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% $Header: /u/gcmpack/manual/part2/time_stepping.tex,v 1.25 2006/06/28 16:55:53 jmc Exp $ |
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% $Name: $ |
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|
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This chapter lays out the numerical schemes that are |
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employed in the core MITgcm algorithm. Whenever possible |
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links are made to actual program code in the MITgcm implementation. |
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The chapter begins with a discussion of the temporal discretization |
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used in MITgcm. This discussion is followed by sections that |
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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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passive and dynamically active tracers are described. |
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|
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\input{part2/notation} |
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|
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\section{Time-stepping} |
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\begin{rawhtml} |
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<!-- CMIREDIR:time-stepping: --> |
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\end{rawhtml} |
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|
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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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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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|
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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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|
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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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|
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|
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\section{Pressure method with rigid-lid} |
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\label{sect:pressure-method-rigid-lid} |
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\begin{rawhtml} |
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<!-- CMIREDIR:pressure_method_rigid_lid: --> |
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\end{rawhtml} |
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|
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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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|
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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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\> MOMENTUM\_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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|
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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} |
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\partial_t u + g \partial_x \eta & = & G_u \\ |
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\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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\end{eqnarray} |
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where we are adopting the oceanic notation for brevity. All terms in |
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the momentum equations, except for surface pressure gradient, are |
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encapsulated in the $G$ vector. The continuity equation, when |
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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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\begin{equation} |
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\partial_x H \widehat{u} + \partial_y H \widehat{v} = 0 |
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\label{eq:rigid-lid-continuity} |
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\end{equation} |
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Here, $H\widehat{u} = \int_H u dz$ is the depth integral of $u$, |
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similarly for $H\widehat{v}$. The rigid-lid approximation sets $w=0$ |
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at the lid so that it does not move but allows a pressure to be |
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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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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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|
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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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\partial_x H \widehat{u^{*}} |
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+ \partial_y H \widehat{v^{*}} \label{eq:elliptic} |
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\\ |
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u^{n+1} & = & u^{*} - \Delta t g \partial_x \eta^{n+1} \label{eq:un+1-rigid-lid}\\ |
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v^{n+1} & = & v^{*} - \Delta t g \partial_y \eta^{n+1} \label{eq:vn+1-rigid-lid} |
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\end{eqnarray} |
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Equations \ref{eq:ustar-rigid-lid} to \ref{eq:vn+1-rigid-lid}, solved |
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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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explicit prediction for the flow would lead to a divergence flow field |
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so a pressure field must be found that keeps the flow non-divergent |
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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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Fig.~\ref{fig:pressure-method-rigid-lid} we depicted as co-located |
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with the future flow field (time level $n+1$) but it could equally |
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have been drawn as staggered in time with the flow. |
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|
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The correspondence to the code is as follows: |
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\begin{itemize} |
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\item |
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the prognostic phase, equations \ref{eq:ustar-rigid-lid} and \ref{eq:vstar-rigid-lid}, |
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stepping forward $u^n$ and $v^n$ to $u^{*}$ and $v^{*}$ is coded in |
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\filelink{TIMESTEP()}{model-src-timestep.F} |
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\item |
