| 13 |
\input{s_algorithm/text/notation} |
\input{s_algorithm/text/notation} |
| 14 |
|
|
| 15 |
\section{Time-stepping} |
\section{Time-stepping} |
| 16 |
|
\label{sec:time_stepping} |
| 17 |
\begin{rawhtml} |
\begin{rawhtml} |
| 18 |
<!-- CMIREDIR:time-stepping: --> |
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| 19 |
\end{rawhtml} |
\end{rawhtml} |
| 51 |
independent of the particular time-stepping scheme chosen we will |
independent of the particular time-stepping scheme chosen we will |
| 52 |
describe first the over-arching algorithm, known as the pressure |
describe first the over-arching algorithm, known as the pressure |
| 53 |
method, with a rigid-lid model in section |
method, with a rigid-lid model in section |
| 54 |
\ref{sect:pressure-method-rigid-lid}. This algorithm is essentially |
\ref{sec:pressure-method-rigid-lid}. This algorithm is essentially |
| 55 |
unchanged, apart for some coefficients, when the rigid lid assumption |
unchanged, apart for some coefficients, when the rigid lid assumption |
| 56 |
is replaced with a linearized implicit free-surface, described in |
is replaced with a linearized implicit free-surface, described in |
| 57 |
section \ref{sect:pressure-method-linear-backward}. These two flavors |
section \ref{sec:pressure-method-linear-backward}. These two flavors |
| 58 |
of the pressure-method encompass all formulations of the model as it |
of the pressure-method encompass all formulations of the model as it |
| 59 |
exists today. The integration of explicit in time terms is out-lined |
exists today. The integration of explicit in time terms is out-lined |
| 60 |
in section \ref{sect:adams-bashforth} and put into the context of the |
in section \ref{sec:adams-bashforth} and put into the context of the |
| 61 |
overall algorithm in sections \ref{sect:adams-bashforth-sync} and |
overall algorithm in sections \ref{sec:adams-bashforth-sync} and |
| 62 |
\ref{sect:adams-bashforth-staggered}. Inclusion of non-hydrostatic |
\ref{sec:adams-bashforth-staggered}. Inclusion of non-hydrostatic |
| 63 |
terms requires applying the pressure method in three dimensions |
terms requires applying the pressure method in three dimensions |
| 64 |
instead of two and this algorithm modification is described in section |
instead of two and this algorithm modification is described in section |
| 65 |
\ref{sect:non-hydrostatic}. Finally, the free-surface equation may be |
\ref{sec:non-hydrostatic}. Finally, the free-surface equation may be |
| 66 |
treated more exactly, including non-linear terms, and this is |
treated more exactly, including non-linear terms, and this is |
| 67 |
described in section \ref{sect:nonlinear-freesurface}. |
described in section \ref{sec:nonlinear-freesurface}. |
| 68 |
|
|
| 69 |
|
|
| 70 |
\section{Pressure method with rigid-lid} |
\section{Pressure method with rigid-lid} |
| 71 |
\label{sect:pressure-method-rigid-lid} |
\label{sec:pressure-method-rigid-lid} |
| 72 |
\begin{rawhtml} |
\begin{rawhtml} |
| 73 |
<!-- CMIREDIR:pressure_method_rigid_lid: --> |
<!-- CMIREDIR:pressure_method_rigid_lid: --> |
| 74 |
\end{rawhtml} |
\end{rawhtml} |
| 205 |
such as the vertical viscosity when using the backward time-stepping scheme |
such as the vertical viscosity when using the backward time-stepping scheme |
| 206 |
(\varlink{implicitViscosity}{implicitViscosity} {\it =.TRUE.}). |
(\varlink{implicitViscosity}{implicitViscosity} {\it =.TRUE.}). |
| 207 |
The method used to solve those implicit terms is provided in |
The method used to solve those implicit terms is provided in |
| 208 |
section \ref{sect:implicit-backward-stepping}, and modifies |
section \ref{sec:implicit-backward-stepping}, and modifies |
