  A new analysis of high-resolution data from the Atacama Large
Millimeter/submillimeter Array (ALMA) for 5 luminous or ultra-luminous infrared
galaxies gives a slope for the Kennicutt-Schmidt (KS) relation equal to
$1.74^{+0.09}_{\rm -0.07}$ for gas surface densities $\Sigma_{\rm
mol}>10^3\;M_\odot$ pc$^{-2}$ and an assumed constant CO-to-H$_2$ conversion
factor. The velocity dispersion of the CO line, $\sigma_v$, scales
approximately as the inverse square root of $\Sigma_{\rm mol}$, making the
empirical gas scale height determined from $H\sim0.5\sigma^2/(\pi G\Sigma_{\rm
mol})$ nearly constant, 150-190 pc, over 1.5 orders of magnitude in
$\Sigma_{\rm mol}$. This constancy of $H$ implies that the average midplane
density, which is presumably dominated by CO-emitting gas for these extreme
star-forming galaxies, scales linearly with the gas surface density, which, in
turn, implies that the gas dynamical rate (the inverse of the free-fall time)
varies with $\Sigma_{\rm mol}^{1/2}$, thereby explaining most of the
super-linear slope in the KS relation. Consistent with these relations, we also
find that the mean efficiency of star formation per free-fall time is roughly
constant, 5%-7%, and the gas depletion time decreases at high $\Sigma_{\rm
mol}$, reaching only $\sim 16$ Myr at $\Sigma_{\rm mol}\sim10^4\;M_\odot$
pc$^{-2}$. The variation of $\sigma_v$ with $\Sigma_{\rm mol}$ and the
constancy of $H$ are in tension with some feedback-driven models, which predict
$\sigma_v$ to be more constant and $H$ to be more variable. However, these
results are consistent with simulations in which large-scale gravity drives
turbulence through a feedback process that maintains an approximately constant
Toomre $Q$ instability parameter.
