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- 2011-11-09T18:07:32+01:00 (12 years ago)
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branches/2011/dev_NOC_2011_MERGE/DOC/TexFiles/Chapters/Chap_ZDF.tex
r2541 r3072 1 1 % ================================================================ 2 % Chapter ÑVertical Ocean Physics (ZDF)2 % Chapter Vertical Ocean Physics (ZDF) 3 3 % ================================================================ 4 4 \chapter{Vertical Ocean Physics (ZDF)} … … 539 539 the clipping factor is of crucial importance for the entrainment depth predicted in 540 540 stably stratified situations, and that its value has to be chosen in accordance 541 with the algebraic model for the turbulent ßuxes. The clipping is only activated541 with the algebraic model for the turbulent fluxes. The clipping is only activated 542 542 if \np{ln\_length\_lim}=true, and the $c_{lim}$ is set to the \np{rn\_clim\_galp} value. 543 543 … … 981 981 reduced as necessary to ensure stability; these changes are not reported. 982 982 983 Limits on the bottom friction coefficient are not imposed if the user has elected to 984 handle the bottom friction implicitly (see \S\ref{ZDF_bfr_imp}). The number of potential 985 breaches of the explicit stability criterion are still reported for information purposes. 986 987 % ------------------------------------------------------------------------------------------------------------- 988 % Implicit Bottom Friction 989 % ------------------------------------------------------------------------------------------------------------- 990 \subsection{Implicit Bottom Friction (\np{ln\_bfrimp}$=$\textit{T})} 991 \label{ZDF_bfr_imp} 992 993 An optional implicit form of bottom friction has been implemented to improve 994 model stability. We recommend this option for shelf sea and coastal ocean applications, especially 995 for split-explicit time splitting. This option can be invoked by setting \np{ln\_bfrimp} 996 to \textit{true} in the \textit{nambfr} namelist. This option requires \np{ln\_zdfexp} to be \textit{false} 997 in the \textit{namzdf} namelist. 998 999 This implementation is realised in \mdl{dynzdf\_imp} and \mdl{dynspg\_ts}. In \mdl{dynzdf\_imp}, the 1000 bottom boundary condition is implemented implicitly. 1001 1002 \begin{equation} \label{Eq_dynzdf_bfr} 1003 \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{mbk} 1004 = \binom{c_{b}^{u}u^{n+1}_{mbk}}{c_{b}^{v}v^{n+1}_{mbk}} 1005 \end{equation} 1006 1007 where $mbk$ is the layer number of the bottom wet layer. superscript $n+1$ means the velocity used in the 1008 friction formula is to be calculated, so, it is implicit. 1009 1010 If split-explicit time splitting is used, care must be taken to avoid the double counting of 1011 the bottom friction in the 2-D barotropic momentum equations. As NEMO only updates the barotropic 1012 pressure gradient and Coriolis' forcing terms in the 2-D barotropic calculation, we need to remove 1013 the bottom friction induced by these two terms which has been included in the 3-D momentum trend 1014 and update it with the latest value. On the other hand, the bottom friction contributed by the 1015 other terms (e.g. the advection term, viscosity term) has been included in the 3-D momentum equations 1016 and should not be added in the 2-D barotropic mode. 1017 1018 The implementation of the implicit bottom friction in \mdl{dynspg\_ts} is done in two steps as the 1019 following: 1020 1021 \begin{equation} \label{Eq_dynspg_ts_bfr1} 1022 \frac{\textbf{U}_{med}-\textbf{U}^{m-1}}{2\Delta t}=-g\nabla\eta-f\textbf{k}\times\textbf{U}^{m}+c_{b} 1023 \left(\textbf{U}_{med}-\textbf{U}^{m-1}\right) 1024 \end{equation} 1025 \begin{equation} \label{Eq_dynspg_ts_bfr2} 1026 \frac{\textbf{U}^{m+1}-\textbf{U}_{med}}{2\Delta t}=\textbf{T}+ 