Idealized cooling

In this section, I discuss how disks with GIs evolve when they do not fragment. An important question in this context is whether the evolution can be adequately approximated by a local a-disk prescription (e.g., Lin & Pringle 1987) or whether GIs are intrinsically a global phenomenon (Laughlin & Rozyczka 1996; Balbus & Papaloizou 1999). To address these points, it is best not to complicate the problem with the details of radiative transport, but to examine the results of simulations with simple, idealized cooling laws. A body of simulations exists where cooling by the disk is assumed to be characterized by requiring that either tcool (Pickett et al. 2003; Mejia et al. 2005a) or tcooln (Gammie 2001; Rice et al. 2003; Lodato & Rice 2004, 2005) has the same constant value everywhere in the disk.

3.1. tcooi = constant

The longest simulation performed with a constant global cooling time is the tcool = 2 orp (orp = outer rotation period) 3D simulation described by Mejia et al. (2005a) for a 0.07 M0 disk stretching from 3 to 40 AU (AU = astronomical unit) around a 0.5 M0 star. The initial equilibrium disk has a surface density profile E — r-1/2 except at the edges, is marginally stable to GIs with a minimum Q of about 1.8 at r = 30 AU prior to the beginning of cooling, and has a meridional aspect ratio of about ten to one. The orp for this disk is defined as the initial rotation period at 33 AU or about 250 yrs. The disk is peppered initially with low amplitude random perturbations to its density.

As illustrated in Figure 4, the evolution exhibits four principal phases:

1. During the cooling phase (not shown), which lasts only a few orps, the disk cools to the instability point and becomes vertically thinner. The disk remains axisymmetric in appearance, but perturbations begin to grow in the outer disk where Q is lowest.

2. Between about four to eight orps, during the burst phase (left panel), a single well-defined multi-armed spiral mode develops, becomes extremely nonlinear, and produces a large pulse of mass redistribution. Spiral arms are ejected outward in a manner similar to the dynamic bar-like instability of rotating stars (Durisen et al. 1986).

3. When these arms fall back partially, very strong shocks develop which heat the disk. It then enters a transition phase, lasting from about 8 to 12 orps. As the disk becomes hotter, Q rises and the nonaxisymmetry washes out.

4. Cooling reasserts itself by 12 orps, and the disk settles into a long-lived asymptotic phase with an overall balance between heating and cooling. Persistent GIs manifest

Figure 4. Equatorial mass density grayscales for the Mejia et al. (2005a) disk evolved with tcool = 2. Left. The Burst Phase at t = 5 orps (outer rotation periods). Strong instability occurs from four to eight orps in a discrete four-armed mode which rapidly redistributes mass in the disk. Center. The Transition Phase at ten orps. Shock heating due to the burst heats the disk, and nonaxisymmetric structure temporarily washes out between 8 and 12 orps. Right. The Asymptotic Phase at t = 20 orps. By about 12 orps, the disk achieves a long-term balance between heating and cooling with sustained self-gravitating turbulence. The panels in this figure are about 170 AU along an edge. Figures are adapted from Mejia et al. (2005a).

themselves as a complex nonlinear system of numerous multi-armed spirals. During the asymptotic phase, the disk maintains a nearly constant value of Q « 1.45 over r = ~12-40 AU, but with spatial and temporal fluctuations about this mean. This phase continues without significant qualitative change until the calculation is stopped arbitrarily at 23.5 orps (5,875 yrs).

Radially inward mass transport peaks at ~10-5 M0/yr in the burst phase, a bit shy of FU Orionis outburst values (Bell et al. 2000). Throughout the asymptotic phase, the disk sustains a highly variable inflow averaging about 5 x 10-7 Mq/yr over the radial range of 10 to 30 AU with a similar outflow rate beyond 30 AU. Fourier decomposition of the density structure into sin(m^) and cos(m^) components, where m corresponds to the number of arms in a spiral disturbance, provides some insight into the source of the transport. For all m's tested (m = 1 to 6) except m =1, there are dozens of coherent modes throughout the disk that span the radii between their inner and outer Lindblad resonances. Although not obvious to the eye in the right panel of Figure 4, there appear to be persistent two-armed (m = 2) modes with corotation radii (CR) near 30 AU during the 11.5 orps of the asymptotic phase that we followed. As expected for gravitational torques exerted by a trailing spiral mode, we see inward transport inside the CR of these two-armed modes and outward transport outside the CR. We conclude that the mass and angular momentum transport is dominated in this case by a few low-order global modes, and direct computation of the gravitational stresses confirms it.

