2.1. Linear regime

Toomre (1964) showed analytically that gravitational instabilities to ring-like modes occur in thin gas disks when the parameter Q = csĀ«/nGS is less than unity. Here cs is the sound speed, k is the epicyclic frequency at which a fluid element perturbed from circular motion will oscillate, G is the gravitational constant, and E is the surface density. High pressure, represented by cs, stabilizes short wavelengths, and a high rotation rate, represented by k, stabilizes long wavelengths. For a Keplerian disk, k = the rotational angular speed Q. The E in the denominator conveys the destabilizing effect of disk self-gravity. A large body of numerical simulations, including those referenced in this review and dating back at least to Papaloizou & Savonije (1991), show that nonaxisymmetric modes leading to spiral structure are unstable for Q < 1.5-1.7. The instability is linear and dynamic, which means that small perturbations grow exponentially on the time scale of a rotation period Prot = 2n/Q (e.g., Laughlin et al. 1998; Pickett et al. 1998). The precise Q-limit for instability depends somewhat on the structure of the disk. The growing multi-arm spirals have a predominantly trailing pattern, the amplification mechanism appears to be swing, and multiple modes with different numbers of arms can grow simultaneously (Nelson et al. 1998; Pickett et al. 1998; Laughlin et al. 1998; Mayer et al. 2004). Because the critical Q for nonaxisymmetric modes is higher than that for axisymmetric modes, these should be the ones encountered in Nature.

The behavior of GIs in the nonlinear regime is difficult to treat analytically, and so the results summarized below come almost entirely from simulations. Recent numerical calculations have differed in geometry and hydrodynamic algorithm. Examples include high-order governing equations in 3D (Laughlin et al. 1998), 2D (thin disk) shearing box local simulations (Gammie 2001), 2D (Nelson et al. 1998), and 3D global grid-based simulations in spherical (Boss 1997 to 2005) and cylindrical (Pickett et al. 1998, 2000, 2003; Mejia et al. 2005a) coordinates, and 2D (Nelson et al. 1998, 2000) and 3D (Mayer et al. 2002, 2004; Rice et al. 2003; Lodato & Rice 2004, 2005) Smoothed Particle Hydrodynamics (SPH) simulations.

As GIs reach nonlinear amplitudes, two major effects control their further development. The first and most important is the balance achieved between the loss of energy by radiative cooling and heating of the disk by dissipation of energy associated with the spiral waves. In many calculations, the latter takes the form of shock heating. That a balance of heating and cooling would control the nonlinear behavior of GIs was anticipated by Goldreich & Lynden-Bell (1965), and this notion has been used as a basis for developing accretion disk models for GI-active disks (e.g., Paczynski 1978; Lin & Pringle 1987). Beginning with Tomley et al. (1991), thermal regulation of GIs has been confirmed by many researchers (Pickett et al. 1998, 2000; Nelson et al. 2000; Gammie 2001; Boss 2001, 2002b, 2003; Rice et al. 2003; Lodato & Rice 2004, 2005; Mejia et al. 2005a). The key parameter controlling the nonlinear amplitude is the cooling rate, i.e., how fast the disk is able to lose the thermal energy pumped into it by GIs. The source of the heating is ultimately gravitational energy. Some comes from the collapse or contraction of material into dense self-gravitating structures, but most is gravitational energy released by net mass transport within the disk.

The second effect that is important at large amplitudes is nonlinear mode coupling, studied extensively by Laughlin and his collaborators (Laughlin et al. 1997, 1998). Power quickly becomes distributed over modes with various wavelengths and number of arms, resulting in a self-gravitating turbulence or gravitoturbulence, in which gravitational torques and Reynold's stresses can be important on a range of scales (Gammie 2001; Lodato & Rice 2004).

Prior to the 1990s, most work on GIs was done in a thin-disk approximation, where the disk is assumed to be hydrostatic in the z-direction. A significant development over the past decade is the recognition that the vertical structure of the disk plays a crucial role, both for cooling and for essential aspects of the dynamics. This is strongly emphasized by Pickett et al. (1998, 2000, 2003), who note an apparent relationship between the spiral modes in disks and the surface or /-modes of stars (see also Pickett et al. 1996; Lubow & Ogilvie 1998). In 3D, the notion of spiral "density" waves is only truly applicable to vertically isothermal disks, where the mode amplitude is actually uniform with height. Otherwise, GIs characteristically have large amplitudes at the surface of the disk. Although shock compression occurs in GI waves, it is accompanied, in disks with vertical temperature stratification, by extremely large surface distortions, strong vertical motions, and disproportionately greater shock heating at high disk altitudes. A dramatic illustration of such behavior will be presented in Section 5.1.

2.3. Fragmentation criteria A consensus answer has emerged over the last few years about when turbulent GI spiral structure may fragment into discrete dense pieces. Let the cooling time tcool be defined as the gas internal energy density e divided by the volumetric cooling rate A. For power-law equations of state and with tcool prescribed to be some value over an annulus of the disk,

the thin shearing box simulations of Gammie (2001) show that fragmentation occurs if and only if tcooln ^^ 3, or, equivalently tcool ^^ Prot/2, where Prot is the disk rotation period. This is confirmed in global SPH simulations by Rice et al. (2003), as shown in Figure 1. The disk in Figure 1 has a disk-to-star mass ratio Md/Ms =0.1, surface density E ~ r-7/4, and an initial Q = 2. When evolved with tcoolQ = 3, it fragments, but, when tcoolil = 5, it does not. Grid-based calculations by Mejia et al. (2005a) also yield fragmentation results consistent with Gammie (2001). The critical value of tcoolil can be somewhat larger than three for more massive and physically thicker disks (Rice et al. 2003). For disks evolved under isothermal conditions, in the sense either that the fluid elements are forced to maintain the same temperature or that the disk temperature is kept constant locally at its initial value, a simple cooling time cannot be defined. In this case, thin shearing box simulations by Johnson & Gammie (2003) give fragmentation if and only if Q < 1.4. This agrees roughly with results from global simulations by other groups taken as a whole (e.g., Boss 2000; Nelson et al. 1998; Pickett et al. 2000, 2003; Mayer et al. 2002, 2004). Classic examples of isothermal disk fragmentation are shown in Figure 2.

