Hydrodynamic instabilities and vorticity generation in protoplanetary disks
Miguel de Val Borro & Pawel Artymowicz
Stockholm Observatory, AlbaNova University Center, 10691 Stockholm, Sweden
Abstract
Instabilities around the planet as well as along the edges of the gap
formed in a protoplanetary disk by growing planets may influence
planet formation by affecting their migration. We study possible
instabilities by comparing the fully nonlinear hydrodynamical solutions (2-D PPM) of disk-planet problem with time-resolved modal
calculations using linear perturbation theory on the numerically obtained radial surface density profile. Our results show that, unlike in
some other recent studies, our polar-grid numerical code does not
generate much vorticity around a Jupiter in a standard solar nebula.
Modal calculations confirm that the obtained radial profiles of density are not susceptible to the growth of linear m=1,. . . ,6 modes on
relevant time scales. This result holds at high resolution runs. Since
differences in numerical algorithms may cause appearance/absence
of vortices, we propose a comparison of different hydrocodes, applied to a common test problem, to settle the issue.
DΣ̃
Σ̃ ∇ ṽ 0
Dt
Dṽ
1
∇P̃ ∇Φ
Dt
Σ̃
D P̃
0
Γ
Dt Σ̃
(1)
(2)
(3)
The perturbed surface density and the other variables in equations 1, 2, 3 have the form Σ̃ Σ δ Σ r φ t . We consider
perturbations which have the dependence on the azimuthal angle
∝ f r exp imφ iω t , where m is the azimuthal mode number.
When the previous equations are linearized, the resulting second
order differential equation for the enthalpy reads:
η B r η C r η 0 (4)
η δ P Σ, and B r C r are coefficients that depend on
thewhere
radial distance.
Numerical models and differences
Among the 100 extrasolar planetary systems, very few resemble
our own. The reasons for this diversity could be planet migration
in the protoplanetary disks and the simultaneous eccentricity evolution. Disk-planet interaction may have been crucial in our solar
system as well. For example, Levison et al. (1998) showed that the
planets in the solar system would tend to have high eccentricity, if
formed in the absence of eccentricity damping by some efficient
mechanism (the disk). Although analytical approaches have been
tried, the problem eventually comes down to numerical simulation,
best reflecting the true nonlinear response of the disk in form of
spiral shocks (wakes), and allowing for the mobility of the planet,
which modifies the gas flow around the planet Artymowicz (2003).
Important differences in the numerical results of different multidimensional hydrocodes are found. Some simulations show smooth
spiral shocks around the planet and little time variability of the flow
in the frame of the planet (e.g., ZEUS-based results of Lubow et al.
1999). Other codes (Figure 1), or the same codes run at higher resolution, produce waves and vortices at the edge of the gap, which
could in principle interact with the wake to cause semi-periodic disturbances propagating away along the shock. Together with nonstationarity of flow near the Roche lobe, these features might affect
the speed or, in some cases, the direction of migration; so it is important to scrutinize the models for any purely numerical instabilities.
We ran two 2-D codes based on the PPM algorithm (Piecewise
Parabolic Method) of Woodward & Collela, using Lagrangian formulation with remap to Eulerian grid. One used inertial Cartesian
coordinates and gave results broadly similar to the fully Eulerian
PPM shown in Figure 1. The other (Figure 4) was a polar-grid
code with variable-resolution mesh following the freely migrating
planet, and maximum resolution ranging between 0.003 and 0.01
of the star-planet distance. It produced much smoother disk density, and almost stationary wake profiles in the typical simulation
period of 100 orbits.
We solve the eigenproblem for the perturbed enthalpy using two
semi-analytical methods. One way of finding the complex eigenfrequency is discretizing the equation on a grid and use boundary
conditions to reduce the problem to finding numerically the roots
of the determinant of a complex tridiagonal matrix (E.g., Laughlin
et al. 1998).
Unstable modes calculation
We do not find any stationary solutions with consistent positive
growth rates, i.e., such that remain for a time of order of the growth
rate, in any of the studied models. This is consistent with the fact
that no vortices appear in the edges of the gap in these simulations
(cf. Figure 4).
In some particular models there are mode solutions that seem
to appear at very late times, near the end of the simulation when
the gap is becoming deeper. At the same time there are indications
that some mild instability is happening in the disk. However,
these are not the fast-growing, intermediate-m modes (m=3-6)
found in other calculations, but probably a growing m=1 disturbance in sufficiently massive disks. The appearance of the
slowly growing modes needs to be studied in detail in the future.
1.025
1.5
1.02
1.4
Omega 1.015
Density
1.3
1.01
1.2
1.005
1.1
1
1
0.4
0.6
0.8
1
Radius
1.2
1.4
1.6
0.4
0.6
0.8
1
Radius
1.2
1.4
1.6
Figure 2. Analytical surface density jump in an homogeneous background disk
as studied by Li et al. (2000).