A remarkable feature of this simulation is the growth of a series of rings near the boundary between the outer GI-active disk (r > 10 AU) and an inner disk that remains too hot to sustain GIs on it own. Figure 5 demonstrates the ring growth. The cause of this phenomenon and its potential significance for planet formation will be discussed in Section 5.1.

It is natural to wonder how these results depend on the somewhat arbitrary choices of simulation parameters. In Mejia et al. (2005a), when the 12 to 18 orp stretch is repeated with tcool = 1 orp, the disk again achieves an overall balance of heating and cooling with a nearly constant Q « 1.45, but the GIs become stronger, as evidenced by an increase in the Fourier amplitudes of the spirals, a doubling of the mass inflow rate, and faster growth of the rings in the inner disk. A comparison of mass inflow rates for tcool = 0.25,

Figure 5. The development of dense rings in the inner disk of a tcool = 2 orp simulation. Left. This equatorial plane density grayscale shows the rings in the innermost disk region near the end of the simulation at 23.5 orps. The panel is about 85 AU along an edge. Right. The mass contained within a cylindrical shell one-sixth of an AU in width is plotted as a function of radius to illustrate the growth of multiple Jupiter mass concentrations of mass. The features at 7 and 10 AU are truly ring-like, while the one at 13 AU has active nonaxisymmetric structure. Figures are provided courtesy of A. C. Mejia. The left panel is from Mejia et al. (2005a).

Figure 5. The development of dense rings in the inner disk of a tcool = 2 orp simulation. Left. This equatorial plane density grayscale shows the rings in the innermost disk region near the end of the simulation at 23.5 orps. The panel is about 85 AU along an edge. Right. The mass contained within a cylindrical shell one-sixth of an AU in width is plotted as a function of radius to illustrate the growth of multiple Jupiter mass concentrations of mass. The features at 7 and 10 AU are truly ring-like, while the one at 13 AU has active nonaxisymmetric structure. Figures are provided courtesy of A. C. Mejia. The left panel is from Mejia et al. (2005a).

1, and 2 orp runs shows that mass inflow rates during the asymptotic phase scale as t^Ooi' in agreement with Tomley et al. (1991). My hydrodynamics group has recently completed two additional tcool = 2 orp runs where the initial disks have E — r-a, where a =1 and 3/2. These evolutions exhibit the same four phases and other behaviors as for a =1/2, except that the burst weakens and the rings become more prominent (Michael & Boley, private communication).

Lodato & Rice (2004, 2005) analyze how GIs behave for a range of Md/Ms from 0.05 to 1.0 when tcoolil is set to the nonfragmenting value of 7.5. Their disk has an initial E — r-1 and a temperature structure T — r-1/2, giving a minimum initial Q « 2. The choice tcoolil = 7.5, equivalently tcool/Prot « 1.2, pins the cooling time everywhere to the local rotation period. For low Md/Ms, Lodato & Rice (2004) find that the evolution of the disk tends to be self-similar. After the disk achieves balance between heating and cooling, tightly wrapped spirals exist everywhere in the disk with similar pitch angles, and the disk settles into sustained instability with Q — 1 to 1.5, as illustrated in the left panel of Figure 6. No pronounced "burst" is reported, and the stresses and resultant transport produced by the GI activity is reasonably well described as a local phenomenon.

However, for the vertically thicker and more massive disks, especially with Md/Ms = 0.5 and 1.0, Lodato & Rice (2005) obtain evolutionary phases similar to those described by Mejia et al. (2005a), with a strong burst in a global mode that transitions into an asymptotic phase. As Md/Ms increases, the amplitudes of low-order (few-armed) modes in the asymptotic gravitoturbulence become more important relative to the higher-order modes. With a wide radial range spanned by the region between their Lindblad resonances, low-order modes communicate torques in a global way. So, very massive disks (Md/Ms ^ 0.25) exhibit some global behavior even when their cooling times are forced to be local. For Md/Ms = 1.0, the disk continues to experience burst-like episodes of global mode behavior rather than settling into an asymptotic phase.

Figure 6. Equatorial plane density grayscales at late times. Left. The Md/Ms = 0.1 disk with tcooi = 1.2Prot and initial £(r) ~ r-1 from Lodato & Rice (2004) has a smooth self-similar appearance with tight spirals. Center. The tcool = 1 orp simulation with Md/Ms = 0.14 and an initial £ ~ r-1/2 from Mejia et al. (2005a) has more open, global spirals. Right. This disk is similar to the one in the center panel except that it has an initial £ ~ r-1 and tcool = 2 orps. All figures have a similar radial scale. Figures are provided courtesy of W. K. Rice, A. C. Mejia, S. Michael, and A. C. Boley.

