Changeset 6625 for branches/UKMO/dev_r5518_v3.4_asm_nemovar_community/DOC/TexFiles/Chapters/Chap_Model_Basics.tex

Timestamp:

2016-05-26T11:08:07+02:00 (8 years ago)

Author:

kingr

Message:

Rolled back to r6613

File:

• branches/UKMO/dev_r5518_v3.4_asm_nemovar_community/DOC/TexFiles/Chapters/Chap_Model_Basics.tex (modified) (10 diffs)

branches/UKMO/dev_r5518_v3.4_asm_nemovar_community/DOC/TexFiles/Chapters/Chap_Model_Basics.tex

@@ -247 +247 @@
 sufficient to solve a linearized version of (\ref{Eq_PE_ssh}), which still allows 
 to take into account freshwater fluxes applied at the ocean surface \citep{Roullet_Madec_JGR00}.
-Nevertheless, with the linearization, an exact conservation of heat and salt contents is lost.
 
 The filtering of EGWs in models with a free surface is usually a matter of discretisation 
-of the temporal derivatives, using a split-explicit method \citep{Killworth_al_JPO91, Zhang_Endoh_JGR92} 
-or the implicit scheme \citep{Dukowicz1994} or the addition of a filtering force in the momentum equation 
-\citep{Roullet_Madec_JGR00}. With the present release, \NEMO offers the choice between 
-an explicit free surface (see \S\ref{DYN_spg_exp}) or a split-explicit scheme strongly 
-inspired the one proposed by \citet{Shchepetkin_McWilliams_OM05} (see \S\ref{DYN_spg_ts}).
-
-%\newpage
-%$\ $\newline    % force a new line
+of the temporal derivatives, using the time splitting method \citep{Killworth_al_JPO91, Zhang_Endoh_JGR92} 
+or the implicit scheme \citep{Dukowicz1994}. In \NEMO, we use a slightly different approach 
+developed by \citet{Roullet_Madec_JGR00}: the damping of EGWs is ensured by introducing an 
+additional force in the momentum equation. \eqref{Eq_PE_dyn} becomes: 
+\begin{equation} \label{Eq_PE_flt}
+\frac{\partial {\rm {\bf U}}_h }{\partial t}= {\rm {\bf M}}
+- g \nabla \left( \tilde{\rho} \ \eta \right) 
+- g \ T_c \nabla \left( \widetilde{\rho} \ \partial_t \eta \right) 
+\end{equation}
+where $T_c$, is a parameter with dimensions of time which characterizes the force, 
+$\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density, and $\rm {\bf M}$ 
+represents the collected contributions of the Coriolis, hydrostatic pressure gradient, 
+non-linear and viscous terms in \eqref{Eq_PE_dyn}.
+
+The new force can be interpreted as a diffusion of vertically integrated volume flux divergence. 
+The time evolution of $D$ is thus governed by a balance of two terms, $-g$ \textbf{A} $\eta$ 
+and $g \, T_c \,$ \textbf{A} $D$, associated with a propagative regime and a diffusive regime 
+in the temporal spectrum, respectively. In the diffusive regime, the EGWs no longer propagate, 
+$i.e.$ they are stationary and damped. The diffusion regime applies to the modes shorter than 
+$T_c$. For longer ones, the diffusion term vanishes. Hence, the temporally unresolved EGWs 
+can be damped by choosing $T_c > \rdt$. \citet{Roullet_Madec_JGR00} demonstrate that 
+(\ref{Eq_PE_flt}) can be integrated with a leap frog scheme except the additional term which 
+has to be computed implicitly. This is not surprising since the use of a large time step has a 
+necessarily numerical cost. Two gains arise in comparison with the previous formulations. 
+Firstly, the damping of EGWs can be quantified through the magnitude of the additional term. 
+Secondly, the numerical scheme does not need any tuning. Numerical stability is ensured as 
+soon as $T_c > \rdt$.
+
+When the variations of free surface elevation are small compared to the thickness of the first 
+model layer, the free surface equation (\ref{Eq_PE_ssh}) can be linearized. As emphasized 
+by \citet{Roullet_Madec_JGR00} the linearization of (\ref{Eq_PE_ssh}) has consequences on the 
+conservation of salt in the model. With the nonlinear free surface equation, the time evolution 
+of the total salt content is 
+\begin{equation} \label{Eq_PE_salt_content}
+    \frac{\partial }{\partial t}\int\limits_{D\eta } {S\;dv} 
+                        =\int\limits_S {S\;(-\frac{\partial \eta }{\partial t}-D+P-E)\;ds}
+\end{equation}
+where $S$ is the salinity, and the total salt is integrated over the whole ocean volume 
+$D_\eta$ bounded by the time-dependent free surface. The right hand side (which is an 
+integral over the free surface) vanishes when the nonlinear equation (\ref{Eq_PE_ssh}) 
+is satisfied, so that the salt is perfectly conserved. When the free surface equation is 
+linearized, \citet{Roullet_Madec_JGR00} show that the total salt content integrated in the fixed 
+volume $D$ (bounded by the surface $z=0$) is no longer conserved:
+\begin{equation} \label{Eq_PE_salt_content_linear}
+         \frac{\partial }{\partial t}\int\limits_D {S\;dv} 
+               = - \int\limits_S {S\;\frac{\partial \eta }{\partial t}ds} 
+\end{equation}
+
+The right hand side of (\ref{Eq_PE_salt_content_linear}) is small in equilibrium solutions 
+\citep{Roullet_Madec_JGR00}. It can be significant when the freshwater forcing is not balanced and 
+the globally averaged free surface is drifting. An increase in sea surface height \textit{$\eta $} 
+results in a decrease of the salinity in the fixed volume $D$. Even in that case though, 
+the total salt integrated in the variable volume $D_{\eta}$ varies much less, since 
+(\ref{Eq_PE_salt_content_linear}) can be rewritten as 
+\begin{equation} \label{Eq_PE_salt_content_corrected}
+\frac{\partial }{\partial t}\int\limits_{D\eta } {S\;dv} 
+=\frac{\partial}{\partial t} \left[ \;{\int\limits_D {S\;dv} +\int\limits_S {S\eta \;ds} } \right]
+=\int\limits_S {\eta \;\frac{\partial S}{\partial t}ds}
+\end{equation}
+
+Although the total salt content is not exactly conserved with the linearized free surface, 
+its variations are driven by correlations of the time variation of surface salinity with the 
+sea surface height, which is a negligible term. This situation contrasts with the case of 
+the rigid lid approximation in which case freshwater forcing is represented by a virtual 
+salt flux, leading to a spurious source of salt at the ocean surface 
+\citep{Huang_JPO93, Roullet_Madec_JGR00}.
+
+\newpage
+$\ $\newline    % force a new ligne
 
