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% $Header$ |
% $Header$ |
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% $Name$ |
% $Name$ |
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The convention used in this section is as follows: |
This chapter lays out the numerical schemes that are |
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Time is ``discretized'' using a time step $\Delta t$ |
employed in the core MITgcm algorithm. Whenever possible |
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and $\Phi^n$ refers to the variable $\Phi$ |
links are made to actual program code in the MITgcm implementation. |
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at time $t = n \Delta t$ . We use the notation $\Phi^{(n)}$ |
The chapter begins with a discussion of the temporal discretization |
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when time interpolation is required to estimate the value of $\phi$ |
used in MITgcm. This discussion is followed by sections that |
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at the time $n \Delta t$. |
describe the spatial discretization. The schemes employed for momentum |
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terms are described first, afterwards the schemes that apply to |
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\section{Time integration} |
passive and dynamically active tracers are described. |
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The discretization in time of the model equations (cf section I ) |
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does not depend of the discretization in space of each |
\section{Time-stepping} |
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term and so this section can be read independently. |
The equations of motion integrated by the model involve four |
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prognostic equations for flow, $u$ and $v$, temperature, $\theta$, and |
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The continuous form of the model equations is: |
salt/moisture, $S$, and three diagnostic equations for vertical flow, |
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$w$, density/buoyancy, $\rho$/$b$, and pressure/geo-potential, |
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$\phi_{hyd}$. In addition, the surface pressure or height may by |
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described by either a prognostic or diagnostic equation and if |
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non-hydrostatics terms are included then a diagnostic equation for |
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non-hydrostatic pressure is also solved. The combination of prognostic |
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and diagnostic equations requires a model algorithm that can march |
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forward prognostic variables while satisfying constraints imposed by |
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diagnostic equations. |
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Since the model comes in several flavors and formulation, it would be |
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confusing to present the model algorithm exactly as written into code |
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along with all the switches and optional terms. Instead, we present |
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the algorithm for each of the basic formulations which are: |
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\begin{enumerate} |
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\item the semi-implicit pressure method for hydrostatic equations |
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with a rigid-lid, variables co-located in time and with |
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Adams-Bashforth time-stepping, \label{it:a} |
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\item as \ref{it:a}. but with an implicit linear free-surface, \label{it:b} |
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\item as \ref{it:a}. or \ref{it:b}. but with variables staggered in time, |
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\label{it:c} |
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\item as \ref{it:a}. or \ref{it:b}. but with non-hydrostatic terms included, |
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\item as \ref{it:b}. or \ref{it:c}. but with non-linear free-surface. |
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\end{enumerate} |
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In all the above configurations it is also possible to substitute the |
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Adams-Bashforth with an alternative time-stepping scheme for terms |
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evaluated explicitly in time. Since the over-arching algorithm is |
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independent of the particular time-stepping scheme chosen we will |
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describe first the over-arching algorithm, known as the pressure |
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method, with a rigid-lid model in section |
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\ref{sect:pressure-method-rigid-lid}. This algorithm is essentially |
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unchanged, apart for some coefficients, when the rigid lid assumption |
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is replaced with a linearized implicit free-surface, described in |
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section \ref{sect:pressure-method-linear-backward}. These two flavors |
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of the pressure-method encompass all formulations of the model as it |
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exists today. The integration of explicit in time terms is out-lined |
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in section \ref{sect:adams-bashforth} and put into the context of the |
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overall algorithm in sections \ref{sect:adams-bashforth-sync} and |
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\ref{sect:adams-bashforth-staggered}. Inclusion of non-hydrostatic |
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terms requires applying the pressure method in three dimensions |
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instead of two and this algorithm modification is described in section |
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\ref{sect:non-hydrostatic}. Finally, the free-surface equation may be |
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treated more exactly, including non-linear terms, and this is |
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described in section \ref{sect:nonlinear-freesurface}. |
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\section{Pressure method with rigid-lid} \label{sect:pressure-method-rigid-lid} |
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\begin{figure} |
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\begin{center} |
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\resizebox{4.0in}{!}{\includegraphics{part2/pressure-method-rigid-lid.eps}} |
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\end{center} |
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\caption{ |
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A schematic of the evolution in time of the pressure method |
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algorithm. A prediction for the flow variables at time level $n+1$ is |
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made based only on the explicit terms, $G^{(n+^1/_2)}$, and denoted |
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$u^*$, $v^*$. Next, a pressure field is found such that $u^{n+1}$, |
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$v^{n+1}$ will be non-divergent. Conceptually, the $*$ quantities |
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exist at time level $n+1$ but they are intermediate and only |
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temporary.} |
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\label{fig:pressure-method-rigid-lid} |
