Hierarchical Structure of Magnetohydrodynamic Turbulence In
  Position-Position-Velocity Space
  Magnetohydrodynamic turbulence is able to create hierarchical structures in
the interstellar medium that are correlated on a wide range of scales via the
energy cascade. We use hierarchical tree diagrams known as dendrograms to
characterize structures in synthetic Position-Position-Velocity (PPV) emission
cubes of optically thin isothermal magnetohydrodynamic turbulence. We show that
the structures and degree of hierarchy observed in PPV space are related to the
physics of the gas, i.e. self-gravity and the global sonic and Alfvenic Mach
number. Simulations with higher Alfvenic Mach number, self-gravity and
supersonic flows display enhanced hierarchical structure. We observed a strong
sonic and Alfvenic dependency when we apply the the statistical moments (i.e.
mean, variance, skewness, kurtosis) to the dendrogram distribution. Larger
magnetic field and sonic Mach number correspond to larger values of the
moments. Application of the dendrogram to 3D density cubes, also known as
Position-Position-Position cubes (PPP), reveals that the dominant emission
contours in PPP and PPV are related for supersonic gas but not for subsonic. We
also explore the effects of smoothing, thermal broadening and velocity
resolution on the dendrograms in order to make our study more applicable to
observational data. These results all point to hierarchical tree diagrams as
being a promising additional tool for studying ISM turbulence and star forming
regions in the direction of obtaining information on the degree of
self-gravity, the Mach numbers and the complicated relationship between PPV and
PPP.