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the vertical integration, $H \widehat{u^*}$ and $H |
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\widehat{v^*}$, divergence and inversion of the elliptic operator in |
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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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\item |
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finally, the new flow field at time level $n+1$ given by equations |
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\ref{eq:un+1-rigid-lid} and \ref{eq:vn+1-rigid-lid} is calculated in |
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\filelink{CORRECTION\_STEP()}{model-src-correction_step.F}. |
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\end{itemize} |
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The calling tree for these routines is given in |
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Fig.~\ref{fig:call-tree-pressure-method}. |
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|
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|
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%\paragraph{Need to discuss implicit viscosity somewhere:} |
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In general, the horizontal momentum time-stepping can contain some terms |
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that are treated implicitly in time, |
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such as the vertical viscosity when using the backward time-stepping scheme |
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(\varlink{implicitViscosity}{implicitViscosity} {\it =.TRUE.}). |
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The method used to solve those implicit terms is provided in |
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section \ref{sect:implicit-backward-stepping}, and modifies |
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equations \ref{eq:discrete-time-u} and \ref{eq:discrete-time-v} to |
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give: |
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\begin{eqnarray} |
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u^{n+1} - \Delta t \partial_z A_v \partial_z u^{n+1} |
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+ \Delta t g \partial_x \eta^{n+1} & = & u^{n} + \Delta t G_u^{(n+1/2)} |
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\\ |
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v^{n+1} - \Delta t \partial_z A_v \partial_z v^{n+1} |
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+ \Delta t g \partial_y \eta^{n+1} & = & v^{n} + \Delta t G_v^{(n+1/2)} |
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\end{eqnarray} |
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|
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|
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\section{Pressure method with implicit linear free-surface} |
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\label{sect:pressure-method-linear-backward} |
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\begin{rawhtml} |
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<!-- CMIREDIR:pressure_method_linear_backward: --> |
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\end{rawhtml} |
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|
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The rigid-lid approximation filters out external gravity waves |
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subsequently modifying the dispersion relation of barotropic Rossby |
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waves. The discrete form of the elliptic equation has some zero |
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eigen-values which makes it a potentially tricky or inefficient |
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problem to solve. |
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|
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The rigid-lid approximation can be easily replaced by a linearization |
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of the free-surface equation which can be written: |
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\begin{equation} |
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\partial_t \eta + \partial_x H \widehat{u} + \partial_y H \widehat{v} = P-E+R |
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\label{eq:linear-free-surface=P-E} |
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\end{equation} |
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which differs from the depth integrated continuity equation with |
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rigid-lid (\ref{eq:rigid-lid-continuity}) by the time-dependent term |
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and fresh-water source term. |
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|
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Equation \ref{eq:discrete-time-cont-rigid-lid} in the rigid-lid |
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pressure method is then replaced by the time discretization of |
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\ref{eq:linear-free-surface=P-E} which is: |
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\begin{equation} |
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\eta^{n+1} |
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+ \Delta t \partial_x H \widehat{u^{n+1}} |
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+ \Delta t \partial_y H \widehat{v^{n+1}} |
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= |
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\eta^{n} |
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+ \Delta t ( P - E ) |
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\label{eq:discrete-time-backward-free-surface} |
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\end{equation} |
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where the use of flow at time level $n+1$ makes the method implicit |
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and backward in time. This is the preferred scheme since it still |
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filters the fast unresolved wave motions by damping them. A centered |
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scheme, such as Crank-Nicholson (see section \ref{sect:freesurf-CrankNick}), |
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would alias the energy of the fast modes onto slower modes of motion. |
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|
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As for the rigid-lid pressure method, equations |
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\ref{eq:discrete-time-u}, \ref{eq:discrete-time-v} and |
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\ref{eq:discrete-time-backward-free-surface} can 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-backward-free-surface} \\ |
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v^{*} & = & v^{n} + \Delta t G_v^{(n+1/2)} \label{eq:vstar-backward-free-surface} \\ |
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\eta^* & = & \epsilon_{fs} \left( \eta^{n} + \Delta t (P-E) \right)- \Delta t |
