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\section{Tracer equations}

The tracer equations are discretized consistantly with the continuity
equation to facilitate conservation properties analogous to the
continuum:
\begin{equation}
{\cal A}_c \Delta r_f h_c \partial_\theta
+ \delta_i U \overline{ \theta }^i
+ \delta_j V \overline{ \theta }^j
+ \delta_k W \overline{ \theta }^k
= {\cal A}_c \Delta r_f h_c {\cal S}_\theta + \theta {\cal A}_c \delta_k (P-E)_{r=0}
\end{equation}
The quantities $U$, $V$ and $W$ are volume fluxes defined:
\marginpar{$U$: {\bf uTrans} }
\marginpar{$V$: {\bf vTrans} }
\marginpar{$W$: {\bf rTrans} }
\begin{eqnarray}
U & = & \Delta y_g \Delta r_f h_w u \\
V & = & \Delta x_g \Delta r_f h_s v \\
W & = & {\cal A}_c w
\end{eqnarray}
${\cal S}$ represents the ``parameterized'' SGS processes and
physics associated with the tracer. For instance, potential
temperature equation in the ocean has is forced by surface and
partially penetrating heat fluxes:
\begin{equation}
{\cal A}_c \Delta r_f h_c {\cal S}_\theta = \frac{1}{c_p \rho_o} \delta_k {\cal A}_c {\cal Q}
\end{equation}
while the salt equation has no real sources, ${\cal S}=0$, which
leaves just the $P-E$ term.

The continuity equation can be recovered by setting ${\cal Q}=0$ and
$\theta=1$. The term $\theta (P-E)_{r=0}$ is required to retain local
conservation of $\theta$. Global conservation is not possible using
the flux-form (as here) and a linearized free-surface
(\cite{Griffies00,Campin02}).