So, all researchers agree that a disk will fragment into dense clumps if it is sufficiently cool and evolves isothermally, or if it cools on a short time scale compared with a rotation period. Where controversy remains is how long-lived these clumps may be. Because the peak clump densities always increase with increasing numerical resolution, and because they always seem to have thermal energies dominated by self-gravitational energies, Boss (2005) concludes that they are permanent bound objects. On the other hand, Pickett et al. (2003) and Mejia et al. (2005a) find, regardless of resolution, that clumps are eventually destroyed by tidal stresses, shears, and collisions. Unfortunately, except for the case shown in Figure 2, Boss's highest-resolution simulations (especially Boss 2005) are usually not integrated for long times, so it remains unclear whether or not his clumps might eventually be disrupted.

Figure 3 presents a sobering result on clump longevity obtained with the Pickett et al. (2003) code. The same low-Q disk is evolved isothermally at high resolution with and without artificial bulk viscosity (ABV) in the momentum equation. Without ABV (left

Figure 2. Equatorial densities for two simulations of isothermally evolved disks that fragment into long-lived multi-Jupiter mass clumps. Left. This grid-based calculation by Boss (2000) shows a 0.9 Mq disk with a 20 AU outer radius and a minimum Q ~ 1.3 orbiting a 1 Mq star after several outer rotations. The dense clump near 12 o'clock can be tracked for many orbits. Right. Long-lived clumps form in an SPH simulation by Mayer et al. (2004) for a 0.1 Mq and Q = 1.38 disk with an initial outer radius of 20 AU orbiting a 1 Mq star. Figures are adapted from Boss (2001) and from Mayer et al. (2004).

Figure 2. Equatorial densities for two simulations of isothermally evolved disks that fragment into long-lived multi-Jupiter mass clumps. Left. This grid-based calculation by Boss (2000) shows a 0.9 Mq disk with a 20 AU outer radius and a minimum Q ~ 1.3 orbiting a 1 Mq star after several outer rotations. The dense clump near 12 o'clock can be tracked for many orbits. Right. Long-lived clumps form in an SPH simulation by Mayer et al. (2004) for a 0.1 Mq and Q = 1.38 disk with an initial outer radius of 20 AU orbiting a 1 Mq star. Figures are adapted from Boss (2001) and from Mayer et al. (2004).

Figure 3. Midplane density grayscales for a Pickett et al. (2000) star/disk model with Q = 1.35 evolved at high resolution (512 azimuthal zones) with the mass distribution of the central star frozen. Left. Without artificial bulk viscosity, fragments form within one outer rotation, but have all been destroyed through violent interactions by 4.9 outer rotations, the time shown. Right. With artificial bulk viscosity, dense fragments still survive at the same evolutionary time as shown in the left panel. The fragment closest to 45 degrees left of the vertical can be traced for at least four rotations, and persists until the simulation is stopped at 7.3 outer rotations. Figures are provided courtesy of M. K. Pickett.

Figure 3. Midplane density grayscales for a Pickett et al. (2000) star/disk model with Q = 1.35 evolved at high resolution (512 azimuthal zones) with the mass distribution of the central star frozen. Left. Without artificial bulk viscosity, fragments form within one outer rotation, but have all been destroyed through violent interactions by 4.9 outer rotations, the time shown. Right. With artificial bulk viscosity, dense fragments still survive at the same evolutionary time as shown in the left panel. The fragment closest to 45 degrees left of the vertical can be traced for at least four rotations, and persists until the simulation is stopped at 7.3 outer rotations. Figures are provided courtesy of M. K. Pickett.

panel), the disk fragments rapidly into many clumps, but the violent interactions of the clumps destroys them all by the time shown in Figure 3. With ABV on in the momentum equation (right panel), clump formation is delayed and is less violent, so that the clumps, once formed, are long lived. One clump in the right panel of Figure 3 can be traced for four complete orbits, and still exists when the simulation is stopped. The point of this demonstration is that clump longevity is sensitive to details of the numerical treatment. ABV is an unavoidable feature of SPH simulations where long-lived clumps are most commonly seen (Mayer et al. 2004). However, most of the simulations by Boss do not have an explicit ABV, and so it is unclear whether Figure 3 has any relevance to clump survival in his code. To complicate matters further, fragmentation can occur for purely numerical reasons (Bate & Burkert 1997; Truelove et al. 1997; Nelson 2003).

It will undoubtedly require a great deal of future effort to determine whether fragments in disks are truly long-lived protoplanets. It may prove more profitable to ascertain instead whether "real" disks ever cool fast enough to fragment in the first place, and to investigate other important effects related to GIs and planet formation that are more tractable. The rest of this review focuses on several recent advances in our understanding of GIs, particularly regarding how GIs interact with, and are affected by, a range of other physical processes, including radiative cooling.

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