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\end{figure} |
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\begin{figure} |
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\begin{center} \fbox{ \begin{minipage}{4.5in} \begin{tabbing} |
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aaa \= aaa \= aaa \= aaa \= aaa \= aaa \kill |
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\filelink{FORWARD\_STEP}{model-src-forward_step.F} \\ |
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\> DYNAMICS \\ |
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\>\> TIMESTEP \` $u^*$,$v^*$ (\ref{eq:ustar-rigid-lid},\ref{eq:vstar-rigid-lid}) \\ |
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\> SOLVE\_FOR\_PRESSURE \\ |
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\>\> CALC\_DIV\_GHAT \` $H\widehat{u^*}$,$H\widehat{v^*}$ (\ref{eq:elliptic}) \\ |
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\>\> CG2D \` $\eta^{n+1}$ (\ref{eq:elliptic}) \\ |
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\> THE\_CORRECTION\_STEP \\ |
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\>\> CALC\_GRAD\_PHI\_SURF \` $\nabla \eta^{n+1}$ \\ |
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\>\> CORRECTION\_STEP \` $u^{n+1}$,$v^{n+1}$ (\ref{eq:un+1-rigid-lid},\ref{eq:vn+1-rigid-lid}) |
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\end{tabbing} \end{minipage} } \end{center} |
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\caption{Calling tree for the pressure method algorithm |
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(\filelink{FORWARD\_STEP}{model-src-forward_step.F})} |
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\label{fig:call-tree-pressure-method} |
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\end{figure} |
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The horizontal momentum and continuity equations for the ocean |
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(\ref{eq:ocean-mom} and \ref{eq:ocean-cont}), or for the atmosphere |
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(\ref{eq:atmos-mom} and \ref{eq:atmos-cont}), can be summarized by: |
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\begin{eqnarray} |
\begin{eqnarray} |
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\partial_t \theta & = & G_\theta |
\partial_t u + g \partial_x \eta & = & G_u \\ |
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\label{eq-tCsC-theta} |
\partial_t v + g \partial_y \eta & = & G_v \\ |
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\\ |
\partial_x u + \partial_y v + \partial_z w & = & 0 |
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\partial_t S & = & G_s |
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\label{eq-tCsC-salt} |
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\\ |
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b' & = & b'(\theta,S,r) |
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\\ |
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\partial_r \phi'_{hyd} & = & -b' |
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\label{eq-tCsC-hyd} |
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\\ |
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\partial_t \vec{\bf v} |
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+ {\bf \nabla}_h b_s \eta |
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+ \epsilon_{nh} {\bf \nabla}_h \phi'_{nh} |
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& = & \vec{\bf G}_{\vec{\bf v}} |
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- {\bf \nabla}_h \phi'_{hyd} |
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\label{eq-tCsC-Hmom} |
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\\ |
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\epsilon_{nh} \frac {\partial{\dot{r}}}{\partial{t}} |
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+ \epsilon_{nh} \partial_r \phi'_{nh} |
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& = & \epsilon_{nh} G_{\dot{r}} |
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\label{eq-tCsC-Vmom} |
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\\ |
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{\bf \nabla}_h \cdot \vec{\bf v} + \partial_r \dot{r} |
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& = & 0 |
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\label{eq-tCsC-cont} |
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\end{eqnarray} |
\end{eqnarray} |
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where |
where we are adopting the oceanic notation for brevity. All terms in |
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\begin{eqnarray*} |
the momentum equations, except for surface pressure gradient, are |
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G_\theta & = & |
encapsulated in the $G$ vector. The continuity equation, when |
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- \vec{\bf v} \cdot {\bf \nabla} \theta + {\cal Q}_\theta |
integrated over the fluid depth, $H$, and with the rigid-lid/no normal |
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\\ |
flow boundary conditions applied, becomes: |
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G_S & = & |
\begin{equation} |
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- \vec{\bf v} \cdot {\bf \nabla} S + {\cal Q}_S |
\partial_x H \widehat{u} + \partial_y H \widehat{v} = 0 |
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\\ |
\label{eq:rigid-lid-continuity} |
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\vec{\bf G}_{\vec{\bf v}} |
\end{equation} |
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& = & |
Here, $H\widehat{u} = \int_H u dz$ is the depth integral of $u$, |
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- \vec{\bf v} \cdot {\bf \nabla} \vec{\bf v} |
similarly for $H\widehat{v}$. The rigid-lid approximation sets $w=0$ |
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- f \hat{\bf k} \wedge \vec{\bf v} |
at the lid so that it does not move but allows a pressure to be |
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+ \vec{\cal F}_{\vec{\bf v}} |
exerted on the fluid by the lid. The horizontal momentum equations and |
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\\ |
vertically integrated continuity equation are be discretized in time |
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G_{\dot{r}} |
and space as follows: |
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\begin{eqnarray} |
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u^{n+1} + \Delta t g \partial_x \eta^{n+1} |
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& = & u^{n} + \Delta t G_u^{(n+1/2)} |
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\label{eq:discrete-time-u} |
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\\ |
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v^{n+1} + \Delta t g \partial_y \eta^{n+1} |
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& = & v^{n} + \Delta t G_v^{(n+1/2)} |
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\label{eq:discrete-time-v} |
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\\ |
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\partial_x H \widehat{u^{n+1}} |
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+ \partial_y H \widehat{v^{n+1}} & = & 0 |
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\label{eq:discrete-time-cont-rigid-lid} |
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\end{eqnarray} |
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As written here, terms on the LHS all involve time level $n+1$ and are |
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referred to as implicit; the implicit backward time stepping scheme is |
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being used. All other terms in the RHS are explicit in time. The |
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thermodynamic quantities are integrated forward in time in parallel |
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with the flow and will be discussed later. For the purposes of |
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describing the pressure method it suffices to say that the hydrostatic |
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pressure gradient is explicit and so can be included in the vector |
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$G$. |
