(ed.). Gravitational waves (IOP, 2001)(422s).

The r-modes 75instability. In an ideal star, it is always possible to find pressure-driven wavesof short enough wavelength around the axis of symmetry (high enough angulareigenvalue m) that satisfy this condition. However, it turns out that even a smallamount of viscosity can damp out the instability in such waves , so it is not clearthat pressure-driven waves will ever be significantly unstable in realistic stars.However, in 1997 Andersson [17] pointed out that there was a class of modescalled r-modes (Rossby modes) that no-one had previously investigated, andthat were formally unstable in all rotating stars. Rossby waves are well knownin oceanography, where they play an important role in energy transport aroundthe Earth’s oceans. They are hard to detect, having long wavelengths and verylow-density perturbations. They are mainly velocity perturbations of the oceans,whose restoring force is the Coriolis effect, and that is their character in neutronstars too. Because they have very small density perturbation, the gravitationalradiation they emit is dominated by the current-quadrupole radiation.For a slowly-rotating, nearly-spherical Newtonian star, the followingvelocity perturbation is characteristic of r-modes:δv a = ς(r)ɛ abc ∇ b r∇ c Y lm , (7.9)where ς(r) is some function of r determined by the mode equations. This velocityis a curl, so it is divergence-free; since it has no radial component, it does notchange the density. If the star is perfectly spherical, these perturbations are simplya small rotation of some of the fluid, and it continues to rotate. They have nooscillation, and have zero frequency.If we consider a star with a small rotational angular velocity , then thefrequency σ is no longer exactly zero and a Newtonian calculation to first orderin shows that there is a mode with pattern speed ω p =−σ/m equal toω p = [1 −2l (l + 1)]. (7.10)These modes are now oscillating currents that move (approximately) along theequipotential surfaces of the rotating star.For l 2, ω p is positive but slower than the speed of the star, so by the CFSmechanism these modes are unstable to the emission of gravitational radiation foran arbitrarily slowly rotating star.The velocity pattern given in equation (7.9) for (l = 2, m = 2) is closelyrelated to the wheel model we described for current-quadrupole radiation infigure 6.1. Take two such wheels and orient their axes along the x- and y-axes,with the star rotating about the z-axis. Choose the sense of rotation so that thewheels at positive-x and positive-y are spinning in the opposite sense at any time,i.e. so that their adjacent edges are always moving in the same direction. Thenthis relationship will be reproduced for all other adjacent pairs of wheels: adjacentedges move together.When seen from above the equatorial plane, the line-of-sight momenta ofthe wheels reinforce each other, and we get the same kind of pattern that we saw