1027 \left(g\nabla\eta^{'}+f\textbf{k}\times\textbf{U}^{'}\right)- 1028 2\Delta t_{bc}c_{b}\left(g\nabla\eta^{'}+f\textbf{k}\times\textbf{u}_{b}\right) 1029 \end{equation} 1030 1031 where $\textbf{T}$ is the vertical integrated 3-D momentum trend. We assume the leap-frog time-stepping 1032 is used here. $\Delta t$ is the barotropic mode time step and $\Delta t_{bc}$ is the baroclinic mode time step. 1033 $c_{b}$ is the friction coefficient. $\eta$ is the sea surface level calculated in the barotropic loops 1034 while $\eta^{'}$ is the sea surface level used in the 3-D baroclinic mode. $\textbf{u}_{b}$ is the bottom 1035 layer horizontal velocity. 1036 1037 1038 1039 983 1040 % ------------------------------------------------------------------------------------------------------------- 984 1041 % Bottom Friction with split-explicit time splitting 985 1042 % ------------------------------------------------------------------------------------------------------------- 986 \subsection{Bottom Friction with split-explicit time splitting }1043 \subsection{Bottom Friction with split-explicit time splitting (\np{ln\_bfrimp}$=$\textit{F})} 987 1044 \label{ZDF_bfr_ts} 988 1045 … … 993 1050 {\key{dynspg\_flt}). Extra attention is required, however, when using 994 1051 split-explicit time stepping (\key{dynspg\_ts}). In this case the free surface 995 equation is solved with a small time step \np{ nn\_baro}*\np{rn\_rdt}, while the three996 dimensional prognostic variables are solved with a longer time step that is a997 multiple of \np{rn\_rdt}. The trend in the barotropic momentum due to bottom1052 equation is solved with a small time step \np{rn\_rdt}/\np{nn\_baro}, while the three 1053 dimensional prognostic variables are solved with the longer time step 1054 of \np{rn\_rdt} seconds. The trend in the barotropic momentum due to bottom 998 1055 friction appropriate to this method is that given by the selected parameterisation 999 1056 ($i.e.$ linear or non-linear bottom friction) computed with the evolving velocities … … 1018 1075 \end{enumerate} 1019 1076 1020 Note that the use of an implicit formulation 1077 Note that the use of an implicit formulation within the barotropic loop 1021 1078 for the bottom friction trend means that any limiting of the bottom friction coefficient 1022 1079 in \mdl{dynbfr} does not adversely affect the solution when using split-explicit time 1023 1080 splitting. This is because the major contribution to bottom friction is likely to come from 1024 the barotropic component which uses the unrestricted value of the coefficient. 1025 1026 The implicit formulation takes the form: 1081 the barotropic component which uses the unrestricted value of the coefficient. However, if the 1082 limiting is thought to be having a major effect (a more likely prospect in coastal and shelf seas 1083 applications) then the fully implicit form of the bottom friction should be used (see \S\ref{ZDF_bfr_imp} ) 1084 which can be selected by setting \np{ln\_bfrimp} $=$ \textit{true}. 1085 1086 Otherwise, the implicit formulation takes the form: 1027 1087 \begin{equation} \label{Eq_zdfbfr_implicitts} 1028 1088 \bar{U}^{t+ \rdt} = \; \left [ \bar{U}^{t-\rdt}\; + 2 \rdt\;RHS \right ] / \left [ 1 - 2 \rdt \;c_b^{u} / H_e \right ] … … 1091 1151 The essential goal of the parameterization is to represent the momentum 1092 1152 exchange between the barotropic tides and the unrepresented internal waves 1093 induced by the tidal ßow over rough topography in a stratified ocean.1153 induced by the tidal flow over rough topography in a stratified ocean. 1094 1154 In the current version of \NEMO, the map is built from the output of 1095 1155 the barotropic global ocean tide model MOG2D-G \citep{Carrere_Lyard_GRL03}.
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