Figure 6. Equatorial plane density grayscales at late times. Left. The Md/Ms = 0.1 disk with tcooi = 1.2Prot and initial £(r) ~ r-1 from Lodato & Rice (2004) has a smooth self-similar appearance with tight spirals. Center. The tcool = 1 orp simulation with Md/Ms = 0.14 and an initial £ ~ r-1/2 from Mejia et al. (2005a) has more open, global spirals. Right. This disk is similar to the one in the center panel except that it has an initial £ ~ r-1 and tcool = 2 orps. All figures have a similar radial scale. Figures are provided courtesy of W. K. Rice, A. C. Mejia, S. Michael, and A. C. Boley.

Lodato & Rice compute the gravitational and hydrodynamic (Reynolds) stresses induced by GIs in their disks. Focusing on the inner to middle disk regions, where there are negative net torques and radial inflow of mass, Lodato & Rice compare the stresses with a prediction by Gammie (2001) that the effective a viscosity should be well approximated by a d lnft d lnr

If equation (3.1) were correct, it would allow long-term evolutions of the radial mass distribution in GI-active disks to be done with a simple a-viscosity prescription (e.g., Yorke & Bodenheimer 1999; Armitage et al. 2001). Contrary to Gammie's thin shearing box local simulations, where he finds that the gravitational and Reynolds stresses are similar, Lodato & Rice always find that gravitational stresses dominate. Nevertheless, for tcoolQ = 7.5, equation (3.1) predicts a = 0.05, which is close to the typical effective a that Lodato & Rice detect in the inflow regions of their disks. This suggests that, on average, equation (3.1) could be a useful approximation for a-disk modeling of GIs. There are, however, substantial (~10%) radial and, presumably, temporal variations about this value. Equation (3.1) is not useful for characterizing the global burst when Md/Ms is large, and deviations from (3.1) do seem systematically larger as Md/Ms increases. However, Lodato & Rice find no evidence at any Md/Ms for significant wave-transport of energy expected when GIs behave globally (Balbus & Papaloizou 1999). Local heating rates seem to match local cooling rates throughout their disks.

3.3. Conclusions

Apparently, when cooling is treated as set by a global time scale (tcool = constant), as in Mejia et al. (2005a), GIs behave in a global manner, whereas, when cooling is treated as a locally determined phenomenon (tcoolQ = constant), as in Lodato & Rice (2004, 2005), GIs behave locally, unless Md/Ms is large. For tcoolQ = constant, a local effective a prescription apparently fits the GI stresses in simulations fairly well. This result is perhaps not too surprising if we recall that tcoolQ = constant is a necessary condition for a steady-state accretion disk. With tcool = constant, on the other hand, Mejia et al. (2005a) find that the effective a for the average inflow during the asymptotic phase

Figure 7. Equatorial plane density grayscales for three calculations with radiative cooling during the asymptotic phase with varying amounts of envelope irradiation. Left. No irradiation. Center. Tirr = 15 K. Right. Tirr = 25 K. The initial equilibrium model is the same as for the simulation in Figure 4. The width of each panel is about 140 AU. Figures are provided courtesy of K. Cai.

Figure 7. Equatorial plane density grayscales for three calculations with radiative cooling during the asymptotic phase with varying amounts of envelope irradiation. Left. No irradiation. Center. Tirr = 15 K. Right. Tirr = 25 K. The initial equilibrium model is the same as for the simulation in Figure 4. The width of each panel is about 140 AU. Figures are provided courtesy of K. Cai.

is a few to many tens of times larger than given by equation (3.1) and is extremely variable, even changing sign for intervals of time. Figure 7 shows a direct comparison of disks evolved under the two types of cooling law. Even visually, it is apparent that the tcooltt = constant disk on the left is more self-similar, while the spirals in the right two panels tend to be more open and irregular with more radially variable structure, including dense rings in the innermost disk.

To summarize all this in one sentence, simulations to date show that, for moderate disk masses, whether GIs are a local or global phenomenon depends on whether the cooling time itself is local or global. As in Pickett et al. (1998, 2000), the lesson is that the behavior of GIs is controlled by the details of the thermal physics. So, let us now consider whether realistic treatments of radiative cooling lead to cooling times that can be characterized as local or global.

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