Special effects

Despite the sorry state of current affairs regarding radiative physics in unstable disks, significant progress is being made on other ways that GIs can affect the appearance and evolution of dusty gas disks.

5.1. Hydraulic jumps

As emphasized by Pickett et al. (1996, 1998, 2000, 2003), GIs produce large spiral corrugations and other more complex surface distortions in gas disks. Pursuant to a suggestion by D. Cox, A. C. Boley and I have been analyzing many of these vertical distortions as hybridized combinations of shocks plus hydraulic jumps, which we call hydraulic shock-jumps or hs-jumps. Martos & Cox (1998) showed that analogs of classic hydraulic jumps occur for compressible fluids in astrophysical disk geometry. Boley et al. (2005) and Boley & Durisen (2006) explain the jumps as follows. Consider the case where disk gas flows into the back of a trailing spiral wave with a pattern speed much slower than the gas orbital speed. Suppose the pre-shock gas is in vertical hydrostatic equilibrium. After passing through a strong adiabatic shock, the vertical pressure gradient of the post-shock gas can easily exceed any increase in vertical gravitational force, so that the gas accelerates upward. As it jumps above the pre-shock disk height, the gas also expands horizontally. This vertically and horizontally expanding plume also tends to stream inward along the spiral due to the reduction in the component of the gas velocity normal to the trailing spiral shock front. The net result is a wave curling back and inward over the spiral shock. Eventually, the vertically "jumping" gas crashes back down on the disk in a huge breaking wave along the spiral arm. The wave produces additional strong shocks at high altitudes in the disk and probably generates turbulence.

Picking this behavior out of the gravitoturbulence of a GI-active disk is daunting, and so Boley et al. (2005) and Boley & Durisen (2006) perform toy calculations that isolate the hs-jumps by stimulating simple well-defined spiral waves. An illustrative case is shown in Figure 9. Although the parameters are not particularly realistic, this simulation provides a clean case with a single two-armed wave. An initially axisymmetric disk is perturbed by a strong cos(2^) potential concentrated at about 5 AU and corotating with the fluid at 5 AU. The intention is to simulate what might happen in the asteroidal region if Jovian mass clumps suddenly appear, due perhaps to the eruption of GIs in a dead zone (see §5.3). The perturbation produces two corotating spiral shocks reaching well into the inner disk, as shown in the upper left panel of Figure 9. The upper right panel shows the corresponding surface disturbance. The curling and breaking character of the surface wave is best seen in the radial cut through the disk in the bottom panel. The vertical shaded region is the shock front along the inner edge of the spiral. The curling and breaking wave and its associated high altitude shocks are evident. Note that the scale of these structures is many tenths of an AU. Fluid elements traced in this flow suffer large radial and vertical excursions.

We are only beginning to explore the consequences of such wave action. For instance, Boley et al. (2005) find that shocks like those shown in Figure 9 have the characteristics, laid out by Desch & Connolly (2002), necessary to produce chondrules. Chondritic material seems to represent the bulk of the solids that condensed in the asteroid region, and the thermodynamics of its origin is a major cosmogonic puzzle. Following Wood (1996), Boss & Durisen (2005a,b) suggest that GI-generated shocks may have played the dominant role in the thermal processing and mixing of primitive material in the Solar Nebula. Time-varying surface distortions should also have observable effects in disks around young stars. In addition to photometric variability, unusual spectroscopic features could be produced as the surface corrugations lift otherwise shaded material into

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Figure 9. Strong spiral shocks produce hydraulic jumps in a massive inner Solar Nebula model. Upper Left. This equatorial plane density grayscale shows the two-armed shocks generated interior to a cos(2^) potential perturbation that rotates with the fluid at 5 AU and is concentrated at that radius. Upper Right. A 3D density contour viewed from above the disk. The density value for the contour is low enough to represent the disk surface. Bottom.. This radial slice illustrates how the vertically jumping material curls inward over the main shock and crashes back down on the pre-shock gas in a breaking wave. The heavy solid lines are density contours, the light-shaded contours show shock heating, and the arrows are meridional velocity vectors of the gas. All figures are from the same time in the simulation. The slice in the bottom panel is roughly at 3 o'clock in the upper left panel. Figures are provided courtesy of A. C. Boley.

Figure 9. Strong spiral shocks produce hydraulic jumps in a massive inner Solar Nebula model. Upper Left. This equatorial plane density grayscale shows the two-armed shocks generated interior to a cos(2^) potential perturbation that rotates with the fluid at 5 AU and is concentrated at that radius. Upper Right. A 3D density contour viewed from above the disk. The density value for the contour is low enough to represent the disk surface. Bottom.. This radial slice illustrates how the vertically jumping material curls inward over the main shock and crashes back down on the pre-shock gas in a breaking wave. The heavy solid lines are density contours, the light-shaded contours show shock heating, and the arrows are meridional velocity vectors of the gas. All figures are from the same time in the simulation. The slice in the bottom panel is roughly at 3 o'clock in the upper left panel. Figures are provided courtesy of A. C. Boley.

the intense radiation field from the star and inner disk. It is even possible that some masers in massive protostars may be due to irradiation of the inner surfaces of spiral arcs (Durisen et al. 2001).

