Changeset 3057 for branches/2011/dev_r2802_NOCL_bfrimp/DOC/TexFiles/Chapters/Chap_ZDF.tex

Timestamp:

2011-11-08T13:39:35+01:00 (12 years ago)

Author:

hliu

Message:

update semi-implicit bottom friction branch, Document has been added in NEMO_book Chapter 10.4.4

File:

• branches/2011/dev_r2802_NOCL_bfrimp/DOC/TexFiles/Chapters/Chap_ZDF.tex (modified) (4 diffs)

branches/2011/dev_r2802_NOCL_bfrimp/DOC/TexFiles/Chapters/Chap_ZDF.tex

@@ -1 +1 @@
 % ================================================================
-% Chapter Ñ Vertical Ocean Physics (ZDF)
+% Chapter  Vertical Ocean Physics (ZDF)
 % ================================================================
 \chapter{Vertical Ocean Physics (ZDF)}
@@ -539 +539 @@
 the clipping factor is of crucial importance for the entrainment depth predicted in 
 stably stratified situations, and that its value has to be chosen in accordance 
-with the algebraic model for the turbulent ßuxes. The clipping is only activated 
+with the algebraic model for the turbulent uxes. The clipping is only activated 
 if \np{ln\_length\_lim}=true, and the $c_{lim}$ is set to the \np{rn\_clim\_galp} value.
 
@@ -982 +982 @@
 
 % -------------------------------------------------------------------------------------------------------------
+%       Implicit Bottom Friction
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Implicit Bottom Friction}
+\label{ZDF_bfr_imp}
+
+An Implicit form of bottom friction has been devised in the NEMO to improve
+model stability. We recommend this option for shelf sea and coastal ocean applications, especially 
+for split-explicit time splitting. This option can be invoked by setting \np{ln\_bfrimp} 
+to \textit{true} in the \textit{nambfr} namelist and \np{ln\_zdfexp} to \textit{false} 
+in the \textit{namzdf} namelist. 
+
+This implementation is realised in \mdl{dynzdf\_imp} and \mdl{dynspg\_ts}. In \mdl{dynzdf\_imp}, the 
+bottom boundary condition is implemented implicitly.
+
+\begin{equation} \label{Eq_dynzdf_bfr}
+\left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{mbk}
+    = \binom{c_{b}^{u}u^{n+1}_{mbk}}{c_{b}^{v}v^{n+1}_{mbk}}
+\end{equation}
+
+where $mbk$ is the layer number of the bottom layer. superscript $n+1$ means the velocity used in the
+friction formulea is to be calculated, so, it is implicit.
+
+If split-explicit time splitting is used, care must be taken to avoid the double counting of
+the bottom friction in the 2-D barotropic momentum equations. As NEMO only updates the barotropic 
+pressure gradient and Coriolis' forcing terms in the 2-D barotropic calculation, we need to remove
+the bottom friction induced by these two terms which has been included in the 3-D momentum trend 
+and update it with the latest value. On the other hand, the bottom friction contributed by the
+other terms (e.g. the advection term, viscosity term) has been included in the 3-D momentum equations
+and should not be added in the 2-D barotropic mode.
+
+The implementation of the implicit bottom friction in \mdl{dynspg\_ts} is done in two steps as the
+following:
+
+\begin{equation} \label{Eq_dynspg_ts_bfr1}
+\frac{\textbf{U}_{med}-\textbf{U}^{m-1}}{2\Delta t}=-g\nabla\eta-f\textbf{k}\times\textbf{U}^{m}+c_{b}
+\left(\textbf{U}_{med}-\textbf{U}^{m-1}\right)
+\end{equation}
+\begin{equation} \label{Eq_dynspg_ts_bfr2}
+\frac{\textbf{U}^{m+1}-\textbf{U}_{med}}{2\Delta t}=\textbf{T}+
+\left(g\nabla\eta^{'}+f\textbf{k}\times\textbf{U}^{'}\right)-
+2\Delta t_{bc}c_{b}\left(g\nabla\eta^{'}+f\textbf{k}\times\textbf{u}_{b}\right)
+\end{equation}
+
+where $\textbf{T}$ is the vertical integrated 3-D momentum trend. We assume the leap-frog time-stepping
+is used here. $\Delta t$ is the barotropic mode time step and $\Delta t_{bc}$ is the baroclinic mode time step.
+ $c_{b}$ is the friction coefficient. $\eta$ is the sea surface level calculated in the barotropic loops
+while $\eta^{'}$ is the sea surface level used in the 3-D baroclinic mode. $\textbf{u}_{b}$ is the bottom
+layer horizontal velocity.
+
+
+
+
+% -------------------------------------------------------------------------------------------------------------
 %       Bottom Friction with split-explicit time splitting
 % -------------------------------------------------------------------------------------------------------------
@@ -1091 +1144 @@
 The essential goal of the parameterization is to represent the momentum 
 exchange between the barotropic tides and the unrepresented internal waves 
-induced by the tidal ßow over rough topography in a stratified ocean. 
+induced by the tidal ow over rough topography in a stratified ocean. 
 In the current version of \NEMO, the map is built from the output of 
 the barotropic global ocean tide model MOG2D-G \citep{Carrere_Lyard_GRL03}.