[cig-commits] r16170 - doc/CitcomS/manual

[tan2 at geodynamics.org]

Author: tan2
Date: 2010-01-25 10:48:01 -0800 (Mon, 25 Jan 2010)
New Revision: 16170

Modified:
   doc/CitcomS/manual/citcoms.lyx
Log:
Fixed definitions of phase change and chemical Rayleigh number. The dominator should be rho_0*alpha_0*Delta-T, not just rho_0.

Added definition of internal heating Rayleigh number.


Modified: doc/CitcomS/manual/citcoms.lyx
===================================================================
--- doc/CitcomS/manual/citcoms.lyx	2010-01-25 03:26:02 UTC (rev 16169)
+++ doc/CitcomS/manual/citcoms.lyx	2010-01-25 18:48:01 UTC (rev 16170)
@@ -1363,7 +1363,7 @@
 
 \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-Ra_{b}\Gamma-Ra_{c}C)g\delta_{ir}=0\label{eq:non-dimensional momentum eqn}\end{equation}
 
 \end_inset
 
@@ -1374,8 +1374,8 @@
 \align right
 \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}\\
--\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}
+\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{Ra_{b}}{Ra}Di\left(T+T_{0}\right)\right)=\bar{\rho}c_{P}\kappa T_{,ii}\\
+-\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{Ra_{b}}{Ra}\right)+\frac{Di}{Ra}\Phi+\bar{\rho}H\end{array}\label{eq:non-dimensional energy eqn}\end{equation}
 
 \end_inset
 
@@ -1418,13 +1418,17 @@
 
  is about a factor of 10 larger than by the usual definition.
  The phase-change Rayleigh number, 
-\begin_inset Formula $Rab$
+\begin_inset Formula $Ra_{b}$
 \end_inset
 
 , the chemical Rayleigh number, 
-\begin_inset Formula $Rac$
+\begin_inset Formula $Ra_{c}$
 \end_inset
 
+, the internal heating Rayleigh number, 
+\begin_inset Formula $Ra_{H}$
+\end_inset
+
 , and the dissipation number 
 \shape italic
 Di
@@ -1435,7 +1439,7 @@
 \begin_layout Standard
 \align right
 \begin_inset Formula \begin{equation}
-Rab=Ra\frac{\delta\rho_{ph}}{\rho_{0}}\label{eq:Rab, the phase-change Rayleigh number}\end{equation}
+Ra_{b}=Ra\frac{\delta\rho_{ph}}{\rho_{0}\alpha_{0}\Delta T}\label{eq:Rab, the phase-change Rayleigh number}\end{equation}
 
 \end_inset
 
@@ -1445,7 +1449,7 @@
 \begin_layout Standard
 \align right
 \begin_inset Formula \begin{equation}
-Rac=Ra\frac{\delta\rho_{ch}}{\rho_{0}}\label{eq:Rac, chemical Rayleigh number}\end{equation}
+Ra_{c}=Ra\frac{\delta\rho_{ch}}{\rho_{0}\alpha_{0}\Delta T}\label{eq:Rac, chemical Rayleigh number}\end{equation}
 
 \end_inset
 
@@ -1453,7 +1457,17 @@
 \end_layout
 
 \begin_layout Standard
+\align right
 \begin_inset Formula \begin{equation}
+Ra_{H}=RaH\frac{R_{0}^{3}-R_{CMB}^{3}}{3R_{0}^{3}}\label{eq:RaH, internal heating Rayleigh number}\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
 Di=\frac{\alpha_{0}g_{0}R_{0}}{c_{P0}}\label{eq:Di, dissipation number}\end{equation}
 
 \end_inset