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\section{Tracer equations} |
\section{Tracer equations} |
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The tracer equations are discretized consistantly with the continuity |
The basic discretization used for the tracer equations is the second |
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equation to facilitate conservation properties analogous to the |
order piece-wise constant finite volume form of the forced |
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continuum: |
advection-diussion equations. There are many alternatives to second |
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order method for advection and alternative parameterizations for the |
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sub-grid scale processes. The Gent-McWilliams eddy parameterization, |
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KPP mixing scheme and PV flux parameterization are all dealt with in |
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separate sections. The basic discretization of the advection-diffusion |
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part of the tracer equations and the various advection schemes will be |
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described here. |
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\subsection{Centered second order advection-diffusion} |
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The basic discretization, centered second order, is the default. It is |
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designed to be consistant with the continuity equation to facilitate |
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conservation properties analogous to the continuum: |
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\begin{equation} |
\begin{equation} |
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{\cal A}_c \Delta r_f h_c \partial_\theta |
{\cal A}_c \Delta r_f h_c \partial_\theta |
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+ \delta_i U \overline{ \theta }^i |
+ \delta_i F_x |
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+ \delta_j V \overline{ \theta }^j |
+ \delta_j F_y |
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+ \delta_k W \overline{ \theta }^k |
+ \delta_k F_r |
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= {\cal A}_c \Delta r_f h_c {\cal S}_\theta + \theta {\cal A}_c \delta_k (P-E)_{r=0} |
= {\cal A}_c \Delta r_f h_c {\cal S}_\theta |
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+ \theta {\cal A}_c \delta_k (P-E)_{r=0} |
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\end{equation} |
\end{equation} |
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where the area integrated fluxes are given by: |
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\begin{eqnarray} |
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F_x & = & U \overline{ \theta }^i |
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- \kappa_h \frac{\Delta y_g \Delta r_f h_w}{\Delta x_c} \delta_i \theta \\ |
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F_y & = & V \overline{ \theta }^j |
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- \kappa_h \frac{\Delta x_g \Delta r_f h_s}{\Delta y_c} \delta_j \theta \\ |
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F_r & = & W \overline{ \theta }^k |
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- \kappa_v \frac{\Delta x_g \Delta y_g}{\Delta r_c} \delta_k \theta |
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\end{eqnarray} |
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The quantities $U$, $V$ and $W$ are volume fluxes defined: |
The quantities $U$, $V$ and $W$ are volume fluxes defined: |
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\marginpar{$U$: {\bf uTrans} } |
\marginpar{$U$: {\bf uTrans} } |
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\marginpar{$V$: {\bf vTrans} } |
\marginpar{$V$: {\bf vTrans} } |
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V & = & \Delta x_g \Delta r_f h_s v \\ |
V & = & \Delta x_g \Delta r_f h_s v \\ |
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W & = & {\cal A}_c w |
W & = & {\cal A}_c w |
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\end{eqnarray} |
\end{eqnarray} |
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${\cal S}$ represents the ``parameterized'' SGS processes and |
${\cal S}$ represents the ``parameterized'' SGS processes and physics |
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physics associated with the tracer. For instance, potential |
and sources associated with the tracer. For instance, potential |
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temperature equation in the ocean has is forced by surface and |
temperature equation in the ocean has is forced by surface and |
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partially penetrating heat fluxes: |
partially penetrating heat fluxes: |
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\begin{equation} |
\begin{equation} |
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while the salt equation has no real sources, ${\cal S}=0$, which |
while the salt equation has no real sources, ${\cal S}=0$, which |
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leaves just the $P-E$ term. |
leaves just the $P-E$ term. |
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|
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The continuity equation can be recovered by setting ${\cal Q}=0$ and |
The continuity equation can be recovered by setting ${\cal Q}=0$, $\kappa_h = \kappa_v = 0$ and |
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$\theta=1$. The term $\theta (P-E)_{r=0}$ is required to retain local |
$\theta=1$. The term $\theta (P-E)_{r=0}$ is required to retain local |
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conservation of $\theta$. Global conservation is not possible using |
conservation of $\theta$. Global conservation is not possible using |
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the flux-form (as here) and a linearized free-surface |
the flux-form (as here) and a linearized free-surface |
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