q_{i,j-1})) \\ d_{i,j} &= \frac{1}{12} (q_{i,j} + (q_{i+1,j-1} + q_{i,j-1})) \end{eqnarray}\end{DoxyItemize}
\begin{DoxyItemize}
\item A\+R\+A\+K\+A\+W\+A\+\_\+\+L\+A\+M\+B\+\_\+\+B\+L\+E\+ND This is a blending of Arakawa and Lamb, Arakawa and Hsu, and the Sadourny Energy scheme. There are weights C\+O\+R\+I\+O\+L\+I\+S\+\_\+\+B\+L\+E\+N\+D\+\_\+\+W\+T\+\_\+\+L\+IN and C\+O\+R\+I\+O\+L\+I\+S\+\_\+\+B\+L\+E\+N\+D\+\_\+\+F\+\_\+\+E\+F\+F\+\_\+\+M\+AX to control this scheme. The equation is the same as for Arakawa and Lamb \ref{eq:Coriolis_abcd}, but the values of $a, b, c, d$ and $ep$ differ when the pure Arakawa and Lamb scheme breaks down due to thickness variations.\end{DoxyItemize}
\begin{DoxyItemize}
\item R\+O\+B\+U\+S\+T\+\_\+\+E\+N\+S\+T\+RO An enstrophy-\/conserving scheme which is robust to vanishing layers.\end{DoxyItemize}
Some of these options also support the B\+O\+U\+N\+D\+\_\+\+C\+O\+R\+I\+O\+L\+IS flag. If true, the Coriolis terms in the $u$ equation are bounded by the four estimates of $\frac{(f+\zeta)}{h}vh$ from the four neighboring $v$ points, and similarly in the $v$ equation. This option would have no effect on the S\+A\+D\+O\+U\+R\+N\+Y75\+\_\+\+E\+N\+E\+R\+GY scheme if it were possible to use centered difference thickness fluxes.\hypertarget{Discrete_Coriolis_Coriolis_BC}{}\subsection{Wall boundary conditions}\label{Discrete_Coriolis_Coriolis_BC}
Two sets of boundary conditions have been coded in the definition of relative vorticity. These are written as\+:

N\+O\+S\+L\+IP defined (in spherical coordinates)\+: \begin{eqnarray} \mbox{relvort} &= dv/dx \mbox{ (east $\&$ west)}, \mbox{ with } v = 0. \\ \mbox{relvort} &= -\sec(\phi) * d(u \cos(\phi))/dy \mbox{ (north $\&$ south)}, \mbox{ with } u = 0. \end{eqnarray}

Free slip (N\+O\+S\+L\+IP not defined)\+: \[ \mbox{relvort} = 0 \mbox{ (all boundaries)} \]

with $\phi$ defined as latitude. The free slip boundary condition is much more natural on a C-\/grid.