Make sure all variables are defined

Author: harpolea

strang/paper.tex

@@ -155,17 +155,20 @@ \section{Introduction}\label{Sec:Introduction}
 \end{align}
 where $\rho$ is the density, $\Ub$ is the velocity vector, $X_k$ are the species mass fractions,
 $\omegadot_k$ are the species creation rates (from reactions), $p$ is the pressure, $E$ is the
-total energy / unit mass, and $\Sdot$ is the nuclear energy generation rate.
+specific total energy, and $\Sdot$ is the nuclear energy generation rate.
 
-When we are reacting, we can look at internal energy:
+When we are reacting, we can look at internal energy
 \begin{equation}
-\rho \frac{De}{Dt} + p \nabla \cdot \Ub = \rho \Sdot
+\rho \frac{De}{Dt} + p \nabla \cdot \Ub = \rho \Sdot,
 \end{equation}
-or alternately, we can evolve the temperature, which we can write as:
+where $e$ is the specific internal energy or alternately, we can evolve the temperature, 
+which we can write as
 \begin{equation}
-\rho c_v \frac{DT}{Dt} = \rho \left (\frac{p}{\rho^2} - e_\rho \right ) \frac{D\rho}{Dt} + \rho \Sdot
+\rho c_v \frac{DT}{Dt} = \rho \left (\frac{p}{\rho^2} - e_\rho \right ) \frac{D\rho}{Dt} + \rho \Sdot,
 \end{equation}
-(this form neglects the change in temperature due to composition
+where $c_v$ is the specific heat capacity at constant volume, $T$ is the temperature and $e_\rho$ is the derivative of 
+the specific internal energy with respect to density (this form neglects the change in 
+temperature due to composition
 changes from reactions, see \citealt{ABNZ:III}, which assumes chemical
 equilibrium). 
 
@@ -269,7 +272,7 @@ \section{Numerical Methodology}
 \centering
 \plotone{test}
 \caption{\label{thefigure} Convergence of fluid quantities as a function of resolution
-for 3 different Strang equation systems: just evolving $X_k$, evolving $(X_k, T)$ with $c_v$ held fixed, and evolving $(X_k, e)$.}
+for 3 different Strang equation systems: just evolving $X_k$, evolving $(X_k, T)$ with $c_v$ held fixed, and evolving $(X_k, e)$. The dotted lines show ideal first and second order convergence.}
 \end{figure}
 
 

xrb2/paper.tex

@@ -392,7 +392,7 @@ \subsection{Effect of Rotation Rate on Flame Structure}\label{ssec:rot_structure
 	L_R \approx \frac{\sqrt{g H_0}}{\Omega},
 \end{equation}
 where $g$ is the gravitational acceleration, $H_0$ is the atmospheric scale height, and $\Omega$ is the 
-neutron star rotation rate. \added{If we take $g \sim 3.6 \times 10^{12}~\mathrm{cm}~\mathrm{s}^{-2}$ and $H_0\sim 10^3~\mathrm{cm}$, then for the $500~\mathrm{Hz}$ system we find the Rossby length to approximately be $L_R \sim 1.2 \times 10^5~\mathrm{cm}$. This estimate is likely to be an overestimate of the true value of the Rossby length in our simulations, as we saw that the flame is confined on a much smaller length scale.} In Figure \ref{fig:time_series_enuc_500} and Figure 
+neutron star rotation rate. \added{If we take $g \sim 1.5 \times 10^{14}~\mathrm{cm}~\mathrm{s}^{-2}$ and $H_0\sim 10^3~\mathrm{cm}$, then for the $500~\mathrm{Hz}$ system we find the Rossby length to approximately be $L_R \sim 1.2 \times 10^5~\mathrm{cm}$. This estimate is likely to be an overestimate of the true value of the Rossby length in our simulations, as we saw that the flame is confined on a much smaller length scale.} In Figure \ref{fig:time_series_enuc_500} and Figure 
 \ref{fig:time_series_enuc_1000}, we use $\dot{e}_\mathrm{nuc}$ measured at $50~\mathrm{ms}$ and $100~\mathrm{ms}$ to discern the horizontal extent of the flame at different rotation rates. Taking the edge at greatest radius of the bright teal/green region where the most significant energy generation is occurring as the leading edge of the flame in each plot, we see that the horizontal extent of the $1000~\mathrm{Hz}$ flame ($\tt{F1000}$) appears to be 
 reduced compared to the lower rotation $500~\mathrm{Hz}$ run ($\tt{F500}$). \added{As the natural aspect ratio makes it hard to see the flame structure, we show the energy generation rate for the $\tt{F1000}$ simulation with the vertical extent stretched in Figure~\ref{fig:flame_stretch}.}  From Equation 
 \ref{eqn:rossby}, we can see that increasing the rotation rate from $500~\mathrm{Hz}$ to