| 516 |
\caption{ |
\caption{ |
| 517 |
A schematic of the explicit Adams-Bashforth and implicit time-stepping |
A schematic of the explicit Adams-Bashforth and implicit time-stepping |
| 518 |
phases of the algorithm but with staggering in time of thermodynamic |
phases of the algorithm but with staggering in time of thermodynamic |
| 519 |
variables with the flow. Explicit thermodynamics tendencies are |
variables with the flow. |
| 520 |
evaluated at time level $n$ as a function of the thermodynamics |
Explicit momentum tendencies are evaluated at time level $n-1/2$ as a |
| 521 |
state at that time level $n$ and flow at time $n+1/2$ (dotted arrow). The |
function of the flow field at that time level $n-1/2$. |
| 522 |
explicit tendency from the previous time level, $n-1$, is used to |
The explicit tendency from the previous time level, $n-3/2$, is used to |
| 523 |
extrapolate tendencies to $n+1/2$ (dashed arrow). This extrapolated |
extrapolate tendencies to $n$ (dashed arrow). |
| 524 |
tendency allows thermo-dynamics variables to be stably integrated |
The hydrostatic pressure/geo-potential $\phi_{hyd}$ is evaluated directly |
| 525 |
forward-in-time to render an estimate ($*$-variables) at the $n+1$ |
at time level $n$ (vertical arrows) and used with the extrapolated tendencies |
| 526 |
time level (solid arc-arrow). The implicit-in-time operator ${\cal |
to step forward the flow variables from $n-1/2$ to $n+1/2$ (solid arc-arrow). |
| 527 |
L_{\theta,S}}$ is solved to yield the thermodynamic variables at time |
The implicit-in-time operator ${\cal L_{u,v}}$ (vertical arrows) is |
| 528 |
level $n+1$. These are then used to calculate the hydrostatic |
then applied to the previous estimation of the the flow field ($*$-variables) |
| 529 |
pressure/geo-potential, $\phi_{hyd}$ (vertical arrows). The |
and yields to the two velocity components $u,v$ at time level $n+1/2$. |
| 530 |
hydrostatic pressure gradient is evaluated directly at time level |
These are then used to calculate the advection term (dashed arc-arrow) |
| 531 |
$n+1$ in stepping forward the flow variables from $n+1/2$ to $n+3/2$ |
of the thermo-dynamics tendencies at time step $n$. |
| 532 |
(solid arc-arrow). } |
The extrapolated thermodynamics tendency, from time level $n-1$ and $n$ |
| 533 |
|
to $n+1/2$, allows thermodynamic variables to be stably integrated |
| 534 |
|
forward-in-time (solid arc-arrow) up to time level $n+1$. |
| 535 |
|
} |
| 536 |
\label{fig:adams-bashforth-staggered} |
\label{fig:adams-bashforth-staggered} |
| 537 |
\end{figure} |
\end{figure} |
| 538 |
|
|
| 639 |
|
|
| 640 |
The only difficulty with this approach is apparent in equation |
The only difficulty with this approach is apparent in equation |
| 641 |
\ref{eq:Gt-n-staggered} and illustrated by the dotted arrow |
\ref{eq:Gt-n-staggered} and illustrated by the dotted arrow |
| 642 |
connecting $u,v^n$ with $G_\theta^{n-1/2}$. The flow used to advect |
connecting $u,v^{n+1/2}$ with $G_\theta^{n}$. The flow used to advect |
| 643 |
tracers around is not naturally located in time. This could be avoided |
tracers around is not naturally located in time. This could be avoided |
| 644 |
by applying the Adams-Bashforth extrapolation to the tracer field |
by applying the Adams-Bashforth extrapolation to the tracer field |
| 645 |
itself and advecting that around but this approach is not yet |
itself and advecting that around but this approach is not yet |
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