 % ================================================================
@@ -713 +773 @@
 \end{equation}
 
-The equations solved by the ocean model \eqref{Eq_PE} in $s-$coordinate can be written as follows (see Appendix~\ref{Apdx_A_momentum}):
+The equations solved by the ocean model \eqref{Eq_PE} in $s-$coordinate can be written as follows:
 
  \vspace{0.5cm}
-$\bullet$ Vector invariant form of the momentum equation :
+* momentum equation:
 \begin{multline} \label{Eq_PE_sco_u}
-\frac{\partial  u   }{\partial t}=
+\frac{1}{e_3} \frac{\partial \left(  e_3\,u  \right) }{\partial t}=
    +   \left( {\zeta +f} \right)\,v                                    
    -   \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left(  u^2+v^2   \right) 
@@ -727 +787 @@
 \end{multline}
 \begin{multline} \label{Eq_PE_sco_v}
-\frac{\partial v }{\partial t}=
+\frac{1}{e_3} \frac{\partial \left(  e_3\,v  \right) }{\partial t}=
    -   \left( {\zeta +f} \right)\,u   
    -   \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left(  u^2+v^2  \right)        
@@ -735 +795 @@
    +  D_v^{\vect{U}}  +   F_v^{\vect{U}} \quad
 \end{multline}
-
- \vspace{0.5cm}
-$\bullet$ Vector invariant form of the momentum equation :
-\begin{multline} \label{Eq_PE_sco_u}
-\frac{1}{e_3} \frac{\partial \left(  e_3\,u  \right) }{\partial t}=
-   +   \left( { f + \frac{1}{e_1 \; e_2 }
-               \left(    v \frac{\partial e_2}{\partial i}
-                  -u \frac{\partial e_1}{\partial j}  \right)}    \right) \, v    \\
-   - \frac{1}{e_1 \; e_2 \; e_3 }   \left( 
-               \frac{\partial \left( {e_2 \, e_3 \, u\,u} \right)}{\partial i}
-      +        \frac{\partial \left( {e_1 \, e_3 \, v\,u} \right)}{\partial j}   \right)
-   - \frac{1}{e_3 }\frac{\partial \left( { \omega\,u} \right)}{\partial k}    \\
-   - \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho _o}    \right)    
-   +  g\frac{\rho }{\rho _o}\sigma _1 
-   +   D_u^{\vect{U}}  +   F_u^{\vect{U}} \quad
-\end{multline}
-\begin{multline} \label{Eq_PE_sco_v}
-\frac{1}{e_3} \frac{\partial \left(  e_3\,v  \right) }{\partial t}=
-   -   \left( { f + \frac{1}{e_1 \; e_2}
-               \left(    v \frac{\partial e_2}{\partial i}
-                  -u \frac{\partial e_1}{\partial j}  \right)}    \right) \, u   \\
-   - \frac{1}{e_1 \; e_2 \; e_3 }   \left( 
-               \frac{\partial \left( {e_2 \; e_3  \,u\,v} \right)}{\partial i}
-      +        \frac{\partial \left( {e_1 \; e_3  \,v\,v} \right)}{\partial j}   \right)
-                 - \frac{1}{e_3 } \frac{\partial \left( { \omega\,v} \right)}{\partial k}    \\
-   -   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}  \right) 
-    +  g\frac{\rho }{\rho _o }\sigma _2   
-   +  D_v^{\vect{U}}  +   F_v^{\vect{U}} \quad
-\end{multline}
-
 where the relative vorticity, \textit{$\zeta $}, the surface pressure gradient, and the hydrostatic 
 pressure have the same expressions as in $z$-coordinates although they do not represent 
 exactly the same quantities. $\omega$ is provided by the continuity equation 
 (see Appendix~\ref{Apdx_A}):
+
 \begin{equation} \label{Eq_PE_sco_continuity}
 \frac{\partial e_3}{\partial t} + e_3 \; \chi + \frac{\partial \omega }{\partial s} = 0   
@@ -778 +809 @@
 
