Rossby wave instability in astrophysical discs

Rossby Wave Instability (RWI) is a concept related to astrophysical discs. In non-self-gravitating discs, for example around newly forming stars, the instability can be triggered by an axisymmetric bump, at some radius r_0, in the disc surface mass-density. It gives rise to exponentially growing non-axisymmetric perturbation [\propto \exp({\rm i}m\phi), m=1,2..] in the vicinity of r_0 consisting of anticyclonic vortices. These vortices are regions of high pressure and consequently act to trap dust particles which in turn can facilitate planetesimal growth in proto-planetary discs. The Rossby vortices in the discs around stars and black holes may cause the observed quasi-periodic modulations of the disc's thermal emission.

The theory of the Rossby wave instability (RWI) in accretion discs was developed by Lovelace et al.[1] and Li et al.[2] for thin Keplerian discs with negligible self-gravity and earlier by Lovelace and Hohlfeld [3] for thin disc galaxies where the self-gravity may or may not be important and where the rotation is in general non-Keplerian. In the first case the instability can occur if there is an axisymmetric bump (as a function of radius) in the inverse potential vorticity


{L}(r) =
{\Sigma ~S^{2/\gamma} \over 2({\mathbf \nabla \times u})\cdot\hat{\mathbf z}}~,

at some radius r_0, where \Sigma is the surface mass density of the disc, {\mathbf u}\approx r\Omega(r)\hat{\phi~} is the flow velocity of the disc, \Omega(r) \approx (GM_*/r^3)^{1/2} is the angular velocity of the flow (with M_* the mass of the central star), S is the specific entropy of the gas, and \gamma is the specific heat ratio. The approximations involve the neglect of the relatively small radial pressure force. Note that { L} is related to the inverse of the {\it vortensity} which is defined as ({\mathbf \nabla \times u})_z/\Sigma. A sketch of a bump in { L}(r) is shown in Figure 1.

Fig.1. Schematic view of the Rossby wave instability with the two propagating regions for the Rossby waves, and in between the evanescent regions close to the inner and outer Lindblad resonant radii, (r_{\rm ILR} and r_{\rm OLR}), respectively, adapted from Ref.12. The radii of these Lindblad resonances are given by the equations \omega = m \Omega(r_{\rm LR})\pm \kappa(r_{\rm LR}), where \kappa(r) is the radial epicyclic frequency which is approximately equal to \Omega(r) for a Keplerian disc. The corotation radius r_{\rm C} is the radius where \omega = m \Omega(r_{\rm C}).

Rossby waves, named after Carl-Gustaf Arvid Rossby are important in planetary atmospheres and oceans and are also known as it planetary waves (see Rossby et al.;[4] Brekhovskikh and Goncharovv;[5] Chelton and Schlax;[6] Lindzen [7]). These waves have a significant role in the transport of heat from equatorial to polar regions of the Earth. They may have a role in the formation of the long-lived (>300 yr) Great Red Spot on Jupiter which is an anticyclonic vortex (e.g., Marcus [8]). The Rossby waves have the notable property of having the phase velocity opposite to the direction of motion of the atmosphere or disc in the comoving frame of the fluid (ref, 5; ref. 1).

Schrödinger-like equation for perturbation

Linearization of the Euler and continuity equations for a thin fluid disc with perturbations proportional to f(r)\exp({\rm i}m\phi-{\rm i }\omega t)(with azimuthal mode number m=1, 2,.. and angular frequency \omega) leads to a Schrödinger-like equation for the enthalpy perturbation \psi=\delta p/\rho,


{d^2 \psi \over dr^2}= V_{\rm eff}(r)~\psi~.

The effective potential well V_{\rm eff} (r) is closely related to {L}(r): If the height of the bump in { L}(r) is too small the potential well is shallow and there are no `bound Rossby wave states' in the well. On the other hand for a sufficiently large bump in { L}(r) the potential V_{\rm eff} is sufficiently deep to have a bound state. The condition for there to be just one bound state allows one to solve for the imaginary part of the wave frequency, \omega_i = \Im(\omega) which is the growth rate of the instability (ref. 1). For moderate strength bumps (with fractional amplitudes \Delta\Sigma/\Sigma \lesssim 0.2), the growth rates are of the order of \omega_i = (0.1-0.2)\Omega(r_0). The real part of the wave frequency \omega_r =\Re(\omega) is approximately m\Omega(r_0).

A more complete analysis (Tagger;[9] Tsang and Lai;[10] Lai and Zhang [11]) reveals that the Rossby wave is not completely trapped in the potential well V_{\rm eff}, but leaks outward across a forbidden region at an outer Lindblad resonance (at r_{\rm OLR} indicated in Figure 1) and inward across another forbidden region at an inner Lindblad resonance (at r_{\rm ILR}). Once the waves cross the forbidden regions they propagate as spiral density wave. The full expression for the effective potential for a thin homentropic (S= const) disc is

V_{\rm eff} = {2m \Omega \over r (\Delta \omega)} {d\over dr}\left[\ln\left({\Omega \Sigma \over \kappa^2-(\Delta \omega)^2}\right)\right]+{m^2 \over r^2}+{\kappa^2-(\Delta \omega)^2 \over c_s^2}

where \Delta \omega \equiv \omega-m\Omega is the Doppler shifted wave frequency in the reference frame moving with the disc matter, c_s is the sound speed in the disc, and \kappa is the radial epicyclic angular frequency, with \kappa^2=r^{-3}d\ell^2/dr and \ell=ru_\phi the specific angular momentum (Meheut et al.[12]).