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\partial_x H \widehat{u^{*}} |
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+ \partial_y H \widehat{v^{*}} |
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\\ |
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\partial_x g H \partial_x \eta^{n+1} |
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& + & \partial_y g H \partial_y \eta^{n+1} |
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- \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2} |
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= |
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- \frac{\eta^*}{\Delta t^2} |
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\label{eq:elliptic-backward-free-surface} |
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\\ |
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u^{n+1} & = & u^{*} - \Delta t g \partial_x \eta^{n+1} \label{eq:un+1-backward-free-surface}\\ |
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v^{n+1} & = & v^{*} - \Delta t g \partial_y \eta^{n+1} \label{eq:vn+1-backward-free-surface} |
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\end{eqnarray} |
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Equations~\ref{eq:ustar-backward-free-surface} |
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to~\ref{eq:vn+1-backward-free-surface}, solved sequentially, represent |
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the pressure method algorithm with a backward implicit, linearized |
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free surface. The method is still formerly a pressure method because |
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in the limit of large $\Delta t$ the rigid-lid method is |
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recovered. However, the implicit treatment of the free-surface allows |
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the flow to be divergent and for the surface pressure/elevation to |
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respond on a finite time-scale (as opposed to instantly). To recover |
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the rigid-lid formulation, we introduced a switch-like parameter, |
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$\epsilon_{fs}$ (\varlink{freesurfFac}{freesurfFac}), |
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which selects between the free-surface and rigid-lid; |
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$\epsilon_{fs}=1$ allows the free-surface to evolve; $\epsilon_{fs}=0$ |
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imposes the rigid-lid. The evolution in time and location of variables |
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is exactly as it was for the rigid-lid model so that |
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Fig.~\ref{fig:pressure-method-rigid-lid} is still |
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applicable. Similarly, the calling sequence, given in |
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Fig.~\ref{fig:call-tree-pressure-method}, is as for the |
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pressure-method. |
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|
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|
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\section{Explicit time-stepping: Adams-Bashforth} |
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\label{sect:adams-bashforth} |
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\begin{rawhtml} |
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<!-- CMIREDIR:adams_bashforth: --> |
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\end{rawhtml} |
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|
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In describing the the pressure method above we deferred describing the |
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time discretization of the explicit terms. We have historically used |
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the quasi-second order Adams-Bashforth method for all explicit terms |
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in both the momentum and tracer equations. This is still the default |
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mode of operation but it is now possible to use alternate schemes for |
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tracers (see section \ref{sect:tracer-advection}). |
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|
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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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FORWARD\_STEP \\ |
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\> THERMODYNAMICS \\ |
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\>\> CALC\_GT \\ |
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\>\>\> GAD\_CALC\_RHS \` $G_\theta^n = G_\theta( u, \theta^n )$ \\ |
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\>either\>\> EXTERNAL\_FORCING \` $G_\theta^n = G_\theta^n + {\cal Q}$ \\ |
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\>\>\> ADAMS\_BASHFORTH2 \` $G_\theta^{(n+1/2)}$ (\ref{eq:adams-bashforth2}) \\ |
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\>or\>\> EXTERNAL\_FORCING \` $G_\theta^{(n+1/2)} = G_\theta^{(n+1/2)} + {\cal Q}$ \\ |
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\>\> TIMESTEP\_TRACER \` $\tau^*$ (\ref{eq:taustar}) \\ |
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\>\> IMPLDIFF \` $\tau^{(n+1)}$ (\ref{eq:tau-n+1-implicit}) |
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\end{tabbing} \end{minipage} } \end{center} |
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\caption{ |
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Calling tree for the Adams-Bashforth time-stepping of temperature with |
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implicit diffusion. |
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(\filelink{THERMODYNAMICS}{model-src-thermodynamics.F}, |
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\filelink{ADAMS\_BASHFORTH2}{model-src-adams_bashforth2.F})} |
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\label{fig:call-tree-adams-bashforth} |
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\end{figure} |
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|
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In the previous sections, we summarized an explicit scheme as: |
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\begin{equation} |
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\tau^{*} = \tau^{n} + \Delta t G_\tau^{(n+1/2)} |
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\label{eq:taustar} |
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\end{equation} |
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where $\tau$ could be any prognostic variable ($u$, $v$, $\theta$ or |
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$S$) and $\tau^*$ is an explicit estimate of $\tau^{n+1}$ and would be |
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exact if not for implicit-in-time terms. The parenthesis about $n+1/2$ |
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indicates that the term is explicit and extrapolated forward in time |