  \vspace{0.5cm}
-$\bullet$ tracer equations:
+* tracer equations:
 \begin{multline} \label{Eq_PE_sco_t}
 \frac{1}{e_3} \frac{\partial \left(  e_3\,T  \right) }{\partial t}=
@@ -992 +1023 @@
 \label{PE_zco_tilde}
 
-The $\tilde{z}$-coordinate has been developed by \citet{Leclair_Madec_OM11}.
-It is available in \NEMO since the version 3.4. Nevertheless, it is currently not robust enough 
-to be used in all possible configurations. Its use is therefore not recommended.
-
+The $\tilde{z}$-coordinate has been developed by \citet{Leclair_Madec_OM10s}.
+It is not available in the current version of \NEMO.
 
 \newpage 
@@ -1128 +1157 @@
 operator acting along $s-$surfaces (see \S\ref{LDF}).
 
-\subsubsection{Lateral Laplacian tracer diffusive operator}
-
-The lateral Laplacian tracer diffusive operator is defined by (see Appendix~\ref{Apdx_B}):
+\subsubsection{Lateral second order tracer diffusive operator}
+
+The lateral second order tracer diffusive operator is defined by (see Appendix~\ref{Apdx_B}):
 \begin{equation} \label{Eq_PE_iso_tensor}
 D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad 
@@ -1151 +1180 @@
 ocean (see Appendix~\ref{Apdx_B}).
 
-For \textit{iso-level} diffusion, $r_1$ and $r_2 $ are zero. $\Re $ reduces to the identity 
-in the horizontal direction, no rotation is applied. 
-
 For \textit{geopotential} diffusion, $r_1$ and $r_2 $ are the slopes between the 
-geopotential and computational surfaces: they are equal to $\sigma _1$ and $\sigma _2$, 
-respectively (see \eqref{Eq_PE_sco_slope} ).
+geopotential and computational surfaces: in $z$-coordinates they are zero 
+($r_1 = r_2 = 0$) while in $s$-coordinate (including $\textit{z*}$ case) they are 
+equal to $\sigma _1$ and $\sigma _2$, respectively (see \eqref{Eq_PE_sco_slope} ).
 
 For \textit{isoneutral} diffusion $r_1$ and $r_2$ are the slopes between the isoneutral 
@@ -1204 +1231 @@
 to zero in the vicinity of the boundaries. The latter strategy is used in \NEMO (cf. Chap.~\ref{LDF}).
 
-\subsubsection{Lateral bilaplacian tracer diffusive operator}
-
-The lateral bilaplacian tracer diffusive operator is defined by:
+\subsubsection{Lateral fourth order tracer diffusive operator}
+
+The lateral fourth order tracer diffusive operator is defined by:
 \begin{equation} \label{Eq_PE_bilapT}
 D^{lT}=\Delta \left( {A^{lT}\;\Delta T} \right) 
 \qquad \text{where} \  D^{lT}=\Delta \left( {A^{lT}\;\Delta T} \right)
  \end{equation}
+
 It is the second order operator given by \eqref{Eq_PE_iso_tensor} applied twice with 
 the eddy diffusion coefficient correctly placed. 
 
-\subsubsection{Lateral Laplacian momentum diffusive operator}
-
-The Laplacian momentum diffusive operator along $z$- or $s$-surfaces is found by 
+
+\subsubsection{Lateral second order momentum diffusive operator}
+
+The second order momentum diffusive operator along $z$- or $s$-surfaces is found by 
 applying \eqref{Eq_PE_lap_vector} to the horizontal velocity vector (see Appendix~\ref{Apdx_B}):
 \begin{equation} \label{Eq_PE_lapU}
@@ -1250 +1279 @@
 of the Equator in a geographical coordinate system \citep{Lengaigne_al_JGR03}.
 
-\subsubsection{lateral bilaplacian momentum diffusive operator}
+\subsubsection{lateral fourth order momentum diffusive operator}
 
 As for tracers, the fourth order momentum diffusive operator along $z$ or $s$-surfaces