/[MITgcm]/manual/s_algorithm/text/mom_fluxform.tex
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revision 1.1 by adcroft, Thu Aug 9 19:48:39 2001 UTC revision 1.2 by adcroft, Tue Sep 25 20:13:42 2001 UTC
# Line 15  dissipation and metric forces: Line 15  dissipation and metric forces:
15  \marginpar{$G_w$: {\bf Gw} }  \marginpar{$G_w$: {\bf Gw} }
16  \begin{eqnarray}  \begin{eqnarray}
17  G_u & = & G_u^{adv} + G_u^{cor} + G_u^{h-diss} + G_u^{v-diss} +  G_u & = & G_u^{adv} + G_u^{cor} + G_u^{h-diss} + G_u^{v-diss} +
18  G_u^{metric} + G_u^{nh-metric} \\  G_u^{metric} + G_u^{nh-metric} \label{eq:gsplit_momu} \\
19  G_v & = & G_v^{adv} + G_v^{cor} + G_v^{h-diss} + G_v^{v-diss} +  G_v & = & G_v^{adv} + G_v^{cor} + G_v^{h-diss} + G_v^{v-diss} +
20  G_v^{metric} + G_v^{nh-metric} \\  G_v^{metric} + G_v^{nh-metric} \label{eq:gsplit_momv} \\
21  G_w & = & G_w^{adv} + G_w^{cor} + G_w^{h-diss} + G_w^{v-diss} +  G_w & = & G_w^{adv} + G_w^{cor} + G_w^{h-diss} + G_w^{v-diss} +
22  G_w^{metric} + G_w^{nh-metric}  G_w^{metric} + G_w^{nh-metric} \label{eq:gsplit_momw}
23  \end{eqnarray}  \end{eqnarray}
24  In the hydrostatic limit, $G_w=0$ and $\epsilon_{nh}=0$, reducing the  In the hydrostatic limit, $G_w=0$ and $\epsilon_{nh}=0$, reducing the
25  vertical momentum to hydrostatic balance.  vertical momentum to hydrostatic balance.
# Line 46  The advective operator is second order a Line 46  The advective operator is second order a
46  {\cal A}_w \Delta r_f h_w G_u^{adv} & = &  {\cal A}_w \Delta r_f h_w G_u^{adv} & = &
47    \delta_i \overline{ U }^i \overline{ u }^i    \delta_i \overline{ U }^i \overline{ u }^i
48  + \delta_j \overline{ V }^i \overline{ u }^j  + \delta_j \overline{ V }^i \overline{ u }^j
49  + \delta_k \overline{ W }^i \overline{ u }^k \\  + \delta_k \overline{ W }^i \overline{ u }^k \label{eq:discrete-momadvu} \\
50  {\cal A}_s \Delta r_f h_s G_v^{adv} & = &  {\cal A}_s \Delta r_f h_s G_v^{adv} & = &
51    \delta_i \overline{ U }^j \overline{ v }^i    \delta_i \overline{ U }^j \overline{ v }^i
52  + \delta_j \overline{ V }^j \overline{ v }^j  + \delta_j \overline{ V }^j \overline{ v }^j
53  + \delta_k \overline{ W }^j \overline{ v }^k \\  + \delta_k \overline{ W }^j \overline{ v }^k \label{eq:discrete-momadvv} \\
54  {\cal A}_c \Delta r_c G_w^{adv} & = &  {\cal A}_c \Delta r_c G_w^{adv} & = &
55    \delta_i \overline{ U }^k \overline{ w }^i    \delta_i \overline{ U }^k \overline{ w }^i
56  + \delta_j \overline{ V }^k \overline{ w }^j  + \delta_j \overline{ V }^k \overline{ w }^j
57  + \delta_k \overline{ W }^k \overline{ w }^k \\  + \delta_k \overline{ W }^k \overline{ w }^k \label{eq:discrete-momadvw}
58  \end{eqnarray}  \end{eqnarray}
59  and because of the flux form does not contribute to the global budget  and because of the flux form does not contribute to the global budget
60  of linear momentum. The quantities $U$, $V$ and $W$ are volume fluxes  of linear momentum. The quantities $U$, $V$ and $W$ are volume fluxes
# Line 63  defined: Line 63  defined:
63  \marginpar{$V$: {\bf vTrans} }  \marginpar{$V$: {\bf vTrans} }
64  \marginpar{$W$: {\bf rTrans} }  \marginpar{$W$: {\bf rTrans} }
65  \begin{eqnarray}  \begin{eqnarray}
66  U & = & \Delta y_g \Delta r_f h_w u \\  U & = & \Delta y_g \Delta r_f h_w u \label{eq:utrans} \\
67  V & = & \Delta x_g \Delta r_f h_s v \\  V & = & \Delta x_g \Delta r_f h_s v \label{eq:vtrans} \\
68  W & = & {\cal A}_c w  W & = & {\cal A}_c w \label{eq:rtrans}
69  \end{eqnarray}  \end{eqnarray}
70  The advection of momentum takes the same form as the advection of  The advection of momentum takes the same form as the advection of
71  tracers but by a translated advective flow. Consequently, the  tracers but by a translated advective flow. Consequently, the
# Line 109  where the Coriolis parameters $f$ and $f Line 109  where the Coriolis parameters $f$ and $f
109  f & = & 2 \Omega \sin{\phi} \\  f & = & 2 \Omega \sin{\phi} \\
110  f' & = & 2 \Omega \cos{\phi}  f' & = & 2 \Omega \cos{\phi}
111  \end{eqnarray}  \end{eqnarray}
112  when using spherical geometry, otherwise the $\beta$-plane definition is used:  where $\phi$ is geographic latitude when using spherical geometry,
113    otherwise the $\beta$-plane definition is used:
114  \begin{eqnarray}  \begin{eqnarray}
115  f & = & f_o + \beta y \\  f & = & f_o + \beta y \\
116  f' & = & 0  f' & = & 0
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