Rossby wave instability in astrophysical discs

File:Rossby wave instability in a Keplerian Disk.png

Fig. 3. Rossby wave instability in a Keplerian Disk.

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. and Li et al. for thin Keplerian discs with negligible self-gravity and earlier by Lovelace and Hohlfeld 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 vortensity which is defined as $({\mathbf \nabla \times u})_z/\Sigma$. A sketch of a bump in ${ L}(r)$ is shown in Figure 1.

Rossby waves, named after Carl-Gustaf Arvid Rossby are important in planetary atmospheres and oceans and are also known as it planetary waves. 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. 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.

Schrödinger-like equation for perturbation

File:Rossby Waves Instability Mechanism.png

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. 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 $\omega = m \Omega(r_{\rm C})$.

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.

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 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.

File:Effective Potential for Rossby Wave Instability.png

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). 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 ($E>0$).

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