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and for this we use the quasi-second order Adams-Bashforth method: |
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\begin{equation} |
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G_\tau^{(n+1/2)} = ( 3/2 + \epsilon_{AB}) G_\tau^n |
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- ( 1/2 + \epsilon_{AB}) G_\tau^{n-1} |
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\label{eq:adams-bashforth2} |
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\end{equation} |
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This is a linear extrapolation, forward in time, to |
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$t=(n+1/2+{\epsilon_{AB}})\Delta t$. An extrapolation to the mid-point |
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in time, $t=(n+1/2)\Delta t$, corresponding to $\epsilon_{AB}=0$, |
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would be second order accurate but is weakly unstable for oscillatory |
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terms. A small but finite value for $\epsilon_{AB}$ stabilizes the |
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method. Strictly speaking, damping terms such as diffusion and |
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dissipation, and fixed terms (forcing), do not need to be inside the |
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Adams-Bashforth extrapolation. However, in the current code, it is |
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simpler to include these terms and this can be justified if the flow |
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and forcing evolves smoothly. Problems can, and do, arise when forcing |
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or motions are high frequency and this corresponds to a reduced |
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stability compared to a simple forward time-stepping of such terms. |
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The model offers the possibility to leave the tracer and momentum |
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forcing terms and the dissipation terms outside the |
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Adams-Bashforth extrapolation, by turning off the logical flags |
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\varlink{forcing\_In\_AB}{forcing_In_AB} |
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(parameter file {\em data}, namelist {\em PARM01}, default value = True). |
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(\varlink{tracForcingOutAB}{tracForcingOutAB}, default=0, |
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\varlink{momForcingOutAB}{momForcingOutAB}, default=0) |
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and \varlink{momDissip\_In\_AB}{momDissip_In_AB} |
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(parameter file {\em data}, namelist {\em PARM01}, default value = True). |
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respectively. |
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|
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A stability analysis for an oscillation equation should be given at this point. |
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\marginpar{AJA needs to find his notes on this...} |
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|
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A stability analysis for a relaxation equation should be given at this point. |
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\marginpar{...and for this too.} |
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|
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\begin{figure} |
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\begin{center} |
379 |
\resizebox{5.5in}{!}{\includegraphics{part2/oscil+damp_AB2.eps}} |
380 |
\end{center} |
381 |
\caption{ |
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Oscillatory and damping response of |
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quasi-second order Adams-Bashforth scheme for different values |
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of the $\epsilon_{AB}$ parameter (0., 0.1, 0.25, from top to bottom) |
385 |
The analytical solution (in black), the physical mode (in blue) |
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and the numerical mode (in red) are represented with a CFL |
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step of 0.1. |
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The left column represents the oscillatory response |
389 |
on the complex plane for CFL ranging from 0.1 up to 0.9. |
390 |
The right column represents the damping response amplitude |
391 |
(y-axis) function of the CFL (x-axis). |
392 |
} |
393 |
\label{fig:adams-bashforth-respons} |
394 |
\end{figure} |
395 |
|
396 |
|
397 |
|
398 |
\section{Implicit time-stepping: backward method} |
399 |
\label{sect:implicit-backward-stepping} |
400 |
\begin{rawhtml} |
401 |
<!-- CMIREDIR:implicit_time-stepping_backward: --> |
402 |
\end{rawhtml} |
403 |
|
404 |
Vertical diffusion and viscosity can be treated implicitly in time |
405 |
using the backward method which is an intrinsic scheme. |
406 |
Recently, the option to treat the vertical advection |
407 |
implicitly has been added, but not yet tested; therefore, |
408 |
the description hereafter is limited to diffusion and viscosity. |
409 |
For tracers, |
410 |
the time discretized equation is: |
411 |
\begin{equation} |
412 |
\tau^{n+1} - \Delta t \partial_r \kappa_v \partial_r \tau^{n+1} = |
413 |
\tau^{n} + \Delta t G_\tau^{(n+1/2)} |
414 |
\label{eq:implicit-diffusion} |
415 |
\end{equation} |
416 |
where $G_\tau^{(n+1/2)}$ is the remaining explicit terms extrapolated |
417 |
using the Adams-Bashforth method as described above. Equation |
418 |
\ref{eq:implicit-diffusion} can be split split into: |
419 |
\begin{eqnarray} |
420 |
\tau^* & = & \tau^{n} + \Delta t G_\tau^{(n+1/2)} |
421 |
\label{eq:taustar-implicit} \\ |
422 |
\tau^{n+1} & = & {\cal L}_\tau^{-1} ( \tau^* ) |
423 |
\label{eq:tau-n+1-implicit} |
424 |
\end{eqnarray} |
425 |
where ${\cal L}_\tau^{-1}$ is the inverse of the operator |
426 |
\begin{equation} |
427 |
{\cal L} = \left[ 1 + \Delta t \partial_r \kappa_v \partial_r \right] |
428 |
\end{equation} |
429 |
Equation \ref{eq:taustar-implicit} looks exactly as \ref{eq:taustar} |
430 |
while \ref{eq:tau-n+1-implicit} involves an operator or matrix |
431 |
inversion. By re-arranging \ref{eq:implicit-diffusion} in this way we |
432 |
have cast the method as an explicit prediction step and an implicit |
433 |
step allowing the latter to be inserted into the over all algorithm |
434 |
with minimal interference. |
435 |
|
436 |
Fig.~\ref{fig:call-tree-adams-bashforth} shows the calling sequence for |
437 |
stepping forward a tracer variable such as temperature. |
438 |
|
439 |
In order to fit within the pressure method, the implicit viscosity |
440 |
must not alter the barotropic flow. In other words, it can only |
441 |
redistribute momentum in the vertical. The upshot of this is that |
442 |
although vertical viscosity may be backward implicit and |
443 |
unconditionally stable, no-slip boundary conditions may not be made |