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Substituting the two momentum equations into the depth integrated |
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continuity equation eliminates $u^{n+1}$ and $v^{n+1}$ yielding an |
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elliptic equation for $\eta^{n+1}$. Equations |
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\ref{eq:discrete-time-u}, \ref{eq:discrete-time-v} and |
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\ref{eq:discrete-time-cont-rigid-lid} can then be re-arranged as follows: |
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\begin{eqnarray} |
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u^{*} & = & u^{n} + \Delta t G_u^{(n+1/2)} \label{eq:ustar-rigid-lid} \\ |
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v^{*} & = & v^{n} + \Delta t G_v^{(n+1/2)} \label{eq:vstar-rigid-lid} \\ |
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\partial_x \Delta t g H \partial_x \eta^{n+1} |
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+ \partial_y \Delta t g H \partial_y \eta^{n+1} |
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& = & |
& = & |
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- \vec{\bf v} \cdot {\bf \nabla} \dot{r} |
\partial_x H \widehat{u^{*}} |
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+ {\cal F}_{\dot{r}} |
+ \partial_y H \widehat{v^{*}} \label{eq:elliptic} |
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\end{eqnarray*} |
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The exact form of all the ``{\it G}''s terms is described in the next |
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section \ref{sect:discrete}. Here its sufficient to mention that they contains |
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all the RHS terms except the pressure/geo-potential terms. |
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The switch $\epsilon_{nh}$ allows one to activate the non-hydrostatic |
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mode ($\epsilon_{nh}=1$) for the ocean model. Otherwise, in the |
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hydrostatic limit $\epsilon_{nh} = 0$ and equation \ref{eq-tCsC-Vmom} |
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is not used. |
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As discussed in section \ref{sect:1.3.6.2}, the equation for $\eta$ is |
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obtained by integrating the continuity equation over the entire depth |
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of the fluid, from $R_{fixed}(x,y)$ up to $R_o(x,y)$. The linear free |
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surface evolves according to: |
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\begin{eqnarray} |
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\epsilon_{fs} \partial_t \eta = |
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\left. \dot{r} \right|_{r=r_{surf}} + \epsilon_{fw} (P-E) = |
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- {\bf \nabla} \cdot \int_{R_{fixed}}^{R_o} \vec{\bf v} dr |
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+ \epsilon_{fw} (P-E) |
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\label{eq-tCsC-eta} |
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\end{eqnarray} |
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Here, $\epsilon_{fs}$ is a flag to switch between the free-surface, |
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$\epsilon_{fs}=1$, and a rigid-lid, $\epsilon_{fs}=0$. The flag |
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$\epsilon_{fw}$ determines whether an exchange of fresh water is |
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included at the ocean surface (natural BC) ($\epsilon_{fw} = 1$) or |
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not ($\epsilon_{fw} = 0$). |
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The hydrostatic potential is found by integrating (equation |
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\ref{eq-tCsC-hyd}) with the boundary condition that |
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$\phi'_{hyd}(r=R_o) = 0$: |
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\begin{eqnarray*} |
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& & |
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\int_{r'}^{R_o} \partial_r \phi'_{hyd} dr = |
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\left[ \phi'_{hyd} \right]_{r'}^{R_o} = |
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\int_{r'}^{R_o} - b' dr |
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\\ |
\\ |
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\Rightarrow & & |
u^{n+1} & = & u^{*} - \Delta t g \partial_x \eta^{n+1} \label{eq:un+1-rigid-lid}\\ |
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\phi'_{hyd}(x,y,r') = \int_{r'}^{R_o} b' dr |
v^{n+1} & = & v^{*} - \Delta t g \partial_y \eta^{n+1} \label{eq:vn+1-rigid-lid} |
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\end{eqnarray*} |
\end{eqnarray} |
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Equations \ref{eq:ustar-rigid-lid} to \ref{eq:vn+1-rigid-lid}, solved |
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\subsection{General method} |
sequentially, represent the pressure method algorithm used in the |
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model. The essence of the pressure method lies in the fact that any |
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An overview of the general method is now presented with explicit |
explicit prediction for the flow would lead to a divergence flow field |
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references to the Fortran code. We often refer to the discretized |
so a pressure field must be found that keeps the flow non-divergent |
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equations of the model that are detailed in the following sections. |
over each step of the integration. The particular location in time of |
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the pressure field is somewhat ambiguous; in |
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The general algorithm consist of a ``predictor step'' that computes |
Fig.~\ref{fig:pressure-method-rigid-lid} we depicted as co-located |
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the forward tendencies ("G" terms") comprising of explicit-in-time |
with the future flow field (time level $n+1$) but it could equally |
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terms and the ``first guess'' values (star notation): $\theta^*, S^*, |
have been drawn as staggered in time with the flow. |
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\vec{\bf v}^*$ (and $\dot{r}^*$ in non-hydrostatic mode). This is done |
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in the two routines {\it THERMODYNAMICS} and {\it DYNAMICS}. |
The correspondence to the code is as follows: |
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\begin{itemize} |
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Terms that are integrated implicitly in time are handled at the end of |
\item |
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the {\it THERMODYNAMICS} and {\it DYNAMICS} routines. Then the |
the prognostic phase, equations \ref{eq:ustar-rigid-lid} and \ref{eq:vstar-rigid-lid}, |
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surface pressure and non hydrostatic pressure are solved for in ({\it |
stepping forward $u^n$ and $v^n$ to $u^{*}$ and $v^{*}$ is coded in |
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SOLVE\_FOR\_PRESSURE}). |
\filelink{TIMESTEP()}{model-src-timestep.F} |
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\item |
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Finally, in the ``corrector step'', (routine {\it |
the vertical integration, $H \widehat{u^*}$ and $H |
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THE\_CORRECTION\_STEP}) the new values of $u,v,w,\theta,S$ are |
\widehat{v^*}$, divergence and inversion of the elliptic operator in |
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determined (see details in \ref{sect:II.1.3}). |
equation \ref{eq:elliptic} is coded in |
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\filelink{SOLVE\_FOR\_PRESSURE()}{model-src-solve_for_pressure.F} |
