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Next: 2.14.5 Lateral dissipation Up: 2.14 Flux-form momentum equations Previous: 2.14.3 Curvature metric terms   Contents

2.14.4 Non-hydrostatic metric terms

For the non-hydrostatic equations, dropping the thin-atmosphere approximation re-introduces metric terms involving $ w$ and are required to conserve angular momentum:

$\displaystyle {\cal A}_w \Delta r_f h_w G_u^{metric}$ $\displaystyle =$ $\displaystyle - \overline{ \frac{ \overline{u}^i \overline{w}^k }{a} {\cal A}_c \Delta r_f h_c }^i$ (2.118)
$\displaystyle {\cal A}_s \Delta r_f h_s G_v^{metric}$ $\displaystyle =$ $\displaystyle - \overline{ \frac{ \overline{v}^j \overline{w}^k }{a} {\cal A}_c \Delta r_f h_c}^j$ (2.119)
$\displaystyle {\cal A}_c \Delta r_c G_w^{metric}$ $\displaystyle =$ $\displaystyle \overline{ \frac{ {\overline{u}^i}^2 + {\overline{v}^j}^2}{a} {\cal A}_c \Delta r_f h_c }^k$ (2.120)

Because we are always consistent, even if consistently wrong, we have, in the past, used a different discretization in the model which is:

$\displaystyle G_u^{metric}$ $\displaystyle =$ $\displaystyle - \frac{u}{a} \overline{w}^{ik}$ (2.121)
$\displaystyle G_v^{metric}$ $\displaystyle =$ $\displaystyle - \frac{v}{a} \overline{w}^{jk}$ (2.122)
$\displaystyle G_w^{metric}$ $\displaystyle =$ $\displaystyle \frac{1}{a} ( {\overline{u}^{ik}}^2 + {\overline{v}^{jk}}^2 )$ (2.123)

\fbox{ \begin{minipage}{4.75in} {\em S/R MOM\_U\_METRIC\_NH} ({\em mom\_u\_metri... ...ic}$, $G_v^{metric}$: {\bf mT} (local to {\em mom\_fluxform.F}) \end{minipage} }


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