444 |
implicit and are thus cast as a an explicit drag term. |
445 |
|
446 |
\section{Synchronous time-stepping: variables co-located in time} |
447 |
\label{sect:adams-bashforth-sync} |
448 |
\begin{rawhtml} |
449 |
<!-- CMIREDIR:adams_bashforth_sync: --> |
450 |
\end{rawhtml} |
451 |
|
452 |
\begin{figure} |
453 |
\begin{center} |
454 |
\resizebox{5.0in}{!}{\includegraphics{part2/adams-bashforth-sync.eps}} |
455 |
\end{center} |
456 |
\caption{ |
457 |
A schematic of the explicit Adams-Bashforth and implicit time-stepping |
458 |
phases of the algorithm. All prognostic variables are co-located in |
459 |
time. Explicit tendencies are evaluated at time level $n$ as a |
460 |
function of the state at that time level (dotted arrow). The explicit |
461 |
tendency from the previous time level, $n-1$, is used to extrapolate |
462 |
tendencies to $n+1/2$ (dashed arrow). This extrapolated tendency |
463 |
allows variables to be stably integrated forward-in-time to render an |
464 |
estimate ($*$-variables) at the $n+1$ time level (solid |
465 |
arc-arrow). The operator ${\cal L}$ formed from implicit-in-time terms |
466 |
is solved to yield the state variables at time level $n+1$. } |
467 |
\label{fig:adams-bashforth-sync} |
468 |
\end{figure} |
469 |
|
470 |
\begin{figure} |
471 |
\begin{center} \fbox{ \begin{minipage}{4.7in} \begin{tabbing} |
472 |
aaa \= aaa \= aaa \= aaa \= aaa \= aaa \kill |
473 |
FORWARD\_STEP \\ |
474 |
\>\> EXTERNAL\_FIELDS\_LOAD\\ |
475 |
\>\> DO\_ATMOSPHERIC\_PHYS \\ |
476 |
\>\> DO\_OCEANIC\_PHYS \\ |
477 |
\> THERMODYNAMICS \\ |
478 |
\>\> CALC\_GT \\ |
479 |
\>\>\> GAD\_CALC\_RHS \` $G_\theta^n = G_\theta( u, \theta^n )$ (\ref{eq:Gt-n-sync})\\ |
480 |
\>\>\> EXTERNAL\_FORCING \` $G_\theta^n = G_\theta^n + {\cal Q}$ \\ |
481 |
\>\>\> ADAMS\_BASHFORTH2 \` $G_\theta^{(n+1/2)}$ (\ref{eq:Gt-n+5-sync}) \\ |
482 |
\>\> TIMESTEP\_TRACER \` $\theta^*$ (\ref{eq:tstar-sync}) \\ |
483 |
\>\> IMPLDIFF \` $\theta^{(n+1)}$ (\ref{eq:t-n+1-sync}) \\ |
484 |
\> DYNAMICS \\ |
485 |
\>\> CALC\_PHI\_HYD \` $\phi_{hyd}^n$ (\ref{eq:phi-hyd-sync}) \\ |
486 |
\>\> MOM\_FLUXFORM or MOM\_VECINV \` $G_{\vec{\bf v}}^n$ (\ref{eq:Gv-n-sync})\\ |
487 |
\>\> TIMESTEP \` $\vec{\bf v}^*$ (\ref{eq:Gv-n+5-sync}, \ref{eq:vstar-sync}) \\ |
488 |
\>\> IMPLDIFF \` $\vec{\bf v}^{**}$ (\ref{eq:vstarstar-sync}) \\ |
489 |
\> UPDATE\_R\_STAR or UPDATE\_SURF\_DR \` (NonLin-FS only)\\ |
490 |
\> SOLVE\_FOR\_PRESSURE \\ |
491 |
\>\> CALC\_DIV\_GHAT \` $\eta^*$ (\ref{eq:nstar-sync}) \\ |
492 |
\>\> CG2D \` $\eta^{n+1}$ (\ref{eq:elliptic-sync}) \\ |
493 |
\> MOMENTUM\_CORRECTION\_STEP \\ |
494 |
\>\> CALC\_GRAD\_PHI\_SURF \` $\nabla \eta^{n+1}$ \\ |
495 |
\>\> CORRECTION\_STEP \` $u^{n+1}$,$v^{n+1}$ (\ref{eq:v-n+1-sync})\\ |
496 |
\> TRACERS\_CORRECTION\_STEP \\ |
497 |
\>\> CYCLE\_TRACER \` $\theta^{n+1}$ \\ |
498 |
\>\> FILTER \` Shapiro Filter, Zonal Filter (FFT) \\ |
499 |
\>\> CONVECTIVE\_ADJUSTMENT \` \\ |
500 |
\end{tabbing} \end{minipage} } \end{center} |
501 |
\caption{ |
502 |
Calling tree for the overall synchronous algorithm using |
503 |
Adams-Bashforth time-stepping. |
504 |
The place where the model geometry |
505 |
({\bf hFac} factors) is updated is added here but is only relevant |
506 |
for the non-linear free-surface algorithm. |
507 |
For completeness, the external forcing, |
508 |
ocean and atmospheric physics have been added, although they are mainly |
509 |
optional} |
510 |
\label{fig:call-tree-adams-bashforth-sync} |
511 |
\end{figure} |
512 |
|
513 |
The Adams-Bashforth extrapolation of explicit tendencies fits neatly |
514 |
into the pressure method algorithm when all state variables are |
515 |
co-located in time. Fig.~\ref{fig:adams-bashforth-sync} illustrates |
516 |
the location of variables in time and the evolution of the algorithm |
517 |
with time. The algorithm can be represented by the sequential solution |
518 |
of the follow equations: |
519 |
\begin{eqnarray} |
520 |
G_{\theta,S}^{n} & = & G_{\theta,S} ( u^n, \theta^n, S^n ) |
521 |
\label{eq:Gt-n-sync} \\ |
522 |
G_{\theta,S}^{(n+1/2)} & = & (3/2+\epsilon_{AB}) G_{\theta,S}^{n}-(1/2+\epsilon_{AB}) G_{\theta,S}^{n-1} |
523 |
\label{eq:Gt-n+5-sync} \\ |
524 |
(\theta^*,S^*) & = & (\theta^{n},S^{n}) + \Delta t G_{\theta,S}^{(n+1/2)} |
525 |
\label{eq:tstar-sync} \\ |
526 |
(\theta^{n+1},S^{n+1}) & = & {\cal L}^{-1}_{\theta,S} (\theta^*,S^*) |
527 |
\label{eq:t-n+1-sync} \\ |
528 |
\phi^n_{hyd} & = & \int b(\theta^n,S^n) dr |
529 |
\label{eq:phi-hyd-sync} \\ |
530 |
\vec{\bf G}_{\vec{\bf v}}^{n} & = & \vec{\bf G}_{\vec{\bf v}} ( \vec{\bf v}^n, \phi^n_{hyd} ) |
531 |
\label{eq:Gv-n-sync} \\ |
532 |
\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} |
533 |
\label{eq:Gv-n+5-sync} \\ |
534 |
\vec{\bf v}^{*} & = & \vec{\bf v}^{n} + \Delta t \vec{\bf G}_{\vec{\bf v}}^{(n+1/2)} |
535 |
\label{eq:vstar-sync} \\ |
536 |
\vec{\bf v}^{**} & = & {\cal L}_{\vec{\bf v}}^{-1} ( \vec{\bf v}^* ) |
537 |
\label{eq:vstarstar-sync} \\ |
538 |
\eta^* & = & \epsilon_{fs} \left( \eta^{n} + \Delta t (P-E) \right)- \Delta t |
539 |
\nabla \cdot H \widehat{ \vec{\bf v}^{**} } |
540 |
\label{eq:nstar-sync} \\ |
541 |
\nabla \cdot g H \nabla \eta^{n+1} & - & \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2} |
542 |
~ = ~ - \frac{\eta^*}{\Delta t^2} |
543 |
\label{eq:elliptic-sync} \\ |
544 |
\vec{\bf v}^{n+1} & = & \vec{\bf v}^{*} - \Delta t g \nabla \eta^{n+1} |
545 |
\label{eq:v-n+1-sync} |
546 |
\end{eqnarray} |
547 |
Fig.~\ref{fig:adams-bashforth-sync} illustrates the location of |
548 |
variables in time and evolution of the algorithm with time. The |
549 |
Adams-Bashforth extrapolation of the tracer tendencies is illustrated |
550 |
by the dashed arrow, the prediction at $n+1$ is indicated by the |
551 |
solid arc. Inversion of the implicit terms, ${\cal |
552 |
L}^{-1}_{\theta,S}$, then yields the new tracer fields at $n+1$. All |
553 |
these operations are carried out in subroutine {\em THERMODYNAMICS} an |
554 |
subsidiaries, which correspond to equations \ref{eq:Gt-n-sync} to |
555 |
\ref{eq:t-n+1-sync}. |
556 |
Similarly illustrated is the Adams-Bashforth extrapolation of |
557 |
accelerations, stepping forward and solving of implicit viscosity and |
558 |
surface pressure gradient terms, corresponding to equations |
559 |
\ref{eq:Gv-n-sync} to \ref{eq:v-n+1-sync}. |
560 |
These operations are carried out in subroutines {\em DYNAMCIS}, {\em |
561 |
SOLVE\_FOR\_PRESSURE} and {\em MOMENTUM\_CORRECTION\_STEP}. This, then, |
562 |
represents an entire algorithm for stepping forward the model one |
563 |
time-step. The corresponding calling tree is given in |
564 |
\ref{fig:call-tree-adams-bashforth-sync}. |
565 |
|
566 |
\section{Staggered baroclinic time-stepping} |
567 |
\label{sect:adams-bashforth-staggered} |
568 |
\begin{rawhtml} |
569 |
<!-- CMIREDIR:adams_bashforth_staggered: --> |
570 |
\end{rawhtml} |
571 |
|
572 |
\begin{figure} |
573 |
\begin{center} |
574 |
\resizebox{5.5in}{!}{\includegraphics{part2/adams-bashforth-staggered.eps}} |
575 |
\end{center} |
576 |
\caption{ |
577 |
A schematic of the explicit Adams-Bashforth and implicit time-stepping |
578 |
phases of the algorithm but with staggering in time of thermodynamic |
579 |
variables with the flow. |
580 |
Explicit momentum tendencies are evaluated at time level $n-1/2$ as a |
581 |