Boley & Durisen (2006) consider it highly likely that the "convective" motions reported by Boss (2004a) are actually dynamic vertical motions similar to those produced by hs-jumps. Boss (2004a) reports that his convective speeds are comparable to the sound speed and correlate with spiral arms, as expected for hs-jumps. Normal thermal convection is a quasistatic process occurring against an otherwise nearly hydrostatic background, and it

Figure 10. The concentration of 50 cm radius solid particles by gas drag in a Gl-active disk. Left. The equatorial plane gas density structure for the Md/Ms = 0.25 and tcoo\Q = 7.5 disk of Lodato & Rice (2004) after about seven outer rotations. Right. The particle-to-gas density ratio for the disk in the left panel for 50 cm particles after the particles have interacted with the gas for only one outer rotation period. The darkest black represents an enhancement of the initial average particle to gas ratio by a factor of about 50. Figures adapted from Rice et al. (2004).

Figure 10. The concentration of 50 cm radius solid particles by gas drag in a Gl-active disk. Left. The equatorial plane gas density structure for the Md/Ms = 0.25 and tcoo\Q = 7.5 disk of Lodato & Rice (2004) after about seven outer rotations. Right. The particle-to-gas density ratio for the disk in the left panel for 50 cm particles after the particles have interacted with the gas for only one outer rotation period. The darkest black represents an enhancement of the initial average particle to gas ratio by a factor of about 50. Figures adapted from Rice et al. (2004).

is difficult to imagine how an analog of convection can occur in a GI-active environment. Boss (2004a) does detect extensive superadiabatic gradients in the vertical direction, but these may be just another consequence of his different radiative boundary conditions. Some superadiabatic gradients are also seen in Mejia/Cai simulations, but it is not at all clear that they are related to thermal convection.

5.2. Interaction of GIs with solids and contaminants

As laid forth in classic papers by Weidenschilling (1977) and Volk et al. (1978), solid particles larger than dust grains experience net drifts in a disk relative to the gas. In a laminar disk with density and pressure decreasing radially and vertically outward, solids tend to orbit with Keplerian speeds, while the gas orbits somewhat more slowly due to pressure gradient forces. The resulting relative motions of the gas and solids induce drag forces that produce both radially inward and vertically downward drifts of the particles relative to the gas. Recently, Haghighipour & Boss (2003a,b) have explored the behavior of particles in a disk with a ring-like density maximum. Their ring, which is intended to represent a dense feature produced by GI activity, has a maximum at about 1 AU and a radial full width at half maximum that is also about an AU. Even for such a broad ring, solids migrate rapidly into the center. The particles with the shortest drift time scales, typically only a few hundred years, have sizes on the order of one meter.

The ring in Haghighipour & Boss (2003a,b) is a static structure. Rice et al. (2004) have injected solid particles into the Md/Ms = 0.25 and tcoolQ = 7.5 simulation of a gravitationally unstable disk by Lodato & Rice (2004). The gas disk is allowed to establish asymptotic behavior before the introduction of the particles. Starting with an initially uniform particle distribution, the gas plus particle disk is then integrated for one more outer rotation, or about 125 years. The particle gravitational forces and the back reaction of drag forces on the gas are not included. Figure 10 illustrates the remarkably effective concentration of 50-cm-radius particles in this short amount of time. The strong

figure 11. Left. Midplane contours of contaminants initially painted on the surface of the disk over a narrow annulus at 9 AU show mixing to the midplane and radial spreading along spiral arms in only 120 years. Right. The line shows the trajectory of a multi-Jupiter mass accreting point mass inserted into a GI-active disk in place of a dense clump. In both panels the square is 40 AU along an edge. Figures adapted from Boss (2004b) and Boss (2005).

figure 11. Left. Midplane contours of contaminants initially painted on the surface of the disk over a narrow annulus at 9 AU show mixing to the midplane and radial spreading along spiral arms in only 120 years. Right. The line shows the trajectory of a multi-Jupiter mass accreting point mass inserted into a GI-active disk in place of a dense clump. In both panels the square is 40 AU along an edge. Figures adapted from Boss (2004b) and Boss (2005).

implication is that the marshalling of particles into dense structures by gas-dynamical GI activity can rapidly accelerate the formation of planetesimals. On the other hand, Rice et al. (2004) also find that an initially thin particulate disk is stirred to a finite thickness by the vertical motions associated with GIs.