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At this point, the regular time stepping process is complete. However, |
\item |
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after the correction step there are optional adjustments such as |
finally, the new flow field at time level $n+1$ given by equations |
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convective adjustment or filters (zonal FFT filter, shapiro filter) |
\ref{eq:un+1-rigid-lid} and \ref{eq:vn+1-rigid-lid} is calculated in |
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that can be applied on both momentum and tracer fields, just prior to |
\filelink{CORRECTION\_STEP()}{model-src-correction_step.F}. |
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incrementing the time level to $n+1$. |
\end{itemize} |
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The calling tree for these routines is given in |
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Since the pressure solver precision is of the order of the ``target |
Fig.~\ref{fig:call-tree-pressure-method}. |
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residual'' and can be lower than the the computer truncation error, |
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and also because some filters might alter the divergence part of the |
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flow field, a final evaluation of the surface r anomaly $\eta^{n+1}$ |
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is performed in {\it CALC\_EXACT\_ETA}. This ensures exact volume |
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conservation. Note that there is no need for an equivalent |
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non-hydrostatic ``exact conservation'' step, since by default $w$ is |
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already computed after the filters are applied. |
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Optional forcing terms (usually part of a physics ``package''), that |
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account for a specific source or sink process (e.g. condensation as a |
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sink of water vapor Q) are generally incorporated in the main |
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algorithm as follows: at the the beginning of the time step, the |
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additional tendencies are computed as a function of the present state |
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(time step $n$) and external forcing; then within the main part of |
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model, only those new tendencies are added to the model variables. |
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[more details needed]\\ |
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The atmospheric physics follows this general scheme. |
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[more about C\_grid, A\_grid conversion \& drag term]\\ |
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\subsection{Standard synchronous time stepping} |
\paragraph{Need to discuss implicit viscosity somewhere:} |
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In the standard formulation, the surface pressure is evaluated at time |
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step n+1 (an implicit method). Equations \ref{eq-tCsC-theta} to |
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\ref{eq-tCsC-cont} are then discretized in time as follows: |
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\begin{eqnarray} |
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\left[ 1 - \partial_r \kappa_v^\theta \partial_r \right] |
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\theta^{n+1} & = & \theta^* |
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\label{eq-tDsC-theta} |
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\\ |
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\left[ 1 - \partial_r \kappa_v^S \partial_r \right] |
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S^{n+1} & = & S^* |
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\label{eq-tDsC-salt} |
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\\ |
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%{b'}^{n} & = & b'(\theta^{n},S^{n},r) |
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%\partial_r {\phi'_{hyd}}^{n} & = & {-b'}^{n} |
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%\\ |
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{\phi'_{hyd}}^{n} & = & \int_{r'}^{R_o} b'(\theta^{n},S^{n},r) dr |
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\label{eq-tDsC-hyd} |
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\\ |
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\vec{\bf v} ^{n+1} |
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+ \Delta t {\bf \nabla}_h b_s {\eta}^{n+1} |
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+ \epsilon_{nh} \Delta t {\bf \nabla} {\phi'_{nh}}^{n+1} |
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- \partial_r A_v \partial_r \vec{\bf v}^{n+1} |
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& = & |
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\vec{\bf v}^* |
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\label{eq-tDsC-Hmom} |
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\\ |
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\epsilon_{fs} {\eta}^{n+1} + \Delta t |
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{\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o} \vec{\bf v}^{n+1} dr |
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& = & |
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\epsilon_{fs} {\eta}^{n} + \epsilon_{fw} \Delta_t (P-E)^{n} |
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\nonumber |
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\\ |
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% = \epsilon_{fs} {\eta}^{n} & + & \epsilon_{fw} \Delta_t (P-E)^{n} |
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\label{eq-tDsC-eta} |
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\\ |
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\epsilon_{nh} \left( \dot{r} ^{n+1} |
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+ \Delta t \partial_r {\phi'_{nh}} ^{n+1} |
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\right) |
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& = & \epsilon_{nh} \dot{r}^* |
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\label{eq-tDsC-Vmom} |
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\\ |
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{\bf \nabla}_h \cdot \vec{\bf v}^{n+1} + \partial_r \dot{r}^{n+1} |
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& = & 0 |
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\label{eq-tDsC-cont} |
|
|
\end{eqnarray} |
|
|
where |
|
| 194 |
\begin{eqnarray} |
\begin{eqnarray} |
| 195 |
\theta^* & = & |
\frac{1}{\Delta t} u^{n+1} - \partial_z A_v \partial_z u^{n+1} |
| 196 |
\theta ^{n} + \Delta t G_{\theta} ^{(n+1/2)} |
+ g \partial_x \eta^{n+1} & = & \frac{1}{\Delta t} u^{n} + |
| 197 |
\\ |
G_u^{(n+1/2)} |
|
S^* & = & |
|
|
S ^{n} + \Delta t G_{S} ^{(n+1/2)} |
|
| 198 |
\\ |
\\ |
| 199 |
\vec{\bf v}^* & = & |
\frac{1}{\Delta t} v^{n+1} - \partial_z A_v \partial_z v^{n+1} |
| 200 |
\vec{\bf v}^{n} + \Delta t \vec{\bf G}_{\vec{\bf v}} ^{(n+1/2)} |
+ g \partial_y \eta^{n+1} & = & \frac{1}{\Delta t} v^{n} + G_v^{(n+1/2)} |
|
+ \Delta t {\bf \nabla}_h {\phi'_{hyd}}^{(n+1/2)} |
|
|
\\ |
|
|
\dot{r}^* & = & |
|
|
\dot{r} ^{n} + \Delta t G_{\dot{r}} ^{(n+1/2)} |
|
| 201 |
\end{eqnarray} |
\end{eqnarray} |
| 202 |
|
|
|
Note that implicit vertical viscosity and diffusivity terms are not |
|
|
considered as part of the ``{\it G}'' terms, but are written |
|
|
separately here. |
|
|
|
|
|
The default time-stepping method is the Adams-Bashforth quasi-second |
|
|
order scheme in which the ``G'' terms are extrapolated forward in time |
|
|
from time-levels $n-1$ and $n$ to time-level $n+1/2$ and provides a |
|
|
simple but stable algorithm: |
|
|
\begin{equation} |
|
|
G^{(n+1/2)} = G^n + (1/2+\epsilon_{AB}) (G^n - G^{n-1}) |
|
|
\end{equation} |
|
|
where $\epsilon_{AB}$ is a small number used to stabilize the time |
|
|
stepping. |
|
|
|