function of the flow field at that time level $n-1/2$. |
582 |
The explicit tendency from the previous time level, $n-3/2$, is used to |
583 |
extrapolate tendencies to $n$ (dashed arrow). |
584 |
The hydrostatic pressure/geo-potential $\phi_{hyd}$ is evaluated directly |
585 |
at time level $n$ (vertical arrows) and used with the extrapolated tendencies |
586 |
to step forward the flow variables from $n-1/2$ to $n+1/2$ (solid arc-arrow). |
587 |
The implicit-in-time operator ${\cal L}_{\bf u,v}$ (vertical arrows) is |
588 |
then applied to the previous estimation of the the flow field ($*$-variables) |
589 |
and yields to the two velocity components $u,v$ at time level $n+1/2$. |
590 |
These are then used to calculate the advection term (dashed arc-arrow) |
591 |
of the thermo-dynamics tendencies at time step $n$. |
592 |
The extrapolated thermodynamics tendency, from time level $n-1$ and $n$ |
593 |
to $n+1/2$, allows thermodynamic variables to be stably integrated |
594 |
forward-in-time (solid arc-arrow) up to time level $n+1$. |
595 |
} |
596 |
\label{fig:adams-bashforth-staggered} |
597 |
\end{figure} |
598 |
|
599 |
For well stratified problems, internal gravity waves may be the |
600 |
limiting process for determining a stable time-step. In the |
601 |
circumstance, it is more efficient to stagger in time the |
602 |
thermodynamic variables with the flow |
603 |
variables. Fig.~\ref{fig:adams-bashforth-staggered} illustrates the |
604 |
staggering and algorithm. The key difference between this and |
605 |
Fig.~\ref{fig:adams-bashforth-sync} is that the thermodynamic variables |
606 |
are solved after the dynamics, using the recently updated flow field. |
607 |
This essentially allows the gravity wave terms to leap-frog in |
608 |
time giving second order accuracy and more stability. |
609 |
|
610 |
The essential change in the staggered algorithm is that the |
611 |
thermodynamics solver is delayed from half a time step, |
612 |
allowing the use of the most recent velocities to compute |
613 |
the advection terms. Once the thermodynamics fields are |
614 |
updated, the hydrostatic pressure is computed |
615 |
to step forwrad the dynamics. |
616 |
Note that the pressure gradient must also be taken out of the |
617 |
Adams-Bashforth extrapolation. Also, retaining the integer time-levels, |
618 |
$n$ and $n+1$, does not give a user the sense of where variables are |
619 |
located in time. Instead, we re-write the entire algorithm, |
620 |
\ref{eq:Gt-n-sync} to \ref{eq:v-n+1-sync}, annotating the |
621 |
position in time of variables appropriately: |
622 |
\begin{eqnarray} |
623 |
\phi^{n}_{hyd} & = & \int b(\theta^{n},S^{n}) dr |
624 |
\label{eq:phi-hyd-staggered} \\ |
625 |
\vec{\bf G}_{\vec{\bf v}}^{n-1/2} & = & \vec{\bf G}_{\vec{\bf v}} ( \vec{\bf v}^{n-1/2} ) |
626 |
\label{eq:Gv-n-staggered} \\ |
627 |
\vec{\bf G}_{\vec{\bf v}}^{(n)} & = & (3/2 + \epsilon_{AB} ) \vec{\bf G}_{\vec{\bf v}}^{n-1/2} - (1/2 + \epsilon_{AB} ) \vec{\bf G}_{\vec{\bf v}}^{n-3/2} |
628 |
\label{eq:Gv-n+5-staggered} \\ |
629 |
\vec{\bf v}^{*} & = & \vec{\bf v}^{n-1/2} + \Delta t \left( \vec{\bf G}_{\vec{\bf v}}^{(n)} - \nabla \phi_{hyd}^{n} \right) |
630 |
\label{eq:vstar-staggered} \\ |
631 |
\vec{\bf v}^{**} & = & {\cal L}_{\vec{\bf v}}^{-1} ( \vec{\bf v}^* ) |
632 |
\label{eq:vstarstar-staggered} \\ |
633 |
\eta^* & = & \epsilon_{fs} \left( \eta^{n-1/2} + \Delta t (P-E)^n \right)- \Delta t |
634 |
\nabla \cdot H \widehat{ \vec{\bf v}^{**} } |
635 |
\label{eq:nstar-staggered} \\ |
636 |
\nabla \cdot g H \nabla \eta^{n+1/2} & - & \frac{\epsilon_{fs} \eta^{n+1/2}}{\Delta t^2} |
637 |
~ = ~ - \frac{\eta^*}{\Delta t^2} |
638 |
\label{eq:elliptic-staggered} \\ |
639 |
\vec{\bf v}^{n+1/2} & = & \vec{\bf v}^{*} - \Delta t g \nabla \eta^{n+1/2} |
640 |
\label{eq:v-n+1-staggered} \\ |
641 |
G_{\theta,S}^{n} & = & G_{\theta,S} ( u^{n+1/2}, \theta^{n}, S^{n} ) |
642 |
\label{eq:Gt-n-staggered} \\ |
643 |
G_{\theta,S}^{(n+1/2)} & = & (3/2+\epsilon_{AB}) G_{\theta,S}^{n}-(1/2+\epsilon_{AB}) G_{\theta,S}^{n-1} |
644 |
\label{eq:Gt-n+5-staggered} \\ |
645 |
(\theta^*,S^*) & = & (\theta^{n},S^{n}) + \Delta t G_{\theta,S}^{(n+1/2)} |
646 |
\label{eq:tstar-staggered} \\ |
647 |
(\theta^{n+1},S^{n+1}) & = & {\cal L}^{-1}_{\theta,S} (\theta^*,S^*) |
648 |
\label{eq:t-n+1-staggered} |
649 |
\end{eqnarray} |
650 |
The corresponding calling tree is given in |
651 |
\ref{fig:call-tree-adams-bashforth-staggered}. |
652 |
The staggered algorithm is activated with the run-time flag |
653 |
{\bf staggerTimeStep}{\em=.TRUE.} in parameter file {\em data}, |
654 |
namelist {\em PARM01}. |
655 |
|
656 |
\begin{figure} |
657 |
\begin{center} \fbox{ \begin{minipage}{4.7in} \begin{tabbing} |
658 |
aaa \= aaa \= aaa \= aaa \= aaa \= aaa \kill |
659 |
FORWARD\_STEP \\ |
660 |
\>\> EXTERNAL\_FIELDS\_LOAD\\ |
661 |
\>\> DO\_ATMOSPHERIC\_PHYS \\ |
662 |
\>\> DO\_OCEANIC\_PHYS \\ |
663 |
\> DYNAMICS \\ |
664 |
\>\> CALC\_PHI\_HYD \` $\phi_{hyd}^n$ (\ref{eq:phi-hyd-staggered}) \\ |
665 |
\>\> MOM\_FLUXFORM or MOM\_VECINV \` $G_{\vec{\bf v}}^{n-1/2}$ |
666 |
(\ref{eq:Gv-n-staggered})\\ |
667 |
\>\> TIMESTEP \` $\vec{\bf v}^*$ (\ref{eq:Gv-n+5-staggered}, |
668 |
\ref{eq:vstar-staggered}) \\ |
669 |
\>\> IMPLDIFF \` $\vec{\bf v}^{**}$ (\ref{eq:vstarstar-staggered}) \\ |
670 |
\> UPDATE\_R\_STAR or UPDATE\_SURF\_DR \` (NonLin-FS only)\\ |
671 |
\> SOLVE\_FOR\_PRESSURE \\ |
672 |
\>\> CALC\_DIV\_GHAT \` $\eta^*$ (\ref{eq:nstar-staggered}) \\ |
673 |
\>\> CG2D \` $\eta^{n+1/2}$ (\ref{eq:elliptic-staggered}) \\ |
674 |
\> MOMENTUM\_CORRECTION\_STEP \\ |
675 |
\>\> CALC\_GRAD\_PHI\_SURF \` $\nabla \eta^{n+1/2}$ \\ |
676 |
\>\> CORRECTION\_STEP \` $u^{n+1/2}$,$v^{n+1/2}$ (\ref{eq:v-n+1-staggered})\\ |
677 |
\> THERMODYNAMICS \\ |
678 |
\>\> CALC\_GT \\ |
679 |
\>\>\> GAD\_CALC\_RHS \` $G_\theta^n = G_\theta( u, \theta^n )$ |
680 |
(\ref{eq:Gt-n-staggered})\\ |
681 |
\>\>\> EXTERNAL\_FORCING \` $G_\theta^n = G_\theta^n + {\cal Q}$ \\ |
682 |
\>\>\> ADAMS\_BASHFORTH2 \` $G_\theta^{(n+1/2)}$ (\ref{eq:Gt-n+5-staggered}) \\ |
683 |
\>\> TIMESTEP\_TRACER \` $\theta^*$ (\ref{eq:tstar-staggered}) \\ |
684 |
\>\> IMPLDIFF \` $\theta^{(n+1)}$ (\ref{eq:t-n+1-staggered}) \\ |
685 |
\> TRACERS\_CORRECTION\_STEP \\ |
686 |
\>\> CYCLE\_TRACER \` $\theta^{n+1}$ \\ |
687 |
\>\> FILTER \` Shapiro Filter, Zonal Filter (FFT) \\ |
688 |
\>\> CONVECTIVE\_ADJUSTMENT \` \\ |
689 |
\end{tabbing} \end{minipage} } \end{center} |
690 |
\caption{ |
691 |
Calling tree for the overall staggered algorithm using |
692 |
Adams-Bashforth time-stepping. |
693 |
The place where the model geometry |
694 |
({\bf hFac} factors) is updated is added here but is only relevant |
695 |
for the non-linear free-surface algorithm. |
696 |
} |
697 |
\label{fig:call-tree-adams-bashforth-staggered} |
698 |
\end{figure} |
699 |
|
700 |
The only difficulty with this approach is apparent in equation |
701 |
\ref{eq:Gt-n-staggered} and illustrated by the dotted arrow |
702 |
connecting $u,v^{n+1/2}$ with $G_\theta^{n}$. The flow used to advect |
703 |
tracers around is not naturally located in time. This could be avoided |
704 |
by applying the Adams-Bashforth extrapolation to the tracer field |
705 |
itself and advecting that around but this approach is not yet |
706 |
available. We're not aware of any detrimental effect of this |
707 |
feature. The difficulty lies mainly in interpretation of what |
