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 {displaystyle r_{0}} , in the disc surface mass-density. It gives rise to exponentially growing non-axisymmetric perturbation [ ∝ exp ⁡ ( i m ϕ ) {displaystyle propto exp({ m {i}}mphi )} , m = 1 , 2 , . . . {displaystyle m=1,2,...} ] in the vicinity of r 0 {displaystyle 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. 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 {displaystyle r_{0}} , in the disc surface mass-density. It gives rise to exponentially growing non-axisymmetric perturbation [ ∝ exp ⁡ ( i m ϕ ) {displaystyle propto exp({ m {i}}mphi )} , m = 1 , 2 , . . . {displaystyle m=1,2,...} ] in the vicinity of r 0 {displaystyle 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 at some radius r 0 {displaystyle r_{0}} , where Σ {displaystyle Sigma } is the surface mass density of the disc, u ≈ r Ω ( r ) ϕ   ^ {displaystyle {mathbf {u} }approx rOmega (r){hat {phi ~}}} is the flow velocity of the disc, Ω ( r ) ≈ ( G M ∗ / r 3 ) 1 / 2 {displaystyle Omega (r)approx (GM_{*}/r^{3})^{1/2}} is the angular velocity of the flow (with M ∗ {displaystyle M_{*}} the mass of the central star), S {displaystyle S} is the specific entropy of the gas, and γ {displaystyle gamma } is the specific heat ratio. The approximations involve the neglect of the relatively small radial pressure force. Note that L {displaystyle {L}} is related to the inverse of the vortensity which is defined as ( ∇ × u ) z / Σ {displaystyle ({mathbf { abla } imes u})_{z}/Sigma } . A sketch of a bump in L ( r ) {displaystyle {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 {displaystyle >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. Linearization of the Euler and continuity equations for a thin fluid disc with perturbations proportional to f ( r ) exp ⁡ ( i m ϕ − i ω t ) {displaystyle f(r)exp({ m {i}}mphi -{ m {i}}omega t)} (with azimuthal mode number m = 1 , 2 , . . {displaystyle m=1,2,..} and angular frequency ω {displaystyle omega } ) leads to a Schrödinger-like equation for the enthalpy perturbation ψ = δ p / ρ {displaystyle psi =delta p/ ho } , The effective potential well V e f f ( r ) {displaystyle V_{ m {eff}}(r)} is closely related to L ( r ) {displaystyle {L}(r)} : If the height of the bump in L ( r ) {displaystyle {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 ) {displaystyle {L}(r)} the potential V e f f {displaystyle V_{ m {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, ω i = ℑ ( ω ) {displaystyle omega _{i}=Im (omega )} which is the growth rate of the instability. For moderate strength bumps (with fractional amplitudes Δ Σ / Σ ≲ 0.2 {displaystyle Delta Sigma /Sigma lesssim 0.2} ), the growth rates are of the order of ω i = ( 0.1 − 0.2 ) Ω ( r 0 ) {displaystyle omega _{i}=(0.1-0.2)Omega (r_{0})} . The real part of the wave frequency ω r = ℜ ( ω ) {displaystyle omega _{r}=Re (omega )} is approximately m Ω ( r 0 ) {displaystyle mOmega (r_{0})} . A more complete analysis reveals that the Rossby wave is not completely trapped in the potential well V e f f {displaystyle V_{ m {eff}}} , but leaks outward across a forbidden region at an outer Lindblad resonance (at r O L R {displaystyle r_{ m {OLR}}} indicated in Figure 1) and inward across another forbidden region at an inner Lindblad resonance (at r I L R {displaystyle r_{ m {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 = {displaystyle S=} const) disc is where Δ ω ≡ ω − m Ω {displaystyle Delta omega equiv omega -mOmega } is the Doppler shifted wave frequency in the reference frame moving with the disc matter, c s {displaystyle c_{s}} is the sound speed in the disc, and κ {displaystyle kappa } is the radial epicyclic angular frequency, with κ 2 = r − 3 d ℓ 2 / d r {displaystyle kappa ^{2}=r^{-3}dell ^{2}/dr} and ℓ = r u ϕ {displaystyle ell =ru_{phi }} the specific angular momentum. Figure 2 shows the effective potential for sample cases. Note that the inward propagating waves with ω r < m Ω ( r ) {displaystyle omega _{r}<mOmega (r)} have negative energy ( E < 0 {displaystyle E<0} ) whereas the outward propagating waves with ω r > m Ω ( r ) {displaystyle omega _{r}>mOmega (r)} have positive energy ( E > 0 {displaystyle E>0} ).