Changeset 3294 for trunk/DOC/TexFiles/Chapters/Chap_ZDF.tex
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- 2012-01-28T17:44:18+01:00 (12 years ago)
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r2541 r3294 1 1 % ================================================================ 2 % Chapter ÑVertical Ocean Physics (ZDF)2 % Chapter Vertical Ocean Physics (ZDF) 3 3 % ================================================================ 4 4 \chapter{Vertical Ocean Physics (ZDF)} … … 100 100 $a=5$ and $n=2$. The last three values can be modified by setting the 101 101 \np{rn\_avmri}, \np{rn\_alp} and \np{nn\_ric} namelist parameters, respectively. 102 103 A simple mixing-layer model to transfer and dissipate the atmospheric 104 forcings (wind-stress and buoyancy fluxes) can be activated setting 105 the \np{ln\_mldw} =.true. in the namelist. 106 107 In this case, the local depth of turbulent wind-mixing or "Ekman depth" 108 $h_{e}(x,y,t)$ is evaluated and the vertical eddy coefficients prescribed within this layer. 109 110 This depth is assumed proportional to the "depth of frictional influence" that is limited by rotation: 111 \begin{equation} 112 h_{e} = Ek \frac {u^{*}} {f_{0}} \\ 113 \end{equation} 114 where, $Ek$ is an empirical parameter, $u^{*}$ is the friction velocity and $f_{0}$ is the Coriolis 115 parameter. 116 117 In this similarity height relationship, the turbulent friction velocity: 118 \begin{equation} 119 u^{*} = \sqrt \frac {|\tau|} {\rho_o} \\ 120 \end{equation} 121 122 is computed from the wind stress vector $|\tau|$ and the reference dendity $ \rho_o$. 123 The final $h_{e}$ is further constrained by the adjustable bounds \np{rn\_mldmin} and \np{rn\_mldmax}. 124 Once $h_{e}$ is computed, the vertical eddy coefficients within $h_{e}$ are set to 125 the empirical values \np{rn\_wtmix} and \np{rn\_wvmix} \citep{Lermusiaux2001}. 102 126 103 127 % ------------------------------------------------------------------------------------------------------------- … … 539 563 the clipping factor is of crucial importance for the entrainment depth predicted in 540 564 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 activated565 with the algebraic model for the turbulent fluxes. The clipping is only activated 542 566 if \np{ln\_length\_lim}=true, and the $c_{lim}$ is set to the \np{rn\_clim\_galp} value. 543 567 … … 981 1005 reduced as necessary to ensure stability; these changes are not reported. 982 1006 1007 Limits on the bottom friction coefficient are not imposed if the user has elected to 1008 handle the bottom friction implicitly (see \S\ref{ZDF_bfr_imp}). The number of potential 1009 breaches of the explicit stability criterion are still reported for information purposes. 1010 1011 % ------------------------------------------------------------------------------------------------------------- 1012 % Implicit Bottom Friction 1013 % ------------------------------------------------------------------------------------------------------------- 1014 \subsection{Implicit Bottom Friction (\np{ln\_bfrimp}$=$\textit{T})} 1015 \label{ZDF_bfr_imp} 1016 1017 An optional implicit form of bottom friction has been implemented to improve 1018 model stability. We recommend this option for shelf sea and coastal ocean applications, especially 1019 for split-explicit time splitting. This option can be invoked by setting \np{ln\_bfrimp} 1020 to \textit{true} in the \textit{nambfr} namelist. This option requires \np{ln\_zdfexp} to be \textit{false} 1021 in the \textit{namzdf} namelist. 1022 1023 This implementation is realised in \mdl{dynzdf\_imp} and \mdl{dynspg\_ts}. In \mdl{dynzdf\_imp}, the 1024 bottom boundary condition is implemented implicitly. 1025 1026 \begin{equation} \label{Eq_dynzdf_bfr} 1027 \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{mbk} 1028 = \binom{c_{b}^{u}u^{n+1}_{mbk}}{c_{b}^{v}v^{n+1}_{mbk}} 1029 \end{equation} 1030 1031 where $mbk$ is the layer number of the bottom wet layer. superscript $n+1$ means the velocity used in the 1032 friction formula is to be calculated, so, it is implicit. 