We present a detailed analysis of the spatially and spectrally resolved 12CO J=2-1 and J=3-2 emission lines from the TW Hya circumstellar disk, based on science verification data from the Atacama Large Millimeter/Submillimeter Array (ALMA). These lines exhibit substantial emission in their high-velocity wings (with projected velocities out to 2.1 km/s, corresponding to intrinsic orbital velocities >20 km/s) that trace molecular gas as close as 2 AU from the central star. However, we are not able to reproduce the intensity of these wings and the general spatio-kinematic pattern of the lines with simple models for the disk structure and kinematics. Using three-dimensional non-local thermodynamic equilibrium molecular excitation and radiative transfer calculations, we construct some alternative models that successfully account for these features by modifying either (1) the temperature structure of the inner disk (inside the dust-depleted disk cavity; r < 4 AU); (2) the intrinsic (Keplerian) disk velocity field; or (3) the distribution of disk inclination angles (a warp). The latter approach is particularly compelling because a representative warped disk model qualitatively reproduces the observed azimuthal modulation of optical light scattered off the disk surface. In any model scenario, the ALMA data clearly require a substantial molecular gas reservoir located inside the region where dust optical depths are known to be substantially diminished in the TW Hya disk, in agreement with previous studies based on infrared spectroscopy. The results from these updated model prescriptions are discussed in terms of their potential physical origins, which might include dynamical perturbations from a low-mass companion with an orbital separation of a few AU.
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