Changeset 4560

2014-03-21T11:54:50+01:00 (7 years ago)

Update time split documentation

1 added
4 edited


  • trunk/DOC/TexFiles/Biblio/Biblio.bib

    r3764 r4560  
    25052505  volume = {87}, 
    25062506  pages = {394--409} 
     2510  author = {A. F. Shchepetkin and J. C. McWilliams}, 
     2511  title = {Handbook of Numerical Analysis, Vol. XIV: Computational Methods for the Ocean and the Atmosphere, pp 121-183}, 
     2512  publisher = {P. G. Ciarlet, editor, R. Temam and J. Tribbia, guest eds., Elsevier Science}, 
     2513  year = {2008}, 
     2514  pages = {784} 
  • trunk/DOC/TexFiles/Chapters/Chap_DYN.tex

    r4147 r4560  
    762762\subsection{Split-Explicit free surface (\key{dynspg\_ts})} 
    765768The split-explicit free surface formulation used in \NEMO (\key{dynspg\_ts} defined), 
    766769also called the time-splitting formulation, follows the one  
    767 proposed by \citet{Griffies_Bk04}. The general idea is to solve the free surface  
     770proposed by \citet{Shchepetkin_McWilliams_OM05}. The general idea is to solve the free surface  
    768771equation and the associated barotropic velocity equations with a smaller time  
    769772step than $\rdt$, the time step used for the three dimensional prognostic  
    771774The size of the small time step, $\rdt_e$ (the external mode or barotropic time step) 
    772775 is provided through the \np{nn\_baro} namelist parameter as:  
    773 $\rdt_e = \rdt / nn\_baro$. 
     776$\rdt_e = \rdt / nn\_baro$. This parameter can be optionally defined automatically (\np{ln\_bt\_nn\_auto}=true)  
     777considering that the stability of the barotropic system is essentially controled by external waves propagation.  
     778Maximum allowed Courant number is in that case time independent, and easily computed online from the input bathymetry. 
     781The barotropic mode solves the following equations: 
     782\begin{subequations} \label{Eq_BT} 
     783  \begin{equation}     \label{Eq_BT_dyn} 
     784\frac{\partial {\rm \overline{{\bf U}}_h} }{\partial t}= 
     785 -f\;{\rm {\bf k}}\times {\rm \overline{{\bf U}}_h}  
     786-g\nabla _h \eta -\frac{c_b^{\textbf U}}{H+\eta} \rm {\overline{{\bf U}}_h} + \rm {\overline{\bf G}} 
     787  \end{equation} 
     789  \begin{equation} \label{Eq_BT_ssh} 
     790\frac{\partial \eta }{\partial t}=-\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\rm{\bf \overline{U}}}_h \,} \right]+P-E 
     791  \end{equation} 
     793where $\rm {\overline{\bf G}}$ is a forcing term held constant, containing coupling term between modes, surface atmospheric forcing as well as slowly varying barotropic terms not explicitly computed to gain efficiency. The third term on the right hand side of \eqref{Eq_BT_dyn} represents the bottom stress (see section \S\ref{ZDF_bfr}), explicitly accounted for at each barotropic iteration. Temporal discretization of the system above follows a three-time step Generalized Forward Backward algorithm detailed in \citet{Shchepetkin_McWilliams_OM05}. AB3-AM4 coefficients used in \NEMO follow the second-order accurate, "multi-purpose" stability compromise as defined in \citet{Shchepetkin_McWilliams_Bk08} (see their figure 12, lower left).  
    776795%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   > 
    777796\begin{figure}[!t]    \begin{center} 
    778 \includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_DYN_dynspg_ts.pdf} 
    779798\caption{  \label{Fig_DYN_dynspg_ts} 
    780799Schematic of the split-explicit time stepping scheme for the external  
    781 and internal modes. Time increases to the right.  
     800and internal modes. Time increases to the right. In this particular exemple,  
     801a boxcar averaging window over $nn\_baro$ barotropic time steps is used ($nn\_bt\_filt=1$) and $nn\_baro=5$. 
    782802Internal mode time steps (which are also the model time steps) are denoted  
    783 by $t-\rdt$, $t, t+\rdt$, and $t+2\rdt$.  
    784 The curved line represents a leap-frog time step, and the smaller time  
    785 steps $N \rdt_e=\frac{3}{2}\rdt$ are denoted by the zig-zag line.  
    786 The vertically integrated forcing \textbf{M}(t) computed at the model time step $t$  
    787 represents the interaction between the external and internal motions.  
    788 While keeping \textbf{M} and freshwater forcing field fixed, a leap-frog  
    789 integration carries the external mode variables (surface height and vertically  
    790 integrated velocity) from $t$ to $t+\frac{3}{2} \rdt$ using N external time  
    791 steps of length $\rdt_e$. Time averaging the external fields over the  
    792 $\frac{2}{3}N+1$ time steps (endpoints included) centers the vertically integrated  
    793 velocity and the sea surface height at the model timestep $t+\rdt$.  
    794 These averaged values are used to update \textbf{M}(t) with both the surface  
    795 pressure gradient and the Coriolis force, therefore providing the $t+\rdt$ 
    796 velocity.  The model time stepping scheme can then be achieved by a baroclinic  
