s_algorithm/text/tracer.tex

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% $Name:  $

\section{Tracer equations}

The basic discretization used for the tracer equations is the second
order piece-wise constant finite volume form of the forced
advection-diussion equations. There are many alternatives to second
order method for advection and alternative parameterizations for the
sub-grid scale processes. The Gent-McWilliams eddy parameterization,
KPP mixing scheme and PV flux parameterization are all dealt with in
separate sections. The basic discretization of the advection-diffusion
part of the tracer equations and the various advection schemes will be
described here.

\subsection{Centered second order advection-diffusion}

The basic discretization, centered second order, is the default. It is
designed to be consistant 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 F_x
+ \delta_j F_y
+ \delta_k F_r
= {\cal A}_c \Delta r_f h_c {\cal S}_\theta
+ \theta {\cal A}_c \delta_k (P-E)_{r=0}
\end{equation}
where the area integrated fluxes are given by:
\begin{eqnarray}
F_x & = & U \overline{ \theta }^i
- \kappa_h \frac{\Delta y_g \Delta r_f h_w}{\Delta x_c} \delta_i \theta \\
F_y & = & V \overline{ \theta }^j
- \kappa_h \frac{\Delta x_g \Delta r_f h_s}{\Delta y_c} \delta_j \theta \\
F_r & = & W \overline{ \theta }^k
- \kappa_v \frac{\Delta x_g \Delta y_g}{\Delta r_c} \delta_k \theta
\end{eqnarray}
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
and sources 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 ${\cal Q}=0$, $\kappa_h = \kappa_v = 0$ 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}).


1	% $Header: /u/gcmpack/mitgcmdoc/part2/tracer.tex,v 1.1 2001/08/09 19:48:39 adcroft Exp $
2	% $Name: $
3
4	\section{Tracer equations}
5
6	The basic discretization used for the tracer equations is the second
7	order piece-wise constant finite volume form of the forced
8	advection-diussion equations. There are many alternatives to second
9	order method for advection and alternative parameterizations for the
10	sub-grid scale processes. The Gent-McWilliams eddy parameterization,
11	KPP mixing scheme and PV flux parameterization are all dealt with in
12	separate sections. The basic discretization of the advection-diffusion
13	part of the tracer equations and the various advection schemes will be
14	described here.
15
16	\subsection{Centered second order advection-diffusion}
17
18	The basic discretization, centered second order, is the default. It is
19	designed to be consistant with the continuity equation to facilitate
20	conservation properties analogous to the continuum:
21	\begin{equation}
22	{\cal A}_c \Delta r_f h_c \partial_\theta
23	+ \delta_i F_x
24	+ \delta_j F_y
25	+ \delta_k F_r
26	= {\cal A}_c \Delta r_f h_c {\cal S}_\theta
27	+ \theta {\cal A}_c \delta_k (P-E)_{r=0}
28	\end{equation}
29	where the area integrated fluxes are given by:
30	\begin{eqnarray}
31	F_x & = & U \overline{ \theta }^i
32	- \kappa_h \frac{\Delta y_g \Delta r_f h_w}{\Delta x_c} \delta_i \theta \\
33	F_y & = & V \overline{ \theta }^j
34	- \kappa_h \frac{\Delta x_g \Delta r_f h_s}{\Delta y_c} \delta_j \theta \\
35	F_r & = & W \overline{ \theta }^k
36	- \kappa_v \frac{\Delta x_g \Delta y_g}{\Delta r_c} \delta_k \theta
37	\end{eqnarray}
38	The quantities $U$, $V$ and $W$ are volume fluxes defined:
39	\marginpar{$U$: {\bf uTrans} }
40	\marginpar{$V$: {\bf vTrans} }
41	\marginpar{$W$: {\bf rTrans} }
42	\begin{eqnarray}
43	U & = & \Delta y_g \Delta r_f h_w u \\
44	V & = & \Delta x_g \Delta r_f h_s v \\
45	W & = & {\cal A}_c w
46	\end{eqnarray}
47	${\cal S}$ represents the ``parameterized'' SGS processes and physics
48	and sources associated with the tracer. For instance, potential
49	temperature equation in the ocean has is forced by surface and
50	partially penetrating heat fluxes:
51	\begin{equation}
52	{\cal A}_c \Delta r_f h_c {\cal S}_\theta = \frac{1}{c_p \rho_o} \delta_k {\cal A}_c {\cal Q}
53	\end{equation}
54	while the salt equation has no real sources, ${\cal S}=0$, which
55	leaves just the $P-E$ term.
56
57	The continuity equation can be recovered by setting ${\cal Q}=0$, $\kappa_h = \kappa_v = 0$ and
58	$\theta=1$. The term $\theta (P-E)_{r=0}$ is required to retain local
59	conservation of $\theta$. Global conservation is not possible using
60	the flux-form (as here) and a linearized free-surface
61	(\cite{Griffies00,Campin02}).
62
63
64
65