Changeset 3294 for trunk/DOC/TexFiles/Chapters/Annex_A.tex

Timestamp:

2012-01-28T17:44:18+01:00 (12 years ago)

Author:

rblod

Message:

Merge of 3.4beta into the trunk

File:

• trunk/DOC/TexFiles/Chapters/Annex_A.tex (modified) (7 diffs, 1 prop)

trunk/DOC/TexFiles/Chapters/Annex_A.tex

@@ -13 +13 @@
 % Chain rule
 % ================================================================
-\section{Chain rule of $s-$coordinate}
+\section{The chain rule for $s-$coordinates}
 \label{Apdx_A_continuity}
 
-In order to establish the set of Primitive Equation in curvilinear $s-$coordinates
+In order to establish the set of Primitive Equation in curvilinear $s$-coordinates
 ($i.e.$ an orthogonal curvilinear coordinate in the horizontal and an Arbitrary Lagrangian 
 Eulerian (ALE) coordinate in the vertical), we start from the set of equations established 
@@ -62 +62 @@
 % continuity equation
 % ================================================================
-\section{Continuity Equation in $s-$coordinate}
+\section{Continuity Equation in $s-$coordinates}
 \label{Apdx_A_continuity}
 
@@ -128 +128 @@
 Here, $w$ is the vertical velocity relative to the $z-$coordinate system. 
 Introducing the dia-surface velocity component, $\omega $, defined as 
-the velocity relative to the moving $s-$surfaces and normal to them:
+the volume flux across the moving $s$-surfaces per unit horizontal area:
 \begin{equation} \label{Apdx_A_w_s}
 \omega  = w - w_s - \sigma _1 \,u - \sigma _2 \,v    \\
@@ -429 +429 @@
 This formulation of the pressure gradient is characterised by the appearance of a term depending on the 
 the sea surface height only (last term on the right hand side of expression \eqref{Apdx_A_grad_p}).
-This term will be abusively named \textit{surface pressure gradient} whereas the first term will be named 
+This term will be loosely termed \textit{surface pressure gradient}
+whereas the first term will be termed the 
 \textit{hydrostatic pressure gradient} by analogy to the $z$-coordinate formulation. 
 In fact, the the true surface pressure gradient is $1/\rho_o \nabla (\rho \eta)$, and 
@@ -451 +452 @@
 To sum up, in a curvilinear $s$-coordinate system, the vector invariant momentum equation 
 solved by the model has the same mathematical expression as the one in a curvilinear 
-$z-$coordinate, but the pressure gradient term :
+$z-$coordinate, except for the pressure gradient term :
 \begin{subequations} \label{Apdx_A_dyn_vect}
 \begin{multline} \label{Apdx_A_PE_dyn_vect_u}
@@ -495 +496 @@
 \end{subequations}
 Both formulation share the same hydrostatic pressure balance expressed in terms of
-hydrostatic pressure and density anmalies, $p_h'$ and $d=( \frac{\rho}{\rho_o}-1 )$:
+hydrostatic pressure and density anomalies, $p_h'$ and $d=( \frac{\rho}{\rho_o}-1 )$:
 \begin{equation} \label{Apdx_A_dyn_zph}
 \frac{\partial p_h'}{\partial k} = - d \, g \, e_3
@@ -502 +503 @@
 It is important to realize that the change in coordinate system has only concerned
 the position on the vertical. It has not affected (\textbf{i},\textbf{j},\textbf{k}), the 
-orthogonal curvilinear set of unit vector. ($u$,$v$) are always horizontal velocities
+orthogonal curvilinear set of unit vectors. ($u$,$v$) are always horizontal velocities
 so that their evolution is driven by \emph{horizontal} forces, in particular 
 the pressure gradient. By contrast, $\omega$ is not $w$, the third component of the velocity,
-but the dia-surface velocity component, $i.e.$ the velocity relative to the moving 
-$s-$surfaces and normal to them. 
+but the dia-surface velocity component, $i.e.$ the volume flux across the moving 
+$s$-surfaces per unit horizontal area.