| 209 |
equations \ref{eq:discrete-time-u} and \ref{eq:discrete-time-v} to |
equations \ref{eq:discrete-time-u} and \ref{eq:discrete-time-v} to |
| 210 |
give: |
give: |
| 211 |
\begin{eqnarray} |
\begin{eqnarray} |
| 218 |
|
|
| 219 |
|
|
| 220 |
\section{Pressure method with implicit linear free-surface} |
\section{Pressure method with implicit linear free-surface} |
| 221 |
\label{sect:pressure-method-linear-backward} |
\label{sec:pressure-method-linear-backward} |
| 222 |
\begin{rawhtml} |
\begin{rawhtml} |
| 223 |
<!-- CMIREDIR:pressure_method_linear_backward: --> |
<!-- CMIREDIR:pressure_method_linear_backward: --> |
| 224 |
\end{rawhtml} |
\end{rawhtml} |
| 254 |
where the use of flow at time level $n+1$ makes the method implicit |
where the use of flow at time level $n+1$ makes the method implicit |
| 255 |
and backward in time. This is the preferred scheme since it still |
and backward in time. This is the preferred scheme since it still |
| 256 |
filters the fast unresolved wave motions by damping them. A centered |
filters the fast unresolved wave motions by damping them. A centered |
| 257 |
scheme, such as Crank-Nicholson (see section \ref{sect:freesurf-CrankNick}), |
scheme, such as Crank-Nicholson (see section \ref{sec:freesurf-CrankNick}), |
| 258 |
would alias the energy of the fast modes onto slower modes of motion. |
would alias the energy of the fast modes onto slower modes of motion. |
| 259 |
|
|
| 260 |
As for the rigid-lid pressure method, equations |
As for the rigid-lid pressure method, equations |
| 263 |
\begin{eqnarray} |
\begin{eqnarray} |
| 264 |
u^{*} & = & u^{n} + \Delta t G_u^{(n+1/2)} \label{eq:ustar-backward-free-surface} \\ |
u^{*} & = & u^{n} + \Delta t G_u^{(n+1/2)} \label{eq:ustar-backward-free-surface} \\ |
| 265 |
v^{*} & = & v^{n} + \Delta t G_v^{(n+1/2)} \label{eq:vstar-backward-free-surface} \\ |
v^{*} & = & v^{n} + \Delta t G_v^{(n+1/2)} \label{eq:vstar-backward-free-surface} \\ |
| 266 |
\eta^* & = & \epsilon_{fs} \left( \eta^{n} + \Delta t (P-E) \right)- \Delta t |
\eta^* & = & \epsilon_{fs} \left( \eta^{n} + \Delta t (P-E) \right) |
| 267 |
\partial_x H \widehat{u^{*}} |
- \Delta t \left( \partial_x H \widehat{u^{*}} |
| 268 |
+ \partial_y H \widehat{v^{*}} |
+ \partial_y H \widehat{v^{*}} \right) |
| 269 |
\\ |
\\ |
| 270 |
\partial_x g H \partial_x \eta^{n+1} |
\partial_x g H \partial_x \eta^{n+1} |
| 271 |
& + & \partial_y g H \partial_y \eta^{n+1} |
& + & \partial_y g H \partial_y \eta^{n+1} |
| 298 |
|
|
| 299 |
|
|
| 300 |
\section{Explicit time-stepping: Adams-Bashforth} |
\section{Explicit time-stepping: Adams-Bashforth} |
| 301 |
\label{sect:adams-bashforth} |
\label{sec:adams-bashforth} |
| 302 |
\begin{rawhtml} |
\begin{rawhtml} |
| 303 |
<!-- CMIREDIR:adams_bashforth: --> |
<!-- CMIREDIR:adams_bashforth: --> |
| 304 |
\end{rawhtml} |
\end{rawhtml} |
| 308 |
the quasi-second order Adams-Bashforth method for all explicit terms |
the quasi-second order Adams-Bashforth method for all explicit terms |
| 309 |
in both the momentum and tracer equations. This is still the default |
in both the momentum and tracer equations. This is still the default |
| 310 |
mode of operation but it is now possible to use alternate schemes for |
mode of operation but it is now possible to use alternate schemes for |
| 311 |
tracers (see section \ref{sect:tracer-advection}). |
tracers (see section \ref{sec:tracer-advection}). |
| 312 |
|
|
| 313 |
\begin{figure} |
\begin{figure} |
| 314 |
\begin{center} \fbox{ \begin{minipage}{4.5in} \begin{tabbing} |
\begin{center} \fbox{ \begin{minipage}{4.5in} \begin{tabbing} |
| 397 |
|
|
| 398 |
|
|
| 399 |
\section{Implicit time-stepping: backward method} |
