Hydrodynamic Instability
in the Similarity Solution of Viscous Accretion Disks
12
3
3
Tomohiro Ono , Takayuki Muto , Hideko Nomura , and Taku Takeuchi
1: Kyoto University, 2:Kogakuin University, 3: Tokyo Institute of Technology

Abstract
Protoplanetary disks are the birthplaces of planets. Studying structures and evolutions of protoplanetary disks is important for planetary science. It isgenerally thought that the magneto-hydrodynamic instability is a dominant mechanism in disk evolution. However, when we consider the region with steeppressure gradient such as a density bump or gap in disks, or an edge of disks, the hydrodynamic instability can be important. Protoplanetary disks evolve
via turbulent viscosity, and their evolution is well represented by the model of Lynden-Bell & Pringle (1974). In this model, the density profile of disks isanalytically derived as the similarity solution. The similarity solution has an exponential cutoff in surface density at the outermost region, and pressure
gradient becomes steep in this region. Therefore, hydrodynamic instability may occur. We first examine the stability of the similarity solution for Rotationalinstability. Rotational instability is local hydrodynamic axisymmetric instability. We find that the similarity solution is unstable against rotational instability(Ono et al., 2014). We also aim to investigate the Rossby wave instability of the similarity solution. Rossby wave instability is global hydrodynamic nonaxisymmetric instability. As a first step, we perform linear perturbation analysis using the method similar to that in Li et al. (2000), and find that the disk isunstable against Rossby wave instability when its surface density profile has a cut-off in the outer region.

IntroductionRossby Wave Instability

The Similarity Solution• The similarity solution, which is derived
analytically by Lynden-Bell & Pringle (1974;LBP74), well represents evolution of surface
density profile of ProtoPlanetary Disks (PPDs)
via turbulent viscosity.
• The surface density decreases in power law
in the inner region, and in exponential law in
The similarity solution of LBP74
the outer region.
• The similarity solution is utilized widely in many works.
• Some observations have suggested that gas surface density profiles ofPPDs are well fitted by the similarity solution (e.g., Hughes et al.2008).

Rossby wave instability is a global non-axisymmetric instability.
This instability is proposed by Lovelace+99 and Li+00.
In this poster, we show the results of some test calculations.
The marginally stableSettings
point for Rayleigh criterion
We set the density profile as below.

Hydrodynamic Instability• In the region where the radial pressure gradient force is strong,such as the outer region of similarity solution,
the disk is unstable for hydrodynamic instability.
• In this work, we investigate the stability of the similarity solution
for hydrodynamic instability.
• We focus on two kinds of hydrodynamic instability as below.
1. Rotational Instability (Ono et al., 2014)
2. Rossby wave Instability

Normalization & Parameter
we define non-dimensional parameters as follows.
・radius:・specific angular momentum:
We assume a power-law temperature profile with respect to radius.
・temperature:(β is power law index.)(subscript “0” means the value at r0)
In this work, we consider the aspect ratio at r0 :=0.1, 0.2, 0.3.
A large value ofmeans PPD is hot or mass of central star is low.
~0.1-0.18: Typical T Tauri disks (e.g., Andrews et al. 2009)~0.3: Disks irradiated by nearby massive stars
in young clusters (e.g., Hillenbrand et al. 1998)
For simplicity, we consider isothermal disks (β=0) in this poster.

Rotational Instability
Rotational Instability is a local axisymmetric instability.
Discriminant for this instability• The Rayleigh criterion
is the discriminant for this instability.
Method• We assume the radial equilibrium and calculate the rotation profile.
• We examine the stability of the similarity solution with Rayleighcriterion.
Result• The outer region of thesimilarity Solution is unstableagainst Rotational instability.
• The marginally stable point
approaches central star withincrease of H0/r0.

This surface density profile has
the marginally stable point forRayleigh criterion and a cut-off in
the outer region, and resembles
the similarity solution.
Method and Results
linear perturbation analysis using the method similar to that in Li+00.
Basic Equaiton
EOC:
EOM:
EOE:

Linearize these equations.
We consider perturbations,.
m is the azimuthal mode number.
ω is the mode frequency.

We can obtain the perturbation equation,
whereis the enthalpy.
B and C are the non-linear functions of ω.
The eigenfrequency ω is in general complex.
To solve this perturbation equation, weneed to obtain the complex roots ofthe determinant of an tridiagonal
matrix.
We calculate the map of determinant.
(color map: low “blue”, high “red”)

singularity of
unstable mode
unstable region

stable region

We look for the regions where roots exist in the map, and thenuse the Muller method to obtain exact values of the roots.
• This method well
We calculate eigenvector and perturbation
reproduces the results ofof density for eigenfrequency with m=3.
Li+00.
• This profile has unstablemodes against Rossby
wave instability (m=3).
• We can infer that thedisk with a cut-ff in theouter region is unstableagainst Rossby wave
instability .

Future Work• Upgrade the method in order to investigate Rossby wave instability ofthe similarity solution.
• Investigate which azimuthal mode has the fastest growth rate ofRossby wave instability.