We checked these two methods on the axisymmetric analytical
step jump profiles in surface density studied by Li et al. (2000),
see Figure 3. We considered azimuthal mode numbers from m=1
to 6 and calculated the growth rates of the unstable modes and the
corresponding eigenfunctions. For a steep density profile , both
our methods agree with those of Li et al. (2000).
Figure 5. Contour lines of the real (red line) and imaginary (green line) parts
of the mismatch of the logarithmic derivative of the wavefunction at the fitting
point of the shooting method, for the underlying axisymmetrized gas density in
Figure 4 (middle panel). Eigenfrequencies are located at the intersections of
the green and red contours. The horizontal and vertical axis are the real and
imaginary parts of the mode frequency in units of the Keplerian frequency at
radius unity. There are no clear stable roots that appear in our analysis during the
first 100 periods of the simulation.
Discussion
The results of our modal calculation are consistent with the absence of rapidly growing waves/vortices near the edge of the gap in
our polar-grid PPM simulations, which is thus not due to any artificial numerical damping of unstable modes. This type of code does
not produce the necessary steepness of the surface density profile
and/or does not support growing nonaxisymmetric perturbations.
Why? While in the case of more diffusive methods like ZEUS or
monotonic transport algorithms this might be due to numerical viscosity or explicit artificial viscosity, both our PPM methods have
very low numerical diffusion, and no explicit Navier-Stokes viscosity. The differences are not primarily due to resolution either. However, we notice that while the Cartesian-grid code does x-y sweeps
of the hydrodynamical solution once per timestep, the polar-grid
code performs these sweeps, as well as gravitational force calculation, twice per timestep, to guarantee 2nd order time integration accuracy (Coriolis terms in equations require accurate estimate
of velocity components perpendicular to that currently advanced).
We found that high-order polynomial interpolation of gravitational
forces is also required for the smoothness of solutions.
We conclude that the vorticity generation at disk edges
is potentially important, but depends on numerical algorithms and implementation details.
Comprehensive testing
of different hydrocodes on a well defined problem posed in
http://www.astro.su.se/ pawel/planets/test.hydro.html
might help resolve the differences. We invite all the interested
modelers to join this comparison.
REFERENCES
Artymowicz, P. 2003, Debris Disks and the Formation of Planets: A symposium
in memory of F. Gillett, 27, 13 pp.
We performed a modal analysis of the numerically computed
disks, in order to see if those codes which do not predict rapid
vorticity generation by a planet do not artificially damp modes,
which might be indicated as growing by linear stability analysis.
Although the perturbations considered are infinitesimal and sinusoidally varying in azimuth, eventually their growth would form
vortices (or Rossby waves) in the nonlinear regime (Li et al. 2001).
We consider non-axisymmetric infinitesimal perturbations sinusoidally varying in azimuth to the inviscid Euler Equations, using a
homentropic equation of state for the disk.
We carried out modal growth analysis on the density profiles obtained with the polar-grid PPM with embedded giant planets ranging from a Neptune to a Jupiter mass, in a disk is with soundspeed profile of a standard, slightly flaring solar nebula (soundspeed/Keplerian velocity = 0.05).
1.6
Figure 1. Gas density in Nelson and Benz (2003) PPM simulation of a low mass
planet (0.3 MJ) in a disk. The gap edge and the whole disk appear wavy (e.g.,
m=5 wave is seen just outside the planet’s orbit).
Modal analysis of PPM simulations
Another approach is the shooting method, where the integration
proceeds from both sides to an intermediate fitting point, where
continuity is required of the eigenfunction and its first derivative. Our implementation uses a leapfrog method to integrate the
equation from the boundaries to the fitting point. The values of
the enthalpy and its derivative at the starting points are specified
with the boundary conditions based on outgoing spiral waves.
Figure 3. Eigenfunction of the problem for an analytical step jump with azimuthal number m=3. The blue, red and green lines correspond to the amplitude,
the real, and the imaginary parts of the eigenfunction respectively. The density
perturbation is proportional to the eigenfunction.
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Levison, H. F., Lissauer, J. J., & Duncan, M. J. 1998, AJ, 116, 1998.
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Figure 4. Polar-grid PPM simulation of a Jupiter mass planet in a standard solar
nebula. Color-coded surface density map is shown for three different times (approx. 6, 48 and 82 orbital periods of the planet). The horizontal axis is the radial
distance with the semi-major axis given by the distance between large ticks. The
vertical axis shows the azimuthal angle covering 360 degrees.
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This manuscript was prepared with the AAS LATEX macros v5.0.