\section{Implicit time-stepping: backward method} |
| 400 |
\label{sect:implicit-backward-stepping} |
\label{sec:implicit-backward-stepping} |
| 401 |
\begin{rawhtml} |
\begin{rawhtml} |
| 402 |
<!-- CMIREDIR:implicit_time-stepping_backward: --> |
<!-- CMIREDIR:implicit_time-stepping_backward: --> |
| 403 |
\end{rawhtml} |
\end{rawhtml} |
| 425 |
\end{eqnarray} |
\end{eqnarray} |
| 426 |
where ${\cal L}_\tau^{-1}$ is the inverse of the operator |
where ${\cal L}_\tau^{-1}$ is the inverse of the operator |
| 427 |
\begin{equation} |
\begin{equation} |
| 428 |
{\cal L} = \left[ 1 + \Delta t \partial_r \kappa_v \partial_r \right] |
{\cal L}_\tau = \left[ 1 + \Delta t \partial_r \kappa_v \partial_r \right] |
| 429 |
\end{equation} |
\end{equation} |
| 430 |
Equation \ref{eq:taustar-implicit} looks exactly as \ref{eq:taustar} |
Equation \ref{eq:taustar-implicit} looks exactly as \ref{eq:taustar} |
| 431 |
while \ref{eq:tau-n+1-implicit} involves an operator or matrix |
while \ref{eq:tau-n+1-implicit} involves an operator or matrix |
| 445 |
implicit and are thus cast as a an explicit drag term. |
implicit and are thus cast as a an explicit drag term. |
| 446 |
|
|
| 447 |
\section{Synchronous time-stepping: variables co-located in time} |
\section{Synchronous time-stepping: variables co-located in time} |
| 448 |
\label{sect:adams-bashforth-sync} |
\label{sec:adams-bashforth-sync} |
| 449 |
\begin{rawhtml} |
\begin{rawhtml} |
| 450 |
<!-- CMIREDIR:adams_bashforth_sync: --> |
<!-- CMIREDIR:adams_bashforth_sync: --> |
| 451 |
\end{rawhtml} |
\end{rawhtml} |
| 565 |
\ref{fig:call-tree-adams-bashforth-sync}. |
\ref{fig:call-tree-adams-bashforth-sync}. |
| 566 |
|
|
| 567 |
\section{Staggered baroclinic time-stepping} |
\section{Staggered baroclinic time-stepping} |
| 568 |
\label{sect:adams-bashforth-staggered} |
\label{sec:adams-bashforth-staggered} |
| 569 |
\begin{rawhtml} |
\begin{rawhtml} |
| 570 |
<!-- CMIREDIR:adams_bashforth_staggered: --> |
<!-- CMIREDIR:adams_bashforth_staggered: --> |
| 571 |
\end{rawhtml} |
\end{rawhtml} |
| 613 |
allowing the use of the most recent velocities to compute |
allowing the use of the most recent velocities to compute |
| 614 |
the advection terms. Once the thermodynamics fields are |
the advection terms. Once the thermodynamics fields are |
| 615 |
updated, the hydrostatic pressure is computed |
updated, the hydrostatic pressure is computed |
| 616 |
to step forwrad the dynamics. |
to step forward the dynamics. |
| 617 |
Note that the pressure gradient must also be taken out of the |
Note that the pressure gradient must also be taken out of the |
| 618 |
Adams-Bashforth extrapolation. Also, retaining the integer time-levels, |
Adams-Bashforth extrapolation. Also, retaining the integer time-levels, |
| 619 |
$n$ and $n+1$, does not give a user the sense of where variables are |
$n$ and $n+1$, does not give a user the sense of where variables are |
| 710 |
|
|
| 711 |
|
|
| 712 |
\section{Non-hydrostatic formulation} |
\section{Non-hydrostatic formulation} |
| 713 |
\label{sect:non-hydrostatic} |
\label{sec:non-hydrostatic} |
| 714 |
\begin{rawhtml} |
\begin{rawhtml} |
| 715 |
<!-- CMIREDIR:non-hydrostatic_formulation: --> |
<!-- CMIREDIR:non-hydrostatic_formulation: --> |
| 716 |
\end{rawhtml} |
\end{rawhtml} |
| 718 |
The non-hydrostatic formulation re-introduces the full vertical |
The non-hydrostatic formulation re-introduces the full vertical |
| 719 |
momentum equation and requires the solution of a 3-D elliptic |
momentum equation and requires the solution of a 3-D elliptic |
| 720 |
equations for non-hydrostatic pressure perturbation. We still |
equations for non-hydrostatic pressure perturbation. We still |
| 721 |
