deletions | additions       diff --git a/The simulation.tex b/The simulation.tex  index a35d819..fa1d8be 100644  --- a/The simulation.tex  +++ b/The simulation.tex 
...
\begin{equation}  \left{|} d {\bf v}_{k}^{2} \right{|} \propto k^-2,  \end{equation}  for wavenumbers $2 < k/2 \pi} < 4$. Similarly to, e.g., \citet{http://adsabs.harvard.edu/doi/10.1093/mnras/stt1644}, \citet{Federrath_2013},  turbulence on smaller length scales was generated self-consistently from the large scale driving. The driving continued until the box reached a saturated state at a Mach number $\scriptstyle{M}$ of roughly 8. During this driving stage we did not include gravitational forces. At this point we scaled the box to a physical length $L = 10$ pc, $H_2$ number density of 100 cm$^{-3}$ at mean molecular weight $\mu=2.33$, and a constant sound speed $c_s$ = 0.2 km s$^{-1}$. After turning off the forcing and turning on self gravity, we turned the simulation over the adaptive mesh refinement (AMR) code RAMSES \citep{2002A&A...385..337T}. The $1024^{3}$ base grid was maintained, along with 2 steps of adaptive refinement triggered when the local Jeans length became shorter than 4 grid cells \citep{1997ApJ...489L.179T}.   We integrated the box through $\sim1.25$ Myr of self gravitation evolution. Following \citet{http://adsabs.harvard.edu/abs/2007ApJ...665..416K}'s \citet{2007ApJ...665..416K}'s  definition of the turbulent turnover time in a box , $T_{turb} = L/(2 c_s \scriptstyle{M})$, we have $T_{turb} \sim 3$ Myr. The free-fall time $T_{ff} = (3\pi / (32 G \rho))^{1/2}$, at the mean density of the simulation, is