  Turbulence is a fundamental parameter in models of grain growth during the
early stages of planet formation. As such, observational constraints on its
magnitude are crucial. Here we self-consistently analyze ALMA CO(2-1), SMA
CO(3-2), and SMA CO(6-5) observations of the disk around TW Hya and find an
upper limit on the turbulent broadening of $<$0.08c$_s$ ($\alpha<$0.007 for
$\alpha$ defined only within 2-3 pressure scale heights above the midplane),
lower than the tentative detection previously found from an analysis of the
CO(2-1) data. We examine in detail the challenges of image plane fitting vs
directly fitting the visibilities, while also considering the role of the
vertical temperature gradient, systematic uncertainty in the amplitude
calibration, and assumptions about the CO abundance, as potential sources of
the discrepancy in the turbulence measurements. These tests result in
variations of the turbulence limit between $<$0.04c$_s$ and $<$0.13c$_s$,
consistently lower than the 0.2-0.4c$_s$ found previously. Having ruled out
numerous factors, we restrict the source of the discrepancy to our assumed
coupling between temperature and density through hydrostatic equilibrium in the
presence of a vertical temperature gradient and/or the confinement of CO to a
thin molecular layer above the midplane, although further work is needed to
quantify the influence of these prescriptions. Assumptions about hydrostatic
equilibrium and the CO distribution are physically motivated, and may have a
small influence on measuring the kinematics of the gas, but they become
important when constraining small effects such as the strength of the
turbulence within a protoplanetary disk.