intergrate vertically for the hydrostatic pressure and solve a 2-D |
integrate vertically for the hydrostatic pressure and solve a 2-D |
| 722 |
elliptic equation for the surface pressure/elevation for this reduces |
elliptic equation for the surface pressure/elevation for this reduces |
| 723 |
the amount of work needed to solve for the non-hydrostatic pressure. |
the amount of work needed to solve for the non-hydrostatic pressure. |
| 724 |
|
|
| 802 |
v^{**} & = & v^{*} - \Delta t g \partial_y \eta^{n+1} \label{eq:vnx-nh}\\ |
v^{**} & = & v^{*} - \Delta t g \partial_y \eta^{n+1} \label{eq:vnx-nh}\\ |
| 803 |
\partial_{xx} \phi_{nh}^{n+1} + \partial_{yy} \phi_{nh}^{n+1} + |
\partial_{xx} \phi_{nh}^{n+1} + \partial_{yy} \phi_{nh}^{n+1} + |
| 804 |
\partial_{rr} \phi_{nh}^{n+1} & = & |
\partial_{rr} \phi_{nh}^{n+1} & = & |
| 805 |
\partial_x u^{**} + \partial_y v^{**} + \partial_r w^{*} \\ |
\partial_x u^{**} + \partial_y v^{**} + \partial_r w^{*} \label{eq:phi-nh}\\ |
| 806 |
u^{n+1} & = & u^{**} - \Delta t \partial_x \phi_{nh}^{n+1} \label{eq:un+1-nh}\\ |
u^{n+1} & = & u^{**} - \Delta t \partial_x \phi_{nh}^{n+1} \label{eq:un+1-nh}\\ |
| 807 |
v^{n+1} & = & v^{**} - \Delta t \partial_y \phi_{nh}^{n+1} \label{eq:vn+1-nh}\\ |
v^{n+1} & = & v^{**} - \Delta t \partial_y \phi_{nh}^{n+1} \label{eq:vn+1-nh}\\ |
| 808 |
\partial_r w^{n+1} & = & - \partial_x u^{n+1} - \partial_y v^{n+1} |
\partial_r w^{n+1} & = & - \partial_x u^{n+1} - \partial_y v^{n+1} |
| 813 |
|
|
| 814 |
|
|
| 815 |
\section{Variants on the Free Surface} |
\section{Variants on the Free Surface} |
| 816 |
\label{sect:free-surface} |
\label{sec:free-surface} |
| 817 |
|
|
| 818 |
We now describe the various formulations of the free-surface that |
We now describe the various formulations of the free-surface that |
| 819 |
include non-linear forms, implicit in time using Crank-Nicholson, |
include non-linear forms, implicit in time using Crank-Nicholson, |
| 926 |
|
|
| 927 |
|
|
| 928 |
\subsection{Crank-Nickelson barotropic time stepping} |
\subsection{Crank-Nickelson barotropic time stepping} |
| 929 |
\label{sect:freesurf-CrankNick} |
\label{sec:freesurf-CrankNick} |
| 930 |
|
|
| 931 |
The full implicit time stepping described previously is |
The full implicit time stepping described previously is |
| 932 |
unconditionally stable but damps the fast gravity waves, resulting in |
unconditionally stable but damps the fast gravity waves, resulting in |
| 950 |
\frac{ \vec{\bf v}^{n+1} }{ \Delta t } |
\frac{ \vec{\bf v}^{n+1} }{ \Delta t } |
| 951 |
+ {\bf \nabla}_h b_s [ \beta {\eta}^{n+1} + (1-\beta) {\eta}^{n} ] |
+ {\bf \nabla}_h b_s [ \beta {\eta}^{n+1} + (1-\beta) {\eta}^{n} ] |
| 952 |
+ \epsilon_{nh} {\bf \nabla}_h {\phi'_{nh}}^{n+1} |
+ \epsilon_{nh} {\bf \nabla}_h {\phi'_{nh}}^{n+1} |
| 953 |
= \frac{ \vec{\bf v}^* }{ \Delta t } |
= \frac{ \vec{\bf v}^{n} }{ \Delta t } |
| 954 |
|
+ \vec{\bf G}_{\vec{\bf v}} ^{(n+1/2)} |
| 955 |
|
+ {\bf \nabla}_h {\phi'_{hyd}}^{(n+1/2)} |
| 956 |
\end{eqnarray*} |
\end{eqnarray*} |
| 957 |
\begin{eqnarray} |
\begin{eqnarray} |
| 958 |
\epsilon_{fs} \frac{ {\eta}^{n+1} - {\eta}^{n} }{ \Delta t} |
\epsilon_{fs} \frac{ {\eta}^{n+1} - {\eta}^{n} }{ \Delta t} |
| 961 |
= \epsilon_{fw} (P-E) |
= \epsilon_{fw} (P-E) |
| 962 |
\label{eq:eta-n+1-CrankNick} |
\label{eq:eta-n+1-CrankNick} |
| 963 |
\end{eqnarray} |
\end{eqnarray} |
| 964 |
where: |
We set |
| 965 |
\begin{eqnarray*} |
\begin{eqnarray*} |
| 966 |
\vec{\bf v}^* & = & |
\vec{\bf v}^* & = & |
| 967 |
\vec{\bf v} ^{n} + \Delta t \vec{\bf G}_{\vec{\bf v}} ^{(n+1/2)} |
\vec{\bf v} ^{n} + \Delta t \vec{\bf G}_{\vec{\bf v}} ^{(n+1/2)} |
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