|
|
In the standard non-staggered formulation, the Adams-Bashforth time |
|
|
stepping is also applied to the hydrostatic pressure/geo-potential |
|
|
gradient, $\nabla_h \Phi'_{hyd}$. Note that presently, this term is in |
|
|
fact incorporated to the $\vec{\bf G}_{\vec{\bf v}}$ arrays ({\bf |
|
|
gU,gV}). |
|
|
\marginpar{JMC: Clarify ``this term''?} |
|
|
|
|
|
\fbox{ \begin{minipage}{4.75in} |
|
|
{\em S/R TIMESTEP} ({\em timestep.F}) |
|
|
|
|
|
$G_u^n$: {\bf Gu} ({\em DYNVARS.h}) |
|
|
|
|
|
$G_u^{n-1}, u^*$: {\bf GuNm1} ({\em DYNVARS.h}) |
|
|
|
|
|
$G_v^n$: {\bf Gv} ({\em DYNVARS.h}) |
|
|
|
|
|
$G_v^{n-1}, v^*$: {\bf GvNm1} ({\em DYNVARS.h}) |
|
|
|
|
|
$G_u^{(n+1/2)}$: {\bf GuTmp} (local) |
|
|
|
|
|
$G_v^{(n+1/2)}$: {\bf GvTmp} (local) |
|
|
|
|
|
\end{minipage} } |
|
| 203 |
|
|
| 204 |
|
\section{Pressure method with implicit linear free-surface} |
| 205 |
|
\label{sect:pressure-method-linear-backward} |
| 206 |
|
|
| 207 |
|
The rigid-lid approximation filters out external gravity waves |
| 208 |
|
subsequently modifying the dispersion relation of barotropic Rossby |
| 209 |
|
waves. The discrete form of the elliptic equation has some zero |
| 210 |
|
eigen-values which makes it a potentially tricky or inefficient |
| 211 |
|
problem to solve. |
| 212 |
|
|
| 213 |
|
The rigid-lid approximation can be easily replaced by a linearization |
| 214 |
|
of the free-surface equation which can be written: |
| 215 |
|
\begin{equation} |
| 216 |
|
\partial_t \eta + \partial_x H \widehat{u} + \partial_y H \widehat{v} = P-E+R |
| 217 |
|
\label{eq:linear-free-surface=P-E+R} |
| 218 |
|
\end{equation} |
| 219 |
|
which differs from the depth integrated continuity equation with |
| 220 |
|
rigid-lid (\ref{eq:rigid-lid-continuity}) by the time-dependent term |
| 221 |
|
and fresh-water source term. |
| 222 |
|
|
| 223 |
|
Equation \ref{eq:discrete-time-cont-rigid-lid} in the rigid-lid |
| 224 |
|
pressure method is then replaced by the time discretization of |
| 225 |
|
\ref{eq:linear-free-surface=P-E+R} which is: |
| 226 |
|
\begin{equation} |
| 227 |
|
\eta^{n+1} |
| 228 |
|
+ \Delta t \partial_x H \widehat{u^{n+1}} |
| 229 |
|
+ \Delta t \partial_y H \widehat{v^{n+1}} |
| 230 |
|
= |
| 231 |
|
\eta^{n} |
| 232 |
|
+ \Delta t ( P - E + R ) |
| 233 |
|
\label{eq:discrete-time-backward-free-surface} |
| 234 |
|
\end{equation} |
| 235 |
|
where the use of flow at time level $n+1$ makes the method implicit |
| 236 |
|
and backward in time. The is the preferred scheme since it still |
| 237 |
|
filters the fast unresolved wave motions by damping them. A centered |
| 238 |
|
scheme, such as Crank-Nicholson, would alias the energy of the fast |
| 239 |
|
modes onto slower modes of motion. |
| 240 |
|
|
| 241 |
|
As for the rigid-lid pressure method, equations |
| 242 |
|
\ref{eq:discrete-time-u}, \ref{eq:discrete-time-v} and |
| 243 |
|
\ref{eq:discrete-time-backward-free-surface} can be re-arranged as follows: |
| 244 |
|
\begin{eqnarray} |
| 245 |
|
u^{*} & = & u^{n} + \Delta t G_u^{(n+1/2)} \label{eq:ustar-backward-free-surface} \\ |
| 246 |
|
v^{*} & = & v^{n} + \Delta t G_v^{(n+1/2)} \label{eq:vstar-backward-free-surface} \\ |
| 247 |
|
\eta^* & = & \epsilon_{fs} \left( \eta^{n} +P-E+R \right)- \Delta t |
| 248 |
|
\partial_x H \widehat{u^{*}} |
| 249 |
|
+ \partial_y H \widehat{v^{*}} |
| 250 |
|
\\ |
| 251 |
|
\partial_x g H \partial_x \eta^{n+1} |
| 252 |
|
+ \partial_y g H \partial_y \eta^{n+1} |
| 253 |
|
- \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2} |
| 254 |
|
& = & |
| 255 |
|
- \frac{\eta^*}{\Delta t^2} |
| 256 |
|
\label{eq:elliptic-backward-free-surface} |
| 257 |
|
\\ |
| 258 |
|
u^{n+1} & = & u^{*} - \Delta t g \partial_x \eta^{n+1} \label{eq:un+1-backward-free-surface}\\ |
| 259 |
|
v^{n+1} & = & v^{*} - \Delta t g \partial_y \eta^{n+1} \label{eq:vn+1-backward-free-surface} |
| 260 |
|
\end{eqnarray} |
| 261 |
|
Equations~\ref{eq:ustar-backward-free-surface} |
| 262 |
|
to~\ref{eq:vn+1-backward-free-surface}, solved sequentially, represent |
| 263 |
|
the pressure method algorithm with a backward implicit, linearized |
| 264 |
|
free surface. The method is still formerly a pressure method because |
| 265 |
|
in the limit of large $\Delta t$ the rigid-lid method is |
| 266 |
|
recovered. However, the implicit treatment of the free-surface allows |
| 267 |
|
the flow to be divergent and for the surface pressure/elevation to |
| 268 |
|
respond on a finite time-scale (as opposed to instantly). To recover |
| 269 |
|
the rigid-lid formulation, we introduced a switch-like parameter, |
| 270 |
|
$\epsilon_{fs}$, which selects between the free-surface and rigid-lid; |
| 271 |
|
$\epsilon_{fs}=1$ allows the free-surface to evolve; $\epsilon_{fs}=0$ |
| 272 |
|
imposes the rigid-lid. The evolution in time and location of variables |
| 273 |
|
is exactly as it was for the rigid-lid model so that |
| 274 |
|
Fig.~\ref{fig:pressure-method-rigid-lid} is still |
| 275 |
|
applicable. Similarly, the calling sequence, given in |
| 276 |
|
Fig.~\ref{fig:call-tree-pressure-method}, is as for the |
| 277 |
|
pressure-method. |
| 278 |
|
|
| 279 |
|
|
| 280 |
|
\section{Explicit time-stepping: Adams-Bashforth} |
| 281 |
|
\label{sect:adams-bashforth} |
| 282 |
|
|
| 283 |
|
In describing the the pressure method above we deferred describing the |
| 284 |
|
time discretization of the explicit terms. We have historically used |
| 285 |
|
the quasi-second order Adams-Bashforth method for all explicit terms |
| 286 |
|
in both the momentum and tracer equations. This is still the default |
| 287 |
|
mode of operation but it is now possible to use alternate schemes for |
| 288 |
|
tracers (see section \ref{sect:tracer-advection}). |
| 289 |
|
|
| 290 |
|
\begin{figure} |
| 291 |
|
\begin{center} \fbox{ \begin{minipage}{4.5in} \begin{tabbing} |
| 292 |
|
aaa \= aaa \= aaa \= aaa \= aaa \= aaa \kill |
| 293 |
|
FORWARD\_STEP \\ |
| 294 |
|
\> THERMODYNAMICS \\ |
| 295 |
|
\>\> CALC\_GT \\ |
| 296 |
|
\>\>\> GAD\_CALC\_RHS \` $G_\theta^n = G_\theta( u, \theta^n )$ \\ |
| 297 |
|
\>\>\> EXTERNAL\_FORCING \` $G_\theta^n = G_\theta^n + {\cal Q}$ \\ |
| 298 |
|
\>\>\> ADAMS\_BASHFORTH2 \` $G_\theta^{(n+1/2)}$ (\ref{eq:adams-bashforth2}) \\ |
| 299 |
|
\>\> TIMESTEP\_TRACER \` $\tau^*$ (\ref{eq:taustar}) \\ |
| 300 |
|
\>\> IMPLDIFF \` $\tau^{(n+1)}$ (\ref{eq:tau-n+1-implicit}) |
| 301 |
|
\end{tabbing} \end{minipage} } \end{center} |
| 302 |
|
\caption{ |
| 303 |
|
Calling tree for the Adams-Bashforth time-stepping of temperature with |
| 304 |
|
implicit diffusion.} |
| 305 |
|
\label{fig:call-tree-adams-bashforth} |
| 306 |
|
\end{figure} |
| 307 |
|
|
| 308 |
|
In the previous sections, we summarized an explicit scheme as: |
| 309 |
|
\begin{equation} |
| 310 |
|
\tau^{*} = \tau^{n} + \Delta t G_\tau^{(n+1/2)} |
| 311 |
|
\label{eq:taustar} |
| 312 |
|
\end{equation} |
| 313 |
|
where $\tau$ could be any prognostic variable ($u$, $v$, $\theta$ or |
| 314 |
|
$S$) and $\tau^*$ is an explicit estimate of $\tau^{n+1}$ and would be |
| 315 |
|
exact if not for implicit-in-time terms. The parenthesis about $n+1/2$ |
| 316 |
|
indicates that the term is explicit and extrapolated forward in time |
| 317 |
|
and for this we use the quasi-second order Adams-Bashforth method: |
| 318 |
|
\begin{equation} |
| 319 |
|
G_\tau^{(n+1/2)} = ( 3/2 + \epsilon_{AB}) G_\tau^n |
| 320 |
|
- ( 1/2 + \epsilon_{AB}) G_\tau^{n-1} |
| 321 |
|
\label{eq:adams-bashforth2} |
| 322 |
|
\end{equation} |
| 323 |
|
This is a linear extrapolation, forward in time, to |
| 324 |
|
$t=(n+1/2+{\epsilon_{AB}})\Delta t$. An extrapolation to the mid-point |
| 325 |
|
in time, $t=(n+1/2)\Delta t$, corresponding to $\epsilon_{AB}=0$, |
| 326 |
|
would be second order accurate but is weakly unstable for oscillatory |
| 327 |
|
terms. A small but finite value for $\epsilon_{AB}$ stabilizes the |
| 328 |
|
method. Strictly speaking, damping terms such as diffusion and |
| 329 |
|
dissipation, and fixed terms (forcing), do not need to be inside the |
| 330 |
|
Adams-Bashforth extrapolation. However, in the current code, it is |
| 331 |
|
simpler to include these terms and this can be justified if the flow |
| 332 |
|
and forcing evolves smoothly. Problems can, and do, arise when forcing |
| 333 |
|
or motions are high frequency and this corresponds to a reduced |
| 334 |
|
stability compared to a simple forward time-stepping of such terms. |
| 335 |
|
|
| 336 |
|