1033 1034 If split-explicit time splitting is used, care must be taken to avoid the double counting of 1035 the bottom friction in the 2-D barotropic momentum equations. As NEMO only updates the barotropic 1036 pressure gradient and Coriolis' forcing terms in the 2-D barotropic calculation, we need to remove 1037 the bottom friction induced by these two terms which has been included in the 3-D momentum trend 1038 and update it with the latest value. On the other hand, the bottom friction contributed by the 1039 other terms (e.g. the advection term, viscosity term) has been included in the 3-D momentum equations 1040 and should not be added in the 2-D barotropic mode. 1041 1042 The implementation of the implicit bottom friction in \mdl{dynspg\_ts} is done in two steps as the 1043 following: 1044 1045 \begin{equation} \label{Eq_dynspg_ts_bfr1} 1046 \frac{\textbf{U}_{med}-\textbf{U}^{m-1}}{2\Delta t}=-g\nabla\eta-f\textbf{k}\times\textbf{U}^{m}+c_{b} 1047 \left(\textbf{U}_{med}-\textbf{U}^{m-1}\right) 1048 \end{equation} 1049 \begin{equation} \label{Eq_dynspg_ts_bfr2} 1050 \frac{\textbf{U}^{m+1}-\textbf{U}_{med}}{2\Delta t}=\textbf{T}+ 1051 \left(g\nabla\eta^{'}+f\textbf{k}\times\textbf{U}^{'}\right)- 1052 2\Delta t_{bc}c_{b}\left(g\nabla\eta^{'}+f\textbf{k}\times\textbf{u}_{b}\right) 1053 \end{equation} 1054 1055 where $\textbf{T}$ is the vertical integrated 3-D momentum trend. We assume the leap-frog time-stepping 1056 is used here. $\Delta t$ is the barotropic mode time step and $\Delta t_{bc}$ is the baroclinic mode time step. 1057 $c_{b}$ is the friction coefficient. $\eta$ is the sea surface level calculated in the barotropic loops 1058 while $\eta^{'}$ is the sea surface level used in the 3-D baroclinic mode. $\textbf{u}_{b}$ is the bottom 1059 layer horizontal velocity. 1060 1061 1062 1063 983 1064 % ------------------------------------------------------------------------------------------------------------- 984 1065 % Bottom Friction with split-explicit time splitting 985 1066 % ------------------------------------------------------------------------------------------------------------- 986 \subsection{Bottom Friction with split-explicit time splitting }1067 \subsection{Bottom Friction with split-explicit time splitting (\np{ln\_bfrimp}$=$\textit{F})} 987 1068 \label{ZDF_bfr_ts} 988 1069 … … 993 1074 {\key{dynspg\_flt}). Extra attention is required, however, when using 994 1075 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 bottom1076 equation is solved with a small time step \np{rn\_rdt}/\np{nn\_baro}, while the three 1077 dimensional prognostic variables are solved with the longer time step 1078 of \np{rn\_rdt} seconds. The trend in the barotropic momentum due to bottom 998 1079 friction appropriate to this method is that given by the selected parameterisation 999 1080 ($i.e.$ linear or non-linear bottom friction) computed with the evolving velocities … … 1018 1099 \end{enumerate} 1019 1100 1020 Note that the use of an implicit formulation 1101 Note that the use of an implicit formulation within the barotropic loop 1021 1102 for the bottom friction trend means that any limiting of the bottom friction coefficient 1022 1103 in \mdl{dynbfr} does not adversely affect the solution when using split-explicit time 1023 1104 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: 1105 the barotropic component which uses the unrestricted value of the coefficient. However, if the 1106 limiting is thought to be having a major effect (a more likely prospect in coastal and shelf seas 1107 applications) then the fully implicit form of the bottom friction should be used (see \S\ref{ZDF_bfr_imp} ) 1108 which can be selected by setting \np{ln\_bfrimp} $=$ \textit{true}. 1109 1110 Otherwise, the implicit formulation takes the form: 1027 1111 \begin{equation} \label{Eq_zdfbfr_implicitts} 1028 1112 \bar{U}^{t+ \rdt} = \; \left [ \bar{U}^{t-\rdt}\; + 2 \rdt\;RHS \right ] / \left [ 1 - 2 \rdt \;c_b^{u} / H_e \right ] … … 1091 1175 The essential goal of the parameterization is to represent the momentum 1092 1176 exchange between the barotropic tides and the unrepresented internal waves 1093 induced by the tidal ßow over rough topography in a stratified ocean.1177 induced by the tidal flow over rough topography in a stratified ocean. 1094 1178 In the current version of \NEMO, the map is built from the output of 1095 1179 the barotropic global ocean tide model MOG2D-G \citep{Carrere_Lyard_GRL03}.
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