Fig. 2. Effective potential for a Gaussian surface density bump of peak amplitude \Delta \Sigma/\Sigma =0.2 and width \Delta r/r = 0.05 for  m = 2 (upper panel) and m = 5 (lower panel) adapted from ref. 12. Waves can propagate only in the regions where V_{\rm eff}(r) < 0. The large positive values of V_{\rm eff} occur at the inner and outer Lindblad resonance radii.

Figure 2 shows the effective potential for sample cases. Note that the inward propagating waves with \omega_r < m \Omega(r) have negative energy (E<0) whereas the outward propagating waves with \omega_r > m \Omega(r) have positive energy (E>0) (ref. 12).

The Rossby wave instability occurs because of the local wave trapping in a disc. It is related to the Papaloizou and Pringle instability;[13][14]) where the wave is trapped between the inner and outer radii of a disc or torus.

Figure 3 shows computer simulations of the formation of a single Rossby vortex.[15]

Figure 3. Rossby wave instability in a Keplerian Disk

More information on the Rossby wave instability in astrophysical discs can be found in Lovelace and Romanova (ref.15).

References

  1. Lovelace, R.V.E., Li, H., Colgate, S.A., \& Nelson, A.F. 1999, "Rossby Wave Instability of Keplerian Accretion Disks", ApJ, 513, 805-810,http://arxiv.org/abs/astro-ph/9809321
  2. Li, H., Finn, J.M., Lovelace, R.V.E., \& Colgate, S.A. 2000, ``Rossby Wave Instability of Thin Accretion Disks. II. Detailed Linear Theory, ApJ, 533, 1023-1034, http://arxiv.org/abs/astro-ph/9907279
  3. Lovelace, R.V.E., \& Hohlfeld, R.G. 1978, "Negative mass instability of flat galaxies", ApJ, 221, 51-61, http://adsabs.harvard.edu/abs/1978ApJ...221...51L
  4. Rossby C.-G. and Collaborators 1939, "Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centers of action", Journal of Marine Research, 2, 38-55, http://www.ingentaconnect.com/content/jmr/jmr/1939/00000002/00000001/art00006
  5. Brekhovskikh, L. M., \& Goncharov, V. 1993, {\it Mechanics of Continua and Wave Dynamics} (Berlin : Springer), pp 246-252, http://adsabs.harvard.edu/abs/1985SSWP....1.....B
  6. Chelton, D.B., and Schlax, M.G. 1996, ``Global Observations of Oceanic Rossby Waves, Science, 272, 234-238, http://www.ocean.washington.edu/courses/oc513/Chelton.Science.1996.pdf
  7. Lindzen, R.S. 1988, "Instability of plane parallel shear flow (toward a mechanistic picture of how it works", Pure Appl. Geophys., 126, 103-121, http://link.springer.com/article/10.100
  8. Marcus, P.S. 1993, ``Jupiter's Great Red Spot and other Vortices, Ann. Rev. Astron. and Astrophys., 31, 523-573, http://adsabs.harvard.edu/abs/1993ARA
  9. Tagger, M., 2001, "On Rossby waves and vortices with differential rotation", A\&A, 380, 750-757, http://arxiv.org/abs/astro-ph/0110298
  10. Tsang, D., and Lai, D., 2008, "Super-reflection in fluid discs: corotation amplifier, corotation resonance, Rossby waves and overstable modes", MNRAS, 387, 446-462, http://arxiv.org/abs/0710.2313
  11. Lai, D., and Tsang ,D., 2009, ``Corotational instability of inertial-acoustic modes in black hole accretion discs and quasi-periodic oscillations, MNRAS, 393, 979-991, http://arxiv.org/abs/0810.0203
  12. Meheut, H., Lovelace, R.V.E., \& Lai, D. 2013, "How Strong are Rossby Vortices", MNRAS, 430, 1988-1993, http://arxiv.org/abs/1301.0689
  13. Papaloizou, J. C. B., \& Pringle, J. E. 1984, "The dynamical stability of differentially rotating discs with constant specific angular momentum", MNRAS, 208, 721-750, http://adsabs.harvard.edu/abs/1984MNRAS.208..721P
  14. Papaloizou, J. C. B., \& Pringle, J. E. 1985, "The dynamical stability of differentially rotating discs. II", MNRAS, 213, 799-820, http://adsabs.harvard.edu/abs/1985MNRAS.213..799P
  15. Lovelace, R.V.E. and Romanova, M.M. 2013,"Rossby Wave Instability in Astrophysical Discs", 13 pages, 4 figures, http://arxiv.org/abs/1312.4572
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