A stability analysis for an oscillation equation should be given at this point. |
| 337 |
|
\marginpar{AJA needs to find his notes on this...} |
| 338 |
|
|
| 339 |
|
A stability analysis for a relaxation equation should be given at this point. |
| 340 |
|
\marginpar{...and for this too.} |
| 341 |
|
|
| 342 |
|
|
| 343 |
|
\section{Implicit time-stepping: backward method} |
| 344 |
|
|
| 345 |
|
Vertical diffusion and viscosity can be treated implicitly in time |
| 346 |
|
using the backward method which is an intrinsic scheme. For tracers, |
| 347 |
|
the time discretized equation is: |
| 348 |
|
\begin{equation} |
| 349 |
|
\tau^{n+1} - \Delta t \partial_r \kappa_v \partial_r \tau^{n+1} = |
| 350 |
|
\tau^{n} + \Delta t G_\tau^{(n+1/2)} |
| 351 |
|
\label{eq:implicit-diffusion} |
| 352 |
|
\end{equation} |
| 353 |
|
where $G_\tau^{(n+1/2)}$ is the remaining explicit terms extrapolated |
| 354 |
|
using the Adams-Bashforth method as described above. Equation |
| 355 |
|
\ref{eq:implicit-diffusion} can be split split into: |
| 356 |
|
\begin{eqnarray} |
| 357 |
|
\tau^* & = & \tau^{n} + \Delta t G_\tau^{(n+1/2)} |
| 358 |
|
\label{eq:taustar-implicit} \\ |
| 359 |
|
\tau^{n+1} & = & {\cal L}_\tau^{-1} ( \tau^* ) |
| 360 |
|
\label{eq:tau-n+1-implicit} |
| 361 |
|
\end{eqnarray} |
| 362 |
|
where ${\cal L}_\tau^{-1}$ is the inverse of the operator |
| 363 |
|
\begin{equation} |
| 364 |
|
{\cal L} = \left[ 1 + \Delta t \partial_r \kappa_v \partial_r \right] |
| 365 |
|
\end{equation} |
| 366 |
|
Equation \ref{eq:taustar-implicit} looks exactly as \ref{eq:taustar} |
| 367 |
|
while \ref{eq:tau-n+1-implicit} involves an operator or matrix |
| 368 |
|
inversion. By re-arranging \ref{eq:implicit-diffusion} in this way we |
| 369 |
|
have cast the method as an explicit prediction step and an implicit |
| 370 |
|
step allowing the latter to be inserted into the over all algorithm |
| 371 |
|
with minimal interference. |
| 372 |
|
|
| 373 |
|
Fig.~\ref{fig:call-tree-adams-bashforth} shows the calling sequence for |
| 374 |
|
stepping forward a tracer variable such as temperature. |
| 375 |
|
|
| 376 |
|
In order to fit within the pressure method, the implicit viscosity |
| 377 |
|
must not alter the barotropic flow. In other words, it can on ly |
| 378 |
|
redistribute momentum in the vertical. The upshot of this is that |
| 379 |
|
although vertical viscosity may be backward implicit and |
| 380 |
|
unconditionally stable, no-slip boundary conditions may not be made |
| 381 |
|
implicit and are thus cast as a an explicit drag term. |
| 382 |
|
|
| 383 |
|
\section{Synchronous time-stepping: variables co-located in time} |
| 384 |
|
\label{sect:adams-bashforth-sync} |
| 385 |
|
|
| 386 |
|
\begin{figure} |
| 387 |
|
\begin{center} |
| 388 |
|
\resizebox{5.0in}{!}{\includegraphics{part2/adams-bashforth-sync.eps}} |
| 389 |
|
\end{center} |
| 390 |
|
\caption{ |
| 391 |
|
A schematic of the explicit Adams-Bashforth and implicit time-stepping |
| 392 |
|
phases of the algorithm. All prognostic variables are co-located in |
| 393 |
|
time. Explicit tendencies are evaluated at time level $n$ as a |
| 394 |
|
function of the state at that time level (dotted arrow). The explicit |
| 395 |
|
tendency from the previous time level, $n-1$, is used to extrapolate |
| 396 |
|
tendencies to $n+1/2$ (dashed arrow). This extrapolated tendency |
| 397 |
|
allows variables to be stably integrated forward-in-time to render an |
| 398 |
|
estimate ($*$-variables) at the $n+1$ time level (solid |
| 399 |
|
arc-arrow). The operator ${\cal L}$ formed from implicit-in-time terms |
| 400 |
|
is solved to yield the state variables at time level $n+1$. } |
| 401 |
|
\label{fig:adams-bashforth-sync} |
| 402 |
|
\end{figure} |
| 403 |
|
|
| 404 |
|
\begin{figure} |
| 405 |
|
\begin{center} \fbox{ \begin{minipage}{4.5in} \begin{tabbing} |
| 406 |
|
aaa \= aaa \= aaa \= aaa \= aaa \= aaa \kill |
| 407 |
|
FORWARD\_STEP \\ |
| 408 |
|
\> THERMODYNAMICS \\ |
| 409 |
|
\>\> CALC\_GT \\ |
| 410 |
|
\>\>\> GAD\_CALC\_RHS \` $G_\theta^n = G_\theta( u, \theta^n )$ (\ref{eq:Gt-n-sync})\\ |
| 411 |
|
\>\>\> EXTERNAL\_FORCING \` $G_\theta^n = G_\theta^n + {\cal Q}$ \\ |
| 412 |
|
\>\>\> ADAMS\_BASHFORTH2 \` $G_\theta^{(n+1/2)}$ (\ref{eq:Gt-n+5-sync}) \\ |
| 413 |
|
\>\> TIMESTEP\_TRACER \` $\tau^*$ (\ref{eq:tstar-sync}) \\ |
| 414 |
|
\>\> IMPLDIFF \` $\tau^{(n+1)}$ (\ref{eq:t-n+1-sync}) \\ |
| 415 |
|
\> DYNAMICS \\ |
| 416 |
|
\>\> CALC\_PHI\_HYD \` $\phi_{hyd}^n$ (\ref{eq:phi-hyd-sync}) \\ |
| 417 |
|
\>\> MOM\_FLUXFORM or MOM\_VECINV \` $G_{\vec{\bf v}}^n$ (\ref{eq:Gv-n-sync})\\ |
| 418 |
|
\>\> TIMESTEP \` $\vec{\bf v}^*$ (\ref{eq:Gv-n+5-sync}, \ref{eq:vstar-sync}) \\ |
| 419 |
|
\>\> IMPLDIFF \` $\vec{\bf v}^{**}$ (\ref{eq:vstarstar-sync}) \\ |
| 420 |
|
\> SOLVE\_FOR\_PRESSURE \\ |
| 421 |
|
\>\> CALC\_DIV\_GHAT \` $\eta^*$ (\ref{eq:nstar-sync}) \\ |
| 422 |
|
\>\> CG2D \` $\eta^{n+1}$ (\ref{eq:elliptic-sync}) \\ |
| 423 |
|
\> THE\_CORRECTION\_STEP \\ |
| 424 |
|
\>\> CALC\_GRAD\_PHI\_SURF \` $\nabla \eta^{n+1}$ \\ |
| 425 |
|
\>\> CORRECTION\_STEP \` $u^{n+1}$,$v^{n+1}$ (\ref{eq:v-n+1-sync}) |
| 426 |
|
\end{tabbing} \end{minipage} } \end{center} |
| 427 |
|
\caption{ |
| 428 |
|
Calling tree for the overall synchronous algorithm using |
| 429 |
|
Adams-Bashforth time-stepping.} |
| 430 |
|
\label{fig:call-tree-adams-bashforth-sync} |
| 431 |
|
\end{figure} |
| 432 |
|
|
| 433 |
|
The Adams-Bashforth extrapolation of explicit tendencies fits neatly |
| 434 |
|
into the pressure method algorithm when all state variables are |
| 435 |
|
co-located in time. Fig.~\ref{fig:adams-bashforth-sync} illustrates |
| 436 |
|
the location of variables in time and the evolution of the algorithm |
| 437 |
|
with time. The algorithm can be represented by the sequential solution |
| 438 |
|
of the follow equations: |
| 439 |
|
\begin{eqnarray} |
| 440 |
|
G_{\theta,S}^{n} & = & G_{\theta,S} ( u^n, \theta^n, S^n ) |
| 441 |
|
\label{eq:Gt-n-sync} \\ |
| 442 |
|
G_{\theta,S}^{(n+1/2)} & = & (3/2+\epsilon_{AB}) G_{\theta,S}^{n}-(1/2+\epsilon_{AB}) G_{\theta,S}^{n-1} |
| 443 |
|
\label{eq:Gt-n+5-sync} \\ |
| 444 |
|
(\theta^*,S^*) & = & (\theta^{n},S^{n}) + \Delta t G_{\theta,S}^{(n+1/2)} |
| 445 |
|
\label{eq:tstar-sync} \\ |
| 446 |
|
(\theta^{n+1},S^{n+1}) & = & {\cal L}^{-1}_{\theta,S} (\theta^*,S^*) |
| 447 |
|
\label{eq:t-n+1-sync} \\ |
| 448 |
|
\phi^n_{hyd} & = & \int b(\theta^n,S^n) dr |
| 449 |
|
\label{eq:phi-hyd-sync} \\ |
| 450 |
|
\vec{\bf G}_{\vec{\bf v}}^{n} & = & \vec{\bf G}_{\vec{\bf v}} ( \vec{\bf v}^n, \phi^n_{hyd} ) |
| 451 |
|
\label{eq:Gv-n-sync} \\ |
| 452 |
|
\vec{\bf G}_{\vec{\bf v}}^{(n+1/2)} & = & (3/2 + \epsilon_{AB} ) \vec{\bf G}_{\vec{\bf v}}^{n} - (1/2 + \epsilon_{AB} ) \vec{\bf G}_{\vec{\bf v}}^{n-1} |
| 453 |
|
\label{eq:Gv-n+5-sync} \\ |
| 454 |
|
\vec{\bf v}^{*} & = & \vec{\bf v}^{n} + \Delta t \vec{\bf G}_{\vec{\bf v}}^{(n+1/2)} |
| 455 |
|
\label{eq:vstar-sync} \\ |
| 456 |
|
\vec{\bf v}^{**} & = & {\cal L}_{\vec{\bf v}}^{-1} ( \vec{\bf v}^* ) |
| 457 |
|
\label{eq:vstarstar-sync} \\ |
| 458 |
|
\eta^* & = & \epsilon_{fs} \left( \eta^{n} +P-E+R \right)- \Delta t |
| 459 |
|
\nabla \cdot H \widehat{ \vec{\bf v}^{**} } |
| 460 |
|
\label{eq:nstar-sync} \\ |
| 461 |
|
\nabla \cdot g H \nabla \eta^{n+1} - \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2} |
| 462 |
|
& = & - \frac{\eta^*}{\Delta t^2} |
| 463 |
|
\label{eq:elliptic-sync} \\ |
| 464 |
|
\vec{\bf v}^{n+1} & = & \vec{\bf v}^{*} - \Delta t g \nabla \eta^{n+1} |
| 465 |
|
\label{eq:v-n+1-sync} |
| 466 |
|
\end{eqnarray} |
| 467 |
|
Fig.~\ref{fig:adams-bashforth-sync} illustrates the location of |
| 468 |
|
variables in time and evolution of the algorithm with time. The |
| 469 |
|
Adams-Bashforth extrapolation of the tracer tendencies is illustrated |
| 470 |
|
by the dashed arrow, the prediction at $n+1$ is indicated by the |
| 471 |
|
solid arc. Inversion of the implicit terms, ${\cal |
| 472 |
|
L}^{-1}_{\theta,S}$, then yields the new tracer fields at $n+1$. All |
| 473 |
|
these operations are carried out in subroutine {\em THERMODYNAMICS} an |
| 474 |
|
subsidiaries, which correspond to equations \ref{eq:Gt-n-sync} to |
| 475 |
|
\ref{eq:t-n+1-sync}. |
| 476 |
|
Similarly illustrated is the Adams-Bashforth extrapolation of |
| 477 |
|
accelerations, stepping forward and solving of implicit viscosity and |
| 478 |
|
surface pressure gradient terms, corresponding to equations |