Another question of interest is how a GI-active disk will affect the orbits of planet-sized objects, either those existing prior to the onset of GIs or formed by the GIs themselves. Although not subject to significant aerodynamic gas drag, planets interact with the surrounding gas through gravitational torques and migrate radially (Armitage, this volume). To study this effect, Boss (2005) assumes that dense clumps formed in his disk simulations are bound protoplanets and replaces them with accreting point masses. As shown in the right panel of Figure 11, an accreting point mass at about 10 AU has a fairly stable orbit radius despite the background of dense time-varying structure and net radial inflow in the disk due to GIs. In other words, protoplanets formed rapidly in a massive GI-active disk do not migrate rapidly inward. This is encouraging for the survival of gas giants, if they can indeed be formed promptly by GIs in a massive disk. Such orbital survival of at least some protoplanets formed directly by GIs was already demonstrated by the SPH simulations of Mayer et al. (2004).

In addition to considering large condensed objects, Boss (2004b) also addresses the question of how contaminants, abundance anomalies, and small entrained particles might be mixed by GI turbulence. In a series of experiments, he creates a narrow region of altered composition and solves a 3D continuity plus diffusion equation for this "color" contaminant simultaneously with the hydrodynamics equations. One such experiment, seen in the left panel of Figure 11, shows that within only 120 years, an anomalous abundance painted at high disk altitude over a 1 AU-wide annulus at 9 AU has propagated to the midplane and spread radially. The contaminants tend to trace the dense spiral arms of the GIs, as would be expected if the mixing flows are hs-jumps (Fig. 9).

So, work to date indicates that GIs are likely to accelerate planetesimal formation by concentrating moderate-sized solids into dense structures, but they do not cause wholesale inward migration of large embedded solids. GIs are also effective at mixing fine-grained and gaseous components of the disk.

5.3. Combined effect of GIs and MRIs

It will eventually be necessary to understand how GIs interact with the presence of other disk transport mechanisms. The magnetorotational instabilities (MRIs), which occur in any disk with sufficient conductivity, are a likely source of turbulence (Balbus & Hawley 1998). It is possible for GIs and MRIs to coexist in disks in separate or even overlapping regions. Gammie (1996) points out that, for disks around young solar-type stars, there could be a region near the midplane beyond the innermost disk, called the "dead" zone, where MRIs do not occur due to the low degree of ionization of this cool and shielded gas. Accretion could continue by MRI turbulence in thin layers near the disk surface, but material in the dead zone would otherwise tend to accumulate until GIs set in. Using a simple a prescription for both MRIs and GIs, Armitage et al. (2001) model the behavior of a disk with a dead zone and find that, for moderate average infall rates onto the disk of about 10-6 M0/yr, the disk exhibits episodic accretion outbursts at intervals of 104 to 105 yrs, with peaks inflow rates reaching 10-5 Mq/yr. The peak accretion rate is similar to the inflow rates seen during the burst phase of tcool = constant simulations (§3.1).

Although the Armitage et al. calculations are suggestive, the adequacy of an a treatment for the behavior of GIs is somewhat questionable (§3.3). A step in the direction of a full blown global treatment has been taken in Fromang et al. (2004), where magnetized and self-gravitating adiabatic disks are evolved in 3D using a magnetohydrodynamics (MHD) code. When both mechanisms operate in their disks, they find significant interactions of global modes which tend to damp the GIs by tens of percents and also produce dynamic oscillations between low and high values of gravitational stress. In addition to strong global modes, the disk becomes suffused with turbulence. Only a limited parameter range is explored so far by Fromang et al. (2004), but there is potential for a rich phenomenology when both processes operate in the same disk. More research along these lines is necessary.

5.4. Hybrid planet formation scenario

Although there may be various ways to make core accretion plus gas capture work faster (Hubickyj, this volume) than in the classic models of Pollack et al. (1996), indications of planet formation at very early times around young stars like CoKu Tau/4 (Quillen et al. 2004) remain challenging for core accretion. When type I migration of growing cores is ignored, probabilistic treatments based on planet growth and disk evolution modeling can be used to fit the correlation of planet detection with central star metallicity in the framework of core accretion (Ida & Lin 2004; Kornet et al. 2005), but rapid type I migration of the cores inward into the star remains a vexing problem (Armitage, this volume).

As discussed in Section 4, it is not clear from currently available simulations that disks ever cool fast enough for direct planet formation by fragmentation; but there is significant agreement (e.g., Boss & Durisen 2005) that GIs will produce dense structures in disks, including spiral arms and arcs. Some of the most long-lived, well-defined features seen in nonfragmenting simulations, with tcool = constant (Pickett et al. 2003; Mejia et al. 2005a) and with the Mejia/Cai treatment of radiative cooling (Mejia et al. 2005b; Cai et al. 2006), are dense gas rings near the boundary between a hot inner disk and an outer GI-active disk. An analysis by Durisen et al. (2005) argues that ring formation is expected to occur near boundaries between GI-active and inactive regions. In Figure 12, ring growth appears to be due to a pattern of overlapping Lindblad resonances in the GI-

Inner

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