[cig-commits] r14573 - doc/CitcomS/manual

[tan2 at geodynamics.org]

Author: tan2
Date: 2009-04-02 14:12:57 -0700 (Thu, 02 Apr 2009)
New Revision: 14573

Modified:
   doc/CitcomS/manual/citcoms.lyx
Log:
Fixes the equations as reported by Dan Bower, and Section 1.5-1.7

Modified: doc/CitcomS/manual/citcoms.lyx
===================================================================
--- doc/CitcomS/manual/citcoms.lyx	2009-04-02 20:47:11 UTC (rev 14572)
+++ doc/CitcomS/manual/citcoms.lyx	2009-04-02 21:12:57 UTC (rev 14573)
@@ -821,7 +821,7 @@
 \begin_layout Standard
 \align right
 \begin_inset Formula \begin{equation}
--P_{,i}+\left(\eta(u_{i,j}+u_{j,i}-\frac{2}{3}u_{k,k}\delta_{ij})\right)_{,i}+\delta\rho g\delta_{ir}=0\label{eq:conservation of momentum}\end{equation}
+-P_{,i}+\left(\eta(u_{i,j}+u_{j,i}-\frac{2}{3}u_{k,k}\delta_{ij})\right)_{,i}-\delta\rho g\delta_{ir}=0\label{eq:conservation of momentum}\end{equation}
 
 \end_inset
 
@@ -831,7 +831,7 @@
 \begin_layout Standard
 \align right
 \begin_inset Formula \begin{equation}
-\rho c_{P}\left(T_{,t}+u_{i}T_{,i}\right)=\rho c_{P}\kappa T_{,ii}+\rho\alpha gu_{r}\left(T+T_{0}\right)+\Phi+\rho\left(Q_{L,t}+u_{i}Q_{L,i}\right)+\rho H\label{eq:conservation of energy}\end{equation}
+\rho c_{P}\left(T_{,t}+u_{i}T_{,i}\right)=\rho c_{P}\kappa T_{,ii}+\rho\alpha gu_{r}T+\Phi+\rho\left(Q_{L,t}+u_{i}Q_{L,i}\right)+\rho H\label{eq:conservation of energy}\end{equation}
 
 \end_inset
 
@@ -883,10 +883,6 @@
 T
 \emph default
  is the temperature, 
-\begin_inset Formula $T_{0}$
-\end_inset
-
- is the temperature at the surface, 
 \begin_inset Formula $c_{P}$
 \end_inset
 
@@ -906,7 +902,7 @@
 \begin_inset Formula $Q_{L}$
 \end_inset
 
- is the latent heat, and 
+ is the latent heat due to phase transitions, and 
 \emph on
 H
 \emph default
@@ -1007,7 +1003,7 @@
 \begin_layout Standard
 \align right
 \begin_inset Formula \begin{equation}
-\pi=\rho g(1-r-d_{ph})-\gamma_{ph}(T-T_{ph})\label{eq:reduced pressure}\end{equation}
+\pi=\bar{\rho}g(1-r-d_{ph})-\gamma_{ph}(T-T_{ph})\label{eq:reduced pressure}\end{equation}
 
 \end_inset
 
@@ -1017,7 +1013,7 @@
 \begin_layout Standard
 \align right
 \begin_inset Formula \begin{equation}
-\Gamma=\frac{1}{2}\left(1+\tanh\left(\frac{\pi}{\rho gw_{ph}}\right)\right)\label{eq:phase function}\end{equation}
+\Gamma=\frac{1}{2}\left(1+\tanh\left(\frac{\pi}{\bar{\rho}gw_{ph}}\right)\right)\label{eq:phase function}\end{equation}
 
 \end_inset
 
@@ -1072,16 +1068,6 @@
 \begin_layout Standard
 \align right
 \begin_inset Formula \begin{equation}
-\rho=\rho_{0}\rho^{'}\label{eq:rho dim}\end{equation}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\align right
-\begin_inset Formula \begin{equation}
 \alpha=\alpha_{0}\alpha^{'}\label{eq:alpha dim}\end{equation}
 
