Changeset 9414 for branches/2017/dev_merge_2017/DOC/tex_sub/chap_model_basics.tex

Timestamp:

2018-03-21T15:39:48+01:00 (6 years ago)

Author:

nicolasmartin

Message:

Fix multiple defined references

File:

• branches/2017/dev_merge_2017/DOC/tex_sub/chap_model_basics.tex (modified) (6 diffs)

branches/2017/dev_merge_2017/DOC/tex_sub/chap_model_basics.tex

@@ -648 +648 @@
 
 In fact one is totally free to choose any space and time vertical coordinate by introducing an arbitrary vertical coordinate :
-\begin{equation} \label{eq:s}
+\begin{equation} \label{eq:PE_s}
 s=s(i,j,k,t)
 \end{equation}
-with the restriction that the above equation gives a single-valued monotonic relationship between $s$ and $k$, when $i$, $j$ and $t$ are held fixed. \autoref{eq:s} is a transformation from the $(i,j,k,t)$ coordinate system with independent variables into the $(i,j,s,t)$ generalised coordinate system with $s$ depending on the other three variables through \autoref{eq:s}.
+with the restriction that the above equation gives a single-valued monotonic relationship between $s$ and $k$, when $i$, $j$ and $t$ are held fixed. \autoref{eq:PE_s} is a transformation from the $(i,j,k,t)$ coordinate system with independent variables into the $(i,j,s,t)$ generalised coordinate system with $s$ depending on the other three variables through \autoref{eq:PE_s}.
 This so-called \textit{generalised vertical coordinate} \citep{Kasahara_MWR74} is in fact an Arbitrary Lagrangian--Eulerian (ALE) coordinate. Indeed, choosing an expression for $s$ is an arbitrary choice that determines which part of the vertical velocity (defined from a fixed referential) will cross the levels (Eulerian part) and which part will be used to move them (Lagrangian part).
 The coordinate is also sometime referenced as an adaptive coordinate \citep{Hofmeister_al_OM09}, since the coordinate system is adapted in the course of the simulation. Its most often used implementation is via an ALE algorithm, in which a pure lagrangian step is followed by regridding and remapping steps, the later step implicitly embedding the vertical advection \citep{Hirt_al_JCP74, Chassignet_al_JPO03, White_al_JCP09}. Here we follow the \citep{Kasahara_MWR74} strategy : a regridding step (an update of the vertical coordinate) followed by an eulerian step with an explicit computation of vertical advection relative to the moving s-surfaces.
@@ -715 +715 @@
  \vspace{0.5cm}
 $\bullet$ Vector invariant form of the momentum equation :
-\begin{multline} \label{eq:PE_sco_u}
+\begin{multline} \label{eq:PE_sco_u_vector}
 \frac{\partial  u   }{\partial t}=
    +   \left( {\zeta +f} \right)\,v                                    
@@ -724 +724 @@
    +   D_u^{\vect{U}}  +   F_u^{\vect{U}} \quad
 \end{multline}
-\begin{multline} \label{eq:PE_sco_v}
+\begin{multline} \label{eq:PE_sco_v_vector}
 \frac{\partial v }{\partial t}=
    -   \left( {\zeta +f} \right)\,u   
@@ -735 +735 @@
 
  \vspace{0.5cm}
-$\bullet$ Vector invariant form of the momentum equation :
-\begin{multline} \label{eq:PE_sco_u}
+$\bullet$ Flux form of the momentum equation :
+\begin{multline} \label{eq:PE_sco_u_flux}
 \frac{1}{e_3} \frac{\partial \left(  e_3\,u  \right) }{\partial t}=
    +   \left( { f + \frac{1}{e_1 \; e_2 }
@@ -749 +749 @@
    +   D_u^{\vect{U}}  +   F_u^{\vect{U}} \quad
 \end{multline}
-\begin{multline} \label{eq:PE_sco_v}
+\begin{multline} \label{eq:PE_sco_v_flux}
 \frac{1}{e_3} \frac{\partial \left(  e_3\,v  \right) }{\partial t}=
    -   \left( { f + \frac{1}{e_1 \; e_2}
@@ -1138 +1138 @@
 rotation between geopotential and $s$-surfaces, while it is only an approximation 
 for the rotation between isoneutral and $z$- or $s$-surfaces. Indeed, in the latter 
-case, two assumptions are made to simplify  \autoref{eq:PE_iso_tensor} \citep{Cox1987}. 
+case, two assumptions are made to simplify \autoref{eq:PE_iso_tensor} \citep{Cox1987}. 
 First, the horizontal contribution of the dianeutral mixing is neglected since the ratio 
 between iso and dia-neutral diffusive coefficients is known to be several orders of