| 479 |
|
\ref{eq:Gv-n-sync} to \ref{eq:v-n+1-sync}. |
| 480 |
|
These operations are carried out in subroutines {\em DYNAMCIS}, {\em |
| 481 |
|
SOLVE\_FOR\_PRESSURE} and {\em THE\_CORRECTION\_STEP}. This, then, |
| 482 |
|
represents an entire algorithm for stepping forward the model one |
| 483 |
|
time-step. The corresponding calling tree is given in |
| 484 |
|
\ref{fig:call-tree-adams-bashforth-sync}. |
| 485 |
|
|
| 486 |
|
\section{Staggered baroclinic time-stepping} |
| 487 |
|
\label{sect:adams-bashforth-staggered} |
| 488 |
|
|
| 489 |
|
\begin{figure} |
| 490 |
|
\begin{center} |
| 491 |
|
\resizebox{5.5in}{!}{\includegraphics{part2/adams-bashforth-staggered.eps}} |
| 492 |
|
\end{center} |
| 493 |
|
\caption{ |
| 494 |
|
A schematic of the explicit Adams-Bashforth and implicit time-stepping |
| 495 |
|
phases of the algorithm but with staggering in time of thermodynamic |
| 496 |
|
variables with the flow. Explicit thermodynamics tendencies are |
| 497 |
|
evaluated at time level $n-1/2$ as a function of the thermodynamics |
| 498 |
|
state at that time level $n$ and flow at time $n$ (dotted arrow). The |
| 499 |
|
explicit tendency from the previous time level, $n-3/2$, is used to |
| 500 |
|
extrapolate tendencies to $n$ (dashed arrow). This extrapolated |
| 501 |
|
tendency allows thermo-dynamics variables to be stably integrated |
| 502 |
|
forward-in-time to render an estimate ($*$-variables) at the $n+1/2$ |
| 503 |
|
time level (solid arc-arrow). The implicit-in-time operator ${\cal |
| 504 |
|
L_{\theta,S}}$ is solved to yield the thermodynamic variables at time |
| 505 |
|
level $n+1/2$. These are then used to calculate the hydrostatic |
| 506 |
|
pressure/geo-potential, $\phi_{hyd}$ (vertical arrows). The |
| 507 |
|
hydrostatic pressure gradient is evaluated directly an time level |
| 508 |
|
$n+1/2$ in stepping forward the flow variables from $n$ to $n+1$ |
| 509 |
|
(solid arc-arrow). } |
| 510 |
|
\label{fig:adams-bashforth-staggered} |
| 511 |
|
\end{figure} |
| 512 |
|
|
| 513 |
|
For well stratified problems, internal gravity waves may be the |
| 514 |
|
limiting process for determining a stable time-step. In the |
| 515 |
|
circumstance, it is more efficient to stagger in time the |
| 516 |
|
thermodynamic variables with the flow |
| 517 |
|
variables. Fig.~\ref{fig:adams-bashforth-staggered} illustrates the |
| 518 |
|
staggering and algorithm. The key difference between this and |
| 519 |
|
Fig.~\ref{fig:adams-bashforth-sync} is that the new thermodynamics |
| 520 |
|
fields are used to compute the hydrostatic pressure at time level |
| 521 |
|
$n+1/2$. The essentially allows the gravity wave terms to leap-frog in |
| 522 |
|
time giving second order accuracy and more stability. |
| 523 |
|
|
| 524 |
|
The essential change in the staggered algorithm is the calculation of |
| 525 |
|
hydrostatic pressure which, in the context of the synchronous |
| 526 |
|
algorithm involves replacing equation \ref{eq:phi-hyd-sync} with |
| 527 |
|
\begin{displaymath} |
| 528 |
|
\phi_{hyd}^n = \int b(\theta^{n+1},S^{n+1}) dr |
| 529 |
|
\end{displaymath} |
| 530 |
|
but the pressure gradient must also be taken out of the |
| 531 |
|
Adams-Bashforth extrapolation. Also, retaining the integer time-levels, |
| 532 |
|
$n$ and $n+1$, does not give a user the sense of where variables are |
| 533 |
|
located in time. Instead, we re-write the entire algorithm, |
| 534 |
|
\ref{eq:Gt-n-sync} to \ref{eq:v-n+1-sync}, annotating the |
| 535 |
|
position in time of variables appropriately: |
| 536 |
|
\begin{eqnarray} |
| 537 |
|
G_{\theta,S}^{n-1/2} & = & G_{\theta,S} ( u^n, \theta^{n-1/2}, S^{n-1/2} ) |
| 538 |
|
\label{eq:Gt-n-staggered} \\ |
| 539 |
|
G_{\theta,S}^{(n)} & = & (3/2+\epsilon_{AB}) G_{\theta,S}^{n-1/2}-(1/2+\epsilon_{AB}) G_{\theta,S}^{n-3/2} |
| 540 |
|
\label{eq:Gt-n+5-staggered} \\ |
| 541 |
|
(\theta^*,S^*) & = & (\theta^{n},S^{n}) + \Delta t G_{\theta,S}^{(n)} |
| 542 |
|
\label{eq:tstar-staggered} \\ |
| 543 |
|
(\theta^{n+1/2},S^{n+1/2}) & = & {\cal L}^{-1}_{\theta,S} (\theta^*,S^*) |
| 544 |
|
\label{eq:t-n+1-staggered} \\ |
| 545 |
|
\phi^{n+1/2}_{hyd} & = & \int b(\theta^{n+1/2},S^{n+1/2}) dr |
| 546 |
|
\label{eq:phi-hyd-staggered} \\ |
| 547 |
|
\vec{\bf G}_{\vec{\bf v}}^{n} & = & \vec{\bf G}_{\vec{\bf v}} ( \vec{\bf v}^n ) |
| 548 |
|
\label{eq:Gv-n-staggered} \\ |
| 549 |
|
\vec{\bf G}_{\vec{\bf v}}^{(n+1/2)} & = & (3/2 + \epsilon_{AB} ) \vec{\bf G}_{\vec{\bf v}}^{n} - (1/2 + \epsilon_{AB} ) \vec{\bf G}_{\vec{\bf v}}^{n-1} |
| 550 |
|
\label{eq:Gv-n+5-staggered} \\ |
| 551 |
|
\vec{\bf v}^{*} & = & \vec{\bf v}^{n} + \Delta t \left( \vec{\bf G}_{\vec{\bf v}}^{(n+1/2)} - \nabla \phi_{hyd}^{n+1/2} \right) |
| 552 |
|
\label{eq:vstar-staggered} \\ |
| 553 |
|
\vec{\bf v}^{**} & = & {\cal L}_{\vec{\bf v}}^{-1} ( \vec{\bf v}^* ) |
| 554 |
|
\label{eq:vstarstar-staggered} \\ |
| 555 |
|
\eta^* & = & \epsilon_{fs} \left( \eta^{n} +P-E+R \right)- \Delta t |
| 556 |
|
\nabla \cdot H \widehat{ \vec{\bf v}^{**} } |
| 557 |
|
\label{eq:nstar-staggered} \\ |
| 558 |
|
\nabla \cdot g H \nabla \eta^{n+1} - \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2} |
| 559 |
|
& = & - \frac{\eta^*}{\Delta t^2} |
| 560 |
|
\label{eq:elliptic-staggered} \\ |
| 561 |
|
\vec{\bf v}^{n+1} & = & \vec{\bf v}^{*} - \Delta t g \nabla \eta^{n+1} |
| 562 |
|
\label{eq:v-n+1-staggered} |
| 563 |
|
\end{eqnarray} |
| 564 |
|
The calling sequence is unchanged from |
| 565 |
|
Fig.~\ref{fig:call-tree-adams-bashforth-sync}. The staggered algorithm |
| 566 |
|
is activated with the run-time flag {\bf staggerTimeStep=.TRUE.} in |
| 567 |
|
{\em PARM01} of {\em data}. |
| 568 |
|
|
| 569 |
|
The only difficulty with this approach is apparent in equation |
| 570 |
|
\ref{eq:Gt-n-staggered} and illustrated by the dotted arrow |
| 571 |
|
connecting $u,v^n$ with $G_\theta^{n-1/2}$. The flow used to advect |
| 572 |
|
tracers around is not naturally located in time. This could be avoided |
| 573 |
|
by applying the Adams-Bashforth extrapolation to the tracer field |
| 574 |
|
itself and advecting that around but this approach is not yet |
| 575 |
|
available. We're not aware of any detrimental effect of this |
| 576 |
|
feature. The difficulty lies mainly in interpretation of what |
| 577 |
|
time-level variables and terms correspond to. |
| 578 |
|
|
| 579 |
|
|
| 580 |
|
\section{Non-hydrostatic formulation} |
| 581 |
|
\label{sect:non-hydrostatic} |
| 582 |
|
|
| 583 |
|
The non-hydrostatic formulation re-introduces the full vertical |
| 584 |
|
momentum equation and requires the solution of a 3-D elliptic |
| 585 |
|
equations for non-hydrostatic pressure perturbation. We still |
| 586 |
|
intergrate vertically for the hydrostatic pressure and solve a 2-D |
| 587 |
|
elliptic equation for the surface pressure/elevation for this reduces |
| 588 |
|
the amount of work needed to solve for the non-hydrostatic pressure. |
| 589 |
|
|
| 590 |
\subsection{Stagger baroclinic time stepping} |
The momentum equations are discretized in time as follows: |
| 591 |
|
\begin{eqnarray} |
| 592 |
An alternative to synchronous time-stepping is to stagger the momentum |
\frac{1}{\Delta t} u^{n+1} + g \partial_x \eta^{n+1} + \partial_x \phi_{nh}^{n+1} |
| 593 |
and tracer fields in time. This allows the evaluation and gradient of |
& = & \frac{1}{\Delta t} u^{n} + G_u^{(n+1/2)} \label{eq:discrete-time-u-nh} \\ |
| 594 |
$\phi'_{hyd}$ to be centered in time with out needing to use the |
\frac{1}{\Delta t} v^{n+1} + g \partial_y \eta^{n+1} + \partial_y \phi_{nh}^{n+1} |
| 595 |
Adams-Bashforth extrapoltion. This option is known as staggered |
& = & \frac{1}{\Delta t} v^{n} + G_v^{(n+1/2)} \label{eq:discrete-time-v-nh} \\ |
| 596 |
baroclinic time stepping because tracer and momentum are stepped |
\frac{1}{\Delta t} w^{n+1} + \partial_r \phi_{nh}^{n+1} |
| 597 |
forward-in-time one after the other. It can be activated by turning |
& = & \frac{1}{\Delta t} w^{n} + G_w^{(n+1/2)} \label{eq:discrete-time-w-nh} \\ |
| 598 |
on a run-time parameter {\bf staggerTimeStep} in namelist ``{\it |
\end{eqnarray} |
| 599 |
PARM01}''. |
which must satisfy the discrete-in-time depth integrated continuity, |
| 600 |
|
equation~\ref{eq:discrete-time-backward-free-surface} and the local continuity equation |
|
The main advantage of staggered time-stepping compared to synchronous, |
|
|
is improved stability to internal gravity wave motions and a very |