 \end_inset
@@ -1244,7 +1230,11 @@
 \begin_inset Formula $R_{0}$
 \end_inset
 
- is the radius of the Earth, and 
+ is the radius of the Earth, 
+\begin_inset Formula $T_{0}$
+\end_inset
+
+ is the temperature at the surface, and 
 \begin_inset Formula $\Delta T$
 \end_inset
 
@@ -1263,7 +1253,7 @@
 
 \begin_layout Standard
 \begin_inset Formula \begin{equation}
--P_{,i}+\left(\eta(u_{i,j}+u_{j,i}-\frac{2}{3}u_{k,k}\delta_{ij})\right)_{,i}+(Ra\bar{\rho}\alpha T+Rab\Gamma-RacC)g\delta_{ir}=0\label{eq:non-dimensional momentum eqn}\end{equation}
+-P_{,i}+\left(\eta(u_{i,j}+u_{j,i}-\frac{2}{3}u_{k,k}\delta_{ij})\right)_{,i}+(Ra\bar{\rho}\alpha T-Rab\Gamma-RacC)g\delta_{ir}=0\label{eq:non-dimensional momentum eqn}\end{equation}
 
 \end_inset
 
@@ -1275,7 +1265,7 @@
 \begin_inset Formula \begin{equation}
 \begin{array}{c}
 \bar{\rho}c_{P}\left(T_{,t}+u_{i}T_{,i}\right)\left(1+2\Gamma\left(1-\Gamma\right)\frac{\gamma_{ph}^{2}}{d_{ph}}\frac{Rab}{Ra}Di\left(T+T_{0}\right)\right)=\bar{\rho}c_{P}\kappa T_{,ii}\\
-+\rho\alpha gu_{r}Di\left(T+T_{0}\right)\left(1+2\Gamma\left(1-\Gamma\right)\frac{\gamma_{ph}}{d_{ph}}\frac{Rab}{Ra}\right)+\frac{Di}{Ra}\Phi+\bar{\rho}H\end{array}\label{eq:non-dimensional energy eqn}\end{equation}
+-\bar{\rho}\alpha gu_{r}Di\left(T+T_{0}\right)\left(1+2\Gamma\left(1-\Gamma\right)\frac{\gamma_{ph}}{d_{ph}}\frac{Rab}{Ra}\right)+\frac{Di}{Ra}\Phi+\bar{\rho}H\end{array}\label{eq:non-dimensional energy eqn}\end{equation}
 
 \end_inset
 
@@ -1381,7 +1371,8 @@
 \end_inset
 
 .
- The energy equation is solved with a Petrov-Galerkin method 
+ The energy equation is solved with a Steamline-Upwind Petrov-Galerkin method
+ 
 \begin_inset LatexCommand cite
 key "Brooks A.N."
 
@@ -1391,8 +1382,8 @@
  Brick elements are used, such as eight velocity nodes with trilinear shape
  functions and one constant pressure node for each element.
  The use of brick elements in 3D (or rectangular elements in 2D) is important
- for accurately determining the pressure, such as dynamic topography, in
- incompressible Stokes flow 
+ for accurately determining the pressure, which controls the dynamic topography,
+ in incompressible Stokes flow 
 \begin_inset LatexCommand cite
 key "Hughes The Finite Element Method"
 
@@ -1824,9 +1815,9 @@
 
 ).
  One would normally associate at least one processor with one cap.
- However, CitcomS can further automatically generate meshes with domain
- decomposition such that additional processors are used to divide caps uniformly
- along the two edges of the caps (Figure 
+ However, CitcomS can further decompose the domain such that additional
+ processors are used to divide caps uniformly along the two edges of the
+ caps (Figure 
 \begin_inset LatexCommand ref
 reference "fig:Orthographic-projection-of"