skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: NON-BAROTROPIC LINEAR ROSSBY WAVE INSTABILITY IN THREE-DIMENSIONAL DISKS

Abstract

Astrophysical disks with localized radial structure, such as protoplanetary disks containing dead zones or gaps due to disk-planet interaction, may be subject to the non-axisymmetric Rossby wave instability (RWI) that leads to vortex formation. The linear instability has recently been demonstrated in three-dimensional (3D) barotropic disks. It is the purpose of this study to generalize the 3D linear problem to include an energy equation, thereby accounting for baroclinity in three dimensions. Linear stability calculations are presented for radially structured, vertically stratified, geometrically thin disks with non-uniform entropy distribution in both directions. Polytropic equilibria are considered but adiabatic perturbations assumed. The unperturbed disk has a localized radial density bump, making it susceptible to the RWI. The linearized fluid equations are solved numerically as a partial differential equation eigenvalue problem. Emphasis on the ease of method implementation is given. It is found that when the polytropic index is fixed and adiabatic index increased, non-uniform entropy has negligible effect on the RWI growth rate, but pressure and density perturbation magnitudes near a pressure enhancement increase away from the midplane. The associated meridional flow is also qualitatively changed from homentropic calculations. Meridional vortical motion is identified in the nonhomentropic linear solution, as well asmore » in a nonlinear global hydrodynamic simulation of the RWI in an initially isothermal disk evolved adiabatically. Numerical results suggest that buoyancy forces play an important role in the internal flow of Rossby vortices.« less

Authors:
 [1]
  1. Canadian Institute for Theoretical Astrophysics, 60 St. George Street, Toronto, ON M5S 3H8 (Canada)
Publication Date:
OSTI Identifier:
22127015
Resource Type:
Journal Article
Resource Relation:
Journal Name: Astrophysical Journal; Journal Volume: 765; Journal Issue: 2; Other Information: Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; ACCRETION DISKS; ASTRONOMY; ASTROPHYSICS; AXIAL SYMMETRY; COMPUTERIZED SIMULATION; DISTURBANCES; EIGENVALUES; ENTROPY; HYDRODYNAMICS; INDEXES; INSTABILITY; PARTIAL DIFFERENTIAL EQUATIONS; PLANETS; PROTOPLANETS; THREE-DIMENSIONAL CALCULATIONS; VORTICES

