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Next: 2.13.4 Non-hydrostatic metric terms Up: 2.13 Flux-form momentum equations Previous: 2.13.2 Coriolis terms   Contents

2.13.3 Curvature metric terms

The most commonly used coordinate system on the sphere is the geographic system $ (\lambda,\varphi)$. The curvilinear nature of these coordinates on the sphere lead to some ``metric'' terms in the component momentum equations. Under the thin-atmosphere and hydrostatic approximations these terms are discretized:

$\displaystyle {\cal A}_w \Delta r_f h_w G_u^{metric}$ $\displaystyle =$ $\displaystyle \overline{ \frac{ \overline{u}^i }{a} \tan{\varphi} {\cal A}_c \Delta r_f h_c \overline{ v }^j }^i$ (2.113)
$\displaystyle {\cal A}_s \Delta r_f h_s G_v^{metric}$ $\displaystyle =$ $\displaystyle - \overline{ \frac{ \overline{u}^i }{a} \tan{\varphi} {\cal A}_c \Delta r_f h_c \overline{ u }^i }^j$ (2.114)
$\displaystyle G_w^{metric}$ $\displaystyle =$ 0 (2.115)

where $ a$ is the radius of the planet (sphericity is assumed) or the radial distance of the particle (i.e. a function of height). It is easy to see that this discretization satisfies all the properties of the discrete Coriolis terms since the metric factor $ \frac{u}{a} \tan{\varphi}$ can be viewed as a modification of the vertical Coriolis parameter: $ f \rightarrow f+\frac{u}{a} \tan{\varphi}$.

However, as for the Coriolis terms, a non-energy conserving form has exclusively been used to date:

$\displaystyle G_u^{metric}$ $\displaystyle =$ $\displaystyle \frac{u \overline{v}^{ij} }{a} \tan{\varphi}$ (2.116)
$\displaystyle G_v^{metric}$ $\displaystyle =$ $\displaystyle \frac{ \overline{u}^{ij} \overline{u}^{ij}}{a} \tan{\varphi}$ (2.117)

where $ \tan{\varphi}$ is evaluated at the $ u$ and $ v$ points respectively.

\fbox{ \begin{minipage}{4.75in} {\em S/R MOM\_U\_METRIC\_SPHERE} ({\em mom\_u\_m... ...c}$, $G_v^{metric}$: {\bf mT} (local to {\em calc\_mom\_rhs.F}) \end{minipage} }

next up previous contents
Next: 2.13.4 Non-hydrostatic metric terms Up: 2.13 Flux-form momentum equations Previous: 2.13.2 Coriolis terms   Contents
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