4. A Hydrodynamic Disk

Cold disks such as protoplanetary disks may lack sufficient ionization to couple to the magnetic fields. What happens in a purely hydrodynamic disk? Since turbulence, angular momentum transport, and net accretion in disks result from the action of the MRI, evidence to date suggests that very little happens in such hydrodynamic disks. There is now a growing body of simulations (Stone & Balbus 1996; Balbus, Hawley & Stone 1996; Hawley, Balbus & Winters 1999; Godon & Livio 1999a) which are consistent with the conclusion that differentially rotating disks are hydrodynamically stable to both linear and finite-amplitude local perturbations. Although there is no particular reason to believe that local hydrodynamic stability properties would be altered in a fully global disk, the hydrodynamic disk nevertheless remains an important limiting test case to consider.

To follow the evolution of an initially turbulent disk in the absence of Lorentz forces, the output from CK5 at time t=3450 is evolved forward in time (to t=4850) purely hydrodynamically (run HK5). It is perhaps a bit self-inconsistent to ask how turbulence that is generated by magnetic fields subsequently evolves hydrodynamically. Of course when there is no self-consistent hydrodynamic turbulence, and one wishes to observe the hydrodynamic decay of turbulence, it is necessary to initialize the turbulence one way or another.

In any event, the results from HK5 are not surprising. After a brief period of readjustment to the loss of magnetic tension and pressure forces, the turbulent kinetic energies drop (Fig. 9). In particular, the vertical kinetic energy declines exponentially and the system became increasingly z-independent. This is consistent with the cylindrical limit and the Taylor-Proudman theorem which holds that in a steady, inviscid flow, slow motions in a rotating fluid should be two-dimensional. All accretion stops, and the inner edge of the disk readjusts slightly, moving outside of R=4. Consistent with past local hydrodynamic simulations, there is no evidence for sustained turbulence or the development of any local hydrodynamic instability. Nor is there evidence for a strongly growing global mode.

Figure 9: Time-history of the radial and vertical kinetic energy in MHD run CK5 (dashed line) and hydrodynamical run HK5 (solid line). HK5 begins from the data of CK5 at time t= 3450. After an initial readjustment to the loss of magnetic force, the kinetic energies in the hydrodynamic run decay away.

Inward and outward propagating pressure waves are present throughout the simulation. These are in the form of tightly-wrapped trailing spiral waves, and they persist with constant or diminishing amplitude. A comparison of the azimuthal fourier power spectrum for density from CK5 and HK5 finds that CK5 has more power for all wavenumbers m. The ratio of power in CK5 to HK5 rises from 2:1 at m=4 (lowest wavenumber) up to 34:1 for m=44. The high-m power in CK5 is driven by the MRI; its amplitude drops promptly with the elimination of the magnetic terms in HK5. The dominant m=4 waves of HK5 appear to be driven mainly from the inner edge of the disk at frequencies close to the local orbital frequency there. The inner edge of the disk has a nonaxisymmetric structure that resembles a surface wave. This type of wave is seen to arise in Keplerian disks with a reflective inner boundary, and it is an example of a Papaloizou-Pringle instability (Godon & Livio 1999b). The pressure waves can, in principle, drive a small net inward drift within the disk. If one averages the accretion rate at each radius over the entire evolution there is a slight average inward accretion in the inner portion of the disk amounting to no more than . A similarly averaged accretion rate in the magnetic disk is 0.075.

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