708 |
time-level variables and terms correspond to. |
709 |
|
710 |
|
711 |
\section{Non-hydrostatic formulation} |
712 |
\label{sect:non-hydrostatic} |
713 |
\begin{rawhtml} |
714 |
<!-- CMIREDIR:non-hydrostatic_formulation: --> |
715 |
\end{rawhtml} |
716 |
|
717 |
The non-hydrostatic formulation re-introduces the full vertical |
718 |
momentum equation and requires the solution of a 3-D elliptic |
719 |
equations for non-hydrostatic pressure perturbation. We still |
720 |
intergrate vertically for the hydrostatic pressure and solve a 2-D |
721 |
elliptic equation for the surface pressure/elevation for this reduces |
722 |
the amount of work needed to solve for the non-hydrostatic pressure. |
723 |
|
724 |
The momentum equations are discretized in time as follows: |
725 |
\begin{eqnarray} |
726 |
\frac{1}{\Delta t} u^{n+1} + g \partial_x \eta^{n+1} + \partial_x \phi_{nh}^{n+1} |
727 |
& = & \frac{1}{\Delta t} u^{n} + G_u^{(n+1/2)} \label{eq:discrete-time-u-nh} \\ |
728 |
\frac{1}{\Delta t} v^{n+1} + g \partial_y \eta^{n+1} + \partial_y \phi_{nh}^{n+1} |
729 |
& = & \frac{1}{\Delta t} v^{n} + G_v^{(n+1/2)} \label{eq:discrete-time-v-nh} \\ |
730 |
\frac{1}{\Delta t} w^{n+1} + \partial_r \phi_{nh}^{n+1} |
731 |
& = & \frac{1}{\Delta t} w^{n} + G_w^{(n+1/2)} \label{eq:discrete-time-w-nh} |
732 |
\end{eqnarray} |
733 |
which must satisfy the discrete-in-time depth integrated continuity, |
734 |
equation~\ref{eq:discrete-time-backward-free-surface} and the local continuity equation |
735 |
\begin{equation} |
736 |
\partial_x u^{n+1} + \partial_y v^{n+1} + \partial_r w^{n+1} = 0 |
737 |
\label{eq:non-divergence-nh} |
738 |
\end{equation} |
739 |
As before, the explicit predictions for momentum are consolidated as: |
740 |
\begin{eqnarray*} |
741 |
u^* & = & u^n + \Delta t G_u^{(n+1/2)} \\ |
742 |
v^* & = & v^n + \Delta t G_v^{(n+1/2)} \\ |
743 |
w^* & = & w^n + \Delta t G_w^{(n+1/2)} |
744 |
\end{eqnarray*} |
745 |
but this time we introduce an intermediate step by splitting the |
746 |
tendancy of the flow as follows: |
747 |
\begin{eqnarray} |
748 |
u^{n+1} = u^{**} - \Delta t \partial_x \phi_{nh}^{n+1} |
749 |
& & |
750 |
u^{**} = u^{*} - \Delta t g \partial_x \eta^{n+1} \\ |
751 |
v^{n+1} = v^{**} - \Delta t \partial_y \phi_{nh}^{n+1} |
752 |
& & |
753 |
v^{**} = v^{*} - \Delta t g \partial_y \eta^{n+1} |
754 |
\end{eqnarray} |
755 |
Substituting into the depth integrated continuity |
756 |
(equation~\ref{eq:discrete-time-backward-free-surface}) gives |
757 |
\begin{equation} |
758 |
\partial_x H \partial_x \left( g \eta^{n+1} + \widehat{\phi}_{nh}^{n+1} \right) |
759 |
+ |
760 |
\partial_y H \partial_y \left( g \eta^{n+1} + \widehat{\phi}_{nh}^{n+1} \right) |
761 |
- \frac{\epsilon_{fs}\eta^*}{\Delta t^2} |
762 |
= - \frac{\eta^*}{\Delta t^2} |
763 |
\end{equation} |
764 |
which is approximated by equation |
765 |
\ref{eq:elliptic-backward-free-surface} on the basis that i) |
766 |
$\phi_{nh}^{n+1}$ is not yet known and ii) $\nabla \widehat{\phi}_{nh} |
767 |
<< g \nabla \eta$. If \ref{eq:elliptic-backward-free-surface} is |
768 |
solved accurately then the implication is that $\widehat{\phi}_{nh} |
769 |
\approx 0$ so that thet non-hydrostatic pressure field does not drive |
770 |
barotropic motion. |
771 |
|
772 |
The flow must satisfy non-divergence |
773 |
(equation~\ref{eq:non-divergence-nh}) locally, as well as depth |
774 |
integrated, and this constraint is used to form a 3-D elliptic |
775 |
equations for $\phi_{nh}^{n+1}$: |
776 |
\begin{equation} |
777 |
\partial_{xx} \phi_{nh}^{n+1} + \partial_{yy} \phi_{nh}^{n+1} + |
778 |
\partial_{rr} \phi_{nh}^{n+1} = |
779 |
\partial_x u^{**} + \partial_y v^{**} + \partial_r w^{*} |
780 |
\end{equation} |
781 |
|
782 |
The entire algorithm can be summarized as the sequential solution of |
783 |
the following equations: |
784 |
\begin{eqnarray} |
785 |
u^{*} & = & u^{n} + \Delta t G_u^{(n+1/2)} \label{eq:ustar-nh} \\ |
786 |
v^{*} & = & v^{n} + \Delta t G_v^{(n+1/2)} \label{eq:vstar-nh} \\ |
787 |
w^{*} & = & w^{n} + \Delta t G_w^{(n+1/2)} \label{eq:wstar-nh} \\ |
788 |
\eta^* ~ = ~ \epsilon_{fs} \left( \eta^{n} + \Delta t (P-E) \right) |
789 |
& - & \Delta t |
790 |
\partial_x H \widehat{u^{*}} |
791 |
+ \partial_y H \widehat{v^{*}} |
792 |
\\ |
793 |
\partial_x g H \partial_x \eta^{n+1} |
794 |
+ \partial_y g H \partial_y \eta^{n+1} |
795 |
& - & \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2} |
796 |
~ = ~ |
797 |
- \frac{\eta^*}{\Delta t^2} |
798 |
\label{eq:elliptic-nh} |
799 |
\\ |
800 |
u^{**} & = & u^{*} - \Delta t g \partial_x \eta^{n+1} \label{eq:unx-nh}\\ |
801 |
v^{**} & = & v^{*} - \Delta t g \partial_y \eta^{n+1} \label{eq:vnx-nh}\\ |
802 |
\partial_{xx} \phi_{nh}^{n+1} + \partial_{yy} \phi_{nh}^{n+1} + |
803 |
\partial_{rr} \phi_{nh}^{n+1} & = & |
804 |
\partial_x u^{**} + \partial_y v^{**} + \partial_r w^{*} \\ |
805 |
u^{n+1} & = & u^{**} - \Delta t \partial_x \phi_{nh}^{n+1} \label{eq:un+1-nh}\\ |
806 |
v^{n+1} & = & v^{**} - \Delta t \partial_y \phi_{nh}^{n+1} \label{eq:vn+1-nh}\\ |
807 |
\partial_r w^{n+1} & = & - \partial_x u^{n+1} - \partial_y v^{n+1} |
808 |
\end{eqnarray} |
809 |
where the last equation is solved by vertically integrating for |
810 |
$w^{n+1}$. |
811 |
|
812 |
|
813 |
|
814 |
\section{Variants on the Free Surface} |
815 |
\label{sect:free-surface} |
816 |
|
817 |
We now describe the various formulations of the free-surface that |
818 |
include non-linear forms, implicit in time using Crank-Nicholson, |
819 |
explicit and [one day] split-explicit. First, we'll reiterate the |
820 |
underlying algorithm but this time using the notation consistent with |
821 |
the more general vertical coordinate $r$. The elliptic equation for |
822 |
free-surface coordinate (units of $r$), corresponding to |
823 |
\ref{eq:discrete-time-backward-free-surface}, and |
824 |
assuming no non-hydrostatic effects ($\epsilon_{nh} = 0$) is: |
825 |
\begin{eqnarray} |
826 |
\epsilon_{fs} {\eta}^{n+1} - |
827 |
{\bf \nabla}_h \cdot \Delta t^2 (R_o-R_{fixed}) {\bf \nabla}_h b_s |
828 |
{\eta}^{n+1} = {\eta}^* |
829 |
\label{eq-solve2D} |
830 |
\end{eqnarray} |
831 |
where |
832 |
\begin{eqnarray} |
833 |
{\eta}^* = \epsilon_{fs} \: {\eta}^{n} - |
834 |
\Delta t {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o} \vec{\bf v}^* dr |
835 |
\: + \: \epsilon_{fw} \Delta t (P-E)^{n} |
836 |
\label{eq-solve2D_rhs} |
837 |
\end{eqnarray} |
838 |
|
839 |
\fbox{ \begin{minipage}{4.75in} |
840 |
{\em S/R SOLVE\_FOR\_PRESSURE} ({\em solve\_for\_pressure.F}) |
841 |
|
842 |
$u^*$: {\bf gU} ({\em DYNVARS.h}) |
843 |
|
844 |
$v^*$: {\bf gV} ({\em DYNVARS.h}) |
845 |
|
846 |
$\eta^*$: {\bf cg2d\_b} (\em SOLVE\_FOR\_PRESSURE.h) |
847 |
|
848 |
$\eta^{n+1}$: {\bf etaN} (\em DYNVARS.h) |
849 |
|
850 |
\end{minipage} } |
851 |
|
852 |
|
853 |
Once ${\eta}^{n+1}$ has been found, substituting into |
854 |
\ref{eq:discrete-time-u}, \ref{eq:discrete-time-v} yields $\vec{\bf v}^{n+1}$ |
855 |
if the model is hydrostatic ($\epsilon_{nh}=0$): |
856 |
$$ |
857 |
\vec{\bf v}^{n+1} = \vec{\bf v}^{*} |
858 |
- \Delta t {\bf \nabla}_h b_s {\eta}^{n+1} |
859 |
$$ |
860 |
|
861 |
This is known as the correction step. However, when the model is |
862 |
non-hydrostatic ($\epsilon_{nh}=1$) we need an additional step and an |
863 |
additional equation for $\phi'_{nh}$. This is obtained by substituting |