Citation Formats

Lin, Min-Kai, E-mail: mklin924@cita.utoronto.ca. NON-BAROTROPIC LINEAR ROSSBY WAVE INSTABILITY IN THREE-DIMENSIONAL DISKS. United States: N. p., 2013. Web. doi:10.1088/0004-637X/765/2/84.
Lin, Min-Kai, E-mail: mklin924@cita.utoronto.ca. NON-BAROTROPIC LINEAR ROSSBY WAVE INSTABILITY IN THREE-DIMENSIONAL DISKS. United States. doi:10.1088/0004-637X/765/2/84.
Lin, Min-Kai, E-mail: mklin924@cita.utoronto.ca. 2013. "NON-BAROTROPIC LINEAR ROSSBY WAVE INSTABILITY IN THREE-DIMENSIONAL DISKS". United States. doi:10.1088/0004-637X/765/2/84.
@article{osti_22127015,
title = {NON-BAROTROPIC LINEAR ROSSBY WAVE INSTABILITY IN THREE-DIMENSIONAL DISKS},
author = {Lin, Min-Kai, E-mail: mklin924@cita.utoronto.ca},
abstractNote = {Astrophysical disks with localized radial structure, such as protoplanetary disks containing dead zones or gaps due to disk-planet interaction, may be subject to the non-axisymmetric Rossby wave instability (RWI) that leads to vortex formation. The linear instability has recently been demonstrated in three-dimensional (3D) barotropic disks. It is the purpose of this study to generalize the 3D linear problem to include an energy equation, thereby accounting for baroclinity in three dimensions. Linear stability calculations are presented for radially structured, vertically stratified, geometrically thin disks with non-uniform entropy distribution in both directions. Polytropic equilibria are considered but adiabatic perturbations assumed. The unperturbed disk has a localized radial density bump, making it susceptible to the RWI. The linearized fluid equations are solved numerically as a partial differential equation eigenvalue problem. Emphasis on the ease of method implementation is given. It is found that when the polytropic index is fixed and adiabatic index increased, non-uniform entropy has negligible effect on the RWI growth rate, but pressure and density perturbation magnitudes near a pressure enhancement increase away from the midplane. The associated meridional flow is also qualitatively changed from homentropic calculations. Meridional vortical motion is identified in the nonhomentropic linear solution, as well as in a nonlinear global hydrodynamic simulation of the RWI in an initially isothermal disk evolved adiabatically. Numerical results suggest that buoyancy forces play an important role in the internal flow of Rossby vortices.},
doi = {10.1088/0004-637X/765/2/84},
journal = {Astrophysical Journal},
number = 2,
volume = 765,
place = {United States},
year = 2013,
month = 3
}
  • Numerical calculations of the linear Rossby wave instability (RWI) in global three-dimensional (3D) disks are presented. The linearized fluid equations are solved for vertically stratified, radially structured disks with either a locally isothermal or polytropic equation of state, by decomposing the vertical dependence of the perturbed hydrodynamic quantities into Hermite and Gegenbauer polynomials, respectively. It is confirmed that the RWI operates in 3D. For perturbations with vertical dependence assumed above, there is little difference in growth rates between 3D and two-dimensional (2D) calculations. Comparison between 2D and 3D solutions of this type suggests the RWI is predominantly a 2D instabilitymore » and that 3D effects, such as vertical motion, can be interpreted as a perturbative consequence of the dominant 2D flow. The vertical flow around corotation, where vortex formation is expected, is examined. In locally isothermal disks, the expected vortex center remains in approximate vertical hydrostatic equilibrium. For polytropic disks, the vortex center has positive vertical velocity, whose magnitude increases with decreasing polytropic index n.« less
  • It has been suggested that the transition between magnetorotationally active and dead zones in protoplanetary disks should be prone to the excitation of vortices via Rossby wave instability (RWI). However, the only numerical evidence for this has come from alpha disk models, where the magnetic field evolution is not followed, and the effect of turbulence is parameterized by Laplacian viscosity. We aim to establish the phenomenology of the flow in the transition in three-dimensional resistive-magnetohydrodynamical models. We model the transition by a sharp jump in resistivity, as expected in the inner dead zone boundary, using the PENCIL CODE to simulatemore » the flow. We find that vortices are readily excited in the dead side of the transition. We measure the mass accretion rate finding similar levels of Reynolds stress at the dead and active zones, at the {alpha} Almost-Equal-To 10{sup -2} level. The vortex sits in a pressure maximum and does not migrate, surviving until the end of the simulation. A pressure maximum in the active zone also triggers the RWI. The magnetized vortex that results should be disrupted by parasitical magneto-elliptic instabilities, yet it subsists in high resolution. This suggests that either the parasitic modes are still numerically damped or that the RWI supplies vorticity faster than they can destroy it. We conclude that the resistive transition between the active and dead zones in the inner regions of protoplanetary disks, if sharp enough, can indeed excite vortices via RWI. Our results lend credence to previous works that relied on the alpha-disk approximation, and caution against the use of overly reduced azimuthal coverage on modeling this transition.« less
  • In an earlier work we identified a global, nonaxisymmetric instability associated with the presence of an extreme in the radial profile of the key function L(r){identical_to}({sigma}{omega}/{kappa}{sup 2})S{sup 2/{gamma}} in a thin, inviscid, nonmagnetized accretion disk. Here {sigma}(r) is the surface mass density of the disk, {omega}(r) is the angular rotation rate, S(r) is the specific entropy, {gamma} is the adiabatic index, and {kappa}(r) is the radial epicyclic frequency. The dispersion relation of the instability was shown to be similar to that of Rossby waves in planetary atmospheres. In this paper, we present the detailed linear theory of this Rossby wavemore » instability and show that it exists for a wider range of conditions, specifically, for the case where there is a ''jump'' over some range of r in {sigma}(r) or in the pressure P(r). We elucidate the physical mechanism of this instability and its dependence on various parameters, including the magnitude of the ''bump'' or ''jump,'' the azimuthal mode number, and the sound speed in the disk. We find a large parameter range where the disk is stable to axisymmetric perturbations but unstable to the nonaxisymmetric Rossby waves. We find that growth rates of the Rossby wave instability can be high, {approx}0.2{omega}{sub K} for relative small jumps or bumps. We discuss possible conditions which can lead to this instability and the consequences of the instability. (c) 2000 The American Astronomical Society.« less
  • We carry out two-fluid, two-dimensional global hydrodynamic simulations to test whether protostellar infall can trigger the Rossby wave instability (RWI) in protoplanetry disks. Our results show that infall can trigger the RWI and generate vortices near the outer edge of the mass landing on the disk (i.e., centrifugal radius). We find that the RWI is triggered under a variety of conditions, although the details depend on the disk parameters and the infall pattern. The common key feature of triggering the RWI is the steep radial gradient of the azimuthal velocity induced by the local increase in density at the outermore » edge of the infall region. Vortices form when the instability enters the nonlinear regime. In our standard model where self-gravity is neglected, vortices merge together to a single vortex within ∼20 local orbital times, and the merged vortex survives for the remaining duration of the calculation (>170 local orbital times). The vortex takes part in outward angular momentum transport, with a Reynolds stress of ≲10{sup −2}. Our two-fluid calculations show that vortices efficiently trap dust particles with stopping times of the order of the orbital time, locally enhancing the dust to gas ratio for particles of the appropriate size by a factor of ∼40 in our standard model. When self-gravity is considered, however, vortices tend to be impeded from merging and may eventually dissipate. We conclude it may well be that protoplanetary disks have favorable conditions for vortex formation during the protostellar infall phase, which might enhance early planetary core formation.« less
  • A fully analytic expression for the linear corotation torque to first order in eccentricity for planets in non-barotropic protoplanetary disks is derived, taking into account the effect of disk entropy gradients. This torque formula is applicable to both the co-orbital, corotation torques and the non-co-orbital, corotation torques—for planets in orbits with non-zero eccentricity—in disks where the thermal diffusivity and viscosity are sufficient to maintain the linearity of these interactions. While the co-orbital, corotation torque is important for migration of planets in Type I migration, the non-co-orbital, corotation torque plays an important role in the eccentricity evolution of giant planets thatmore » have opened gaps in the disk. The presence of an entropy gradient in the disk can significantly modify the corotation torque in both these cases.« less