    797 leap-frog time step that carries the surface height from $t-\rdt$ to $t+\rdt$.  } 
     803by $t-\rdt$, $t$ and $t+\rdt$. Variables with $k$ superscript refer to instantaneous barotropic variables,  
     804$< >$ and $<< >>$ operator refer to time filtered variables using respectively primary (red vertical bars) and secondary weights (blue vertical bars).  
     805The former are used to obtain time filtered quantities at $t+\rdt$ while the latter are used to obtain time averaged  
     806transports to advect tracers. 
     807a) Forward time integration: \np{ln\_bt\_fw}=true, \np{ln\_bt\_ave}=true.  
     808b) Centred time integration: \np{ln\_bt\_fw}=false, \np{ln\_bt\_ave}=true.  
     809c) Forward time integration with no time filtering (POM-like scheme): \np{ln\_bt\_fw}=true, \np{ln\_bt\_ave}=false. } 
    798810\end{center}    \end{figure} 
    799811%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   > 
    801 The split-explicit formulation has a damping effect on external gravity waves,  
    802 which is weaker damping than that for the filtered free surface but still significant, as  
    803 shown by \citet{Levier2007} in the case of an analytical barotropic Kelvin wave.  
     813In the default case (\np{ln\_bt\_fw}=true), the external mode is integrated  
     814between \textit{now} and  \textit{after} baroclinic time-steps (Fig.~\ref{Fig_DYN_dynspg_ts}a). To avoid aliasing of fast barotropic motions into three dimensional equations, time filtering is eventually applied on barotropic  
     815quantities (\np{ln\_bt\_ave}=true). In that case, the integration is extended slightly beyond  \textit{after} time step to provide time filtered quantities.  
     816These are used for the subsequent initialization of the barotropic mode in the following baroclinic step.  
     817Since external mode equations written at baroclinic time steps finally follow a forward time stepping scheme,  
     818asselin filtering is not applied to barotropic quantities. \\ 
     819Alternatively, one can choose to integrate barotropic equations starting  
     820from \textit{before} time step (\np{ln\_bt\_fw}=false). Although more computationaly expensive ( \np{nn\_baro} additional iterations are indeed necessary), the baroclinic to barotropic forcing term given at \textit{now} time step  
     821become centred in the middle of the integration window. It can easily be shown that this property  
     822removes part of splitting errors between modes, which increases the overall numerical robustness. 
     823%references to Patrick Marsaleix' work here. Also work done by SHOM group. 
     827As far as tracer conservation is concerned, barotropic velocities used to advect tracers must also be updated  
     828at \textit{now} time step. This implies to change the traditional order of computations in \NEMO: most of momentum   
     829trends (including the barotropic mode calculation) updated first, tracers' after. This \textit{de facto} makes semi-implicit hydrostatic  
     830pressure gradient (see section \S\ref{DYN_hpg_imp}) and time splitting not compatible.  
     831Advective barotropic velocities are obtained by using a secondary set of filtering weights, uniquely defined from the filter  
     832coefficients used for the time averaging (\citet{Shchepetkin_McWilliams_OM05}). Consistency between the time averaged continuity equation and the time stepping of tracers is here the key to obtain exact conservation. 
     836One can eventually choose to feedback instantaneous values by not using any time filter (\np{ln\_bt\_ave}=false).  
     837In that case, external mode equations are continuous in time, ie they are not re-initialized when starting a new  
     838sub-stepping sequence. This is the method used so far in the POM model, the stability being maintained by refreshing at (almost)  
     839each barotropic time step advection and horizontal diffusion terms. Since the latter terms have not been added in \NEMO for  
     840computational efficiency, removing time filtering is not recommended except for debugging purposes.  
     841This may be used for instance to appreciate the damping effect of the standard formulation on external gravity waves in idealized or weakly non-linear cases. Although the damping is lower than for the filtered free surface, it is still significant as shown by \citet{Levier2007} in the case of an analytical barotropic Kelvin wave. 
  • trunk/DOC/TexFiles/Figures/Fig_DYN_dynspg_ts.pdf

    • Property dummyproperty set to 1
  • trunk/DOC/TexFiles/Namelist/namdom

    r4147 r4560  
    1010                           ! 
    1111   rn_rdt      = 5760.     !  time step for the dynamics (and tracer if nn_acc=0) 
    12    nn_baro     =   64      !  number of barotropic time step            ("key_dynspg_ts") 
    1312   rn_atfp     =    0.1    !  asselin time filter parameter 
    1413   nn_acc      =    0      !  acceleration of convergence : =1      used, rdt < rdttra(k) 
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