864 |
\ref{eq:discrete-time-u-nh}, \ref{eq:discrete-time-v-nh} and \ref{eq:discrete-time-w-nh} |
865 |
into continuity: |
866 |
\begin{equation} |
867 |
\left[ {\bf \nabla}_h^2 + \partial_{rr} \right] {\phi'_{nh}}^{n+1} |
868 |
= \frac{1}{\Delta t} \left( |
869 |
{\bf \nabla}_h \cdot \vec{\bf v}^{**} + \partial_r \dot{r}^* \right) |
870 |
\end{equation} |
871 |
where |
872 |
\begin{displaymath} |
873 |
\vec{\bf v}^{**} = \vec{\bf v}^* - \Delta t {\bf \nabla}_h b_s {\eta}^{n+1} |
874 |
\end{displaymath} |
875 |
Note that $\eta^{n+1}$ is also used to update the second RHS term |
876 |
$\partial_r \dot{r}^* $ since |
877 |
the vertical velocity at the surface ($\dot{r}_{surf}$) |
878 |
is evaluated as $(\eta^{n+1} - \eta^n) / \Delta t$. |
879 |
|
880 |
Finally, the horizontal velocities at the new time level are found by: |
881 |
\begin{equation} |
882 |
\vec{\bf v}^{n+1} = \vec{\bf v}^{**} |
883 |
- \epsilon_{nh} \Delta t {\bf \nabla}_h {\phi'_{nh}}^{n+1} |
884 |
\end{equation} |
885 |
and the vertical velocity is found by integrating the continuity |
886 |
equation vertically. Note that, for the convenience of the restart |
887 |
procedure, the vertical integration of the continuity equation has |
888 |
been moved to the beginning of the time step (instead of at the end), |
889 |
without any consequence on the solution. |
890 |
|
891 |
\fbox{ \begin{minipage}{4.75in} |
892 |
{\em S/R CORRECTION\_STEP} ({\em correction\_step.F}) |
893 |
|
894 |
$\eta^{n+1}$: {\bf etaN} (\em DYNVARS.h) |
895 |
|
896 |
$\phi_{nh}^{n+1}$: {\bf phi\_nh} (\em NH\_VARS.h) |
897 |
|
898 |
$u^*$: {\bf gU} ({\em DYNVARS.h}) |
899 |
|
900 |
$v^*$: {\bf gV} ({\em DYNVARS.h}) |
901 |
|
902 |
$u^{n+1}$: {\bf uVel} ({\em DYNVARS.h}) |
903 |
|
904 |
$v^{n+1}$: {\bf vVel} ({\em DYNVARS.h}) |
905 |
|
906 |
\end{minipage} } |
907 |
|
908 |
|
909 |
|
910 |
Regarding the implementation of the surface pressure solver, all |
911 |
computation are done within the routine {\it SOLVE\_FOR\_PRESSURE} and |
912 |
its dependent calls. The standard method to solve the 2D elliptic |
913 |
problem (\ref{eq-solve2D}) uses the conjugate gradient method (routine |
914 |
{\it CG2D}); the solver matrix and conjugate gradient operator are |
915 |
only function of the discretized domain and are therefore evaluated |
916 |
separately, before the time iteration loop, within {\it INI\_CG2D}. |
917 |
The computation of the RHS $\eta^*$ is partly done in {\it |
918 |
CALC\_DIV\_GHAT} and in {\it SOLVE\_FOR\_PRESSURE}. |
919 |
|
920 |
The same method is applied for the non hydrostatic part, using a |
921 |
conjugate gradient 3D solver ({\it CG3D}) that is initialized in {\it |
922 |
INI\_CG3D}. The RHS terms of 2D and 3D problems are computed together |
923 |
at the same point in the code. |
924 |
|
925 |
|
926 |
|
927 |
\subsection{Crank-Nickelson barotropic time stepping} |
928 |
\label{sect:freesurf-CrankNick} |
929 |
|
930 |
The full implicit time stepping described previously is |
931 |
unconditionally stable but damps the fast gravity waves, resulting in |
932 |
a loss of potential energy. The modification presented now allows one |
933 |
to combine an implicit part ($\beta,\gamma$) and an explicit part |
934 |
($1-\beta,1-\gamma$) for the surface pressure gradient ($\beta$) and |
935 |
for the barotropic flow divergence ($\gamma$). |
936 |
\\ |
937 |
For instance, $\beta=\gamma=1$ is the previous fully implicit scheme; |
938 |
$\beta=\gamma=1/2$ is the non damping (energy conserving), unconditionally |
939 |
stable, Crank-Nickelson scheme; $(\beta,\gamma)=(1,0)$ or $=(0,1)$ |
940 |
corresponds to the forward - backward scheme that conserves energy but is |
941 |
only stable for small time steps.\\ |
942 |
In the code, $\beta,\gamma$ are defined as parameters, respectively |
943 |
{\bf implicSurfPress}, {\bf implicDiv2DFlow}. They are read from |
944 |
the main parameter file "{\em data}" and are set by default to 1,1. |
945 |
|
946 |
Equations \ref{eq:ustar-backward-free-surface} -- |
947 |
\ref{eq:vn+1-backward-free-surface} are modified as follows: |
948 |
\begin{eqnarray*} |
949 |
\frac{ \vec{\bf v}^{n+1} }{ \Delta t } |
950 |
+ {\bf \nabla}_h b_s [ \beta {\eta}^{n+1} + (1-\beta) {\eta}^{n} ] |
951 |
+ \epsilon_{nh} {\bf \nabla}_h {\phi'_{nh}}^{n+1} |
952 |
= \frac{ \vec{\bf v}^* }{ \Delta t } |
953 |
\end{eqnarray*} |
954 |
\begin{eqnarray} |
955 |
\epsilon_{fs} \frac{ {\eta}^{n+1} - {\eta}^{n} }{ \Delta t} |
956 |
+ {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o} |
957 |
[ \gamma \vec{\bf v}^{n+1} + (1-\gamma) \vec{\bf v}^{n}] dr |
958 |
= \epsilon_{fw} (P-E) |
959 |
\label{eq:eta-n+1-CrankNick} |
960 |
\end{eqnarray} |
961 |
where: |
962 |
\begin{eqnarray*} |
963 |
\vec{\bf v}^* & = & |
964 |
\vec{\bf v} ^{n} + \Delta t \vec{\bf G}_{\vec{\bf v}} ^{(n+1/2)} |
965 |
+ (\beta-1) \Delta t {\bf \nabla}_h b_s {\eta}^{n} |
966 |
+ \Delta t {\bf \nabla}_h {\phi'_{hyd}}^{(n+1/2)} |
967 |
\\ |
968 |
{\eta}^* & = & |
969 |
\epsilon_{fs} {\eta}^{n} + \epsilon_{fw} \Delta t (P-E) |
970 |
- \Delta t {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o} |
971 |
[ \gamma \vec{\bf v}^* + (1-\gamma) \vec{\bf v}^{n}] dr |
972 |
\end{eqnarray*} |
973 |
\\ |
974 |
In the hydrostatic case ($\epsilon_{nh}=0$), allowing us to find |
975 |
${\eta}^{n+1}$, thus: |
976 |
$$ |
977 |
\epsilon_{fs} {\eta}^{n+1} - |
978 |
{\bf \nabla}_h \cdot \beta\gamma \Delta t^2 b_s (R_o - R_{fixed}) |
979 |
{\bf \nabla}_h {\eta}^{n+1} |
980 |
= {\eta}^* |
981 |
$$ |
982 |
and then to compute ({\em CORRECTION\_STEP}): |
983 |
$$ |
984 |
\vec{\bf v}^{n+1} = \vec{\bf v}^{*} |
985 |
- \beta \Delta t {\bf \nabla}_h b_s {\eta}^{n+1} |
986 |
$$ |
987 |
|
988 |
%The non-hydrostatic part is solved as described previously. |
989 |
|
990 |
\noindent |
991 |
Notes: |
992 |
\begin{enumerate} |
993 |
\item The RHS term of equation \ref{eq:eta-n+1-CrankNick} |
994 |
corresponds the contribution of fresh water flux (P-E) |
995 |
to the free-surface variations ($\epsilon_{fw}=1$, |
996 |
{\bf useRealFreshWater}{\em=TRUE} in parameter file {\em data}). |
997 |
In order to remain consistent with the tracer equation, specially in |
998 |
the non-linear free-surface formulation, this term is also |
999 |
affected by the Crank-Nickelson time stepping. The RHS reads: |
1000 |
$\epsilon_{fw} ( \gamma (P-E)^{n+1/2} + (1-\gamma) (P-E)^{n-1/2} )$ |
1001 |
\item The non-hydrostatic part of the code has not yet been |
1002 |
updated, and therefore cannot be used with $(\beta,\gamma) \neq (1,1)$. |
1003 |
\item The stability criteria with Crank-Nickelson time stepping |
1004 |
for the pure linear gravity wave problem in cartesian coordinates is: |
1005 |
\begin{itemize} |
1006 |
\item $\beta + \gamma < 1$ : unstable |
1007 |
\item $\beta \geq 1/2$ and $ \gamma \geq 1/2$ : stable |
1008 |
\item $\beta + \gamma \geq 1$ : stable if |
1009 |
$$ |
1010 |
c_{max}^2 (\beta - 1/2)(\gamma - 1/2) + 1 \geq 0 |
1011 |
$$ |
1012 |
$$ |
1013 |
\mbox{with }~ |
1014 |
%c^2 = 2 g H {\Delta t}^2 |
1015 |
%(\frac{1-cos 2 \pi / k}{\Delta x^2} |
1016 |
%+\frac{1-cos 2 \pi / l}{\Delta y^2}) |
1017 |
%$$ |
1018 |
%Practically, the most stringent condition is obtained with $k=l=2$ : |
1019 |
%$$ |
1020 |
c_{max} = 2 \Delta t \: \sqrt{g H} \: |
1021 |
\sqrt{ \frac{1}{\Delta x^2} + \frac{1}{\Delta y^2} } |
1022 |
$$ |
1023 |
\end{itemize} |
1024 |
\end{enumerate} |
1025 |
|