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2.17.1 Centered second order advection-diffusion

The basic discretization, centered second order, is the default. It is designed to be consistent with the continuity equation to facilitate conservation properties analogous to the continuum. However, centered second order advection is notoriously noisy and must be used in conjunction with some finite amount of diffusion to produce a sensible solution.

The advection operator is discretized:

$\displaystyle {\cal A}_c \Delta r_f h_c G_{adv}^\tau = \delta_i F_x + \delta_j F_y + \delta_k F_r$ (2.171)

where the area integrated fluxes are given by:
$\displaystyle F_x$ $\displaystyle =$ $\displaystyle U \overline{ \tau }^i$ (2.172)
$\displaystyle F_y$ $\displaystyle =$ $\displaystyle V \overline{ \tau }^j$ (2.173)
$\displaystyle F_r$ $\displaystyle =$ $\displaystyle W \overline{ \tau }^k$ (2.174)

The quantities $ U$ , $ V$ and $ W$ are volume fluxes defined:
$\displaystyle U$ $\displaystyle =$ $\displaystyle \Delta y_g \Delta r_f h_w u$ (2.175)
$\displaystyle V$ $\displaystyle =$ $\displaystyle \Delta x_g \Delta r_f h_s v$ (2.176)
$\displaystyle W$ $\displaystyle =$ $\displaystyle {\cal A}_c w$ (2.177)

For non-divergent flow, this discretization can be shown to conserve the tracer both locally and globally and to globally conserve tracer variance, $ \tau^2$ . The proof is given in Adcroft et al. [1997]; Adcroft [1995].

\fbox{ \begin{minipage}{4.75in}
{\em S/R GAD\_C2\_ADV\_X} ({\em gad\_c2\_adv\_x....
...bf rTrans} (argument)
\par
$\tau$: {\bf tracer} (argument)
\par
\end{minipage} }


next up previous contents
Next: 2.17.2 Third order upwind Up: 2.17 Linear advection schemes Previous: 2.17 Linear advection schemes   Contents
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