|
|
natural implementation of a 2nd order in time hydrostatic |
|
|
pressure/geo-potential gradient term. However, synchronous |
|
|
time-stepping might be better for convection problems, time dependent |
|
|
forcing and diagnosing the model are also easier and it allows a more |
|
|
efficient parallel decomposition. |
|
|
|
|
|
The staggered baroclinic time-stepping scheme is equations |
|
|
\ref{eq-tDsC-theta} to \ref{eq-tDsC-cont} except that \ref{eq-tDsC-hyd} is replaced with: |
|
| 601 |
\begin{equation} |
\begin{equation} |
| 602 |
{\phi'_{hyd}}^{n+1/2} = \int_{r'}^{R_o} b'(\theta^{n+1/2},S^{n+1/2},r) |
\partial_x u^{n+1} + \partial_y v^{n+1} + \partial_r w^{n+1} = 0 |
| 603 |
dr |
\label{eq:non-divergence-nh} |
| 604 |
\end{equation} |
\end{equation} |
| 605 |
and the time-level for tracers has been shifted back by half: |
As before, the explicit predictions for momentum are consolidated as: |
| 606 |
\begin{eqnarray*} |
\begin{eqnarray*} |
| 607 |
\theta^* & = & |
u^* & = & u^n + \Delta t G_u^{(n+1/2)} \\ |
| 608 |
\theta ^{(n-1/2)} + \Delta t G_{\theta} ^{(n)} |
v^* & = & v^n + \Delta t G_v^{(n+1/2)} \\ |
| 609 |
\\ |
w^* & = & w^n + \Delta t G_w^{(n+1/2)} |
|
S^* & = & |
|
|
S ^{(n-1/2)} + \Delta t G_{S} ^{(n)} |
|
|
\\ |
|
|
\left[ 1 - \partial_r \kappa_v^\theta \partial_r \right] |
|
|
\theta^{n+1/2} & = & \theta^* |
|
|
\\ |
|
|
\left[ 1 - \partial_r \kappa_v^S \partial_r \right] |
|
|
S^{n+1/2} & = & S^* |
|
| 610 |
\end{eqnarray*} |
\end{eqnarray*} |
| 611 |
|
but this time we introduce an intermediate step by splitting the |
| 612 |
|
tendancy of the flow as follows: |
| 613 |
|
\begin{eqnarray} |
| 614 |
|
u^{n+1} = u^{**} - \Delta t \partial_x \phi_{nh}^{n+1} |
| 615 |
|
& & |
| 616 |
|
u^{**} = u^{*} - \Delta t g \partial_x \eta^{n+1} \\ |
| 617 |
|
v^{n+1} = v^{**} - \Delta t \partial_y \phi_{nh}^{n+1} |
| 618 |
|
& & |
| 619 |
|
v^{**} = v^{*} - \Delta t g \partial_y \eta^{n+1} |
| 620 |
|
\end{eqnarray} |
| 621 |
|
Substituting into the depth integrated continuity |
| 622 |
|
(equation~\ref{eq:discrete-time-backward-free-surface}) gives |
| 623 |
|
\begin{equation} |
| 624 |
|
\partial_x H \partial_x \left( g \eta^{n+1} + \widehat{\phi}_{nh}^{n+1} \right) |
| 625 |
|
+ |
| 626 |
|
\partial_y H \partial_y \left( g \eta^{n+1} + \widehat{\phi}_{nh}^{n+1} \right) |
| 627 |
|
- \frac{\epsilon_{fs}\eta^*}{\Delta t^2} |
| 628 |
|
= - \frac{\eta^*}{\Delta t^2} |
| 629 |
|
\end{equation} |
| 630 |
|
which is approximated by equation |
| 631 |
|
\ref{eq:elliptic-backward-free-surface} on the basis that i) |
| 632 |
|
$\phi_{nh}^{n+1}$ is not yet known and ii) $\nabla \widehat{\phi}_{nh} |
| 633 |
|
<< g \nabla \eta$. If \ref{eq:elliptic-backward-free-surface} is |
| 634 |
|
solved accurately then the implication is that $\widehat{\phi}_{nh} |
| 635 |
|
\approx 0$ so that thet non-hydrostatic pressure field does not drive |
| 636 |
|
barotropic motion. |
| 637 |
|
|
| 638 |
|
The flow must satisfy non-divergence |
| 639 |
|
(equation~\ref{eq:non-divergence-nh}) locally, as well as depth |
| 640 |
|
integrated, and this constraint is used to form a 3-D elliptic |
| 641 |
|
equations for $\phi_{nh}^{n+1}$: |
| 642 |
|
\begin{equation} |
| 643 |
|
\partial_{xx} \phi_{nh}^{n+1} + \partial_{yy} \phi_{nh}^{n+1} + |
| 644 |
|
\partial_{rr} \phi_{nh}^{n+1} = |
| 645 |
|
\partial_x u^{**} + \partial_y v^{**} + \partial_r w^{*} |
| 646 |
|
\end{equation} |
| 647 |
|
|
| 648 |
|
The entire algorithm can be summarized as the sequential solution of |
| 649 |
|
the following equations: |
| 650 |
|
\begin{eqnarray} |
| 651 |
|
u^{*} & = & u^{n} + \Delta t G_u^{(n+1/2)} \label{eq:ustar-nh} \\ |
| 652 |
|
v^{*} & = & v^{n} + \Delta t G_v^{(n+1/2)} \label{eq:vstar-nh} \\ |
| 653 |
|
w^{*} & = & w^{n} + \Delta t G_w^{(n+1/2)} \label{eq:wstar-nh} \\ |
| 654 |
|
\eta^* & = & \epsilon_{fs} \left( \eta^{n} +P-E+R \right)- \Delta t |
| 655 |
|
\partial_x H \widehat{u^{*}} |
| 656 |
|
+ \partial_y H \widehat{v^{*}} |
| 657 |
|
\\ |
| 658 |
|
\partial_x g H \partial_x \eta^{n+1} |
| 659 |
|
+ \partial_y g H \partial_y \eta^{n+1} |
| 660 |
|
- \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2} |
| 661 |
|
& = & |
| 662 |
|
- \frac{\eta^*}{\Delta t^2} |
| 663 |
|
\label{eq:elliptic-nh} |
| 664 |
|
\\ |
| 665 |
|
u^{**} & = & u^{*} - \Delta t g \partial_x \eta^{n+1} \label{eq:unx-nh}\\ |
| 666 |
|
v^{**} & = & v^{*} - \Delta t g \partial_y \eta^{n+1} \label{eq:vnx-nh}\\ |
| 667 |
|
\partial_{xx} \phi_{nh}^{n+1} + \partial_{yy} \phi_{nh}^{n+1} + |
| 668 |
|
\partial_{rr} \phi_{nh}^{n+1} & = & |
| 669 |
|
\partial_x u^{**} + \partial_y v^{**} + \partial_r w^{*} \\ |
| 670 |
|
u^{n+1} & = & u^{**} - \Delta t \partial_x \phi_{nh}^{n+1} \label{eq:un+1-nh}\\ |
| 671 |
|
v^{n+1} & = & v^{**} - \Delta t \partial_y \phi_{nh}^{n+1} \label{eq:vn+1-nh}\\ |
| 672 |
|
\partial_r w^{n+1} & = & - \partial_x u^{n+1} - \partial_y v^{n+1} |
| 673 |
|
\end{eqnarray} |
| 674 |
|
where the last equation is solved by vertically integrating for |
| 675 |
|
$w^{n+1}$. |
| 676 |
|
|
| 677 |
|
|
| 678 |
|
|
| 679 |
\subsection{Surface pressure} |
\section{Variants on the Free Surface} |
| 680 |
|
|
| 681 |
Substituting \ref{eq-tDsC-Hmom} into \ref{eq-tDsC-cont}, assuming |
We now describe the various formulations of the free-surface that |
| 682 |
$\epsilon_{nh} = 0$ yields a Helmholtz equation for ${\eta}^{n+1}$: |
include non-linear forms, implicit in time using Crank-Nicholson, |
| 683 |
|
explicit and [one day] split-explicit. First, we'll reiterate the |
| 684 |
|
underlying algorithm but this time using the notation consistent with |
| 685 |
|
the more general vertical coordinate $r$. The elliptic equation for |
| 686 |
|
free-surface coordinate (units of $r$), corresponding to |
| 687 |
|
\ref{eq:discrete-time-backward-free-surface}, and |
| 688 |
|
assuming no non-hydrostatic effects ($\epsilon_{nh} = 0$) is: |
| 689 |
\begin{eqnarray} |
\begin{eqnarray} |
| 690 |
\epsilon_{fs} {\eta}^{n+1} - |
\epsilon_{fs} {\eta}^{n+1} - |
| 691 |
{\bf \nabla}_h \cdot \Delta t^2 (R_o-R_{fixed}) |
{\bf \nabla}_h \cdot \Delta t^2 (R_o-R_{fixed}) {\bf \nabla}_h b_s |
| 692 |
{\bf \nabla}_h b_s {\eta}^{n+1} |
{\eta}^{n+1} = {\eta}^* |
|
= {\eta}^* |
|
| 693 |
\label{eq-solve2D} |
\label{eq-solve2D} |
| 694 |
\end{eqnarray} |
\end{eqnarray} |
| 695 |
where |
where |
| 715 |
|
|
| 716 |
|
|
| 717 |
Once ${\eta}^{n+1}$ has been found, substituting into |
Once ${\eta}^{n+1}$ has been found, substituting into |
| 718 |
\ref{eq-tDsC-Hmom} yields $\vec{\bf v}^{n+1}$ if the model is |
\ref{eq:discrete-time-u,eq:discrete-time-v} yields $\vec{\bf v}^{n+1}$ if the model is |
| 719 |
hydrostatic ($\epsilon_{nh}=0$): |
hydrostatic ($\epsilon_{nh}=0$): |
| 720 |
$$ |
$$ |
| 721 |
\vec{\bf v}^{n+1} = \vec{\bf v}^{*} |
\vec{\bf v}^{n+1} = \vec{\bf v}^{*} |
| 725 |
This is known as the correction step. However, when the model is |
This is known as the correction step. However, when the model is |
| 726 |
non-hydrostatic ($\epsilon_{nh}=1$) we need an additional step and an |
non-hydrostatic ($\epsilon_{nh}=1$) we need an additional step and an |
| 727 |
additional equation for $\phi'_{nh}$. This is obtained by substituting |
additional equation for $\phi'_{nh}$. This is obtained by substituting |
| 728 |
\ref{eq-tDsC-Hmom} and \ref{eq-tDsC-Vmom} into |
\ref{eq:discrete-time-u-nh}, \ref{eq:discrete-time-v-nh} and \ref{eq:discrete-time-w-nh} |
| 729 |
\ref{eq-tDsC-cont}: |
into continuity: |
| 730 |
\begin{equation} |
\begin{equation} |
| 731 |
\left[ {\bf \nabla}_h^2 + \partial_{rr} \right] {\phi'_{nh}}^{n+1} |
\left[ {\bf \nabla}_h^2 + \partial_{rr} \right] {\phi'_{nh}}^{n+1} |
| 732 |
= \frac{1}{\Delta t} \left( |
= \frac{1}{\Delta t} \left( |
| 806 |
{\it implicSurfPress}, {\it implicDiv2DFlow}. They are read from |
{\it implicSurfPress}, {\it implicDiv2DFlow}. They are read from |
| 807 |
the main data file "{\it data}" and are set by default to 1,1. |
the main data file "{\it data}" and are set by default to 1,1. |
| 808 |
|
|
| 809 |
Equations \ref{eq-tDsC-Hmom} and \ref{eq-tDsC-eta} are modified as follows: |
Equations \ref{eq:ustar-backward-free-surface} -- |
| 810 |
|
\ref{eq:vn+1-backward-free-surface} are modified as follows: |
| 811 |
$$ |
$$ |
| 812 |
\frac{ \vec{\bf v}^{n+1} }{ \Delta t } |
\frac{ \vec{\bf v}^{n+1} }{ \Delta t } |
| 813 |
+ {\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} ] |
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