Measuring Turbulence in TW Hya with ALMA: Methods and Limitations
  We obtain high spatial and spectral resolution images of the CO J=2-1, CN
N=2-1 and CS J=5-4 emission with ALMA in Cycle~2. The radial distribution of
the turbulent broadening is derived with three approaches: two `direct' and one
modelling. The first requires a single transition and derives \Tex{} directly
from the line profile, yielding a \vturb{}. The second assumes two different
molecules are co-spatial thus their relative linewidths allow for a calculation
of \Tkin{} and \vturb{}. Finally we fit a parametric disk model where physical
properties of the disk are described by power laws, to compare our `direct'
methods with previous values. The two direct methods were limited to the outer
$r > 40$~au disk due to beam smear. The direct method found \vturb{} ranging
from $\approx$~\vel{130} at 40~au, dropping to $\approx$~\vel{50} in the outer
disk, qualitatively recovered with the parametric model fitting. This
corresponds to roughly $0.2 - 0.4~c_s$. CN was found to exhibit strong non-LTE
effects outside $r \approx 140$~au, so \vturb{} was limited to within this
radius. The assumption that CN and CS are co-spatial is consistent with
observed linewidths only within $r \lesssim 100$~au, within which \vturb{} was
found to drop from \vel{100} ($\approx~0.4~c_s$) to nothing at 100~au. The
parametric model yielded a near constant \vel{50} for CS ($0.2 - 0.4~c_s$). We
demonstrate that absolute flux calibration is and will be the limiting factor
in all studies of turbulence using a single molecule. The magnitude of the
dispersion is comparable with or below that predicted by the magneto-rotational
instability theory. A more precise comparison would require to reach an
absolute calibration precision of order 3\%, or to find a suitable combination
of light and heavy molecules which are co-located in the disk.
