We present an 11 arcsec resolution map of the central two parsecs of the Galaxy in the CO J =7->6 rotational transition. The CO emission shows rotation about Sgr A*, but also evidence for non-circular turbulent motion and a clumpy morphology. We combine our dataset with available CO measurements to model the physical conditions in the disk. We find that the molecular gas in the region is both warm and dense, with T~200-300 K, n_H2~50,000-70,000 cm^-3. The mass of warm molecular gas we measure in the central two parsecs is at least 2000 M_solar, about 20 times the UV-excited atomic gas mass, ruling out an UV heating scenario for the molecular material. We compare the available spectral tracers with theoretical models and conclude that molecular gas is heated with magneto-hydrodynamic shocks with v~10-20 kms and B~0.3-0.5 mG. Using the conditions derived with the CO analysis, we include the other important coolants--neutral oxygen and molecular hydrogen--to estimate the total cooling budget of the molecular material. We derive a mass to luminosity ratio of 2-3 M_solar/ L_solar, which is consistent with the total power dissipated via turbulent decay in 0.1 pc cells with v_rms~15 kms. These size and velocity scales are comparable to the observed clumping scale and the velocity dispersion. At this rate, the material near Sgr A* its dissipating its orbital energy on an orbital timescale, and cannot last for more than a few orbits. Our conclusions support a scenario in which the features near Sgr A* such as the CND and northern arm are generated by infalling clouds with low specific angular momentum.
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